arXiv subjects

Gengsheng Wang

Publications and source records attributed to Gengsheng Wang.

41 records · Page 3Linked to original sources

arXiv2012

Advantages for controls imposed in a proper subset

In this paper, we study time optimal control problems for heat equations on $Ω\times \mathbb{R}^+$. Two properties under consideration are the existence and the bang-bang properties of time optimal controls. It is proved that those two properties hold when controls are imposed on some proper subsets of $Ω$; while they do not stand when controls are active on the whole $Ω$. Besides, a new property for eigenfunctions of $-Δ$ with Dirichlet boundary condition is revealed.

math.OC↗

arXiv2011

Equivalence of three different kinds of optimal control problems for heat equations and its applications

This paper presents an equivalence theorem for three different kinds of optimal control problems, which are optimal target control problems, optimal norm control problems and optimal time control problems. Controlled systems in this study are internally controlled heat equations. With the aid of this theorem, we establish an optimal norm feedback law and build up two algorithms for optimal norms (together with optimal norm controls) and optimal time (along with optimal time controls), respectively.

math.OC↗

arXiv2011

An observability for parabolic equations from a measurable set in time

This paper presents a new observability estimate for parabolic equations in $Ω\times(0,T)$, where $Ω$ is a convex domain. The observation region is restricted over a product set of an open nonempty subset of $Ω$ and a subset of positive measure in $(0,T)$. This estimate is derived with the aid of a quantitative unique continuation at one point in time. Applications to the bang-bang property for norm and time optimal control problems are provided.

math.AP↗

arXiv2006

A Bang-Bang Principle of Time Optimal Internal Controls of the Heat Equation

In this paper, we study a time optimal internal control problem governed by the heat equation in $Ω\times [0,\infty)$. In the problem, the target set $S$ is nonempty in $L^2(Ω)$, the control set $U$ is closed, bounded and nonempty in $L^2(Ω)$ and control functions are taken from the set $\uad=\{u(\cdot, t): [0,\infty)\ra L^2(Ω) {measurable}; u(\cdot, t)\in U, {a.e. in t} \}$. We first establish a certain null controllability for the heat equation in $Ω\times [0,T]$, with controls restricted to a product set of an open nonempty subset in $Ω$ and a subset of positive measure in the interval $[0,T]$. Based on this, we prove that each optimal control $u^*(\cdot, t)$ of the problem satisfies necessarily the bang-bang property: $u^*(\cdot, t)\in \p U$ for almost all $t\in [0, T^*]$, where $\p U$ denotes the boundary of the set $U$ and $T^*$ is the optimal time. We also obtain the uniqueness of the optimal control when the target set $S$ is convex and the control set $U$ is a closed ball.

math.OC↗

arXiv2005

Decomposition of vector-valued divergence free Sobolev functions and shape optimization for stationary Navier-Stokes equations

We establish a divergence free partition for vector-valued Sobolev functions with free divergence in ${\bf R}^n, n\geq 1$. We prove that for any domain $\om$ of class $\cal C$ in ${\bf R}^n,n=2,3$, the space $D_0^1(\om)\equiv\{\mathbf{v} \in H^1_0(Ω)^n ; {div}{\mathbf{v}}=0\}$ and the space $H_{0,σ}^1(\om)\equiv \bar{\{\mathbf{v}\in C^{\infty}_0(Ω)^n;{div}{\mathbf v}=0\}}^{\|\cdot\|_{H^1(\om)^n}}$, which is the completion of $\{\mathbf{v} \in C^{\infty}_0(Ω)^n; {div}{\mathbf v}=0\}$ in the $H^1(Ω)^n$-norm, are identical. We will also prove that $H_{0,σ}^1(D\setminus\bar\om)=\{{\mathbf v}\in H_{0,σ}^1(D); {\mathbf v}=0 {a.e. in} \om\}$, where $D$ is a bounded Lipschitz domain such that $\om\subset\subset D$. These results, together with properties for domains of class $\mathcal C$, are used to solve an existence problem in the shape optimization theory of the stationary Navier-Stokes equations.

math.OC↗