Optimal $H_{\infty}$ control based on stable manifold of discounted Hamilton-Jacobi-Isaacs equation

TL;DR

This paper develops a novel approach to solve discounted infinite-horizon H-infinity control problems by linking the stable manifold of the associated contact Hamiltonian system with deep learning techniques, demonstrated on the Allen-Cahn equation.

Contribution

It introduces a precise discount factor estimate for existence of solutions, connects the stable manifold to the control problem, and proposes a deep learning method to approximate optimal controllers.

Findings

Existence of a stabilizing solution linked to the stable manifold.

Approximate controllers achieve near-optimal L2-gain performance.

Successful application to the Allen-Cahn equation demonstrates effectiveness.

Abstract

The optimal \(H_{\infty}\) control problem over an infinite time horizon, which incorporates a performance function with a discount factor \(e^{-\alpha t}\) (\(\alpha > 0\)), is important in various fields. Solving this optimal \(H_{\infty}\) control problem is equivalent to addressing a discounted Hamilton-Jacobi-Isaacs (HJI) partial differential equation. In this paper, we first provide a precise estimate for the discount factor \(\alpha\) that ensures the existence of a nonnegative stabilizing solution to the HJI equation. This stabilizing solution corresponds to the stable manifold of the characteristic system of the HJI equation, which is a contact Hamiltonian system due to the presence of the discount factor. Secondly, we demonstrate that approximating the optimal controller in a natural manner results in a closed-loop system with a finite \(L_2\)-gain that is nearly less than the gain of the original system. Thirdly, based on the theoretical results obtained, we propose a deep learning algorithm to approximate the optimal controller using the stable manifold of the contact Hamiltonian system associated with the HJI equation. Finally, we apply our method to the \(H_{\infty}\) control of the Allen-Cahn equation to illustrate its effectiveness.