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• W3126613760 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 $mathcal J_T(f):=frac12int_{(0;T)times Omega} u_f^2$ or, for some $epsilon>0$, $mathcal J_T^epsilon (f):=frac12int_{(0;T)times Omega} u_f^2+epsilon int_Omega u_f^2(T,cdot)$ 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 $tin (0;T)$, an $L^1$ constraint. In this case, assuming $epsilon>0$, we prove an estimate of the form $mathcal J_T^epsilon (f^*)-mathcal J_T^epsilon (f)gtrsimint_0^T a_epsilon (t) Vert f(t,cdot)-f^*(t,cdot)Vert_{L^1(Omega)}^2$ where $a_epsilon (t)>0$ for any $tin (0;T)$. The proof of this result relies on a uniform bathtub principle." @default.
• W3126613760 created "2021-02-15" @default.
• W3126613760 creator A5043933436 @default.
• W3126613760 date "2021-02-10" @default.
• W3126613760 modified "2023-09-27" @default.
• W3126613760 title "Quantitative estimates for parabolic optimal control problems under $L^infty$ and $L^1$ constraints in the ball:Quantifying parabolic isoperimetric inequalities" @default.
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• W3126613760 hasPublicationYear "2021" @default.
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