论文标题
半线性方程\ emph {via}浓度的线性最佳控制的本地最佳条件，用于振荡线性抛物线方程溶液的浓度结果
Localising optimality conditions for the linear optimal control of semilinear equations \emph{via} concentration results for oscillating solutions of linear parabolic equations
论文作者
论文摘要
我们对二阶最佳条件进行了精细分析，以相对于初始条件对半线性抛物线方程的最佳控制。更准确地说，我们研究以下问题：相对于$ y \在l^\ infty（{（0; t）\ timesω}）$成本功能$ j（y）= \ iint _ {（0; t）\ timesω} U-ΔU= F（T，X，U）+Y \，U（0，\ CDOT）= U_0 $具有某些经典边界条件，在$-κ__0\ leq y \ leq y \ leq y \ leq leqκ_1\ text {a.e。} \ e。这类问题在几个应用程序字段中产生。这些问题的一个具有挑战性的特征是对所谓的异常集合$ \ { - κ__0<y^*<κ_1\} $，其中$ y^*$是一个优化者。该集合通常是非空的，重要的（例如，对于数值应用程序），了解此集合中$ y^*$的行为：哪些值可以$ y^*$占用？在本文中，我们介绍了一种拉普拉斯型方法，以提供一些问题的答案。这种拉普拉斯类型的方法具有独立的兴趣。
We propose a fine analysis of second order optimality conditions for the optimal control of semi-linear parabolic equations with respect to the initial condition. More precisely, we investigate the following problem: maximise with respect to $y\in L^\infty({(0;T)\times Ω})$ the cost functional $J(y)=\iint_{(0;T)\times Ω}j_1(t,x,u)+\int_Ωj_2(x,u(T,\cdot))$ where $\partial_t u-Δu=f(t,x,u)+y\,, u(0,\cdot)=u_0$ with some classical boundary conditions, under constraints of the form $-κ_0\leq y\leq κ_1\text{ a.e.}\,, \int_Ωy(t,\cdot)=V_0$. This class of problems arises in several application fields. A challenging feature of these problems is the study of the so-called abnormal set $ \{-κ_0<y^*<κ_1\}$ where $y^*$ is an optimiser. This set is in general non-empty and it is important (for instance for numerical applications) to understand the behaviour of $y^*$ in this set: which values can $ y^*$ take? In this paper, we introduce a Laplace-type method to provide some answers to this question. This Laplace type method is of independent interest.
