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A numerical approach to the infinite horizon problem of deterministic control theory. (English) Zbl 0715.49023
The author considers the closed loop control of the infinite horizon problem of deterministic control theory. A time discretization of the related Hamilton-Jacobi (HJ) equation was introduced by I. C. Dolcetta [ibid. 10, 367-377 (1983; Zbl 0582.49019)]. At this stage, the author introduces a discretization of the state variable using finite element techniques. The convergence of the whole approximation process to the viscosity solution of the HJ equation is demonstrated. A relaxation type algorithm proposed by R. L. Gonzalez and E. Rofman [SIAM J. Control Optimization 23, 242-266, 267-285 (1985; Zbl 0563.49024, Zbl 0563.49025, resp.)] is used to obtain an approximate solution of the HJ equation. It is shown that it can be reinterpreted as an acceleration method for a sequence generated by a contracting operator. This allows an error estimate for each step of the algorithm to be obtained.

MSC:
49L20 Dynamic programming in optimal control and differential games
49M20 Numerical methods of relaxation type
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[1] Aubin JP, Cellina A (1984) Differential Inclusions. Springer-Verlag, Berlin, Heidelberg, New York · Zbl 0538.34007
[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 · doi:10.1007/BF01448394
[3] Capuzzo Dolcetta I, Ishii H (1984) Approximate solutions of the Bellman equation of deterministic control theory. Appl Math Optim 11:161-181 · Zbl 0553.49024 · doi:10.1007/BF01442176
[4] Crandall MG, Lions PL (1983) Viscosity solutions of Hamilton-Jacobi equations. Trans Amer Math Soc 277:1-42 · Zbl 0599.35024 · doi:10.1090/S0002-9947-1983-0690039-8
[5] Crandall MG, Lions PL (1984) Two approximations of solutions of Hamilton-Jacobi equations. Math Comp 43:1-19 · Zbl 0557.65066 · doi:10.1090/S0025-5718-1984-0744921-8
[6] Crandall MG, Evans LC, Lions PL (1984) Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans Amer Math Soc 282:487-502 · Zbl 0543.35011 · doi:10.1090/S0002-9947-1984-0732102-X
[7] Falcone M (1985) Numerical solution of deterministic continuous control problems. Proceedings of the International Symposium on Numerical Analysis, Madrid, September 1985
[8] Falcone M (1986) (forthcoming)
[9] Fleming WH, Rishel RW (1975) Deterministic and Stochastic Optimal Control. Springer-Verlag, Berlin, Heidelberg, New York
[10] Glowinski R, Lions JL, Trémolières R (1976) Analyse Numerique des Inéquations Variationnelles, vols 1 and 2. Dunod, Paris
[11] Gonzales R, Rofman E (1985) On deterministic control problems: an approximation procedure for the optimal cost, I and II. SIAM J Control Optim 23:242-285 · Zbl 0563.49024 · doi:10.1137/0323018
[12] Lions PL (1982) Generalized Solutions of Hamilton-Jacobi Equations. Pitman, London · Zbl 0497.35001
[13] Lions PL, Mercier B (1980) Approximation numerique des equations de Hamilton-Jacobi-Bellman. RAIRO Anal Numér 14:369-393 · Zbl 0469.65041
[14] Quadrat JP (1975) Analyse Numerique de l’Equation de Bellman Stochastique. Rapport INRIA no 140
[15] Rofman E (1985) Approximation of Hamilton-Jacobi-Bellman equation in deterministic control theory. An application to energy production systems. In: Capuzzo Dolcetta I, Fleming WH, Zolezzi T (eds) Recent Mathematical Methods in Dynamic Programming. Lecture Notes in Mathematics 1119. Springer-Verlag, Berlin, Heidelberg, New York
[16] Souganidis PE (1985) Approximation schemes for viscosity solutions of Hamilton-Jacobi equations. J. Differential Equations 59:1-43 · Zbl 0566.70022 · doi:10.1016/0022-0396(85)90136-6
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