Analysis Seminar

Existence of optimal shapes in optimal control theory

by Prof. Idriss Mazari (Université de Paris Dauphine)

Sala Riunioni (Dipartimento di Matematica)

Sala Riunioni

Dipartimento di Matematica


In this talk, we present several recent contributions (in collaboration with G. Nadin and Y. Privat) on the question of the existence of optimal shapes in optimal control theory for bilinear models. Motivated by applications in spatial ecology, we investigate the following problem: consider a parabolic or elliptic equation
$Lu=mu+F(u)$ where $L$ is a parabolic or elliptic operator, $m$ is the control and $F$ is a given non-linearity. The goal is to solve the optimisation problem
$$Max_m \int_\Omega j(x;u)\,dx$$
where $j$ is simply a cost functional, and $m$ is an admissible control that satisfies $L^1$ and $L^\infty$ bounds. In other words, we assume $0\le m\le1$ almost everywhere, and $\int_\Omega m\,dx=V_0$ where $V_0$ is a fixed volume constraint. A basic property for such problem is to obtain the bang-bang property for maximisers. In other words, are optimal control characteristic functions of subsets of the domain on which the equation is set? Put otherwise, can optimisers be identified with subsets of the domain? What we explain in this talk is that for bilinear control problems, the answer is analogous to the Buttazzo-DalMaso theorem: if the functional we want to optimise is monotonic, then the answer to this question is positive. Our result relies on new oscillatory techniques.