Approximate solutions of the Bellman equation of deterministic control theory. (English) Zbl 0553.49024

This paper considers an infinite horizon discounted optimal control problem and its time discretized approximation. The rate of convergence of approximate solutions to the exact solution is of order \(\gamma\) /2, assuming the exact solution is Hölder continuous with exponent \(0<\gamma \leq 1\). The notion of viscosity solution for the optimal control problem is used in making these estimates. The convergence rate of the approximate solutions is shown to be of order 1 provided these approximation solutions satisfy a semi-concavity assumption. The optimal controls of the approximate problem converge to an optimal relaxed control for the original problem.
Reviewer: S.Lenhart


49M25 Discrete approximations in optimal control
35D99 Generalized solutions to partial differential equations
49L20 Dynamic programming in optimal control and differential games
49J20 Existence theories for optimal control problems involving partial differential equations
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[1] Berkovitz LD (1974) Optimal control theory. Springer-Verlag, New York · Zbl 0295.49001
[2] Capuzzo Dolcetta I (1983) On a discrete approximation of the Hamilton-Jacobi equation of dynamic programming. Appl Math Optim 10:367-377 · Zbl 0582.49019
[3] Capuzzo Dolcetta I, Evans LC (To appear) Optimal switching for ordinary differential equations. SIAM J Control Optim · Zbl 0641.49017
[4] Capuzzo Dolcetta I, Matzeu M (1981) On the dynamic programming inequalities associated with the optimal stopping problem in discrete and continuous time. Numer Funct Anal Op 3:425-450 · Zbl 0476.49021
[5] Capuzzo Dolcetta I, Matzeu M (To appear) A constructive approach to the deterministic stopping time problem. Control and Cybernetics · Zbl 0492.49013
[6] Capuzzo Dolcetta I, Matzeu M, Menaldi JL (To appear) On a system of first order quasi-variational inequalities connected with the optimal switching problem. Systems and Control Letters · Zbl 0521.49008
[7] Crandall MG, Evans LC, Lions PL (To appear) Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans AMS
[8] Crandall MG, Lions PL (1983) Viscosity solutions of Hamilton-Jacobi equations. Trans AMS 277:1-42 · Zbl 0599.35024
[9] Crandall MG, Lions PL (To appear) Two approximations of solutions of Hamilton-Jacobi equations.
[10] Cullum J (1969) Discrete approximations to continuous optimal control problems. SIAM J Control 7:32-49 · Zbl 0175.10502
[11] Cullum J (1971) An explicit procedure for discretizing continuous optimal control problems. J Op Theory Appl 8(1):15-34 · Zbl 0215.21907
[12] Douglis A (1961) The continuous dependence of generalized solutions of nonlinear partial differential equations upon initial data. Comm Pure Appl math 14:267-284 · Zbl 0117.31102
[13] Edwards RE (1965) Functional analysis, theory and applications. Holt, Rinehart and Winston, New York · Zbl 0182.16101
[14] Fleming WH, Rishel R (1975) Deterministic and stochastic optimal control. Springer-Verlag, New York · Zbl 0323.49001
[15] Henrici P (1962) Discrete variable methods in ordinary differential equations. J. Wiley, New York · Zbl 0112.34901
[16] Hrustalev MM (9173) Necessary and sufficient optimality conditions in the form of Bellman’s equation. Soviet Math Dokl 19:1262-1266
[17] Kruzkov SN (1975) Generalized solutions of Hamilton-Jacobi equations of eikonal type. Math USSR Sbornik 27:406-446 · Zbl 0369.35012
[18] Lee EB, Markus L (1967) Foundations of optimal control theory. J. Wiley, New York · Zbl 0159.13201
[19] Lions PL (1982) Generalized solutions of Hamilton-Jacobi equations. Pitman, London · Zbl 0497.35001
[20] Malanowski K (1979) On convergence of finite difference approximation to optimal control problems for systems with control appearing linearly. Archiwum Automatyki i Telemachaniki 24(2):155-170 · Zbl 0408.49038
[21] Souganidis PE (1983) PhD thesis. University of Wisconsin
[22] Souganidis PE (To appear) Existence of viscosity solutions of Hamilton-Jacobi equations
[23] Tartar L (1979) Compensated compactness and applications to partial differential equations. In: Knops RJ (ed) Nonlinear analysis and mechanics. Heriot-Watt Symposium, vol. 4. Pitman, London · Zbl 0437.35004
[24] Warga J (1972) Optimal control of differential and functional equations. Academic Press, New York · Zbl 0253.49001
[25] Young LC (1969) Lectures on the calculus of variations and optimal control theory. W.B. Saunders, Philadelphia · Zbl 0177.37801
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