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

This paper proposes an efficient symbolic-numeric method to compute integrals in the successive Galerkin approximation (SGA) of the Hamilton-Jacobi-Bellman (HJB) equation. By approximating its solution with a linear combination of basis functions, the HJB equation is reduced to a linear equation comprising integrals that include the basis functions. By choosing the Hermite polynomials as the basis functions, their recursive structure is inherited by the integrals. The recurrence relations of the integrals are computed using the symbolic computation and Mellin transform of differential operators. The integrals can then be computed using recursive substitutions, which are more accurate and require less computational cost than numerical integrations. A numerical example is provided to demonstrate the efficiency of the proposed method compared to other numerical integration methods.