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

Guoyuan Chen; Yi Wang; Qinglong Zhou

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arXiv · 2410.02272

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

Guoyuan Chen·Yi Wang·Qinglong Zhou

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Abstract

The optimal $H_{\infty}$ control problem over an infinite time horizon, which incorporates a performance function with a discount factor $e^{-αt}$ ($α> 0$), 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 $α$ 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.

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BibTeX RIS

Guoyuan Chen, Yi Wang, Qinglong Zhou. 2024-10-03. Optimal $H_{\infty}$ control based on stable manifold of discounted Hamilton-Jacobi-Isaacs equation. https://arxiv.org/abs/2410.02272

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