Numerical methods for construction of value functions in optimal control problems on an infinite horizon

Abstract

This article deals with the optimal control problem on an infinite horizon, the quality functional of which is contained in the integrand index and the discounting factor. A special feature of this formulation of the problem is the assumption of the possible unboundedness of the integrand index. The problem reduces to an equivalent optimal control problem with a stationary value function as a generalized (minimax, viscosity) solution of the Hamilton–Jacobi equation satisfying the Hölder condition and the condition of linear growth. The article describes the backward procedure on an infinite horizon. It is the method of numerical approximation of the generalized solution of the Hamilton–Jacobi equation. The main result of the article is an estimate of the accuracy of approximation of a backward procedure for solving the original problem. Problems of the analyzed type are related to modeling processes of economic growth and to problems of stabilizing dynamic systems. The results obtained can be used to construct numerical finite-difference schemes for calculating the value function of optimal control problems or differential games. © 2019 Udmurt State University. All right reserved.