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.

翻译：本文提出一种高效的符号-数值方法，用于计算Hamilton-Jacobi-Bellman (HJB)方程逐次Galerkin逼近 (SGA)中的积分。首先利用Hermite多项式的线性组合对HJB方程的解进行近似，随后通过迭代求解线性方程组计算该组合的系数，其中线性方程组包含Hermite多项式与非线性函数乘积的积分。Hermite多项式的递推结构被继承至积分中，通过微分算子的符号计算可求得其递推关系。利用该递推关系，所有积分均可由其中部分数值求积的积分计算得出。数值算例表明，与标准数值积分方法相比，本文方法具有高效性。