Quantitative estimates for parabolic optimal control problems under $L^{\infty }$ and $L^1$ constraints in the ball:Quantifying parabolic isoperimetric inequalities

Abstract

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 $(u_f{)}_t-\Delta u_f=f$, $f$ being the control. We seek to maximise the functional ${\mathcal{J}}_T(f):=\frac{1}{2}{\int }_{(0;T)\times \Omega }{u}_f^2$ or, for some $\epsilon >0$, ${\mathcal{J}}_T^{\epsilon }(f):=\frac{1}{2}{\int }_{(0;T)\times \Omega }{u}_f^2+\epsilon {\int }_{\Omega }{u}_f^2(T,\cdot )$ and to obtain quantitative estimates for these maximisation problems. We offer two approaches in the case where the domain $\Omega$ 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 ${\mathcal{J}}_T(f^{\ast })-{\mathcal{J}}_T(f)\gtrsim \Vert f-f^{\ast }{\Vert }_{L^1(\Omega )}^2$, $f^{\ast }$ 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 $\epsilon >0$, we prove an estimate of the form ${\mathcal{J}}_T^{\epsilon }(f^{\ast })-{\mathcal{J}}_T^{\epsilon }(f)\gtrsim {\int }_0^T{a}_{\epsilon }(t)\Vert f(t,\cdot )-f^{\ast }(t,\cdot ){\Vert }_{L^1(\Omega )}^2$ where $a_{\epsilon }(t)>0$ for any $t\in (0;T)$. The proof of this result relies on a uniform bathtub principle.

53 pages

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• Faber-Krahn inequalities in sharp quantitative form
• Insensitizing exact controls for the scalar wave equation and exact controllability of $2$-coupled cascade systems of PDE's by a single control
• Large-time asymptotics in deep learning