Optimal $H_\infty$ control based on stable manifold of discounted Hamilton-Jacobi-Isaacs equation
<|reference_start|>Optimal $H_\infty$ control based on stable manifold of discounted Hamilton-Jacobi-Isaacs equation: 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.<|reference_end|>
@article{chen2024optimal,
    title={Optimal $H_{\infty}$ control based on stable manifold of discounted
  Hamilton-Jacobi-Isaacs equation},
    author={Guoyuan Chen, Yi Wang and Qinglong Zhou},
    journal={arXiv preprint arXiv:2410.02272},
    year={2024},
    archivePrefix={arXiv},
    eprint={2410.02272},
    primaryClass={math.OC cs.SY eess.SY}
}