Symbolic-Numeric Computation of Integrals in Successive Galerkin Approximation of Hamilton-Jacobi-Bellman Equation

Abstract

This paper proposes an efficient symbolic-numeric method to compute the integrals in the successive Galerkin approximation (SGA) of the Hamilton-Jacobi-Bellman (HJB) equation. A solution of the HJB equation is first approximated with a linear combination of the Hermite polynomials. The coefficients of the combination are then computed by iteratively solving a linear equation, which consists of the integrals of the Hermite polynomials multiplied by nonlinear functions. The recursive structure of the Hermite polynomials is inherited by the integrals, and their recurrence relations can be computed by using the symbolic computation of differential operators. By using the recurrence relations, all the integrals can be computed from a part of them that are numerically evaluated. A numerical example is provided to show the efficiency of the proposed method compared to a standard numerical integration method.