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

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.

Figures (3)

• Figure 1: The dynamics of the NN controlled system with $d(x,t)=0.3\sin t$. The first one is the trajectory from $\gamma=1.2$, and the second one is from $\gamma=+\infty$.
• Figure 2: The norm of $u$ and $w$ with $w(x,t)=0.3\sin t$. The first one (resp. second one) gives the norms or the case $\gamma=1.2$ (resp. $\gamma=+\infty$).
• Figure 3: Tracking trajectory of $r(x,t)=\sin t$.

Theorems & Definitions (33)

• Remark 2.1
• Definition 2.1: Finite $L_2$-gain
• Remark 2.2
• Proposition 2.1
• proof
• Remark 3.1
• Lemma 3.1
• proof
• Definition 3.1: Stabilizing solution of HJI equation
• Proposition 3.1