Global minima for semilinear optimal control problems

关于PDE约束形状优化中半线性椭圆方程的$W^{1,\infty}$-最陡下降法的全局收敛性

Global convergence of $W^{1,\infty}$-steepest descent for PDE constrained shape optimisation with semilinear elliptic equations in function space

本文研究了PDE约束形状优化中半线性椭圆方程的$W^{1,\infty}$-最陡下降法的全局收敛性，证明了在函数空间中的收敛性及二维空间中的条件收敛结果。

半离散随机半线性抛物型方程的零能控性

Null controllability for semi-discrete stochastic semilinear parabolic equations

本文研究空间有限差分离散的随机半线性抛物型方程零能控性，通过显式给出Carleman参数与离散化参数并结合Banach不动点方法证明其成立。

Global exponential stability and Input-to-State Stability of semilinear hyperbolic systems for the $L^{2}$ norm

In this paper we study the global exponential stability in the $L^{2}$ norm of semilinear $1$-$d$ hyperbolic systems on a bounded domain, when the source term and the nonlinear boundary conditions are Lipschitz. We exhibit two sufficient stability conditions: an internal condition and a boundary condition. This result holds also when the source term is nonlocal. Finally, we show its robustness by extending it to global Input-to State Stability in the $L^{2}$ norm with respect to both interior and boundary disturbances.

Global Output Feedback Stabilization of Semilinear Reaction-Diffusion PDEs

This paper addresses the topic of global output feedback stabilization of semilinear reaction-diffusion PDEs. The semilinearity is assumed to be confined into a sector condition. We consider two different types of actuation configurations, namely: bounded control operator and right Robin boundary control. The measurement is selected as a left Dirichlet trace. The control strategy is finite dimensional and is designed based on a linear version of the plant. We derive a set of sufficient conditions ensuring the global exponential stabilization of the semilinear reaction-diffusion PDE. These cond…

半线性波方程解析性的全局传播与唯一延拓

Global propagation of analyticity and unique continuation for semilinear waves

本文结合控制论与Galerkin逼近，提出新方法证明半线性波型方程解析性的全局传播与唯一延拓，并在次临界散焦情形下得到几何控制条件最优时间内的可观测性估计。

A note on the global controllability of the semilinear wave equation

We study the internal controllability of the semilinear wave equation $$v_{tt}(x,t)-Δv(x,t) + f(x,v(x,t))= \Un_ω u(x,t)$$ for some nonlinearities $f$ which can produce several non-trivial steady states. One of the usual hypotheses to get global controllability, is to assume that $f(x,v)v\geq 0$. In this case, a stabilisation term $u=γ(x)v_t$ makes any solution converging to zero. The global controllability then follows from a theorem of local controllability and the time reversibility of the equation. In this paper, the nonlinearity $f$ can be more general, so that the solutions of the damped …

Controllability under positivity constraints of semilinear heat equations

In many practical applications of control theory some constraints on the state and/or on the control need to be imposed. In this paper, we prove controllability results for semilinear parabolic equations under positivity constraints on the control, when the time horizon is long enough. As we shall see, in fact, the minimal controllability time turns out to be strictly positive. More precisely, we prove a global steady state constrained controllability result for a semilinear parabolic equation with $C^1$ nonlinearity, without sign or globally Lipschitz assumptions on the nonlinear term. Then, …

A uniform bound on costs of controlling semilinear heat equations on a sequence of increasing domains and its application

In this paper, we first prove a uniform upper bound on costs of null controls for semilinear heat equations with globally Lipschitz nonlinearity on a sequence of increasing domains, where the controls are acted on an equidistributed set that spreads out in the whole Euclidean space R^N . As an application, we then show the exactly null controllability for this semilinear heat equation in R^N . The main novelty here is that the upper bound on costs of null controls for such kind of equations in large but bounded domains can be made uniformly with respect to the sizes of domains under considerat…

基于小增益的边界反馈设计用于1-D半线性抛物型PDEs的全局指数稳定

Small-Gain-Based Boundary Feedback Design for Global Exponential Stabilization of 1-D Semilinear Parabolic PDEs

本文提出了一种用于1-D半线性抛物型PDEs边界反馈稳定器设计的新方法，基于小增益论证，适用于满足线性增长条件的非线性项，包括非局部项，并展示了反馈设计在抛物型情况下的根本限制。

分数半线性PDEs的最优控制

Optimal control of fractional semilinear PDEs

本文研究了具有谱和积分分数扩散算子的半线性分数PDEs的最优控制问题，通过证明解的有界性、非线性增长条件以及最优控制问题的解的存在性，推导了最优性条件。