On the circle criterion for boundary control systems in factor form: Lyapunov stability and Lur'e equations

¹ Institute of Automatics, AGH University of Science and Technology, avenue A. Mickiewicz 30, B1, rm.314, 30-059 Cracow, Poland; pgrab@ia.agh.edu.pl
² University of Namur (FUNDP), Department of Mathematics, Rempart de la Vierge 8, 5000 Namur, Belgium; frank.callier@fundp.ac.be

Received: 1 April 2004
Revised: 6 January 2005
Revised: 23 February 2005

Abstract

A Lur'e feedback control system consisting of a linear, infinite-dimensional system of boundary control in factor form and a nonlinear static sector type controller is considered. A criterion of absolute strong asymptotic stability of the null equilibrium is obtained using a quadratic form Lyapunov functional. The construction of such a functional is reduced to solving a Lur'e system of equations. A sufficient strict circle criterion of solvability of the latter is found, which is based on results by Oostveen and Curtain [Automatica 34 (1998) 953–967]. All the results are illustrated in detail by an electrical transmission line example of the distortionless loaded -type. The paper uses extensively the philosophy of reciprocal systems with bounded generating operators as recently studied and used by Curtain in (2003) [Syst. Control Lett. 49 (2003) 81–89; SIAM J. Control Optim. 42 (2003) 1671–1702].