### Thursday, May 7

## IP3

Linear Operator Inequalities and the Positive-Real Lemma for Infinite-Dimensional Linear Systems

2:00 PM-3:00 PM

*Chair: Belinda King, Oregon State University*

*Room: Salon D*

It is well known that control problems for linear systems with a quadratic cost functional are related to the existence of solutions to Riccati equations. In the singular case, the latter reduce to linear operator (in)equalities, the most famous example being the positive-real lemma.

The operator equalities are essentially constrained Lyapunov equations and have been applied to establish stability of nonlinear differential equations, and the positive-real lemma is the key tool in establishing stability for adaptive controllers. While the theory of the existence of solutions to Riccati equations is well established, little has been done for the singular case. In this presentation, the speaker will discuss some new results on the singular version for infinite-dimensional linear systems using the powerful spectral factorization approach introduced recently by Staffans and Weiss. She will highlight the special case of the positive-real lemma and apply it to design adaptive compensators for a class infinite-dimensional linear systems with a structured perturbation involving an unknown constant parameter.

**Ruth F. Curtain**

*Mathematics Department*

*University of Groningen, The Netherlands*

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*tjf, 11/17/97*

*MMD, 11/25/97*