Some Recent Developments in Systems and Control Theory on Infinite Dimensional Banach Spaces: Part 1

Abstract

The objective of this paper is to present some recent developments in systems and control theory on infinite dimensional spaces. Since this work is based on semigroup theory, for the general audience, it was considered proper to introduce some fundamental results from semigroup theory. We start with some examples where semigroup theory plays a central role and then present some key results from semigroup theory. Next we consider evolution equations and inclusions and discuss various notions of solution such as classical, strong, weak, mild and finally measure solutions and present some basic results on existence and regularity properties of solutions. Following this we consider non convex control problems and present some results on the question of existence of optimal relaxed controls for evolution equations and differential inclusions. Next we consider the problems of optimal feedback control for stochastic systems and present methods of their construction. Here we study HJB equation on Hilbert space and present some results on existence and regularity of solutions leading to the construction of optimal feedback control law. Then we use a new and direct approach where we consider optimization on a topological space of operator valued functions as output feedback control laws subject to measurement uncertainty. We present necessary conditions of optimality.