Linear Operator Inequalities for Strongly Stable Weakly Regular Linear Systems

We consider the question of the existence of solutions to certain linear operator inequalities (Lur'e equations) for strongly stable, weakly regular linear systems with generating operators A, B, C, 0. These operator inequalities are related to the spectral factorization of an associated Popov function and to singular optimal control problems with a nonnegative definite quadratic cost functional. We split our problem into two subproblems: the existence of spectral factors of the nonnegative Popov function and the existence of a certain extended output map. Sufficient conditions for the solvability of the first problem are known and for the case that A has compact resolvent and its eigenvectors form a Riesz basis for the state space, we give an explicit solution to the second problem in terms of A, B, C and the spectral factor. The applicability of these results is demonstrated by various heat equation examples satisfying a positive-real condition. If (A, B) is approximately controllable, we obtain an alternative criterion for the existence of an extended output operator which is applicable to retarded systems. The above results have been used to design adaptive observers for positive-real infinite-dimensional systems. Date received: July 25, 1997. Date revised: February 10, 2001.