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  • Articles  (44)
  • Optimal control  (44)
  • 2020-2023
  • 1990-1994  (44)
  • Mathematics  (43)
  • Energy, Environment Protection, Nuclear Power Engineering  (1)
  • 1
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    Annals of operations research 54 (1994), S. 237-262 
    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.
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  • 2
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    Annals of operations research 54 (1994), S. 217-235 
    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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  • 3
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    Applied mathematics & optimization 30 (1994), S. 1-14 
    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.
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  • 4
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    Applied mathematics & optimization 30 (1994), S. 113-126 
    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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  • 5
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    Applied mathematics & optimization 29 (1994), S. 285-307 
    ISSN: 1432-0606
    Keywords: Optimal control ; Optimality conditions ; State constraints ; Lagrange multipliers ; Parabolic systems ; Penalization ; 49B22
    Source: Springer Online Journal Archives 1860-2000
    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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  • 6
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    Applied mathematics & optimization 29 (1994), S. 309-329 
    ISSN: 1432-0606
    Keywords: Optimal control ; Parabolic equation ; Nonlinear boundary condition ; State constraint ; Ritz-Galerkin approximation ; 49K20 ; 65K10 ; 65M60
    Source: Springer Online Journal Archives 1860-2000
    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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  • 7
    ISSN: 1432-1009
    Keywords: Cement dust ; Atmospheric particulates pollution ; Optimal control ; Economic assessment
    Source: Springer Online Journal Archives 1860-2000
    Topics: Energy, Environment Protection, Nuclear Power Engineering
    Notes: Abstract At present, many rural enterprises in China's mountainous areas are developing rapidly, and due to poor planning and improper management, in an uncontrolled manner. These small enterprises are making atmospheric pollution far more serious and more difficult to control than before. Thus, as is the case with most developing countries, China is facing the challenge of managing the increased environmental pollution that is accompanying its economic development. This paper examines the case of cement dust pollution in the town of Wenquan in Sichuan province in order to determine how to adjust the relationship between development of rural enterprises and atmospheric environmental pollution in mountainous areas. Using the single objective linear programming method, and based on the principle of overall optimization, an optimal control plan for different pollution sources was worked out and an economic assessment on reclamation of cement dust was completed. According to our analysis, after implementation of this plan for four years, two months, the concentrations of suspended particles in Wenquan will achieve the requirement of the national third-order ambient air quality standard; the current serious dust pollution will be completely controlled; and the reclamation of cement dust will totally compensate for the cost of dust control and will result in 92,000 yuan of pure profit per year. At that time, the economy and environment will be in harmony.
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  • 8
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    Journal of optimization theory and applications 83 (1994), S. 489-509 
    ISSN: 1573-2878
    Keywords: Optimal control ; dynamics of the firm ; investment policy ; pollution standards
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The purpose of this paper is to determine the effects of different pollution standards on the firm's resource allocation decisions. To do so, a dynamic model of the firm is developed in which it is assumed that production causes pollution as an inevitable byproduct. Concerning its investment policy, we suppose that the firm can choose between investing in productive capital goods and investing in abatement efforts. It is shown that, in some cases, future abatement expenses have a negative impact on the present level of productive investment, even if the pollution standard is not binding at the moment. This implies a really dynamic optimal investment policy for the firm, which cannot be obtained within a comparative static analysis.
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  • 9
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    Journal of optimization theory and applications 80 (1994), S. 227-260 
    ISSN: 1573-2878
    Keywords: Optimal control ; maximum principle ; sufficiency field theorem ; brachistochrone problem ; classical differential geometry ; normal and abnormal controls
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The Frenet-Serret equations of classical differential geometry are used to describe the quickest descent tunneling path problem. The optimal tunnel is shown to have a constant turn rate with zero torsion and is equivalent to Edelbaum's hypocycloid solution. The solutions are obtained using the maximum principle and singular arc conditions. The optimal curvature is a first-order singular arc and the optimal torsion is a second-order singular arc. Our treatment includes both the normal and the abnormal optimal control problems. Our problem is abnormal for the case where the final speed is zero. Analytical solutions for the optimal time histories are derived for all states and all adjoint states. One of Leitmann's sufficiency field theorems is used to establish optimality of the solutions.
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  • 10
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    Journal of optimization theory and applications 82 (1994), S. 295-313 
    ISSN: 1573-2878
    Keywords: Optimal control ; delay-differential equations ; continuous inequality state constraints ; terminal inequality state constraints ; terminal equality state constraints ; control parametrization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A computational algorithm for solving a class of optimal control problems involving terminal and continuous state constraints of inequality type was developed in Ref. 1. In this paper, we extend the results of Ref. 1 to a more general class of constrained time-delayed optimal control problems, which involves terminal state equality constraints as well as terminal state inequality constraints and continuous state constraints. Two examples have been solved to illustrate the efficiency of the method.
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  • 11
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    Applied mathematics & optimization 28 (1993), S. 113-132 
    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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  • 12
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    Applied mathematics & optimization 28 (1993), S. 225-257 
    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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  • 13
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    Applied mathematics & optimization 27 (1993), S. 35-56 
    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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  • 14
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    Mathematics of control, signals, and systems 6 (1993), S. 180-193 
    ISSN: 1435-568X
    Keywords: Optimal control ; Dynamic programming ; Free endtime ; Hamilton-Jacobi theory
    Source: Springer Online Journal Archives 1860-2000
    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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  • 15
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    Journal of optimization theory and applications 77 (1993), S. 31-50 
    ISSN: 1573-2878
    Keywords: Optimal control ; nonlinear parabolic systems ; discretization ; approximation ; relaxed controls ; distributed systems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    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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  • 16
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    Journal of optimization theory and applications 79 (1993), S. 581-597 
    ISSN: 1573-2878
    Keywords: Optimal control ; heat equation ; convective boundary condition ; optimality system
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We consider the problem of controlling the solution of the heat equation with the convective boundary condition taking the heat transfer coefficient as the control. We take as our cost functional the sum of theL 2-norms of the control and the difference between the temperature attained and the desired temperature. We establish the existence of solutions of the underlying initial boundary-value problem and of an optimal control that minimizes the cost functional. We derive an optimality system by formally differentiating the cost functional with respect to the control and evaluating the result at an optimal control. We show how the solution depends in a differentiable way on the control using appropriate a priori estimates. We establish existence and uniqueness of the solution of the optimality system, and thus determine the unique optimal control in terms of the solution of the optimality system.
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  • 17
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    Journal of optimization theory and applications 77 (1993), S. 421-438 
    ISSN: 1573-2878
    Keywords: Optimal control ; investment policy ; adjustment costs ; installation experience
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    Topics: Mathematics
    Notes: Abstract This paper analyzes the consequences of incorporating a learning-by-doing effect in the firm's adjustment cost function. The hypothesis is that, the larger the existing capital stock, the larger the installation experience gained, and therefore the smaller the cost of installing an additional unit of capital stock. The implications of the hypothesis are investigated in an optimal control model for the determination of the firm's optimal investment policy over an infinite planning period.
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  • 18
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    Journal of optimization theory and applications 78 (1993), S. 493-521 
    ISSN: 1573-2878
    Keywords: Optimal control ; Lagrange multipliers ; augmented Lagrangians
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We consider state-constrained optimal control problems governed by elliptic equations. Doing Slater-like assumptions, we know that Lagrange multipliers exist for such problems, and we propose a decoupled augmented Lagrangian method. We present the algorithm with a simple example of a distributed control problem.
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  • 19
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    Annals of operations research 37 (1992), S. 375-401 
    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.
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  • 20
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    Applied mathematics & optimization 25 (1992), S. 109-126 
    ISSN: 1432-0606
    Keywords: Hamilton-Jacobi equation ; Viscosity solution ; Optimal control ; Neumann problem ; 35F20 ; 49B10 ; 49CO5
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    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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  • 21
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    Applied mathematics & optimization 26 (1992), S. 139-169 
    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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  • 22
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    Journal of optimization theory and applications 72 (1992), S. 319-332 
    ISSN: 1573-2878
    Keywords: Optimal control ; method of region analysis ; economics ; ecology
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    Topics: Mathematics
    Notes: Abstract A macroeconomic model on the relation between economic growth and the environment is investigated with the aim of illustrating how the method of region analysis can be used in solving practical problems formulated as regular optimal control problems with state constraints.
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  • 23
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    Journal of optimization theory and applications 73 (1992), S. 577-599 
    ISSN: 1573-2878
    Keywords: Optimal control ; inventory ; stochastic processes ; contraction mapping ; dynamic programming ; discounted dynamic programming ; Markovian decision processes ; periodic arrivals
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    Topics: Mathematics
    Notes: Abstract The system investigated consists of a stochastic periodic stream of raw material, a continuous processing operation with controllable deterministic service rates, and a storage facility. The arrival stream is periodically interrupted and divided into alternating on-off intervals of fixed length. The processing facility is allowed to operate during the off-interval. Superimposed on this system is a cost structure composed of processing and holding costs. Such operations may be found in manufacturing as well as service systems (for example, dry cleaners, machine shops, repair and maintenance shops, printers, information processing centers, etc). A service rate control rule that minimizes the infinite-horizon discounted expected total cost is found. Existence and uniqueness of long-term optimal cost and policy functions is shown. Since the optimal policy cannot be expressed explicitly, an approximate solution was obtained. An error bound on the optimal cost associated with this solution is exhibited. The approximate solution is characterized by a service rate control rule that is a linear function of the level of inventory at the start of each on-interval and a piecewise linear function of inventory at the start of each off-interval. The optimal discounted expected total cost is quadratic in the inventory level at the start of each interval. Computational results indicate relative cost errors in the order of 2–3 percent.
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    Journal of optimization theory and applications 75 (1992), S. 369-388 
    ISSN: 1573-2878
    Keywords: Optimal control ; life-cycle models ; Hopf bifurcation theory ; stable limit cycles ; human capital
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, we analyze the joint determination of optimal consumption and allocation of time between learning (accumulation of human capital), working for wages (used for consumption and accumulation of financial capital), and leisure. Using Hopf bifurcation theory, we are able to show that cyclical training, working, leisuring, and consumption are optimal under certain constellations of parameters.
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  • 25
    ISSN: 1572-9338
    Keywords: Optimal control ; Markov chains ; partial observability ; average cost ; optimality equation ; structured optimal policies
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    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.
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    Applied mathematics & optimization 24 (1991), S. 289-316 
    ISSN: 1432-0606
    Keywords: Optimal control ; Stochastic control ; Sampled-data controllers ; Digital control ; Optimal regulators ; Optimal tracking
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    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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    Journal of optimization theory and applications 71 (1991), S. 599-611 
    ISSN: 1573-2878
    Keywords: Optimal control ; feedback control ; trajectory optimization
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    Topics: Mathematics
    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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    Journal of optimization theory and applications 70 (1991), S. 607-618 
    ISSN: 1573-2878
    Keywords: Optimal control ; maximum principle ; piecewise deterministic processes ; recursive utility maximization
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    Topics: Mathematics
    Notes: Abstract We consider an optimal control problem in which the time horizon is a random variable and the discount factor may depend on the past state and control values. This problem combines features of controlled piecewise deterministic processes and recursive utility maximization. Applying a simple transformation and a refined version of Halkin's proof of the maximum principle for optimal control problems on unbounded time intervals (Ref. 1), we obtain the maximum principle for the problem under consideration. Our assumptions are weaker than those of related results in the literature.
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    Journal of optimization theory and applications 71 (1991), S. 485-501 
    ISSN: 1573-2878
    Keywords: Optimal control ; partially observed diffusions ; minimum principle
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    Topics: Mathematics
    Notes: Abstract The optimal control of a partially observed diffusion is discussed when the control parameter is present in both the drift and diffusion coefficients. Using a differentiation result of Blagovescenskii and Freidlin, and adapting techniques of Bensoussan, we obtain a stochastic minimum principle.
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    Journal of optimization theory and applications 69 (1991), S. 343-350 
    ISSN: 1573-2878
    Keywords: Optimal control ; maximum principle ; integrodifferential equations ; continuous leads ; economic applications
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    Topics: Mathematics
    Notes: Abstract A sufficiency theorem is provided for the optimal control of systems with continuous leads wherein the motion of today's state is governed by the future trajectory of the control and the state. An application to the economics of dynamic limit pricing is given.
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    Journal of optimization theory and applications 69 (1991), S. 589-603 
    ISSN: 1573-2878
    Keywords: Optimal control ; relaxed control ; state-space constraints ; discretization ; convergence
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    Topics: Mathematics
    Notes: Abstract A class of relaxed optimal control problems for ordinary differential equations with a state-space constraint is considered. The discretization by the control parametrization method, formerly proposed by Teo and Goh (Refs. 1, 2), is modified by admitting a tolerance in the state constraint, which enables one to prove a conditional convergence under certain additional qualification on the dynamics. Also, a counterexample is constructed, showing that the original, nonmodified discretization need not approximate the continuous problem.
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    Journal of optimization theory and applications 70 (1991), S. 1-23 
    ISSN: 1573-2878
    Keywords: Optimal control ; Chebyshev-type optimal control problems ; minimax optimal control problems ; optimal trajectories ; state constraints ; state constraints of third order ; bang-bang controls ; singular controls ; multipoint boundary-value problems ; multiple shooting methods ; flight mechanics ; landing ; abort landing ; windshear problems
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    Topics: Mathematics
    Notes: Abstract The landing of a passenger aircraft in the presence of windshear is a threat to aviation safety. The present paper is concerned with the abort landing of an aircraft in such a serious situation. Mathematically, the flight maneuver can be described by a minimax optimal control problem. By transforming this minimax problem into an optimal control problem of standard form, a state constraint has to be taken into account which is of order three. Moreover, two additional constraints, a first-order state constraint and a control variable constraint, are imposed upon the model. Since the only control variable appears linearly, the Hamiltonian is not regular. Thus, well-known existence theorems about the occurrence of boundary arcs and boundary points cannot be applied. Numerically, this optimal control problem is solved by means of the multiple shooting method in connection with an appropriate homotopy strategy. The solution obtained here satisfies all the sharp necessary conditions including those depending on the sign of certain multipliers. The trajectory consists of bang-bang and singular subarcs, as well as boundary subarcs induced by the two state constraints. The occurrence of boundary arcs is known to be impossible for regular Hamiltonians and odd-ordered state constraints if the order exceeds two. Additionally, a boundary point also occurs where the third-order state constraint is active. Such a situation is known to be the only possibility for odd-ordered state constraints to be active if the order exceeds two and if the Hamiltonian is regular. Because of the complexity of the optimal control, this single problem combines many of the features that make this kind of optimal control problems extremely hard to solve. Moreover, the problem contains nonsmooth data arising from the approximations of the aerodynamic forces and the distribution of the wind velocity components. Therefore, the paper can serve as some sort of user's guide to solve inequality constrained real-life optimal control problems by multiple shooting.
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    Journal of optimization theory and applications 70 (1991), S. 223-254 
    ISSN: 1573-2878
    Keywords: Optimal control ; multiple shooting ; multipoint boundary-value problems ; homotopy ; minimax optimal control problems ; state constraints ; bang-bang controls ; singular controls ; flight mechanics ; landing ; abort landing ; windshear problems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In Part 1 of the paper (Ref. 2), we have shown that the necessary conditions for the optimal control problem of the abort landing of a passenger aircraft in the presence of windshear result in a multipoint boundary-value problem. This boundary-value problem is especially well suited for numerical treatment by the multiple shooting method. Since this method is basically a Newton iteration, initial guesses of all variables are needed and assumptions about the switching structure have to be made. These are big obstacles, but both can be overcome by a so-called homotopy strategy where the problem is imbedded into a one-parameter family of subproblems in such a way that (at least) the first problem is simple to solve. The solution data to the first problem may serve as an initial guess for the next problem, thus resulting in a whole chain of problems. This process is to be continued until the objective problem is reached. Techniques are presented here on how to handle the various changes of the switching structure during the homotopy run. The windshear problem, of great interest for safety in aviation, also serves as an excellent benchmark problem: Nearly all features that can arise in optimal control appear when solving this problem. For example, the candidate for an optimal trajectory of the minimax optimal control problem shows subarcs with both bang-bang and singular control functions, boundary arcs and touch points of two state constraints, one being of first order and the other being of third order, etc. Therefore, the results of this paper may also serve as some sort of user's guide for the solution of complicated real-life optimal control problems by multiple shooting. The candidate found for an optimal trajectory is discussed and compared with an approximate solution already known (Refs. 3–4). Besides the known necessary conditions, additional sharp necessary conditions based on sign conditions of certain multipliers are also checked. This is not possible when using direct methods.
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    Journal of optimization theory and applications 71 (1991), S. 205-235 
    ISSN: 1573-2878
    Keywords: Optimal control ; optimal distribution of enzymes ; Pontryagin's maximum principle
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    Topics: Mathematics
    Notes: Abstract Some optimization problems concerning a substrate in a fluid are considered. The concentration of the substrate is affected by diffusion, convection, and elimination by enzymes, and the problem is to find the optimal distribution of enzymes. In this paper, the rate of elimination and the transmission coefficient are optimized. Mathematically, these problems are optimal control problems, and they are analyzed by means of Pontryagin's maximum principle.
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    Journal of optimization theory and applications 64 (1990), S. 557-571 
    ISSN: 1573-2878
    Keywords: Optimal control ; deterministic systems ; discrete incomplete information ; sufficient conditions for optimality ; feedback systems
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    Topics: Mathematics
    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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    Journal of optimization theory and applications 65 (1990), S. 129-138 
    ISSN: 1573-2878
    Keywords: Optimal control ; mathematical modelling ; study strategies
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    Topics: Mathematics
    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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    Journal of optimization theory and applications 65 (1990), S. 395-407 
    ISSN: 1573-2878
    Keywords: Optimal control ; nonlinear systems ; discretization ; nonconvexity ; relaxed controls ; approximation
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    Notes: Abstract We consider a general nonlinear optimal control problem for systems governed by ordinary differential equations with terminal state constraints. No convexity assumptions are made. The problem, in its so-called relaxed form, is discretized and necessary conditions for discrete relaxed optimality are derived. We then prove that discrete optimality [resp., extremality] in the limit carries over to continuous optimality [resp., extremality]. Finally, we prove that limits of sequences of Gamkrelidze discrete relaxed controls can be approximated by classical controls.
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    Journal of optimization theory and applications 66 (1990), S. 255-273 
    ISSN: 1573-2878
    Keywords: Optimal control ; state constraints ; production problems
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    Topics: Mathematics
    Notes: Abstract This paper deals with a production plant in which two different products can be produced. The plant consists of three subsystemsS i . Before or after a phase of separate processing in subsystemsS 1 andS 2, the two products have to be processed in subsystemS 3. Each of these subsystems has a limited capacity. In the first part, we assume empty stocks at the beginning; at a fixed timeT in the future, certain quantitiesX i of the two products have to be delivered to the customers. Facing linear holding costs, convex production costs, and stringent capacity constraints, the problem is to decide when to produce which product at what rate. It is shown that the optimal solution consists of up to six different regimes and that the time paths of the production rates need not be monotonic. These results, which can be obtained analytically, are also illustrated in several numerical examples. Finally, the case is considered where the terminal demand at timeT is replaced by a continuous and seasonally fluctuating demand rate. It is demonstrated that the optimal production rates show an interesting and nontrivial behavior. In particular, it may happen that, on intervals where the demand for the one product increases, the optimal production rate decreases. This is also demonstrated by computer plots in some numerical examples.
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    Journal of optimization theory and applications 66 (1990), S. 443-454 
    ISSN: 1573-2878
    Keywords: Optimal control ; heat transfer ; singularly perturbed integral equations
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    Topics: Mathematics
    Notes: Abstract The problem considered is that of maintaining the end temperature of a long rod near a prescribed level over a fixed time interval. Control is achieved via the heat flux at the near end, and it is optimal in the sense that it minimizes a given performance index of quadratic form. The performance index contains a penalty parameter associated with the magnitude of the control. Particular attention is given to the determination of the optimal control when the penalty parameter is small (i.e., cheap control). This gives rise to a singularly perturbed integral equation, which is solved asymptotically by a methodology which has recently been developed for a related class of problems.
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    Journal of optimization theory and applications 67 (1990), S. 17-41 
    ISSN: 1573-2878
    Keywords: Optimal control ; existence ; numerical approximation ; piston rings ; journal bearings ; hydrodynamic lubrication theory
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    Topics: Mathematics
    Notes: Abstract The present paper is concerned with the study of controlling the motion of a bearing so that the thin film of lubricant separating it from its container will have the largest possible thickness. Explicit equations are derived, and the control problem is solved explicitly in some simple cases.
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    Journal of optimization theory and applications 67 (1990), S. 259-277 
    ISSN: 1573-2878
    Keywords: Optimal control ; time-axis decomposition ; Stackelberg games ; bilevel programming
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    Topics: Mathematics
    Notes: Abstract Continuous-time optimal control problems can rarely be solved directly but have to be approximated with discrete analogues. Shorter time steps lead to more accurate approximations, but result in formulations that are often too big for computer memory. This paper presents a technique for decomposing the problem along the time axis and iterating toward a solution in a leader-follower framework. In the model, the leader controls a set of coordination parameters, which he passes to the followers, who then solve their individual subproblems. State and sensitivity information is returned to the leader, who attempts to minimize an unconstrained problem in the coordination space. Parameters are updated and the process continues until improvement ceases. Two advantages of this technique are that feasible solutions to the original problem are available at each iteration and that the optimal coordination parameters obtained provide some measure of feedback control. Computational results are presented for a comprehensive set of test problems.
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    Journal of optimization theory and applications 67 (1990), S. 567-585 
    ISSN: 1573-2878
    Keywords: Optimal control ; parabolic partial differential equations ; time-delay systems ; necessary conditions ; sufficient conditions ; maximum principle
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    Topics: Mathematics
    Notes: Abstract An optimal control problem with a prescribed performance index for parabolic systems with time delays is investigated. A necessary condition for optimality is formulated and proved in the form of a maximum principle. Under additional conditions, the maximum principle gives sufficient conditions for optimality. It is also shown that the optimal control is unique. As an illustration of the theoretical consideration, an analytic solution is obtained for a time-delayed diffusion system.
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    Journal of optimization theory and applications 65 (1990), S. 363-373 
    ISSN: 1573-2878
    Keywords: Optimal control ; maximum principle ; dynamic programming ; viscosity solutions ; superdifferential ; subdifferential
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Two major tools for studying optimally controlled systems are Pontryagin's maximum principle and Bellman's dynamic programming, which involve the adjoint function, the Hamiltonian function, and the value function. The relationships among these functions are investigated in this work, in the case of deterministic, finite-dimensional systems, by employing the notions of superdifferential and subdifferential introduced by Crandall and Lions. Our results are essentially non-smooth versions of the classical ones. The connection between the maximum principle and the Hamilton-Jacobi-Bellman equation (in the viscosity sense) is thereby explained by virtue of the above relationship.
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    Journal of optimization theory and applications 65 (1990), S. 375-384 
    ISSN: 1573-2878
    Keywords: Optimal control ; time-varying systems ; Fourier series operational matrix ; Fourier series approximation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The optimal control of linear time-varying systems with quadratic cost functional is obtained by Fourier series approximation. The properties of Fourier series are first briefly presented and the operational matrix of backward integration together with the product operational matrix are utilized to reduce the optimal control problem to a set of simultaneous linear algebraic equations. An illustrative example is included to demonstrate the validity and applicability of the technique.
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