Hybrid optimal control problems for partial differential equations

Hybrid optimal control problems consist in optimizing a control function and some parameters modeling the qualitative behaviour of the whole system. These parameters can be for example switching times, at which the dynamics of the system changes, or at which an actuatuor is activated or disactivated. In general, these parameters lead to a lack of regularity for the system. We develop specific methods in order to circumvent this difficulty, both from a theoretical point of view (for the derivation of optimality conditions) and from a numerical point of view.

In a recent work, control problems of semilinear parabolic partial differential equations with switching times have been investigated. Problems with an L^infinity cost have also been studied, in a general framework. We currently aim at developing techniques for control problems of systems governed by conservation laws with space parameters. The applications are various, and range from the determination of the optimal moment at which a maximum has to be reached, to the creation in a basin of the highest wave possible, whose the location is let free.

The motivation of this approach lies in cardiac electrophysiology: We aim at maximizing a mechanical pressure inside the tissue of the heart, in order to improve the efficiency of defibrillation, with some electric device whose activation has to be done at an optimal moment. This development is still a work in progress.