Résumé
The first part is devoted to the study of strong minima in the setting of optimal control problems governed by semi-linear elliptic and parabolic equations with integral constraints on the final state. The second part studies the minimum time problem for a system describing a chemostat model under a ''triangle'' inputs constraint on the control. An optimal control problem for a chemostat model with two competitive species is also studied. In the third part, we provide an optimal feedback control for a system describing a fed-batch bioreactor with impulsive control. A minimal time problem with a saturating singular arc is also investigated. The fourth part deals with two optimal control problems with periodic constraints on the state. The last chapter is devoted to geometric shape optimization problems under a convexity constraint. Thanks to positive polynomials that can be written as a sum of square, the convexity constraint is transformed into an SDP-constraint which allows to solve the problem using semi-definite optimization.