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  • 1
    Electronic Resource
    Electronic Resource
    On the average cost optimality equation and the structure of optimal policies for partially observable Markov decision processes (1991)
    Springer
    Annals of operations research 29 (1991), S. 439-469 
    Details
    ISSN: 1572-9338
    Keywords: Optimal control ; Markov chains ; partial observability ; average cost ; optimality equation ; structured optimal policies
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract We consider partially observable Markov decision processes with finite or countably infinite (core) state and observation spaces and finite action set. Following a standard approach, an equivalent completely observed problem is formulated, with the same finite action set but with anuncountable state space, namely the space of probability distributions on the original core state space. By developing a suitable theoretical framework, it is shown that some characteristics induced in the original problem due to the countability of the spaces involved are reflected onto the equivalent problem. Sufficient conditions are then derived for solutions to the average cost optimality equation to exist. We illustrate these results in the context of machine replacement problems. Structural properties for average cost optimal policies are obtained for a two state replacement problem; these are similar to results available for discount optimal policies. The set of assumptions used compares favorably to others currently available.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02283610
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  • 2
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    A Technique for Stochastic Control Problems with Unbounded Control Set (1999)
    Springer
    Journal of theoretical probability 12 (1999), S. 255-270 
    Details
    ISSN: 1572-9230
    Keywords: Optimal control ; stochastic differential equations ; convergence in law ; unbounded control set ; suboptimal control
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We describe a change of time technique for stochastic control problems with unbounded control set. We demonstrate the technique on a class of maximization problems that do not have optimal controls. Given such a problem, we introduce an extended problem which has the same value function as the original problem and for which there exist optimal controls that are expressible in simple terms. This device yields a natural sequence of suboptimal controls for the original problem. By this we mean a sequence of controls for which the payoff functions approach the value function.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1021761030407
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  • 3
    Electronic Resource
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    Opticon: An algorithm for the optimal control of nonlinear stochastic models (1992)
    Springer
    Annals of operations research 37 (1992), S. 375-401 
    Details
    ISSN: 1572-9338
    Keywords: Optimal control ; stochastic control ; dynamic systems ; nonlinear systems ; control algorithm ; optimal economic policies
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract In this paper we describe the algorithm OPTCON which has been developed for the optimal control of nonlinear stochastic models. It can be applied to obtain approximate numerical solutions of control problems where the objective function is quadratic and the dynamic system is nonlinear. In addition to the usual additive uncertainty, some or all of the parameters of the model may be stochastic variables. The optimal values of the control variables are computed in an iterative fashion: First, the time-invariant nonlinear system is linearized around a reference path and approximated by a time-varying linear system. Second, this new problem is solved by applying Bellman's principle of optimality. The resulting feedback equations are used to project expected optimal state and control variables. These projections then serve as a new reference path, and the two steps are repeated until convergence is reached. The algorithm has been implemented in the statistical programming system GAUSS. We derive some mathematical results needed for the algorithm and give an overview of the structure of OPTCON. Moreover, we report on some tentative applications of OPTCON to two small macroeconometric models for Austria.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02071066
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  • 4
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    Exploitation of natural resources and pollution (1994)
    Springer
    Annals of operations research 54 (1994), S. 237-262 
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    ISSN: 1572-9338
    Keywords: Optimal control ; differential games ; environmental policy ; JEL C61 ; C73 ; Q28
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract This paper provides some differential game models of natural resource exploitation when environmental pollution takes place. The “classical” approach to determine optimal harvest rates of renewable resources utilizes Optimal Control models, i.e. there is either a monopolistic market structure or there is pure competition. In case of pollution, however, all agents can be put together, forming the groups of the resource harvesters on one side and of polluters on the other side. So differential games can be used to analyze environmental problems. The models introduced in this paper are put together in order to showdifferent problems that can all be analyzed using differential games.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02031736
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  • 5
    Electronic Resource
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    Optimal budgetary and monetary policies under uncertainty: A stochastic control approach (1995)
    Springer
    Annals of operations research 58 (1995), S. 379-402 
    Details
    ISSN: 1572-9338
    Keywords: Optimal control ; stochastic control ; dynamic systems ; economics ; public-sector applications ; optimization ; budgetary policies ; monetary policy
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract In this paper, we determine optimal budgetary and monetary policies for Austria using a small macroeconometric model. We use a Keynesian model of the Austrian economy, called FINPOL1, estimated by ordinary least squares, which relates the main objective variables of Austrian economic policies, such as the growth rate of real gross domestic product, the rate of unemployment, the rate of inflation, the balance of payments, and the ratio of the federal budget deficit to GDP, to fiscal and monetary policy instruments, namely expenditures and revenues of the federal budget and money supply. Optimal fiscal and monetary policies are calculated for the model under a quadratic objective function using the algorithm OPTCON for the optimum control of nonlinear stochastic dynamic systems. Several control experiments are performed in order to assess the influence of different kinds of uncertainty on optimal budgetary and monetary policies. Apart from deterministic optimization runs, different assumptions about parameter uncertainties are introduced; the results of these different stochastic optimum control experiments are compared and interpreted.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02038862
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  • 6
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    Optimal pollution control, irreversibilities, and the value of future information (1994)
    Springer
    Annals of operations research 54 (1994), S. 217-235 
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    ISSN: 1572-9338
    Keywords: Optimal control ; growth paths ; exhaustible resources ; JEL C61 ; O41 ; Q23
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract This paper focuses on the issue of optimal pollution control when either pollution itself is irreversible or when some characteristic of the environmental resource is irreversibly destroyed in the course of growing pollution. It is shown that exhausting the assimilative capacity through too heavy pollution is never optimal unless the rate of social time preference is sufficiently high. The paper also investigates the case that decisions about irreversible developments have to be made under uncertainty today when the decision maker faces the prospect of better information about the irreversible damage at some future point in time. A non-negative quasi-option value is shown to exist as in the Arrow-Fisher-Henry model that relates to natural resource deletion by projects of industrial development.
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    URL: http://dx.doi.org/10.1007/BF02031735
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  • 7
    Electronic Resource
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    Sensitivity, controllability, and necessary conditions of optimal control problems governed by integral equations (1995)
    Springer
    Applied mathematics & optimization 32 (1995), S. 73-97 
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    ISSN: 1432-0606
    Keywords: Optimal control ; State constraint ; Volterra integral equation ; Necessary conditions ; Controllability ; Sensitivity ; Value function ; Proximal normal analysis ; Proximal analysis ; 49J22 ; 49K22 ; 93B05 ; 93B06 ; 93B35
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Infinite-dimensional perturbations in all constraints of an optimal control problem governed by a Volterra integral equation with the presence of a state constraint are considered. These perturbations give rise to a value function, whose analysis through the proximal normal technique provides sensitivity, controllability, and even necessary conditions for the basic problem. Actually all information about the value function is contained in Clarke's normal cone of its epigraph, which can be characterized by the proximal normal formula.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01189904
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  • 8
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    Value function and optimality condition for semilinear control problems. II: Parabolic case (1996)
    Springer
    Applied mathematics & optimization 33 (1996), S. 1-33 
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    ISSN: 1432-0606
    Keywords: Optimal control ; Value function ; Analytic semigroups ; Semiconcave functions ; Distributed parameter systems ; 49K20 ; 49L20 ; 47B44 ; 49J52
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper we continue to study properties of the value function and of optimal solutions of a semilinear Mayer problem in infinite dimensions. Applications concern systems governed by a state equation of parabolic type. In particular, the issues of the joint Lipschitz continuity and semiconcavity of the value function are treated in order to investigate the differentiability of the value function along optimal trajectories.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01187960
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  • 9
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    The optimal control approach to generalized multiprocessor scheduling (1996)
    Springer
    Algorithmica 15 (1996), S. 17-49 
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    ISSN: 1432-0541
    Keywords: Scheduling ; Multiprocessor scheduling ; Parallel algorithms ; NP-completeness ; Theoretical computer science ; Operations research ; Optimal control
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper we present several new results in the theory of homogeneous multiprocessor scheduling. We start with some assumptions about the behavior of tasks, with associated precedence constraints, as processor power is applied. We assume that as more processors are applied to a task, the time taken to compute it decreases, yielding some speedup. Because of communication, synchronization, and task scheduling overhead, this speedup increases less than linearly with the number of processors applied. We also assume that the number of processors which can be assigned to a task is a continuous variable, with a view to exploiting continuous mathematics. The optimal scheduling problem is to determine the number of processors assigned to each task, and task sequencing, to minimize the finishing time. These assumptions allow us to recast the optimal scheduling problem in a form which can be addressed by optimal control theory. Various theorems can be proven which characterize the optimal scheduling solution. Most importantly, for the special case where the speedup function of each task isp α , wherep is the amount of processing power applied to the task, we can directly solve our equations for the optimal solution. In this case, for task graphs formed from parallel and series connections, the solution can be derived by inspection. The solution can also be shown to be shortest path from the initial to the final state, as measured by anl 1/α distance metric, subject to obstacle constraints imposed by the precedence constraints.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01942605
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  • 10
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    Provably good approximation algorithms for optimal kinodynamic planning: Robots with decoupled dynamics bounds (1995)
    Springer
    Algorithmica 14 (1995), S. 443-479 
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    ISSN: 1432-0541
    Keywords: Robot motion planning ; Optimal control ; Polynomial-timeɛ-approximation algorithm ; Time-optimal trajectory ; Shortest path ; Kinodynamics ; Polyhedral obstacles
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We consider the following problem: given a robot system, find a minimal-time trajectory that goes from a start state to a goal state while avoiding obstacles by a speed-dependent safety margin and respecting dynamics bounds. In [1] we developed a provably good approximation algorithm for the minimum-time trajectory problem for a robot system with decoupled dynamics bounds (e.g., a point robot in ℝ3). This algorithm differs from previous work in three ways. It is possible (1) to bound the goodness of the approximation by an error termɛ; (2) to bound the computational complexity of our algorithm polynomially; and (3) to express the complexity as a polynomial function of the error term. Hence, given the geometric obstacles, dynamics bounds, and the error termɛ, the algorithm returns a solution that isɛ-close to optimal and requires only a polynomial (in (1/ɛ)) amount of time. We extend the results of [1] in two ways. First, we modify it to halve the exponent in the polynomial bounds from 6d to 3d, so that the new algorithm isO(c d N 1/ɛ)3d ), whereN is the geometric complexity of the obstacles andc is a robot-dependent constant. Second, the new algorithm finds a trajectory that matches the optimal in time with anɛ factor sacrificed in the obstacle-avoidance safety margin. Similar results hold for polyhedral Cartesian manipulators in polyhedral environments. The new results indicate that an implementation of the algorithm could be reasonable, and a preliminary implementation has been done for the planar case.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01586636
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  • 11
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    Provably good approximation algorithms for optimal kinodynamic planning for Cartesian robots and open-chain manipulators (1995)
    Springer
    Algorithmica 14 (1995), S. 480-530 
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    ISSN: 1432-0541
    Keywords: Robot motion planning ; Optimal control ; Polynomial-timeɛ-approximation algorithm ; Time-optimal trajectory ; Full dynamics ; Shortest path ; Kinodynamics ; Polyhedral obstacles
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Inoptimal kinodynamic planning, given a robot system, we must find a minimal-time trajectory that goes from a start state to a goal state while avoiding obstacles by a speed-dependent safety margin and respecting dynamics bounds. With Canny and Reif [1], we approached this problem from anɛ-approximation standpoint and introduced a provably good approximation algorithm for optimal kinodynamic planning for a robot obeying particle dynamics. If a solution exists, this algorithm returns a trajectoryɛ-close to optimal in time polynomial in both (1/ɛ) and the geometric complexity. We extend [1] and [2] tod-link three-dimensional robots with full rigid-body dynamics amidst obstacles. Specifically, we describe polynomial-time approximation algorithms for Cartesian robots obeyingL 2 dynamic bounds and for open-kinematic-chain manipulators with revolute and prismatic joints. The latter class includes many industrial manipulators. The correctness and complexity of these algorithms rely on new trajectory tracking lemmas for robots with coupled dynamics bounds.
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    URL: http://dx.doi.org/10.1007/BF01586637
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  • 12
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    A minimax problem for semilinear nonlocal competitive systems (1993)
    Springer
    Applied mathematics & optimization 28 (1993), S. 113-132 
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    ISSN: 1432-0606
    Keywords: Elliptic systems ; Nonlinear competitive interactions ; Game theory ; Optimal control ; 49A45 ; 35K10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A two-sided game for the control of a stationary semilinear competitive system with autonomous sources is considered, where the controls are the kernels of the nonlocal interaction terms. The saddle point (the optimal solution of the game) is characterized as the unique solution of the associated optimality system, which is solved by an iterative scheme.
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    URL: http://dx.doi.org/10.1007/BF01182976
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  • 13
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    Optimal control problems for distributed parameter systems in Banach spaces (1993)
    Springer
    Applied mathematics & optimization 28 (1993), S. 225-257 
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    ISSN: 1432-0606
    Keywords: Lagrange multiplier rule ; Kuhn-Tucker conditions ; Maximum principle ; Optimal control ; 93E20 ; 93E25
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We consider the infinite-dimensional nonlinear programming problem of minimizing a real-valued functionf 0 (u) defined in a metric spaceV subject to the constraintf(u) ε Y, wheref(u) is defined inV and takes values in a Banach spaceE and Y is a subset ofE. We derive and use a theorem of Kuhn-Tucker type to obtain Pontryagin's maximum principle for certain semilinear parabolic distributed parameter systems. The results apply to systems described by nonlinear heat equations and reaction-diffusion equations inL 1 andL ∞ spaces.
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    URL: http://dx.doi.org/10.1007/BF01200380
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  • 14
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    Second-order necessary conditions for differential inclusion problems (1994)
    Springer
    Applied mathematics & optimization 30 (1994), S. 1-14 
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    ISSN: 1432-0606
    Keywords: Optimal control ; Unbounded differential inclusion ; Second-order necessary condition ; Second variation ; 49K24
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We study second-order necessary conditions for optimality in the unbounded differential inclusion control problem and recover the accessory problem in optimal control theory.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01261988
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  • 15
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    On unbounded solutions of Bellman's equation associated with optimal switching control problems with state constraints (1995)
    Springer
    Applied mathematics & optimization 31 (1995), S. 1-17 
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    ISSN: 1432-0606
    Keywords: Bellman equation ; Hamilton-Jacobi equation ; Quasi-variational inequality ; Viscosity solution ; Unbounded solutions ; Optimal switching ; Optimal control ; State constraints ; Production engineering ; 49C05 ; 49C20 ; 49B10 ; 49A36
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We study a quasi-variational inequality system with unbounded solutions. It represents the Bellman equation associated with an optimal switching control problem with state constraints arising from production engineering. We show that the optimal cost is the unique viscosity solution of the system.
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    URL: http://dx.doi.org/10.1007/BF01182554
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  • 16
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    Constrained retarded nonlinear optimal control problems in Banach state space (1996)
    Springer
    Applied mathematics & optimization 34 (1996), S. 191-230 
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    ISSN: 1432-0606
    Keywords: Optimal control ; Retarded systems ; Control in infinite-dimensional state space ; Primary 49K25 ; Secondary 34K35
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Necessary conditions are proved for the optimal control of solutions of ordinary and retarded differential equations in a Banach state space, with mixed and pure state restrictions. The treatment includes the possibility of point targets. A generalization of earlier results for finite-dimensional or Hilbert state spaces is obtained.
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    URL: http://dx.doi.org/10.1007/BF01182624
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  • 17
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    Boundary optimal control of MHD flows (1995)
    Springer
    Applied mathematics & optimization 32 (1995), S. 143-162 
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    ISSN: 1432-0606
    Keywords: Optimal control ; Magnetohydrodynamics ; Electrically conducting fluid ; Primary 49J20 ; 49K20 ; Secondary 35Q35 ; 49M05 ; 76W05 ; 93C10 ; 93C20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper deals with some optimal control problems associated with the equations of steady-state, incompressible magnetohydrodynamics. These problems have direct applications to nuclear reactor technology, magnetic propulsion devices, and design of electromagnetic pumps. These problems are first put into an appropriate mathematical formulation. Then the existence of optimal solutions is proved. The use of Lagrange multiplier techniques is justified and an optimality system of equations is derived. The theory is applied to an example.
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    URL: http://dx.doi.org/10.1007/BF01185228
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  • 18
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    Optimal control of a fine structure (1994)
    Springer
    Applied mathematics & optimization 30 (1994), S. 113-126 
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    ISSN: 1432-0606
    Keywords: Nonconvex variational problems ; Relaxation ; Optimal control ; 49J27 ; 35B25 ; 65N30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An optimal control problem for a multivalued system governed by a nonconvex variational problem, involving a regularization parameter ɛ〉0, is proposed and studied. The solution to the variational problem exhibits typically rapid oscillations (a so-called fine structure) corresponding to a multiphase state of the material. We want to control only this fine structure. Existence of an optimal control is proved. Its convergence with ɛ→0 is studied by means of an optimal control problem for a relaxed variational problem involving (suitably generalized) Young measures. The uniqueness of the solution to the relaxed variational problem, which is nontrivial but is very important in the context of optimal control, is studied in special cases. A finite-element approximation is proposed.
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    URL: http://dx.doi.org/10.1007/BF01189449
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  • 19
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    Generalized sentinels defined via least squares (1995)
    Springer
    Applied mathematics & optimization 31 (1995), S. 189-218 
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    ISSN: 1432-0606
    Keywords: Least squares ; Sentinels ; Optimal control ; Regularization ; Duality ; 93A30 ; 49K27
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We address the problem of monitoring a linear functional (c, x)Eof an unknown vectorx of a Hilbert spaceE, the available data being the observationz, in a Hilbert spaceF, of a vectorAx depending linearly onx through some known operatorAεℒ(E; F). WhenE=E 1×E 2,c=(c 1 0), andA is injective and defined through the solution of a partial differential equation, Lions ([6]–[8]) introduced sentinelssεF such that (s, Ax)Fis sensitive to x1 εE 1 but insensitive to x2 ε E2. In this paper we prove the existence, in the general case, of (i) a generalized sentinel (s, σ) ε ℱ ×E, where ℱ ⊃F withF dense in 80, such that for anya priori guess x0 ofx, we have 〈s, Ax〉ℱℱ + (σ, x0)E=(c, x)E, where x is the least-squares estimate ofx closest tox 0, and (ii) a family of regularized sentinels (s n , σ n ) ε F×E which converge to (s, σ). Generalized sentinels unify the least-squares approach (by construction !) and the sentinel approach (whenA is injective), and provide a general framework for the construction of “sentinels with special sensitivity” in the sense of Lions [8]).
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    URL: http://dx.doi.org/10.1007/BF01182788
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  • 20
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    An SQP method for optimal control of weakly singular Hammerstein integral equations (1996)
    Springer
    Applied mathematics & optimization 33 (1996), S. 227-252 
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    ISSN: 1432-0606
    Keywords: Lagrange-Newton method ; Sequential quadratic programming ; Optimal control ; Weakly singular Hammerstein integral equations ; Control constraints ; 49M05 ; 49K22
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We investigate local convergence of an SQP method for nonlinear optimal control of weakly singular Hammerstein integral equations. Sufficient conditions for local quadratic convergence of the method are discussed.
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    URL: http://dx.doi.org/10.1007/BF01204703
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  • 21
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    Discrete and sampled-data stochastic control problems with complete and incomplete state information (1991)
    Springer
    Applied mathematics & optimization 24 (1991), S. 289-316 
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    ISSN: 1432-0606
    Keywords: Optimal control ; Stochastic control ; Sampled-data controllers ; Digital control ; Optimal regulators ; Optimal tracking
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An unconstrained stochastic optimization problem involving a discrete-time linear process with a normally distributed initial condition and subject to additive gaussian state and measurement noise is formulated in terms of a quite general finite horizon, discrete-time quadratic cost criterion and solved when there is either complete or incomplete state information. It is shown that both the stochastic sampled-data optimal tracker and the stochastic sampled-data optimal regulator are special cases of this problem. A breakdown of the minimum cost for both sampled-data controllers is given.
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    URL: http://dx.doi.org/10.1007/BF01447747
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  • 22
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    Viscosity solutions for the dynamic programming equations (1992)
    Springer
    Applied mathematics & optimization 25 (1992), S. 109-126 
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    ISSN: 1432-0606
    Keywords: Hamilton-Jacobi equation ; Viscosity solution ; Optimal control ; Neumann problem ; 35F20 ; 49B10 ; 49CO5
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In a previous paper the author has introduced a new notion of a (generalized) viscosity solution for Hamilton-Jacobi equations with an unbounded nonlinear term. It is proved here that the minimal time function (resp. the optimal value function) for time optimal control problems (resp. optimal control problems) governed by evolution equations is a (generalized) viscosity solution for the Bellman equation (resp. the dynamic programming equation). It is also proved that the Neumann problem in convex domains may be viewed as a Hamilton-Jacobi equation with a suitable unbounded nonlinear term.
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    URL: http://dx.doi.org/10.1007/BF01182476
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  • 23
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    Value function and optimality conditions for semilinear control problems (1992)
    Springer
    Applied mathematics & optimization 26 (1992), S. 139-169 
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    ISSN: 1432-0606
    Keywords: Optimal control ; Distributed parameter systems ; Dynamic programming ; Optimality conditions ; Strongly continuous semigroups ; Nonsmooth analysis ; 49A22 ; 49B22 ; 49A52 ; 47D05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper is concerned with optimal control problems of Mayer and Bolza type for systems governed by a semilinear state equationx′(t)=Ax(t) + f(t, x(t), u(t)), u(t) ε U, whereA is the infinitesimal generator of a strongly continuous semigroup in a Banach spaceX. We prove necessary and sufficient conditions for optimality and then use these conditions to investigate properties of the value function related to superdifferentials. Conversely, we use the value function to obtain criteria for optimality and feedback systems.
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    URL: http://dx.doi.org/10.1007/BF01189028
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  • 24
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    Optimal control of semilinear elliptic equations with pointwise constraints on the gradient of the state (1993)
    Springer
    Applied mathematics & optimization 27 (1993), S. 35-56 
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    ISSN: 1432-0606
    Keywords: Optimal control ; State constraints ; Semilinear elliptic equations ; Optimality conditions ; Lagrange multipliers ; 49K20 ; 49J20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper we are concerned with optimal control problems governed by an elliptic semilinear equation, the control being distributed in Ω. The existence of constraints on the control as well as pointwise constraints on the gradient of the state is assumed. A convenient choice of the control space permits us to derive the optimality conditions and study the adjoint state equation, which has derivatives of measures as data. In order to carry out this study, we prove a trace theorem and state Green's formula by using the transposition method.
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    URL: http://dx.doi.org/10.1007/BF01182597
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  • 25
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    Optimal control of parabolic problems with state constraints: a penalization method for optimality conditions (1994)
    Springer
    Applied mathematics & optimization 29 (1994), S. 285-307 
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    ISSN: 1432-0606
    Keywords: Optimal control ; Optimality conditions ; State constraints ; Lagrange multipliers ; Parabolic systems ; Penalization ; 49B22
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    Topics: Mathematics
    Notes: Abstract In this paper state constrained optimal control problems governed by parabolic evolution equations are studied. Our purpose is to obtain a (first-order) decoupled optimality system (that ensures the Lagrange multipliers existence). In a first step we are led to Slater-like assumptions and we are then allowed to extend the application field of the decoupled system we obtain. With a weaker assumption the existence of Lagrange multipliers (that are measures) for nonqualified problems may be established.
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    Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems—Strong convergence of optimal controls (1994)
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    Applied mathematics & optimization 29 (1994), S. 309-329 
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    ISSN: 1432-0606
    Keywords: Optimal control ; Parabolic equation ; Nonlinear boundary condition ; State constraint ; Ritz-Galerkin approximation ; 49K20 ; 65K10 ; 65M60
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    Topics: Mathematics
    Notes: Abstract A class of optimal control problems for a parabolic equation with nonlinear boundary condition and constraints on the control and the state is considered. Associated approximate problems are established, where the equation of state is defined by a semidiscrete Ritz-Galerkin method. Moreover, we are able to allow for the discretization of admissible controls. We show the convergence of the approximate controls to the solution of the exact control problem, as the discretization parameter tends toward zero. This result holds true under the assumption of a certain sufficient second-order optimality condition.
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    Analysis of a sterile insect release model with predation (1982)
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    Journal of mathematical biology 16 (1982), S. 33-48 
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    ISSN: 1432-1416
    Keywords: Sterile insect release ; Predation ; Stability ; Limit cycles ; Optimal control
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    Topics: Biology , Mathematics
    Notes: Abstract A model for the sterile insect release method of pest control in which the target species is under predatory or parasitic regulation is analyzed. The equations are nondimensionalized and the rescaled parameters are interpreted. There are four types of equilibria, whose existence and stability depend on which of ten regions of parameter space contain the rescaled parameters, and in turn give minimal release rates to achieve eradication of the pest. In at least one region, Hopf bifurcation theory shows the existence of limit cycles, but they are found to be unstable. In addition, the optimal release rate to minimize a total cost functional for pest control by the sterile release method is studied. Both approaches show that when predation accounts for a large fraction of the natural deaths, the necessary release rate and total cost are higher than for weak predation. If the predators are removed without being replaced by any other source of mortality, the cost rises in all cases but rises much more dramatically for cases with strong predation. A definite danger of the sterile release method when some predatory control exists is that the predators are frequently driven extinct before the prey, so that the target species could explode to much higher levels and be more difficult to eradicate again after the sterile release is terminated.
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    Pontryagin's principle for control problems in age-dependent population dynamics (1985)
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    Journal of mathematical biology 23 (1985), S. 75-101 
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    ISSN: 1432-1416
    Keywords: Population dynamics ; Optimal harvesting ; Optimal control ; Hyperbolic systems ; Pontryagin's principle
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    Topics: Biology , Mathematics
    Notes: Abstract In this paper, Pontryagin's principle is proved for a fairly general problem of optimal control of populations with continuous time and age variable. As a consequence, maximum principles are developed for an optimal harvesting problem and a problem of optimal birth control.
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    Optimal control of the chemotherapy of HIV (1997)
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    Journal of mathematical biology 35 (1997), S. 775-792 
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    ISSN: 1432-1416
    Keywords: Key words: Chemotherapy ; HIV ; Optimal control ; Ordinary differential equation system
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    Topics: Biology , Mathematics
    Notes: Abstract.  Using an existing ordinary differential equation model which describes the interaction of the immune system with the human immunodeficiency virus (HIV), we introduce chemotherapy in an early treatment setting through a dynamic treatment and then solve for an optimal chemotherapy strategy. The control represents the percentage of effect the chemotherapy has on the viral production. Using an objective function based on a combination of maximizing benefit based on T cell counts and minimizing the systemic cost of chemotherapy (based on high drug dose/strength), we solve for the optimal control in the optimality system composed of four ordinary differential equations and four adjoint ordinary differential equations.
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    Optimal control for a tandem network of queues with blocking (1997)
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    Acta mathematicae applicatae sinica 13 (1997), S. 425-437 
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    ISSN: 1618-3932
    Keywords: Optimal control ; tandem queueing system
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    Topics: Mathematics
    Notes: Abstract We consider a two-station tandem queue with no intermediate buffer. Jobs at the first station may be blocked when the following station is occupied by another job. The objective is to control the arrival and departure processes, subject to some capacity limits, so that an expected discounted profit function is maximized. We prove that the optimal control policy is of a threshold type, and the characterization of the threshold is provided.
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    Neural network controller for a multivariable model of submarine dynamics (1998)
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    Neural computing & applications 7 (1998), S. 295-308 
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    ISSN: 1433-3058
    Keywords: Adaptive ; Backpropagation ; Multivariable ; Neural networks ; Optimal control ; Submarine
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    Topics: Computer Science , Mathematics
    Notes: Abstract Recently, there have been many attempts to use neural networks as a feedback controller. However, most of the reported cases seek to control Single-Input Single-Output (SISO) systems using some sort of adaptive strategy. In this paper, we demonstrate that neural networks can be used for the control of complex multivariable, rather than simply SISO, systems. A modified direct control scheme using a neural network architecture is used with backpropagation as the adaptive algorithm. The proposed algorithm is designed for Multi-Input Multi-Output (MIMO) systems, and is similar to that proposed by Saerens and Soquet [1] and Goldenthal and Farrell [2] for (SISO) systems, and differs only in the form of the gradient approximation. As an example of the application of this approach, we investigate the control of the dynamics of a submarine vehicle with four inputs and four outputs, in which the differential stern, bow and rudder control surfaces are dynamically coordinated to cause the submarine to follow commanded changes in roll, yaw rate, depth rate and pitch attitude. Results obtained using this scheme are compared with those obtained using optimal linear quadratic control.
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    The linear quadratic optimal control problem for linear descriptor systems with variable coefficients (1997)
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    Mathematics of control, signals, and systems 10 (1997), S. 247-264 
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    ISSN: 1435-568X
    Keywords: Descriptor systems ; Differential algebraic equations ; Optimal control ; Strangeness index ; Riccati differential-algebraic equations ; Euler-Lagrange equations ; Linear feedback
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    Topics: Electrical Engineering, Measurement and Control Technology , Mathematics , Technology
    Notes: Abstract We study linear quadratic optimal control problems for linear variable coefficient descriptor systems. Generalization from the case of standard control problems leads to several difficulties. We discuss a behavioral approach that solves some of these difficulties. Furthermore, an analysis of general rectangular systems is given and generalized Euler-Lagrange equations and Riccati differential algebraic equations are discussed.
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    Dynamic programming for free-time problems with endpoint constraints (1993)
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    Mathematics of control, signals, and systems 6 (1993), S. 180-193 
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    ISSN: 1435-568X
    Keywords: Optimal control ; Dynamic programming ; Free endtime ; Hamilton-Jacobi theory
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    Topics: Electrical Engineering, Measurement and Control Technology , Mathematics , Technology
    Notes: Abstract If we are able to find a local verification function associated with an admissible trajectory x(.), then x(.) is a local minimizer. It is of interest therefore to know when such local verification functions exist. In this paper it is shown that the existence of a local verification function is necessary for x(.) to be a local minimizer, under a normality hypothesis. The novelty of these results is that they treat problems with a general endpoint constraint and where the endtime is a choice variable. Here the value function of the original problem does not serve as a local verification function; instead it must be constructed from some derived problem. The data are allowed to be measurable in the time variable, and the normality hypothesis is expressed in terms of recent free-endtime necessary conditions of optimality for problems with measurable time dependence.
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    Disturbance rejecting optimal regulation of hyperbolic systems (1995)
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    Mathematics of control, signals, and systems 8 (1995), S. 241-256 
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    ISSN: 1435-568X
    Keywords: Robust control ; H ∞ control ; Disturbance rejection ; Hyperbolic systems ; Optimal control
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    Topics: Electrical Engineering, Measurement and Control Technology , Mathematics , Technology
    Notes: Abstract Optimal regulation of hyperbolic systems in the presence of unknown exogenous and initial disturbances is considered. Necessary conditions for determining the optimal control that tracks a desired trajectory in the presence of disturbances are developed. These necessary conditions have the form of a twopoint boundary-value problem along with certain equality and inequality conditions. The results also characterize the worst possible disturbances that are accommodated by the optimum controller without any serious performance degradation. Numerical results on the control of a vibrating beam are presented.
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    Optimal Control of Linear Time-Varying Systems via Haar Wavelets (1999)
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    Journal of optimization theory and applications 103 (1999), S. 641-655 
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    ISSN: 1573-2878
    Keywords: Optimal control ; Haar wavelets
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    Notes: Abstract This paper introduces the application of Haar wavelets to the optimal control synthesis for linear time-varying systems. Based upon some useful properties of Haar wavelets, a special product matrix, a related coefficient matrix, and an operational matrix of backward integration are proposed to solve the adjoint equation of optimization. The results obtained by the proposed Haar approach are almost the same as those obtained by the conventional Riccati method.
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    Sufficient conditions for the optimal control of a class of systems with continuous lags (1974)
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    Journal of optimization theory and applications 13 (1974), S. 545-552 
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    Keywords: Optimal control ; maximum principle ; sufficient conditions ; integro-differential equations ; path constraints ; economic applications
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    Notes: Abstract Sufficient conditions in the form of a maximum principle are obtained for the optimal control of a system described by integro-differential equations and subject to some specified path constraints. The conditions are relaxed to allow for jumps in the adjoint variables at the junction points, provided a certain convexity hypothesis is satisfied for the constraint set at these points.
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    Quadratic control problems (1975)
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    Journal of optimization theory and applications 17 (1975), S. 1-42 
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    Keywords: Optimal control ; calculus of variations ; quadratic control problems ; linear spaces ; conjugate points ; focal points ; Bolza problem ; Hilbert space techniques
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    Notes: Abstract The present paper is concerned with the study of quadratic control problems on linear spaces. In particular, we are concerned with the conditions under which a quadratic criterion function is positive on certain linear spaces. This involves the elementary theory of conjugate and focal points, the existence of a conjugate system with a nonvanishing determinant, and the existence of extremal fields. The results given are in part a translation into control language of known theory for the problem of Bolza. The method used is based on the Hilbert space techniques developed earlier by the author.
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    On the existence of optimal controls (1975)
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    Journal of optimization theory and applications 17 (1975), S. 273-278 
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    ISSN: 1573-2878
    Keywords: Optimal control ; existence theorems ; control theory
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    Topics: Mathematics
    Notes: Abstract Using a recent result due to Berkovitz, we prove the existence of an optimal control in a broad class of problems, under relatively mild conditions.
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    Remarks on bang-bang control in a Hilbert space (1977)
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    Journal of optimization theory and applications 21 (1977), S. 51-57 
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    ISSN: 1573-2878
    Keywords: Optimal control ; linear systems ; linear-quadratic problems ; bang-bang control ; Hilbert spaces
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    Topics: Mathematics
    Notes: Abstract The control of a linear system, whose performance index is the sum of a linear term and a quadratic term, is considered. A necessary and sufficient condition is given for the optimal control to be bang-bang, and this is used to extend and clarify the results of Refs. 1–2. As an illustration, an application to an elliptic boundary-value problem is given.
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    Forward differential dynamic programming (1977)
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    Journal of optimization theory and applications 21 (1977), S. 487-504 
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    ISSN: 1573-2878
    Keywords: Optimal control ; forward dynamic programming ; differential dynamic programming
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    Notes: Abstract The dynamic programming formulation of the forward principle of optimality in the solution of optimal control problems results in a partial differential equation with initial boundary condition whose solution is independent of terminal cost and terminal constraints. Based on this property, two computational algorithms are described. The first-order algorithm with minimum computer storage requirements uses only integration of a system of differential equations with specified initial conditions and numerical minimization in finite-dimensional space. The second-order algorithm is based on the differential dynamic programming approach. Either of the two algorithms may be used for problems with nondifferentiable terminal cost or terminal constraints, and the solution of problems with complicated terminal conditions (e.g., with free terminal time) is greatly simplified.
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    Some remarks on the existence of optimal controls for quasilinear systems (1981)
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    Journal of optimization theory and applications 33 (1981), S. 393-399 
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    ISSN: 1573-2878
    Keywords: Optimal control ; nonlinear systems ; existence theorems
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    Notes: Abstract Let a quasilinear control system having the state space $$\bar X \subseteq R^n $$ be governed by the vector differential equation $$\dot x = G(u(t))x,$$ wherex(0) =x 0 andU is the family of all bounded measurable functions from [0,T] intoU, a compact and convex subset ofR m.LetG:U ⇑R be a bounded measurable nonlinear function, such thatG(U) is compact and convex.G −1 can be convex onG(U) or concave. The main results of the paper establish the existence of a controlu ∈U which minimizes the cost functional $$I(u) = \int_0^T {L(u(t))x(t)dt,} $$ whereL(·) is convex. A practical example of application for chemical reactions is worked out in detail.
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    On the optimal control of a single-server queueing system: Reply (1978)
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    Journal of optimization theory and applications 26 (1978), S. 463-464 
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    ISSN: 1573-2878
    Keywords: Optimal control ; queueing systems
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    Topics: Mathematics
    Notes: Abstract This comment replies to a criticism due to Klein and Gruver (Ref. 1) of our earlier paper (Ref. 2) on the application of control theory to a queueing system. The criticism concerns the state-space diagram and the table which we inadvertently gave for the terminal-reward problem, albeit incorrectly labeled, rather than for the free-endpoint problem considered in our paper. We show that the solution given by Klein and Gruver is itself incorrect and nonoptimal.
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    Sequential gradient-restoration algorithm for optimal control problems with general boundary conditions (1978)
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    Journal of optimization theory and applications 26 (1978), S. 395-425 
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    ISSN: 1573-2878
    Keywords: Optimal control ; numerical methods ; computing methods ; gradient methods ; gradient-restoration algorithms ; sequential gradient-restoration algorithms ; general boundary conditions ; nondifferential constraints ; bounded control ; bounded state
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    Topics: Mathematics
    Notes: Abstract This paper considers the numerical solution of two classes of optimal control problems, called Problem P1 and Problem P2 for easy identification. Problem P1 involves a functionalI subject to differential constraints and general boundary conditions. It consists of finding the statex(t), the controlu(t), and the parameter π so that the functionalI is minimized, while the constraints and the boundary conditions are satisfied to a predetermined accuracy. Problem P2 extends Problem P1 to include nondifferential constraints to be satisfied everywhere along the interval of integration. Algorithms are developed for both Problem P1 and Problem P2. The approach taken is a sequence of two-phase cycles, composed of a gradient phase and a restoration phase. The gradient phase involves one iteration and is designed to decrease the value of the functional, while the constraints are satisfied to first order. The restoration phase involves one or more iterations and is designed to force constraint satisfaction to a predetermined accuracy, while the norm squared of the variations of the control, the parameter, and the missing components of the initial state is minimized. The principal property of both algorithms is that they produce a sequence of feasible suboptimal solutions: the functions obtained at the end of each cycle satisfy the constraints to a predetermined accuracy. Therefore, the values of the functionalI corresponding to any two elements of the sequence are comparable. The stepsize of the gradient phase is determined by a one-dimensional search on the augmented functionalJ, while the stepsize of the restoration phase is obtained by a one-dimensional search on the constraint errorP. The gradient stepsize and the restoration stepsize are chosen so that the restoration phase preserves the descent property of the gradient phase. Therefore, the value of the functionalI at the end of any complete gradient-restoration cycle is smaller than the value of the same functional at the beginning of that cycle. The algorithms presented here differ from those of Refs. 1 and 2, in that it is not required that the state vector be given at the initial point. Instead, the initial conditions can be absolutely general. In analogy with Refs. 1 and 2, the present algorithms are capable of handling general final conditions; therefore, they are suited for the solution of optimal control problems with general boundary conditions. Their importance lies in the fact that many optimal control problems involve initial conditions of the type considered here. Six numerical examples are presented in order to illustrate the performance of the algorithms associated with Problem P1 and Problem P2. The numerical results show the feasibility as well as the convergence characteristics of these algorithms.
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    Sufficient conditions for the optimal control of a class systems with continuous lags (1982)
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    Journal of optimization theory and applications 38 (1982), S. 153-154 
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    Keywords: Optimal control ; maximum principle ; sufficient conditions ; integro-differential equations ; path constraints ; economic applications
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    Topics: Mathematics
    Notes: Abstract Notational errors in Theorem 5.2 of Ref. 1 are corrected.
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    On the optimal control problem of a single-server queueing system (1983)
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    Journal of optimization theory and applications 41 (1983), S. 317-325 
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    Keywords: Optimal control ; queueing theory ; phase type distributions
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    Notes: Abstract For a single-server queueing system (with a finite waiting room) with phase type arrivals and exponential service times, an optimal control for the service rate is derived. This generalizes the result of Scott and Jefferson for theM/M/1/1 queueing model.
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    A transformation technique for optimal control problems with partially linear state inequality constraints (1979)
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    Journal of optimization theory and applications 28 (1979), S. 185-212 
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    ISSN: 1573-2878
    Keywords: Optimal control ; numerical methods ; computing methods ; transformation techniques ; sequential gradient-restoration algorithm ; nondifferential constraints ; state inequality constraints ; linear state inequality constraints ; partially linear state inequality constraints
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    Topics: Mathematics
    Notes: Abstract This paper considers optimal control problems involving the minimization of a functional subject to differential constraints, terminal constraints, and a state inequality constraint. The state inequality constraint is of a special type, namely, it is linear in some or all of the components of the state vector. A transformation technique is introduced, by means of which the inequality-constrained problem is converted into an equality-constrained problem involving differential constraints, terminal constraints, and a control equality constraint. The transformation technique takes advantage of the partial linearity of the state inequality constraint so as to yield a transformed problem characterized by a new state vector of minimal size. This concept is important computationally, in that the computer time per iteration increases with the square of the dimension of the state vector. In order to illustrate the advantages of the new transformation technique, several numerical examples are solved by means of the sequential gradient-restoration algorithm for optimal control problems involving nondifferential constraints. The examples show the substantial savings in computer time for convergence, which are associated with the new transformation technique.
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    A combination of penalty function and multiplier methods for solving optimal control problems (1979)
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    Journal of optimization theory and applications 28 (1979), S. 303-329 
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    Keywords: Optimal control ; multiplier methods ; penalty functions ; Riccati equation ; convergence rate
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    Notes: Abstract The properties of combined multiplier and penalty function methods are investigated using a second-order expansion and results known for the Riccati equation. It is shown that the lower bound of the values of the penalty constant necessary to obtain a minimum is given by a certain Riccati equation. The convergence rate of a common updating rule for the multipliers is shown to be linear.
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    A minimax optimal control problem (1979)
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    Journal of optimization theory and applications 28 (1979), S. 391-410 
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    Keywords: Optimal control ; minimax problems ; necessary conditions ; maximum principle
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    Notes: Abstract A control system x=f(t,x,u) is considered, and a cost functional ess supT 0≤t≤T 1 G(t, x(t),u(t)) is to be minimized. Necessary conditions for optimality (maximum principle and transversality conditions) are derived. It is also shown that an optimal control is optimal for the corresponding problem on a subinterval of [T 0,T 1], if a certain controllability condition is satisfied.
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    On the optimal control of a single-server queueing system: Further remarks (1979)
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    Journal of optimization theory and applications 29 (1979), S. 155-158 
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    Keywords: Optimal control ; queueing systems ; single-server queueing systems
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    Topics: Mathematics
    Notes: Abstract This comment is in response to a reply by Scott and Jefferson (Ref. 3) concerning the application of control theory to a queueing problem.
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    Optimal advertising policy with the contagion model (1979)
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    Journal of optimization theory and applications 29 (1979), S. 615-627 
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    Keywords: Optimal control ; advertising ; Green's theorem approach ; nearest feasible path ; infinite horizon ; optimal stationary equilibria ; economic application
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    Notes: Abstract This paper considers an optimal control problem for the dynamics of a contagion model, the optimal control being the rate of advertising expenditure that maximizes the present value of net profit streams over an infinite horizon. By using a Green's theorem approach, it is shown that there are multiple optimal stationary equilibria and that the optimal path from any given initial condition is a nearest feasible path to one of these equilibria.
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    Near-optimum design of nonstationary linear systems with state and control delays (1980)
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    Journal of optimization theory and applications 30 (1980), S. 73-88 
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    Keywords: Optimal control ; suboptimal control ; time-delay systems ; linear systems
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    Topics: Mathematics
    Notes: Abstract A computationally efficient method based on a sensitivity approach is developed to obtain suboptimal control for nonstationary linear systems with multiple state and control delays and with quadratic cost. The proposed suboptimal control includes some truncated series whose terms are calculated in a recursive manner from nondelay system optimization.
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    Suboptimal control of systems with multiple delays (1980)
    Springer
    Journal of optimization theory and applications 30 (1980), S. 621-633 
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    ISSN: 1573-2878
    Keywords: Optimal control ; time-delay systems ; linear systems ; multiple-delay systems
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    Notes: Abstract The application of Pontryagin's maximum principle to the optimization of linear systems with time delays results in a system of coupled two-point boundary-value problems involving both delay and advance terms. The exact solution of this system of TPBV problems is extremely difficult, if not impossible. In this paper, a fast-converging iterative approach is developed for obtaining the suboptimal control for nonstationary linear systems with multiple state and control delays and with quadratic cost. At each step of the proposed method, a linear nondelay system with an extra perturbing input must be optimized. The procedure can be extended for the optimization of nonlinear systems with multiple time-varying delays, provided that some of the nonlinearities satisfy the Lipschitz condition.
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    Solution of nonlinear optimal control problems in Hilbert spaces by means of linear programming techniques (1980)
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    Journal of optimization theory and applications 30 (1980), S. 643-661 
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    ISSN: 1573-2878
    Keywords: Optimal control ; measures ; Hilbert spaces ; linear programming ; approximation techniques
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    Topics: Mathematics
    Notes: Abstract Optimal control problems in Hilbert spaces are considered in a measure-theoretical framework. Instead of minimizing a functional defined on a class of admissible trajectory-control pairs, we minimize one defined on a set of measures; this set is defined by the boundary conditions and the differential equation of the problem. The new problem is an infinite-dimensionallinear programming problem; it is shown that it is possible to approximate its solution by that of a finite-dimensional linear program of sufficiently high dimensions, while this solution itself can be approximated by a trajectory-control pair. This pair may not be strictly admissible; if the dimensionality of the finite-dimensional linear program and the accuracy of the computations are high enough, the conditions of admissibility can be said to be satisfied up to any given accuracy. The value given by this pair to the functional measuring the performance criterion can be about equal to theglobal infimum associated with the classical problem, or it may be less than this number. It appears that this method may become a useful technique for the computation of optimal controls, provided the approximations involved are acceptable.
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    Junction phenomena for optimal control with state-variable inequality constraints of third order (1980)
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    Journal of optimization theory and applications 31 (1980), S. 85-99 
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    Keywords: Optimal control ; junction conditions ; calculus of variations ; inequality constraints
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    Notes: Abstract It is known that extremal arcs governed by inequality constraints of third order (constraint relations that must be differentiated three times to generate a control equation) cannot join an unconstrained arc, except in special cases. But a control problem is exhibited, for which every extremal includes a constrained arc of third order. The constrained arc joins the end of an infinite sequence of consecutive unconstrained arcs of finite total duration. Evidence (but not proof) is given that this phenomenon is typical, rather than exceptional. An analogous phenomenon is well known for optimal control problems with singular arcs of second order.
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    Existence and global asymptotic stability of optimal trajectories for a class of infinite-horizon, nonconvex systems (1980)
    Springer
    Journal of optimization theory and applications 31 (1980), S. 515-533 
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    ISSN: 1573-2878
    Keywords: Optimal control ; economic systems ; stability ; existence theory ; turnpike property
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    Notes: Abstract Sufficient conditions for the existence of optimal trajectories and for the global asymptotic stability of these trajectories are given for a class of nonconvex and nonautonomous systems controlled over an infinite-time horizon. The concept ofG-supported trajectory is introduced. It is shown that, under some assumptions, aG-supported trajectory is overtaking and is globally asymptotically stable. The concept of overtaking trajectory has been previously defined as a notion of optimality on an infinite-time domain. For autonomous systems, under weaker conditions, one guarantees the existence of weakly overtaking trajectories. Finally, it is shown howG-supported trajectories can be obtained, and an application to the study of a pre-predator ecosystem optimally harvested is sketched.
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    On optimal control problems with state constraints (1980)
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    Journal of optimization theory and applications 32 (1980), S. 75-80 
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    Keywords: Optimal control ; state constraints ; multipliers ; perturbation theory
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    Topics: Mathematics
    Notes: Abstract This paper is concerned with necessary conditions for a general optimal control problem developed by Russak and Tan. It is shown that, in most cases, a further relation between the multipliers holds. This result is of interest in particular for the investigation of perturbations of the state constraint.
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    A property of an autonomous minimax optimal control problem (1980)
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    Journal of optimization theory and applications 32 (1980), S. 81-87 
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    Keywords: Optimal control ; minimax problems ; maximum principle ; Hamiltonian function
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    Notes: Abstract A control system $$\dot x = f\left( {x,u} \right)$$ ,u) with cost functional $$\mathop {ess \sup }\limits_{T0 \leqslant t \leqslant T1} G\left( {x\left( t \right),u\left( t \right)} \right)$$ is considered. For an optimal pair $$\left( {\bar x\left( \cdot \right),\bar u\left( \cdot \right)} \right)$$ ,ū(·)), there is a maximum principle of the form $$\eta \left( t \right)f\left( {\bar x\left( t \right),\bar u\left( t \right)} \right) = \mathop {\max }\limits_{u \in \Omega \left( t \right)} \eta \left( t \right)f\left( {\bar x\left( t \right),u} \right).$$ By means of this fact, it is shown that $$\eta \left( t \right)f\left( {\bar x\left( t \right),\bar u\left( t \right)} \right)$$ is equal to a constant almost everywhere.
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    An exact penalty function algorithm for optimal control problems with control and terminal equality constraints, part 1 (1980)
    Springer
    Journal of optimization theory and applications 32 (1980), S. 211-246 
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    ISSN: 1573-2878
    Keywords: Optimal control ; control constraints ; terminal equality constraints ; exact penalty function ; convergence
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    Topics: Mathematics
    Notes: Abstract The presence of control constraints, because they are nondifferentiable in the space of control functions, makes it difficult to cope with terminal equality constraints in optimal control problems. Gradient-projection algorithms, for example, cannot be employed easily. These difficulties are overcome in this paper by employing an exact penalty function to handle the cost and terminal equality constraints and using the control constraints to define the space of permissible search directions in the search-direction subalgorithm. The search-direction subalgorithm is, therefore, more complex than the usual linear program employed in feasible-directions algorithms. The subalgorithm approximately solves a convex optimal control problem to determine the search direction; in the implementable version of the algorithm, the accuracy of the approximation is automatically increased to ensure convergence.
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    Sufficient conditions for asymptotic optimal control (1980)
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    Journal of optimization theory and applications 32 (1980), S. 307-325 
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    ISSN: 1573-2878
    Keywords: Optimal control ; asymptotic optimal control ; calculus of variations ; Lagrange problems ; global minima ; absolute minima ; necessary conditions ; sufficient conditions
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    Notes: Abstract For a selected family of Lagrange-type control problems involving a nonnegative integral costJ T (y,u) over the interval [0,T], 0〈T〈∞, with system conditions consisting of differential inequalities and/or equalities, the following material is treated: (i) a resumé of relevant necessary conditions and sufficient conditions for a pair (y T ,u T ) to minimizeJ T (y,u); (ii) conditions sufficient for the convergence asT→∞ of minimizing pairs (y T ,u T ) over [0,T] to a limit pair (y ∞,u ∞) over the infinite-time interval [0, ∞); (iii) conditions sufficient for (y ∞,u ∞) to minimize the costJ ∞(y,u) over [0, ∞); and (iv) conditions sufficient for the optimal cost per unit timeJ T (y T ,u T )/T to have a limit asT→∞.
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    An exact penalty function algorithm for optimal control problems with control and terminal equality constraints, part 2 (1980)
    Springer
    Journal of optimization theory and applications 32 (1980), S. 345-364 
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    ISSN: 1573-2878
    Keywords: Optimal control ; control constraints ; terminal equality constraints ; exact penalty functions ; convergence ; relaxed control
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    Topics: Mathematics
    Notes: Abstract In Part 1 of this paper, implementable and conceptual versions of an algorithm for optimal control problems with control constraints and terminal equality constraints were presented. It was shown that anyL ∞ accumulation points of control sequences generated by the algorithms satisfy necessary conditions of optimality. Since such accumulation points need not exist, it is shown in this paper that control sequences generated by the algorithms always have accumulation points in the sense of control measure, and these accumulation points satisfy optimality conditions for the corresponding relaxed control problem.
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    On the order of singular optimal control problems (1980)
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    Journal of optimization theory and applications 32 (1980), S. 479-489 
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    Keywords: Optimal control ; singular control ; junction conditions ; singular control order
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    Topics: Mathematics
    Notes: Abstract In singular optimal control problems, the functional form of the optimal control function is usually determined by solving the algebraic equation which results by successively differentiating the switching function until the control appears explicitly. This process defines the order of the singular problem. Order-related results are developed for singular linear-quadratic problems and for a bilinear example which gives new insights into the relationship between singular problem order and singular are order.
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    On the general inverse problem of optimal control theory (1980)
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    Journal of optimization theory and applications 32 (1980), S. 491-497 
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    Keywords: Optimal control ; inverse problems ; dynamic programming
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    Notes: Abstract The general inverse problem of optimal control is considered from a dynamic programming point of view. Necessary and sufficient conditions are developed which two integral criteria must satisfy if they are to yield the same optimal feedback law, the dynamics being fixed. Specializing to the linear-quadratic case, it is shown how the general results given here recapture previously obtained results for quadratic criteria with linear dynamics.
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    Realistic feedback control of two interconnected turbogenerators (1984)
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    Journal of optimization theory and applications 42 (1984), S. 103-141 
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    Keywords: Optimal control ; realistic feedback control ; interconnected turbogenerators ; torque and voltage control
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    Notes: Abstract In this paper, the optimal control of a system with two identical interconnected turbogenerators, which are connected to an infinite bus, is considered. The alternators are controlled through a linear feedback of the state variables. The feedback parameters are obtained by solving a nonlinear, two-point boundary-value problem. The values obtained for these parameters depend on the strength and duration of the disturbance, since the model is nonlinear, contrary to the usual feedback control of a linear model. In contrast to the model used in Ref. 1, the model used here includes the transfer function of the governors, the turbines, and the voltage regulators.
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    Convergence of a strong variational algorithm for relaxed controls involving a class of hyperbolic systems (1984)
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    Journal of optimization theory and applications 42 (1984), S. 467-485 
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    Keywords: Optimal control ; hyperbolic partial differential equations ; Darboux boundary conditions ; strong variational algorithm ; convergence ; relaxed control
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    Notes: Abstract In this paper, we consider a class of optimal control problems involving linear hyperbolic partial differential equations with Darboux boundary conditions. A strong variational algorithm has been obtained for solving this class of optimal control problems in a previous paper by the third and the first authors. It was also shown that anyL ∞ accumulation points of control sequences generated by the algorithm satisfy a necessary condition for optimality. Since such accumulation points need not exist, it is shown in this paper that the control sequences generated by the algorithm always have accumulation points in the sense of control measure, and these accumulation points satisfy a necessary condition for optimality for the corresponding relaxed control problems.
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    Necessary and sufficient conditions for optimal control of quasilinear partial differential systems (1984)
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    Journal of optimization theory and applications 43 (1984), S. 89-101 
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    Keywords: Optimal control ; distributed-parameter systems ; control of partial differential equations ; sufficiency conditions ; Green's formula
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    Notes: Abstract First-order necessary and sufficient conditions are obtained for the following quasilinear distributed-parameter optimal control problem: $$max\left\{ {J(u) = \int_\Omega {F(x,u,t) d\omega + } \int_{\partial \Omega } {G(x,t) \cdot d\sigma } } \right\},$$ subject to the partial differential equation $$A(t)x = f(x,u,t),$$ wheret,u,G are vectors andx,F are scalars. Use is made of then-dimensional Green's theorem and the adjoint problem of the equation. The second integral in the objective function is a generalized surface integral. Use of then-dimensional Green's theorem allows simple generalization of single-parameter methods. Sufficiency is proved under a concavity assumption for the maximized Hamiltonian $$H^\circ (x,\lambda ,t) = \max \{ H(x,u,\lambda ,t):u\varepsilon K\} $$ .
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    Existence of an optimal control with sparse jumps in the state variable (1985)
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    Journal of optimization theory and applications 45 (1985), S. 265-293 
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    ISSN: 1573-2878
    Keywords: Optimal control ; control differential systems involving impulses ; jumps in the state variables
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    Notes: Abstract The existence of an optimal control is proved in a problem where the criterion functional and the equation of motion contain a control that is a Lebesgue-Stieltjes measure. Due to nonlinearities in the problem, it is necessary to postulate a condition implying that large atoms of the control measures are sparse and that their derivatives, wherever they exist, have a uniform bound.
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    Optimal control of a rotary crane (1981)
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    Journal of optimization theory and applications 35 (1981), S. 535-557 
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    Keywords: Optimal control ; rotary crane ; nonlinear systems ; computational algorithms
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    Notes: Abstract This paper is concerned with the optimal control of a rotary crane, which makes two kinds of motion (rotation and hoisting) at the same time. The optimal control which transfers a load to a desired place as fast as possible and minimizes the swing of the load during the transfer, as well as the swing at the end of transfer, is calculated on the basis of a dynamic model. A new computational technique is employed for computing the optimal control, and several numerical results are presented.
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    A class of nondifferentiable control problems (1988)
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    Journal of optimization theory and applications 56 (1988), S. 227-243 
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    Keywords: Optimal control ; nondifferentiable control problems ; optimality conditions ; Fritz John optimality conditions ; duality ; converse duality ; continuous programming ; mathematical programming
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    Topics: Mathematics
    Notes: Abstract Optimality conditions and duality results are obtained for a class of control problems having a nondifferentiable term in the integrand of the objective functional. These results generalize many well-known results in optimal control theory involving differentiable functions, and also provide a relationship with certain nondifferentiable mathematical programming problems. Some extensions concerning the unified treatment of optimal control theory and continuous programming are also mentioned. Finally, a control problem containing an arbitrary norm, along with its appropriate norm, is given.
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    Optimal feedback control laws using nonlinear programming (1991)
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    Journal of optimization theory and applications 71 (1991), S. 599-611 
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    Keywords: Optimal control ; feedback control ; trajectory optimization
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    Notes: Abstract A procedure of parametrizing feedback controls when solving the optimal control problem using nonlinear programming is considered. The maximum principle is utilized to determine the forms of the parametrized feedback control. Applications are demonstrated by numerical examples.
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    Approximation of relaxed nonlinear parabolic optimal control problems (1993)
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    Journal of optimization theory and applications 77 (1993), S. 31-50 
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    Keywords: Optimal control ; nonlinear parabolic systems ; discretization ; approximation ; relaxed controls ; distributed systems
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    Notes: Abstract We consider a relaxed optimal control problem for systems defined by nonlinear parabolic partial differential equations with distributed control. The problem is completely discretized by using a finite-element approximation scheme with piecewise linear states and piecewise constant controls. Existence of optimal controls and necessary conditions for optimality are derived for both the continuous and the discrete problem. We then prove that accumulation points of sequences of discrete optimal [resp. extremal] controls are optimal [resp. extremal] for the continuous problem.
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    Optimal control of continuous-time deterministic systems with incomplete discrete feedback (1990)
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    Journal of optimization theory and applications 64 (1990), S. 557-571 
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    Keywords: Optimal control ; deterministic systems ; discrete incomplete information ; sufficient conditions for optimality ; feedback systems
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    Notes: Abstract Sufficient conditions for optimality are obtained for controls that depend on the current time and values of known functions of the state vector at finite points of the time interval. The equations for finding the required control laws are derived. An example is given for which an exact solution of the problem can be obtained.
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    On a student-related optimal control problem (1990)
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    Journal of optimization theory and applications 65 (1990), S. 129-138 
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    Keywords: Optimal control ; mathematical modelling ; study strategies
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    Notes: Abstract In a recent paper (Ref. 1), Cheng and Teo discussed some further extensions of a student-related optimal control problem which was originally proposed by Raggettet al. (Ref. 2) and later on modified by Parlar (Ref. 3). In this paper, we treat further extensions of the problem.
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    Boundaries of the receding horizon control for interconnected systems (1995)
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    Journal of optimization theory and applications 84 (1995), S. 251-271 
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    Keywords: Optimal control ; decentralized control ; interconnected systems ; receding horizon control ; rolling horizon control ; predictive control ; quadratic optimization
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    Topics: Mathematics
    Notes: Abstract In recent years, the finite-horizon quadratic minimization problem has become popular in process control, where the horizon is constantly rolled back. In this paper, this type of control, which is also called the receding horizon control, is considered for interconnected systems. First, the receding horizon control equations are formulated; then, some stability conditions depending on the interconnection norms and the horizon lengths are presented. For ∈-coupled systems, stability results similar to centralized systems are obtained. For interconnected systems which are not ∈-coupled, the existence of a horizon length and a corresponding stabilizing receding horizon control are derived. Finally, the performance of a locally computed receding horizon control for time-invariant and time-varying systems with different updating intervals is examined in an example.
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    Time-optimal extension and retraction of robots: Numerical analysis of the switching structure (1995)
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    Journal of optimization theory and applications 84 (1995), S. 589-616 
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    Keywords: Optimal control ; robotics ; path planning ; maximum principle ; multi-point boundary value problems ; multiple shooting
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    Topics: Mathematics
    Notes: Abstract The problem of the time-optimal control of robot manipulators is of importance because of its potential for increasing the productivity of assembly lines. This work is part of a series of papers by the authors on this topic using direct and indirect methods of optimization. A cylindrical robot or a spherical polar robot constrained to the horizontal plane is considered, and optimal solutions for radial maneuvers are generated. Indirect methods are employed in order to establish the switching structure of the solutions. The results show that even such apparently simple maneuvers as extension or retraction of a robot with a prismatic joint can produce very complex optimal solutions. Time-optimal retraction can exhibit ten different switching structures with eight switching points and two singular arcs.
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    Second-order sufficient conditions for control problems with mixed control-state constraints (1995)
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    Journal of optimization theory and applications 86 (1995), S. 649-667 
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    Keywords: Optimal control ; mixed control-state constraints ; Hamilton-Jacobi inequality ; second-order sufficient conditions ; parametric optimization ; Riccati equations
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    Topics: Mathematics
    Notes: Abstract References 1–4 develop second-order sufficient conditions for local minima of optimal control problems with state and control constraints. These second-order conditions tighten the gap between necessary and sufficient conditions by evaluating a positive-definiteness criterion on the tangent space of the active constraints. The purpose of this paper is twofold. First, we extend the methods in Refs. 3, 4 and include general boundary conditions. Then, we relate the approach to the two-norm approach developed in Ref. 5. A direct sufficiency criterion is based on a quadratic function that satisfies a Hamilton-Jacobi inequality. A specific form of such a function is obtained by applying the second-order sufficient conditions to a parametric optimization problem. The resulting second-order positive-definiteness conditions can be verified by solving Riccati equations.
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    Use of Augmented Lagrangian Methods for the Optimal Control of Obstacle Problems (1997)
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    Journal of optimization theory and applications 95 (1997), S. 101-126 
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    Keywords: Optimal control ; Lagrange multipliers ; augmented Lagrangians ; variational inequalities
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    Notes: Abstract We investigate optimal control problems governed by variational inequalities involving constraints on the control, and more precisely the example of the obstacle problem. In this paper, we discuss some augmented Lagrangian algorithms to compute the solution.
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    Optimal Control of a Chemical Vapor Deposition Reactor (1998)
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    Journal of optimization theory and applications 97 (1998), S. 623-644 
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    Keywords: Optimal control ; bang-bang control ; free boundary problems ; parabolic equations ; homogenizations ; optimality conditions
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    Notes: Abstract We study a simple model of chemical vapor deposition on a silicon wafer. The control is the flux of chemical species, and the objective is to grow the semiconductor film so that its surface attains a prescribed profile as nearly as possible. The surface is spatially fast oscillating due to the small feature scale, and therefore the problem is formulated in terms of its homogenized approximation. We prove that the optimal control is bang-bang, and we use this information to develop a numerical scheme for computing the optimal control.
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    Tracking Demands in Optimal Control of Managerial Systems with Continuously-Divisible, Doubly Constrained Resources (1998)
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    Journal of global optimization 13 (1998), S. 43-59 
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    ISSN: 1573-2916
    Keywords: Resource constrained scheduling ; renewable and nonrenewable resources ; Optimal control
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The paper addresses problems of allocating continuously divisible resources among multiple production activities. The resources are allowed to be doubly constrained, so that both usage at every point of time and cumulative consumption over a planning horizon are limited as it is often the case in project and production scheduling. The objective is to track changing in time demands for the activities as closely as possible. We propose a general continuous-time model that states the problem in a form of the optimal control problem with non-linear speed-resource usage functions. With the aid of the maximum principle, properties of the solutions are derived to characterize optimal resource usage policies. On the basis of this analytical investigation, numerical scheduling methods are suggested and computationally studied.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008256009222
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    Spectral analysis on a small computer (1973)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 6 (1973), S. 395-411 
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    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The design and implementation of software for real-time spectral analysis on an IBM 1130 with 8k core is described. The procedures for estimation of coherence and cross-spectra are discussed in detail.
    Additional Material: 8 Ill.
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    URL: http://dx.doi.org/10.1002/nme.1620060310
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    Application of the Fast Fourier Transform to linear systems in civil engineering (1973)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 6 (1973), S. 381-394 
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    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The Fast Fourier Transform is employed as a method of Laplace transform inversion to solve problems in the civil engineering fields of visco-elasticity and hydrology. When these problems are accurately represented by a linear time-invariant model, it is shown that the Fast Fourier Transform inversion procedure is often more accurate than standard convolution and inversion techniques. The speed and accuracy of solution resulting from this application of the Fast Fourier Transform is illustrated by referring to case studies solved on an IBM 1130 computer, model 3D processor with 32K memory.
    Additional Material: 13 Ill.
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    URL: http://dx.doi.org/10.1002/nme.1620060309
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    Determination of stress intensity factors in cracked plates by the finite element method (1973)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 6 (1973), S. 427-439 
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    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Stress fields near crack tips in an elastic body can be specified by the stress intensity factors which are closely related to the stress singularities arising from the crack tips. These singularities, however, cannot be represented exactly by conventional finite element models. A new method for the analysis of stresses around cracks is proposed in this paper on the basis of the superposition of analytical and finite element solutions. This method is applied to several two-dimensional problems whose solutions are obtained analytically, and it is shown that their numerical results are in excellent agreement with analytical ones. Sufficiently accurate results can be obtained by the conventional finite element analysis with rather coarse mesh subdivision. Computational efforts are then considerably reduced compared with other methods.
    Additional Material: 8 Ill.
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    URL: http://dx.doi.org/10.1002/nme.1620060312
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    Reply to comment By R. G. Vos (1973)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 6 (1973), S. 447-447 
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    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620060316
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  • 83
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    Letters to the editor (1973)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 6 (1973), S. 454-456 
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    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620060320
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    Advanced engineering mathematics, Erwin Kreyszig, Wiley, London, 1972. No. of pages: 898. Price: £6.25 (1973)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 6 (1973), S. 456-456 
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    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620060321
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  • 85
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    A mixed finite element model for use in potential flow problems (1973)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 6 (1973), S. 467-473 
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    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A generalized variational principle is presented, which leads to a modified finite element approach for three-dimensional field problems. Both, potential function and velocity field, are approximated by expansions which are continuous across the inter-element boundaries. The rapid convergence of this mixed model is shown by two examples.
    Additional Material: 7 Ill.
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    URL: http://dx.doi.org/10.1002/nme.1620060403
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    Computer program for solution of large, sparse, unsymmetric systems of linear equations (1973)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 6 (1973), S. 497-509 
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    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A computer program for the in-core solution of large, sparse, unsymmetric systems of linear equations is presented in this paper. The program employs elimination techniques for solution of systems of linear equations. A limited number of zeros is stored and trivial arithmetic is by-passed to preserve computer storage and to reduce the time required for solution. Several techniques for selecting the pivotal elements are discussed and their effect on accuracy and computational time are examined.
    Additional Material: 1 Ill.
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    URL: http://dx.doi.org/10.1002/nme.1620060406
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  • 87
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    Impulsively started viscous flow past a finite flat plate with and without an applied magnetic field (1973)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 6 (1973), S. 521-527 
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    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Numerical methods are used to investigate the two-dimensional motion of a viscous incompressible fluid impulsively started past a flat plate of finite breadth at zero incidence to the uniform motion of the fluid at large distances from the plate. A step by step integration in time of Helmholtz's vorticity equation is used for Reynolds numbers 10-500.The magnetohydrodynamic case is also considered with the applied magnetic field at infinity parallel to the uniform stream and the non-conducting plate. Results for the Magnetic Reynolds number 50 and infinite, Viscous Reynolds number 50 and 0≤β≤2, where β is the ratio of the square of the Alfvén speed to the square of the main stream velocity, are presented.
    Additional Material: 4 Ill.
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    URL: http://dx.doi.org/10.1002/nme.1620060408
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    Handbook of industrial control computers, T. J. Harrison, Wiley Interscience, New york, 1972. No. of pages: 1,056. Price: £17.80 (1973)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 6 (1973), S. 608-608 
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    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620060424
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    Triangular finite elements for the generalized Bessel equation of order m (1973)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 7 (1973), S. 43-55 
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    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A new family of triangular finite elements is described, useful for solving the axisymmetric vector Helmholtz equation, and a variety of scalar Helmholtz equation problems which lead to generalized Bessel equations of some order m. This family is similar in principle to the scalar axisymmetric Helmholtz elements derived earlier, but requires both reformulation of its describing equations and corresponding new universal element matrices, for successful computational implementation. The necessary formulation is given in this paper. Matrix elements to the sixth-order inclusive have been calculated and extensively tested computationally.
    Additional Material: 4 Ill.
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    URL: http://dx.doi.org/10.1002/nme.1620070104
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    Stability criteria for explicit finite difference solutions of the parabolic diffusion equation with non-linear boundary conditions (1973)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 7 (1973), S. 57-67 
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    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The criteria for stability of the explicit finite difference solution of the one-dimensional, transient, conduction heat transfer problem with both radiant and convection heat transfer at the boundaries are considered in this paper. These criteria are governed by an inequality set from a functional relationship between the newly calculated and the old temperature at each node.From the node with the most stringent criteria, it is shown that setting the coefficient of the old temperature equal to zero in the governing difference equation is not sufficient for a general criterion. On the other hand, setting the derivative of the new temperature with respect to the old temperature equal to zero in the governing difference equation presents a simple, straightforward technique for obtaining a sufficient condition for a stable system. It is further shown that the second law of thermodynamics, written in explicit finite difference form, does present a necessary criterion for stability. However, the second law, because it is in the form of an inequality, does not present as simple a criterion as the derivative method does.The specific problem studied is a finite thickness slab, initially at a uniform temperature, but instantaneously subjected to both radiation and convection on its two surfaces. Temperature profiles were calculated on a digital computer and are presented in dimensionless graphical form over a range of five dimensionless parameters. A plot that relates stability to the maximum time-step size for the entire range of practical conditions of radiation numbers is also presented.
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    URL: http://dx.doi.org/10.1002/nme.1620070105
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    Masthead (1973)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 7 (1973) 
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    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620070201
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    A characterization of stiffly stable linear multistep methodsThis research was partially supported by the Old Dominion University Research Foundation, Old Dominion University, Norfolk, Virginia, U.S.A. (1973)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 7 (1973), S. 117-124 
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    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A theorem expressing geometric conditions necessary and sufficient for the stiff stability of a linear multistep method is obtained. A corollary to this theorem provides sufficient conditions for stiff stability, for the class of algorithms characterized by a single essential root. By applying the Schur-Cohn criterion and employing a computerized algebraic manipulation program, the result provides a first step towards an exhaustive search technique for determining a continuum of members of this class.
    Additional Material: 4 Ill.
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    URL: http://dx.doi.org/10.1002/nme.1620070203
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    An iterative method for large systems of linear structural equations (1973)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 7 (1973), S. 175-183 
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    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A method is described for the solution of large sets of sparse equations arising in structural analysis. This method, called partial elimination, combines the concepts of elimination and iteration in such a way that good convergence rates can be obtained using a computer storage space not much greater than that required for other iterative methods.
    Additional Material: 5 Ill.
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    URL: http://dx.doi.org/10.1002/nme.1620070207
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    Computing with sparse matrices (1973)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 7 (1973), S. 211-223 
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    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A survey of recent developments in sparse matrix technology is presented. Two fundamental areas are reviewed: 1.Sorting and reordering techniques by which the non-zero elements of a given sparse matrix can be rearranged to obtain a form which leads to more efficient computations.2.Direct methods for solving systems of linear equations and computing inverses.
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    URL: http://dx.doi.org/10.1002/nme.1620070210
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    Finite element method applied to the problem of stability of a non-conservative system (1971)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 63-87 
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    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The finite element method is applied to the stability analysis of structural systems subject to non-conservative forces. The development of the method is general, but the specific application considered here is the stability of thin-walled members subject to follower forces. The method predicts the type of instability, whether it be buckling or flutter. Example problems, for which exact solutions are known, illustrate the accuracy and convergence characteristic of the finite element formulation.
    Additional Material: 10 Ill.
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    URL: http://dx.doi.org/10.1002/nme.1620030110
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    Announcement (1973)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 7 (1973), S. 240-240 
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    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620070216
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    Comparison of some penalty function based optimization procedures for the synthesis of a planar truss (1973)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 7 (1973), S. 155-173 
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    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The present investigation examines the multibar truss optimization problem in the context of a general class of unconstrained optimization procedures in conjunction with various types of penalty function transformations. Specifically, the problem is transformed into a series of unconstrained minimization problems using the penalty function techniques of Heaviside and SUMT. These are solved using the methods of Rosenbrock (orthogonal directions), Powell (conjugate directions) and Nelder-Mead (Simplex). This resulted in many cases in substantial improvements being recorded over previously reported data. The paper includes a comparative study of the synthesis based on these procedures.
    Additional Material: 16 Ill.
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    URL: http://dx.doi.org/10.1002/nme.1620070206
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    Dynamic analysis of structures using the rank force method (1971)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 103-117 
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    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The application of the finite element rank force method for the dynamic analysis of redundant structures is presented. The applied loading consists of discrete and distributed loads. Both forced and free vibrations are considered, the latter yielding latent vectors corresponding to dynamic redundancies.It is shown that the dynamic properties of an element are given by a static flexibility matrix, an inverse mass matrix, a damping parameter and a displacement vector representing the effect of the applied distributed loading.
    Additional Material: 5 Ill.
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    URL: http://dx.doi.org/10.1002/nme.1620030112
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    An annular segment finite element for plate bending (1971)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 119-129 
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    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A stiffness matrix for a finite element having the planform of an annular segment is derived using the displacement approach. Numerical problems involved in the derivation are discussed and rapid convergence to exact solutions is demonstrated on three sample problems. It is concluded that the new element will be of great value to engineers concerned with the analysis of slabs of bridge decks curved in plan.
    Additional Material: 8 Ill.
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    URL: http://dx.doi.org/10.1002/nme.1620030113
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    Stress and deformation analysis of fibrous composite materials by point matching (1971)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 89-101 
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    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The point matching numerical method and its generalization, the method of boundary point least squares, have been successfully applied to numerous boundary value and eigenvalue problems. The present paper demonstrates the application of these techniques to problems in the micromechanics of fibrous composite materials, i.e. determination of elastic moduli and stress concentrations for parallel-fibre materials which are loaded transversely with respect to the fibres. The solution technique utilizes exact solutions of the governing equations of plane elasticity for each component fibre and its surrounding matrix material in a typical repeating section of the composite material. The continuity conditions for stresses and displacements between fibre and matrix and the repeatability conditions at the boundary of the repeating section are satisfied approximately in a pointwise manner. Some special numerical techniques which were found to be particularly useful in applying the point matching method to these problems are delineated. The method is demonstrated for composite materials having circular, elliptical and square fibres in regular, staggered arrays. Numerical results are given which show the accuracy of the method as well as stress concentration and composite elastic moduli data.
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    URL: http://dx.doi.org/10.1002/nme.1620030111
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