Quantitative estimates for parabolic optimal control problems under L∞ and L1 constraints

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 − ∆u f = f , f being the control. We seek to maximise the functional JT (f ) := 1 2 ˜(0;T )×Ω u 2 f or, for some ε > 0 ,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 JT, 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 ε > 0, we prove an estimate of the formwhere aε(t) > 0 for any t ∈ (0; T ). The proof of this result relies on a uniform bathtub principle.

Spatial ecology, optimal control and game theoretical fishing problems

“…Originating in the seminal [72], in the case of the Schwarz rearrangement for Dirichlet boundary conditions, these inequalities aim at comparing the solution u of a Poisson equation of the form −∆u = f with Dirichlet boundary conditions with the solution v of a symmetrised equation. Among the many results related to possible extensions and to the qualitative analysis of these inequalities to other operators [1,3,7,55,62,69] let us focus on the results of [45]. To use them, we need to recall the rearrangement order on L 1 (0; 1): for any two non-negative functions f, g ∈ L 1 (0; 1), we say that f dominates g in the sense of rearrangements and we write f ≺ g if, and only if,…”

Optimisation of the total population size with respect to the initial condition for semilinear parabolic equations: two-scale expansions and symmetrisations

“…It should be noted that, since we are working with Neumann boundary conditions, it is not possible to use directly well-known parabolic isoperimetric inequalities [5,6,34]. We refer to [6,28,41] and the references therein for an introduction to parabolic isoperimetric inequalities, and only underline here that the most precise results available in the literature only encompass the case of Dirichlet boundary conditions. For Neumann boundary conditions, a large literature [11,14,25] is devoted to such questions.…”

A note on the rearrangement of functions in time and on the parabolic Talenti inequality

“…the Schwarz rearrangement) of source terms in elliptic equations improved "concentration"-like properties. Before we make this statement more precise let us note that this work of Talenti has sparked an immense interest from the calculus of variations and optimisation community, leading to major developments, whether in calculus of variations, in optimal control or in fine comparison relations for parabolic and elliptic partial differential equations [1,2,3,6,5,4,7,8,9,12,11,13,15,16,17,18,21,22]. For the time being we refer to the monograph [10] and to the survey of Talenti himself [20].…”