arXiv subjects
Publications and source records attributed to Donghui Yang.
In this paper, we consider a two-person game problem governed by a linear heat equation. The existence of Nash equilibrium for this problem is considered. Moreover, the bang-bang property of Nash equilibrium is discussed.
In this paper, we study the approximate controllability for the stochastic heat equation over measurable sets, and the optimal actuator location of the minimum norm controls. We formulate a relaxed optimization problem for both actuator location and its corresponding minimum norm control into a two-person zero sum game problem and develop a sufficient and necessary condition for the optimal solution via Nash equilibrium. At last, we prove that the relaxed optimal solution is an optimal actuator location for the classical problem.
In this paper, we first design a time optimal control problem for the heat equation with sampled-data controls, and then use it to approximate a time optimal control problem for the heat equation with distributed controls. Our design is reasonable from perspective of sampled-data controls. And it might provide a right way for the numerical approach of a time optimal distributed control problem, via the corresponding semi-discretized (in time variable) time optimal control problem. The study of such a time optimal sampled-data control problem is not easy, because it may have infinitely many optimal controls. We find connections among this problem, a minimal norm sampled-data control problem and a minimization problem. And obtain some properties on these problems. Based on these, we not only build up error estimates for optimal time and optimal controls between the time optimal sampled-data control problem and the time optimal distributed control problem, in terms of the sampling period, but also prove that such estimates are optimal in some sense.
This paper aims to provide directly the observability inequality of backward stochastic heat equations for measurable sets. As an immediate application, the null controllability of the forward heat equations is obtained. Moreover, an interesting relaxed optimal actuator location problem is formulated, and the existence of its solution is proved. Finally, the solution is characterized by a Nash equilibrium of the associated game problem.
A notion of $L^p$-exact controllability is introduced for linear controlled (forward) stochastic differential equations, for which several sufficient conditions are established. Further, it is proved that the $L^p$-exact controllability, the validity of an observability inequality for the adjoint equation, the solvability of an optimization problem, and the solvability of an $L^p$-type norm optimal control problem are all equivalent.
In this paper we obtain a new class of open sets, and we prove the class is compact under the Hausdorff distance, then we prove the existence of solutions of some shape optimization for elliptic equations.
In this paper we will prove the nodal line $N$ of the second eigenfunction of the Laplacian over some simply connected concave domain $Ω$ in $\mathbb{R}^2$ must intersect the boundary $\partialΩ$ at exactly two points.
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.