Résumé
This presentation is about periodic optimal control problems governed by a system, linear w.r.t. the control u, under an integral constraint on u. In the one-dimensional setting, we give conditioon on the value of the cost function at steady state with a constant control \bar u can be improved by considering periodic controls u(.) with average value equal to \bar u. This leads to the so-called "over-yielding" met in several applications such as in resource-consumer models.With the use of the Pontryagin Maximum Principle, we provide the optimal synthesis of periodic strategies under the integral constraint. The results will be illustrated on a single population model in order to study the effect of periodic inputs on the utility of the stock of resource.