The Infinite-Dimensional Riccati Equation for Systems Defined by Evolution Operators

Abstract

In the paper we consider the linear, quadratic control and filtering problems for systems defined by integral equations given in terms of evolution operators. We impose very weak conditions on the evolution operators and prove that the solution to both problems leads to an integral Riccati equation which possesses a unique solution. By imposing more structure on the evolution operator we show that the integral Riccati equation can be differentiated, and finally by considering an even smaller class of evolution operators we are able to prove that the differentiated version has a unique solution. The motivation for the study of such systems is that they enable us to consider wide classes of differential delay equations and partial differential equations in the same formulation. We derive new results for such a system and show how all of the existing results can be obtained directly by our methods.