球内$L^\infty$和$L^1$约束下抛物型最优控制问题的定量估计：抛物型等周不等式的量化
Quantitative estimates for parabolic optimal control problems under $L^\infty$ and $L^1$ constraints in the ball:Quantifying parabolic isoperimetric inequalities
摘要：
在这篇文章中，我们提出了两种不同的方法来获得定量不等式的上下文中的抛物型最优控制问题。我们的模型由一个线性控制的热方程Dirichlet边界条件$(u_f)_t-Δu_f=f$，$f$是控制。我们寻求最大化的功能$\mathcal J_T(f):=\frac12\int_{(0;T)\times Ω} u_f^2$或，对于一些$ε>0$，$\mathcal J_T^ε(f):=\frac12\int_{(0;T)\times Ω} u_f^2+ε\int_Ωu_f^2(T,\cdot)$，并获得这些最大化问题的定量估计。我们提供了两种方法的情况下，域$Ω$是一个球。在这种情况下，如果$f$满足$L^1$和$L^\infty$的约束，并不依赖于时间，我们提出了一个形状衍生物的方法，表明，对于任何竞争对手$f=f(x)$满足相同的约束，我们有$\mathcal J_T(f^*)-\mathcal J_T(f)\gtrsim \Vert f-f^*\Vert_{L^1(Ω)}^2$，$f^*$是最大化。通过我们的时间独立的情况下，我们还展示了如何获得的形状hessians在这样的抛物型优化问题的bavity规范。我们还考虑的情况下，$f=f(t,x)$满足全球$L^\infty$约束，并为每个$t\in (0;T)$，$L^1$约束。在这种情况下，假设$ε>0$，我们证明了估计的形式$\mathcal J_T^ε(f^*)-\mathcal J_T^ε(f)\gtrsim\int_0^T a_ε(t) \Vert f(t,\cdot)-f^*(t,\cdot)\Vert_{L^1(Ω)}^2$其中$a_ε(t)>0$为任何$t\in (0;T)$。这个结果的证明依赖于一个统一的浴缸原理。
英文摘要：
In this article, we present two different approaches for obtaining quantitative inequalities in the context of parabolic optimal control problems. Our model consists of a linearly controlled heat equation with Dirichlet boundary condition $(u_f)_t-Δu_f=f$, $f$ being the control. We seek to maximise the functional $\mathcal J_T(f):=\frac12\int_{(0;T)\times Ω} u_f^2$ or, for some $ε>0$, $\mathcal J_T^ε(f):=\frac12\int_{(0;T)\times Ω} u_f^2+ε\int_Ωu_f^2(T,\cdot)$ and to obtain quantitative estimates for these maximisation problems. We offer two approaches in the case where the domain $Ω$ is a ball. In that case, if $f$ satisfies $L^1$ and $L^\infty$ constraints and does not depend on time, we propose a shape derivative approach that shows that, for any competitor $f=f(x)$ satisfying the same constraints, we have $\mathcal J_T(f^*)-\mathcal J_T(f)\gtrsim \Vert f-f^*\Vert_{L^1(Ω)}^2$, $f^*$ being the maximiser. Through our proof of this time-independent case, we also show how to obtain coercivity norms for shape hessians in such parabolic optimisation problems. We also consider the case where $f=f(t,x)$ satisfies a global $L^\infty$ constraint and, for every $t\in (0;T)$, an $L^1$ constraint. In this case, assuming $ε>0$, we prove an estimate of the form $\mathcal J_T^ε(f^*)-\mathcal J_T^ε(f)\gtrsim\int_0^T a_ε(t) \Vert f(t,\cdot)-f^*(t,\cdot)\Vert_{L^1(Ω)}^2$ where $a_ε(t)>0$ for any $t\in (0;T)$. The proof of this result relies on a uniform bathtub principle.
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