On the Circle Criterion for Feedback Systems with both Unbounded Observation and Control

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 J.C. Oostveen and R.F. Curtain (1998). The paper uses extensively the philosophy of reciprocal systems with bounded generating operators as recently studied and used by R.F. Curtain (2003)