Abstract
An approximation of the Hamilton-Jacobi-Bellman equation connected with the infinite horizon optimal control problem with discount is proposed. The approximate solutions are shown to converge uniformly to the viscosity solution, in the sense of Crandall-Lions, of the original problem. Moreover, the approximate solutions are interpreted as value functions of some discrete time control problem. This allows to construct by dynamic programming a minimizing sequence of piecewise constant controls.
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References
Bertsekas DP, Shreve SE (1978) Stochastic optimal control: The discrete time case. Academic Press, New York
Capuzzo Dolcetta I, Evans LC (to appear) Optimal switching for ordinary differential equations. SIAM J Control
Capuzzo Dolcetta I, Matzeu M (1981) On the dynamic programming inequalities associated with the deterministic optimal stopping problem in discrete and continuous time. Num Funct Anal Optim 3:425–450
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
Crandall MG, Evans LC, Lions PL (to appear) Some properties of the viscosity solutions of Hamilton-Jacobi equations. Trans Amer Math Soc
Crandall MG, Lions PL (to appear) Viscosity solutions of Hamilton-Jacobi equations. Trans Amer Math Soc
Evans LC (1980) On solving certain nonlinear partial differential equations by accretive operator methods. Israel J Math 36:365–389
Fleming WH, Rishel RW (1975) Deterministic and stochastic optimal control. Springer-Verlag, Berlin-Heidelberg-New York
Gawronski M (1982) Dissertation. Istituto Matematico, Università di Roma, Rome
Goletti F (1981) Dissertation. Istituto Matematico, Università di Roma, Rome
Henrici P (1962) Discrete variable methods in ordinary differential equations. J. Wiley, New York
Lions PL (1982) Generalized solutions of Hamilton-Jacobi equations. Pitman, London
Menaldi JL (1982) Le problème de temps d'arret optimal déterministe et l'inéquation variationnelle du premier ordre associée. Appl Math Optim 8:131–158
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Communicated by A. V. Balakrishnan
Supported in part by a CNR-NATO grant during a visit at the University of Maryland.
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Dolcetta, I.C. On a discrete approximation of the Hamilton-Jacobi equation of dynamic programming. Appl Math Optim 10, 367–377 (1983). https://doi.org/10.1007/BF01448394
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DOI: https://doi.org/10.1007/BF01448394