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Guoyuan Chen

Publications and source records attributed to Guoyuan Chen.

11 recordsLinked to original sources
arXiv2024

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 \(e^{-αt}\) (\(α> 0\)), 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.

math.OC↗
arXiv2024

Deep neural network approximations for the stable manifolds of the Hamilton-Jacobi-Bellman equations

For an infinite-horizon control problem, the optimal control can be represented by the stable manifold of the characteristic Hamiltonian system of Hamilton-Jacobi-Bellman (HJB) equation in a semiglobal domain. In this paper, we first theoretically prove that if an approximation is sufficiently close to the exact stable manifold of the HJB equation in a certain sense, then the control derived from this approximation stabilizes the system and is nearly optimal. Then, based on the theoretical result, we propose a deep learning algorithm to approximate the stable manifold and compute optimal feedback control numerically. The algorithm relies on adaptive data generation through finding trajectories randomly within the stable manifold. Such kind of algorithm is grid-free basically, making it potentially applicable to a wide range of high-dimensional nonlinear systems. We demonstrate the effectiveness of our method through two examples: stabilizing the Reaction Wheel Pendulums and controlling the parabolic Allen-Cahn equation.

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arXiv2021

Symplectic algorithms for stable manifolds in control theory

In this note, we propose a symplectic algorithm for the stable manifolds of the Hamilton-Jacobi equations combined with an iterative procedure in [Sakamoto-van~der Schaft, IEEE Transactions on Automatic Control, 2008]. Our algorithm includes two key aspects. The first one is to prove a precise estimate for radius of convergence and the errors of local approximate stable manifolds. The second one is to extend the local approximate stable manifolds to larger ones by symplectic algorithms which have better long-time behaviors than general-purpose schemes. Our approach avoids the case of divergence of the iterative sequence of approximate stable manifolds, and reduces the computation cost. We illustrate the effectiveness of the algorithm by an optimal control problem with exponential nonlinearity.

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arXiv2020

Nondegeneracy of ground states and multiple semiclassical solutions of the Hartree equation for general dimensions

We study nondegeneracy of ground states of the Hartree equation $$ -Δu+u=(I_{2}\ast u^2)u\quad\mbox{ in }\mathbb R^n $$ where $n=3,4,5$ and $I_2$ is the Newton potential. As an application of the nondegeneracy result, we use a Lyapunov-Schmidt reduction argument to construct multiple semiclassical solutions to the following Hartree equation with an external potential $$-\varepsilon^2Δu+u+V(x)u=\varepsilon^{-2}(I_{2}\ast u^2)u\quad \mbox{ in }\mathbb R^n.$$

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arXiv2018

Nondegeneracy of harmonic maps from $\mathbb R^2$ to $\mathbb S^2$

We prove that all harmonic maps from $\mathbb R^2$ to $\mathbb S^2$ with finite energy are nondegenerate. That is, for any harmonic map $u$ from $\mathbb R^2$ to $\mathbb S^2$ of degree $m$ (in $\mathbb Z$), all bounded kernel maps of the linearized operator $L_u$ at $u$ are generated by these harmonic maps near $u$ and hence the real dimension of bounded kernel space of $L_u$ is $4|m|+2$.

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arXiv2016

Singularly perturbed Neumann problem for fractional Schrödinger equations

This paper is concerned with a Neumann type problem for singularly perturbed fractional nonlinear Schrödinger equations with subcritical exponent. For some smooth bounded domain $Ω\subset \mathbf R^n$, our boundary condition is given by \begin{equation*} \int_Ω\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy=0\quad\mbox{for }x\in \mathbf R^n\setminus\barΩ. \end{equation*} We establish existence of nonnegative small energy solutions, and also investigate the integrability of the solutions on $\mathbf R^n$.

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arXiv2015

Fractional nonlinear Schrödinger equations with singular potential in $\mathbf R^n$

We are interested in nonlinear fractional Schrödinger equations with singular potential of form \begin{equation*} (-Δ)^su=\fracλ{|x|^α}u+|u|^{p-1}u,\quad \mathbf R^n\setminus\{0\}, \end{equation*} where $s\in (0,1)$, $α>0$, $p\ge1$ and $λ\in \mathbf R$. Via Caffarelli-Silvestre extension method, we obtain existence, nonexistence, regularity and symmetry properties of solutions to this equation for various $α$, $p$ and $λ$.

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arXiv2014

Multiple Semiclassical Standing Waves for Fractional Nonlinear Schrödinger Equations

Via a Lyapunov-Schmidt reduction, we obtain multiple semiclassical solutions to a class of fractional nonlinear Schrödinger equations. Precisely, we consider \begin{equation*} \varepsilon^{2s}(-Δ)^{s}u+u+V(x)u=|u|^{p-1}u,\quad u\in H^s(\mathbf R^n), \end{equation*} where $0 4-4s$, $1 2s$) and $1<p<\infty$ (if $n\le 2s$), $V(x)$ is a non-negative potential function. If $V$ is a sufficiently smooth bounded function with a non-degenerate compact critical manifold $M$, then, when $\varepsilon$ is sufficiently small, there exist at least $l(M)$ semiclassical solutions, where $l(M)$ is the cup length of $M$.

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arXiv2013

Concentration phenomenon for fractional nonlinear Schrödinger equations

We study the concentration phenomenon for solutions of the fractional nonlinear Schrödinger equation, which is nonlocal. We mainly use the Lyapunov-Schmidt reduction method. Precisely, consider the nonlinear equation \begin{equation}\label{e:abstract} (-\varepsilon^2Δ)^sv+Vv-|v|^αv=0\quad\mbox{in}\quad\mathbf R^n, \end{equation} where $n =1, 2, 3$, $\max\{\frac{1}{2}, \frac{n}{4}\}< s < 1$, $1 \leq α< α_*(s,n)$, $V\in C^3_{b}(\mathbf{R}^n)$. Here the exponent $α_*(s,n)=\frac{4s}{n-2s}$ for $0 < s < \frac{n}{2}$ and $α_*(s,n)=\infty$ for $s \geq\frac{n}{2}$. Then for each non-degenerate critical point $z_0$ of $V$, there is a nontrivial solution of equation (\ref{e:abstract}) concentrating to $z_0$ as $\varepsilon\to 0$.

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arXiv2011

Perturbation of Sectorial Projections of Elliptic Pseudo-differential Operators

Over a closed manifold, we consider the sectorial projection of an elliptic pseudo-differential operator A of positive order with two rays of minimal growth. We show that it depends continuously on A when the space of pseudo-differential operators is equipped with a certain topology which we explicitly describe. Our main application deals with a continuous curve of arbitrary first order linear elliptic differential operators over a compact manifold with boundary. Under the additional assumption of the weak inner unique continuation property, we derive the continuity of a related curve of Calderon projections and hence of the Cauchy data spaces of the original operator curve. In the Appendix, we describe a topological obstruction against a verbatim use of R. Seeley's original argument for the complex powers, which was seemingly overlooked in previous studies of the sectorial projection.

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