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

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 $\mathcal J_T(f):=\frac12\int_{(0;T)\times \Omega} u_f^2$ or, for some $\epsilon>0$, $\mathcal J_T^\epsilon (f):=\frac12\int_{(0;T)\times \Omega} u_f^2+\epsilon \int_\Omega u_f^2(T,\cdot)$ 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 $\mathcal J_T(f^*)-\mathcal J_T(f)\gtrsim \Vert f-f^*\Vert_{L^1(\Omega)}^2$, $f^*$ 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 $\epsilon>0$, we prove an estimate of the form $\mathcal J_T^\epsilon (f^*)-\mathcal J_T^\epsilon (f)\gtrsim\int_0^T a_\epsilon (t) \Vert f(t,\cdot)-f^*(t,\cdot)\Vert_{L^1(\Omega)}^2$ where $a_\epsilon (t)>0$ for any $t\in (0;T)$. The proof of this result relies on a uniform bathtub principle.