Abstract (EN)

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 (uf)t−Δuf=f, f being the control. We seek to maximise the functional JT(f):=12∫(0;T)×Ωu2f or, for some ϵ>0, JϵT(f):=12∫(0;T)×Ωu2f+ϵ∫Ωu2f(T,⋅) 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 L1 and L∞ 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 JT(f∗)−JT(f)≳∥f−f∗∥2L1(Ω), 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∞ constraint and, for every t∈(0;T), an L1 constraint. In this case, assuming ϵ>0, we prove an estimate of the form JϵT(f∗)−JϵT(f)≳∫T0aϵ(t)∥f(t,⋅)−f∗(t,⋅)∥2L1(Ω) where aϵ(t)>0 for any t∈(0;T). The proof of this result relies on a uniform bathtub principle.