Computation of variations and optimal control with deviated arguments.

Summary This thesis is dedicated to the study of some problems in the computation of variations of deviated argument functionals involved for instance in optimal control problems of deviated argument differential equations and in variational problems with deviated arguments. We use the direct method to show the existence of the deviated argument problem in dimension n > 1 in a Sobolev type functional space with weight related to the deviation. Then, we put forward the necessary conditions of optimality based on the area formula. We obtain an equivalence between a problem without deviation and a problem with deviation in a convex framework. We also show a form of Pontryagin's principle for a class of optimal control problems governed by an equation with a memory. Some application examples are considered. Infin we complete with some existence and uniqueness results for some nonlocal elliptic equations known as deviated argument, encountered in the literature.