OPTIMAL-CONTROL OF STABLE WEAKLY REGULAR LINEAR-SYSTEMS

The paper extends quadratic optimal control theory to weakly regular l inear systems, a rather broad class of infinite-dimensional systems wi th unbounded control and observation operators. We assume that the sys tem is stable (in a sense to be defined) and that the associated Popov function is bounded From below. We study the properties of the optima lly controlled system, of the optimal cost operator X, and the various Riccati equations which are satisfied by X. We introduce the concept of an optimal state feedback operator, which is an observation operato r for the open-loop system, and which produces the optimal feedback sy stem when its output is connected to the input of the system. We show that if the spectral factors of the Popov function are regular, then a (unique) optimal state feedback operator exists, and we give its form ula in terms of X. Most of the formulas are quite reminiscent of the c lassical formulas from the finite-dimensional theory. However, an unex pected factor appears both in the formula of the optimal state feedbac k operator as well as in the main Riccati equation We apply our theory to an extensive example.