Compactification of calculation domains in approximation schemes for value functions of optimal control problems with infinite horizon

Abstract

The paper is devoted to analysis of the structure of approximation schemes for value functions in optimal control problems with the infinite time horizon. The problem peculiarity is a quality functional with an unbounded integrand, which is discounted by an exponential index. The main result is reduction of the optimal control problem to an equivalent problem with the value function that has the compact definition domain and the compact value range. This goal is reached by a series of nonlinear changes of variables and the transformations of the system dynamics and the Hamilton-Jacobi equations. The proposed method is resulted, first, in obtaining the strongly invariant calculation domains of approximation schemes and, second, in compactification of ranges for the value functions. The first property provides the opportunity to implement approximation schemes in the form of numerical finite difference schemes for the value functions in compact definition domains. The second property guarantees convergence of approximation schemes to the value functions.