Optimal control of differential equations with - or without - memory.

Summary The thesis deals with optimal control problems where the dynamics are given by differential equations with memory. For these optimization problems, optimality conditions are established. Second order optimality conditions constitute an important part of the results of the thesis. In the case - without memory - of ordinary differential equations, the standard optimality conditions are strengthened by involving only the Lagrange multipliers for which the Pontryaguine principle is satisfied. This restriction to a subset of the multipliers represents a challenge in establishing the necessary conditions and allows the sufficient conditions to ensure local optimality in a stronger sense. The standard conditions are further extended to the case - with memory - of integral equations. The pure constraints on the state of the previous problem have been preserved and require a specific study of the integral dynamics. Another form of memory in the equation of state of an optimal control problem comes from a modeling work with therapeutic optimization as a medical application in mind. The population dynamics of cancer cells under the action of a treatment is reduced to differential equations with time delays. The asymptotic behavior in long time of the age-structured model is also studied.