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.


49L20 Dynamic programming in optimal control and differential games
49M20 Numerical methods of relaxation type
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