Quantitative estimates for parabolic optimal control problems under L∞ and L1 constraints in the ball: Quantifying parabolic isoperimetric inequalities. (February 2022) - Mazari, Idriss
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 − Δ u f = f, f being the control. We seek to maximise the functional J T ( f ) ≔ 1 2 ∬ ( 0 ; T ) × Ω u f 2 or, for some ɛ > 0, J T ɛ ( f ) ≔ 1 2 ∬ ( 0 ; T ) × Ω u f 2 + ɛ ∫ Ω u f 2 ( 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 L 1 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 J T ( f ∗ ) − J T ( f ) ≳ ‖ f − f ∗ ‖ 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 ∞ constraint and, for every t ∈ ( 0 ; T ), an L 1 constraint. In this case, assuming ɛ > 0, we prove an estimate of the form J T ɛ ( f ∗ ) − J T ɛ ( f ) ≳ ∫ 0 T a ɛ ( t ) ‖ f ( t, ⋅ ) − f ∗ ( t, ⋅ ) ‖ L 1 ( Ω ) 2 where a ɛ ( t ) > 0 for any t ∈ ( 0 ; T ) . The proof of this result relies on a uniform bathtub principle.