Necessary optimality conditions in control theory

Abstract

We discuss the Bolza and the Mayer problems arising in optimal control which are of great interest in all models involving dynamic optimization. The main frame is to minimize a functional under a dynamic constraint which involves controls, that is parameters depending on time. Such models do arise in biodiversity, economics, insurance, finance, population dynamics, etc. It is well known that every strong local minimizer of these problems satisfies the Pontryagin maximum principle. In the absence of constraints qualifications the maximum principle may be abnormal, that is, not involving the cost functions and therefore it provides useless information. We provide sufficient conditions for normality and apply them to guarantee the non occurrence of the so-called Lavrentieff phenomenon. When an optimal control is singular, second-order optimality conditions are necessary in order to identify optimal trajectories. To this aim we provide a generalization of the Goh and the Legendre-Clebsch conditions.