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  • Articles  (19)
  • optimal control  (14)
  • AMS(MOS): 65H10
  • Springer  (19)
  • American Chemical Society
  • Annual Reviews
  • Blackwell Publishing Ltd
  • Elsevier
  • Wiley
  • 2005-2009
  • 1990-1994  (9)
  • 1980-1984  (10)
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  • 1991  (9)
  • 1982  (10)
  • Mathematics  (19)
  • Computer Science  (1)
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  • Articles  (19)
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  • Springer  (19)
  • American Chemical Society
  • Annual Reviews
  • Blackwell Publishing Ltd
  • Elsevier
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  • 2005-2009
  • 1990-1994  (9)
  • 1980-1984  (10)
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  • Mathematics  (19)
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  • 1
    Electronic Resource
    Electronic Resource
    Optimal schedule for cancer chemotherapy (1991)
    Springer
    Mathematical programming 52 (1991), S. 11-17 
    Details
    ISSN: 1436-4646
    Keywords: Modeling ; cancer ; optimization ; optimal control ; drug delivery
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper we consider the problems of modeling the tumor growth and optimize the chemotherapy treatment. A biologically based model is used with the goal of solving an optimization problem involving discrete delivery of antineoplastic drugs. Our model is formulated via compartmental analysis in order to take into account the cell cycle. The cost functional measures not only the final size of the tumor but also the total amount of drug delivered. We propose an algorithm based on the discrete maximum principle to solve the optimal drug schedule problem. Our numerical results show nice interpretations from the medical point of view.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01582876
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  • 2
    Electronic Resource
    Electronic Resource
    Some convergence results for asynchronous algorithms (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 325-340 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Description / Table of Contents: Summary In this paper we present convergence results for the asynchronous algorithms based essentially on the notion of classical contraction. We generalize, in particular, all convergence results for those algorithms which are based on the vectorial norm hypothesis, in wide spread use recently. Certain problems, for which the vectorial norm hypothesis can be difficult or even impossible to verify, can nontheless be tackled within the scope of the classical contraction that we adopte.
    Notes: Resumé Nous présentons dans cet article des résultats de convergence des algorithmes asynchrones basés essentiellement sur la notion classique de contraction. Nous généralisons, en particulier, tous les résultats de convergence de ces algorithmes qui font l'hypothèse de contraction en norme vectorielle qui récemment a été très souvant utilisée. Par ailleurs, l'hypothèse de contraction en norme vectorielle peut se trouver difficile, voire impossible à vérifier pour certains problèmes qui peuvent être cependant abordés dans le cadre de la contraction classique que nous adoptons.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01407866
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  • 3
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    Bisection is optimal (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 111-117 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We seek an approximation to a zero of a continuous functionf:[a,b]→ℝ such thatf(a)≦0 andf(b)≧0. It is known that the bisection algorithm makes optimal use ofn function evaluations, i.e., yields the minimal error which is (b−a)/2 n+1, see e.g. Kung [2]. Traub and Wozniakowski [5] proposed using more general information onf by permitting the adaptive evaluations ofn arbitrary linear functionals. They conjectured [5, p. 170] that the bisection algorithm remains optimal even if these general evaluations are permitted. This paper affirmatively proves this conjecture. In fact we prove optimality of the bisection algorithm even assuming thatf is infinitely many times differentiable on [a, b] and has exactly one simple zero.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01459080
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  • 4
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    Aitken's and Steffensen's accelerations in several variables (1991)
    Springer
    Numerische Mathematik 59 (1991), S. 295-310 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: G1.5
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Aitken's acceleration of scalar sequences extends to sequences of vectors that behave asymptotically as iterations of a linear transformation. However, the minimal and characteristic polynomials of that transformation must coincide (but the initial sequence of vectors need not converge) for a numerically stable convergence of Aitken's acceleration to occur. Similar results hold for Steffensen's acceleration of the iterations of a function of several variables. First, the iterated function need not be a contracting map in any neighbourhood of its fixed point. Instead, the second partial derivatives need only remain bounded in such a neighbourhood for Steffensen's acceleration to converge quadratically, even if ordinary iterations diverge. Second, at the fixed point the minimal and characteristic polynomials of the Jacobian matrix must coincide to ensure a numerically stable convergence. By generalizing the work that Noda did on the subject between 1981 and 1986, the results presented here explain the numerical observations reported by Henrici in 1964 and 1982.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385782
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  • 5
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    Some supplementary results on the 1+ $$\sqrt 2 $$ order method for the solution of nonlinear equations (1982)
    Springer
    Numerische Mathematik 38 (1982), S. 383-392 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Recently an iterative method for the solution of systems of nonlinear equations having at leastR-order 1+ $$\sqrt 2 $$ for simple roots has been investigated by the author [7]; this method uses as many function evaluations per step as the classical Newton method. In the present note we deal with several properties of the method such as monotone convergence, asymptotic inclusion of the solution and convergence in the case of multiple roots.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396439
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  • 6
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    Partitioned variable metric updates for large structured optimization problems (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 119-137 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper presents a minimization method based on the idea of partitioned updating of the Hessian matrix in the case where the objective function can be decomposed in a sum of convex “element” functions. This situation occurs in a large class of practical problems including nonlinear finite elements calculations. Some theoretical and algorithmic properties of the update are discussed and encouraging numerical results are presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01399316
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  • 7
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    Algebraic structure of some stochastic discrete event systems, with applications (1991)
    Springer
    Discrete event dynamic systems 1 (1991), S. 7-35 
    Details
    ISSN: 1573-7594
    Keywords: antimatroid ; generalized semi-Markov processes ; infinitesimal perburtation analysis ; optimal control ; stochastic Petri nets
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Generalized semi-Markov processes (GSMPs) and stochastic Petri nets (SPNs) are generally regarded as performance models (as opposed to logical models) of discrete event systems. Here we take the view that GSMPs and SPNS are essentially automata (generators) driven by input sequences that determine the timing of events. This view combines the deterministic, logical aspects and the stochastic, timed aspects of the two models. We focus on two conditions, (M) and (CX) (which we previously developed to study monotonicity and convexity properties of GSMPs), and the antimatroid and lattice structure they imply for the language generated by a GSMP or SPN. We illustrate applications of these structural properties in the areas of derivative estimation, simulation variance reduction, parallel simulation, and optimal control.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01797141
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  • 8
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    Minimum-time 180° turns of aircraft (1982)
    Springer
    Journal of optimization theory and applications 38 (1982), S. 83-96 
    Details
    ISSN: 1573-2878
    Keywords: Numerical methods ; multiple shooting method ; optimal control ; aircraft trajectories ; flight mechanics
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Three-dimensional minimum-time 180° turns of a fighter aircraft are computed for several initial velocitiesV 0 and altitudesh 0. It is shown that the optimum turns consist of split -S maneuvers forV 0≦V 10, three-dimensional maneuvers forV 10〈V 0〈V 20, and half-loops forV 0≧V 20, withV 10,V 20 being functions of altitude or thrust/weight ratio.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00934324
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  • 9
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    Constructive method for solving a linear minimax problem of optimal control (1991)
    Springer
    Journal of optimization theory and applications 71 (1991), S. 255-275 
    Details
    ISSN: 1573-2878
    Keywords: Constructive methods ; minimax problems ; optimal control ; optimality criteria ; implicit function theorem
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, we propose a constructive method for solving a linear minimax problem of optimal control. Following the Gabasov-Kirillova approach, we introduce the concept of so-called support control. After establishing an optimality criterion for the support control, we describe a scheme for reducing the initial infinite-dimensional problem to a finite-dimensional one, which can be solved numerically by the methods of linear programming. At the end, we give an illustrative example.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00939920
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  • 10
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    New Taylor series approach to state-space analysis and optimal control of linear systems (1991)
    Springer
    Journal of optimization theory and applications 71 (1991), S. 315-340 
    Details
    ISSN: 1573-2878
    Keywords: Linear systems ; Taylor series ; state-space analysis ; optimal control ; estimation of the approximation error
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A new Taylor series approach is presented which reduces the problem of determining the state vector coefficient matrixX for time-invariant systems to an expression involving multiplications of matrices of small dimensions. This approach is numerically superior to known techniques and is extended to cover the time-varying case, wherein analogous expressions are derived. Furthermore, the optimal control problem is solved using the same technique. Finally, an expression is derived for the computation of the approximation error involved in computingX, prior to determiningX.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00939923
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  • 11
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    Some mathematical aspects of the political business cycle (1982)
    Springer
    Journal of optimization theory and applications 37 (1982), S. 251-275 
    Details
    ISSN: 1573-2878
    Keywords: Economics ; political cycles ; discrete dynamic systems ; dynamic programming ; optimal control ; Poincaré mapping ; Stieltjes matrix ; optimization in Hilbert space ; infinite horizon ; turnpike theorem
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Under the hypothesis of a myopic electorate, vote-loss-minimizing behavior by the party in power, subject to a dynamic inflation-unemployment relation, is shown to generate an attractive, stable electoral policy cycle. The model presented is derived, with some improvements, from the analogous models of MacRae and Nordhaus. Furthermore, an attempt is made to specify the mathematical aspects of the problem by the Poincaré mapping.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00934770
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  • 12
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    Numerical solution of minimax problems of optimal control, part 2 (1982)
    Springer
    Journal of optimization theory and applications 38 (1982), S. 111-135 
    Details
    ISSN: 1573-2878
    Keywords: Minimax problems ; minimax function ; minimax function depending on the state ; minimax function depending on the control ; optimal control ; minimax optimal control ; numerical methods ; computing methods ; transformation techniques ; gradient-restoration algorithms ; sequential gradient-restoration algorithms
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In a previous paper (Part 1), we presented general transformation techniques useful to convert minimax problems of optimal control into the Mayer-Bolza problem of the calculus of variations [Problem (P)]. We considered two types of minimax problems: minimax problems of Type (Q), in which the minimax function depends on the state and does not depend on the control; and minimax problems of Type (R), in which the minimax function depends on both the state and the control. Both Problem (Q) and Problem (R) can be reduced to Problem (P). In this paper, the transformation techniques presented in Part 1 are employed in conjunction with the sequential gradient-restoration algorithm for solving optimal control problems on a digital computer. Both the single-subarc approach and the multiple-subarc approach are employed. Three test problems characterized by known analytical solutions are solved numerically. It is found that the combination of transformation techniques and sequential gradient-restoration algorithm yields numerical solutions which are quite close to the analytical solutions from the point of view of the minimax performance index. The relative differences between the numerical values and the analytical values of the minimax performance index are of order 10−3 if the single-subarc approach is employed. These relative differences are of order 10−4 or better if the multiple-subarc approach is employed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00934326
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  • 13
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    Numerical solution of minimax problems of optimal control, part 1 (1982)
    Springer
    Journal of optimization theory and applications 38 (1982), S. 97-109 
    Details
    ISSN: 1573-2878
    Keywords: Minimax problems ; minimax function ; minimax function depending on the state ; minimax function depending on the control ; optimal control ; minimax optimal control ; numerical methods ; computing methods ; transformation techniques ; gradient-restoration algorithms ; sequential gradient-restoration algorithms
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper contains general transformation techniques useful to convert minimax problems of optimal control into the Mayer-Bolza problem of the calculus of variations [Problem (P)]. We consider two types of minimax problems: minimax problems of Type (Q), in which the minimax function depends on the state and does not depend on the control; and minimax problems of Type (R), in which the minimax function depends on both the state and the control. Both Problem (Q) and Problem (R) can be reduced to Problem (P). For Problem (Q), we exploit the analogy with a bounded-state problem in combination with a transformation of the Jacobson type. This requires the proper augmentation of the state vectorx(t), the control vectoru(t), and the parameter vector π, as well as the proper augmentation of the constraining relations. As a result of the transformation, the unknown minimax value of the performance index becomes a component of the parameter vector being optimized. For Problem (R), we exploit the analogy with a bounded-control problem in combination with a transformation of the Valentine type. This requires the proper augmentation of the control vectoru(t) and the parameter vector π, as well as the proper augmentation of the constraining relations. As a result of the transformation, the unknown minimax value of the performance index becomes a component of the parameter vector being optimized. In a subsequent paper (Part 2), the transformation techniques presented here are employed in conjunction with the sequential gradient-restoration algorithm for solving optimal control problems on a digital computer; both the single-subarc approach and the multiple-subarc approach are discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00934325
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  • 14
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    Hyperplane method for reachable state estimation for linear time-invariant systems (1991)
    Springer
    Journal of optimization theory and applications 69 (1991), S. 555-588 
    Details
    ISSN: 1573-2878
    Keywords: Reachable sets ; simplicial approximation ; linear systems ; optimal control
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A numerical algorithm is presented for generating inner and outer approximations for the set of reachable states for linear time-invariant systems. The algorithm is based on analytical results characterizing the solutions to a class of optimization problems which determine supporting hyperplanes for the reachable set. Explicit bounds on the truncation error for the finite-time case yield a set of so-called ε-supporting hyperplanes which can be generated to approximate the infinite-time reachable set within an arbitrary degree of accuracy. At the same time, an inner approximation is generated as the convex hull of points on the boundary of the finite-time reachable set. Numerical results are presented to illustrate the hyperplane method. The concluding section discusses directions for future work and applications of the method to problems in trajectory planning in servo systems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00940689
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  • 15
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    Optimal control of the multidimensional diffusion equation (1991)
    Springer
    Journal of optimization theory and applications 70 (1991), S. 191-209 
    Details
    ISSN: 1573-2878
    Keywords: Diffusion equation ; boundary control ; optimal control ; Radon measures ; linear programming ; approximations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The existence and numerical estimation of a boundary control for then-dimensional linear diffusion equation is considered. The problem is modified into one consisting of the minimization of a linear functional over a set of Radon measures. The existence of an optimal measure corresponding to the above problem is shown, and the optimal measure is approximated by a finite convex combination of atomic measures. This construction gives rise to a finite-dimensional linear programming problem, whose solution can be used to construct the combination of atomic measures, and thus a piecewise-constant control function which approximates the action of the optimal measure, so that the final state corresponding to the above control function is close to the desired final state, and the value it assigns to the performance criterion is close to the corresponding infimum. A numerical procedure is developed for the estimation of these controls, entailing the solution of large, finite-dimensional linear programming problems. This procedure is illustrated by several examples.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00940511
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  • 16
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    Local stability of solutions to differentiable optimization problems in banach spaces (1991)
    Springer
    Journal of optimization theory and applications 70 (1991), S. 443-466 
    Details
    ISSN: 1573-2878
    Keywords: Nonlinear optimization ; parametric programming ; stability of solutions ; optimal control
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper considers a class of nonlinear differentiable optimization problems depending on a parameter. We show that, if constraint regularity, a second-order sufficient optimality condition, and a stability condition for the Lagrange multipliers hold, then for sufficiently smooth perturbations of the constraints and the objective function the optimal solutions locally obey a type of Lipschitz condition. The results are applied to finite-dimensional problems, equality constrained problems, and optimal control problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00941297
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  • 17
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    Optimal control of a dynamical system representing a gantry crane (1982)
    Springer
    Journal of optimization theory and applications 36 (1982), S. 409-417 
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    ISSN: 1573-2878
    Keywords: Optimal design ; optimal control ; dynamic systems ; gantry cranes
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Problems arising in the optimal control of gantry crane instaliations are considered. Continuous controls to minimize a control squared objective function are obtained. The amplitude of in-plane oscillations of the suspended mass is assumed small. The optimal controls are sufficiently simple for practical realization.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00934354
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  • 18
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    Global optimality of extremals: An example (1982)
    Springer
    Journal of optimization theory and applications 36 (1982), S. 521-534 
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    ISSN: 1573-2878
    Keywords: Darboux points ; global optimality ; conjugate points ; optimal control
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The question of the existence and the location of Darboux points (beyond which global optimality is lost) is crucial for minimal sufficient conditions for global optimality and for computation of optimal trajectories. Here, we investigate numerically the Darboux points and their relationship with conjugate points for a problem of minimum fuel, constant velocity, horizontal aircraft turns to capture a line. This simple second-order optimal control problem shows that ignoring the possible existence of Darboux points may play havoc with the computation of optimal trajectories.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00940545
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  • 19
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    Pointwise quasi-Newton method for unconstrained optimal control problems, II (1991)
    Springer
    Journal of optimization theory and applications 71 (1991), S. 535-547 
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    ISSN: 1573-2878
    Keywords: Quasi-Newton methods ; optimal control
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, the necessary optimality conditions for an unconstrained optimal control problem are used to derive a quasi-Newton method where the update involves only second-order derivative terms. A pointwise update which was presented in a previous paper by the authors is changed to allow for more general second-order sufficiency conditions in the control problem. In particular, pointwise versions of the Broyden, PSB, and SR1 update are considered. A convergence rate theorem is given for the Broyden and PSB versions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00941402
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