The Riccati Integral Equations for Optimal Control Problems on Hilbert Spaces

Abstract

The two Riccati integral equations for linear-quadratic control problems involving evolution operators on Hilbert spaces are derived and shown to have a common solution, which yields the closed-loop structure of the optimal control. Riccati integral equations, instead of differential equations, arise because evolution operators are used to represent system dynamics. The operator representing the closed-loop control perturbs the evolution operator representing the uncontrolled system to produce a second evolution operator, representing the optimally controlled system, hence the two Riccati integral equations in terms of these two evolution operators, respectively.Having both Riccati integral equations facilitates the extension of the analysis of optimal control on finite time intervals to the analysis of optimal control on infinite time intervals, and then existence, uniqueness, and stability results for periodic solutions of the Riccati equations are obtained. Finally, sufficient conditions are given for convergence of approximate solutions of optimal control problems on both finite and infinite time intervals.