A numerical approach to the infinite horizon problem of deterministic control theory. (English) Zbl 0715.49023

Falcone, M.

Appl. Math. Optimization 15, No. 1, 1-13 (1987).

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

Keywords:

time discretization; Hamilton-Jacobi (HJ) equation; finite element techniques; viscosity solution

Citations:

Zbl 0582.49019; Zbl 0563.49024; Zbl 0563.49025

References:

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