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Issue ESAIM: COCV
Volume 12, Number 1, January 2006
Page(s) 169 - 197
DOI 10.1051/cocv:2005027
Published online 15 December 2005

ESAIM: COCV, January 2006, Vol. 12, pp. 169-197
DOI: 10.1051/cocv:2005027

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

Piotr Grabowski1 and Frank M. Callier2

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 April 1, 2004. Revised January 6 and February 23, 2005. / Published online: 15 December 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 $\mathfrak{RLCG} $-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].


Mathematics Subject Classification. 34G, 35A, 47D, 93B

Key words: Infinite-dimensional control systems, semigroups, Lyapunov functionals, circle criterion.


© EDP Sciences, SMAI 2006


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