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

Grabowski, Piotr; Callier, Frank M.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 1, pp. 169-197.

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 $\mathrm{\Re 𝔏ℭ𝔊}$-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].

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