Linear operator inequalities for stable weakly regular linear systems
Ruth F. Curtain · NCSU Libraries Repository (North Carolina State University Libraries) · 1997
We consider 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 de nite 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.Sucient conditions for the rst 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.Although delay equations do not satisfy the above criterion, if the closed span of the eigenvectors equals the state space, a general approach is proposed that shows promise for retarded systems.The above results have been used to design adaptive observers for positive-real in nite-dimensional systems.