Solving Hamilton-Jacobi-Bellman equations for optimal control problems using a modified Adomian decomposition method based on an inverse operator

Ordinary differential equations (ODEs) and partial differential equations (PDEs) with initial conditions can be effectively solved using the Adomian Decomposition Method (ADM). However, optimal control problems involve Hamilton-Jacobi-Bellman (HJB) equations with terminal conditions, requiring a backward computational framework. This study develops a Modified Adomian Decomposition Method (MADM) based on a backward inverse operator for solving terminal-value problems. The proposed formulation incorporates the terminal condition directly into the decomposition process, eliminating the need for time transformations. A numerical algorithm is presented to compute the value function, control variable, and performance index. Convergence is established using the Banach fixed point theorem. The method is validated against backward finite difference and backward fourth-order Runge-Kutta schemes. Numerical experiments for linear and nonlinear examples demonstrate the effectiveness and reliability of the proposed framework for solving HJB equations with terminal conditions.