Optimal nonpermanent control problems

Abstract

The Pontryagin Maximum Principle (PMP in short) is a fundamental result of optimal control theory established in the 1950s which provides first-order necessary optimality conditions for control systems described by ordinary differential equations. Soon afterwards and even nowadays, the PMP has been adapted to many situations: for control systems of different natures, with various constraints, etc. In the classical version of the PMP, the control is assumed to be permanent, in the sense that the value of the control is authorized to be modified at any real time. As a consequence, in numerous problems, achieving the optimal trajectory requires a permanent modification of the value of the control. However such a request is not feasible in practical situations, neither for human beings, nor for mechanical or numerical devices. For this reason, piecewise constant controls (also called sampled-data controls in the literature), whose number of authorized modifications is finite, are widely used in Automatic and Engineering. Sampled-data controls constitute a first example of nonpermanent controls. Other examples come from control systems whose trajectories cross shadow zones (such as a mobile phone or a GPS device passing under a tunnel for illustration). In this talk, we will discuss the general analysis of optimal nonpermanent control problems and the new mathematical phenomena that can be observed in them.