Closed Form HJB Solution for Continuous-Time Optimal Control of a Non-Linear Input-Affine System

TL;DR

This paper presents a novel analytical method to derive closed-form solutions for the Hamilton-Jacobi-Bellman equation in continuous-time nonlinear input-affine systems, eliminating the need for iterative learning and improving computational efficiency.

Contribution

It introduces a new analytical framework that provides explicit solutions to the HJB equation for certain nonlinear systems, avoiding iterative reinforcement learning methods.

Findings

Closed-form HJB solutions for a class of nonlinear systems

Proven asymptotic stability of the closed-loop system

Enhanced computational efficiency and optimal control performance

Abstract

Designing optimal controllers for nonlinear dynamical systems often relies on reinforcement learning and adaptive dynamic programming (ADP) to approximate solutions of the Hamilton Jacobi Bellman (HJB) equation. However, these methods require iterative training and depend on an initially admissible policy. This work introduces a new analytical framework that yields closed-form solutions to the HJB equation for a class of continuous-time nonlinear input-affine systems with known dynamics. Unlike ADP-based approaches, it avoids iterative learning and numerical approximation. Lyapunov theory is used to prove the asymptotic stability of the resulting closed-loop system, and theoretical guarantees are provided. The method offers a closed-form control policy derived from the HJB framework, demonstrating improved computational efficiency and optimal performance on state-of-the-art optimal control problems in the literature.