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 $\mathrm{\Re \U0001d50f\u212d\U0001d50a}$-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].

Keywords: infinite-dimensional control systems, semigroups, Lyapunov functionals, circle criterion

@article{COCV_2006__12_1_169_0, author = {Grabowski, Piotr and Callier, Frank M.}, title = {On the circle criterion for boundary control systems in factor form : {Lyapunov} stability and {Lur'e} equations}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {169--197}, publisher = {EDP-Sciences}, volume = {12}, number = {1}, year = {2006}, doi = {10.1051/cocv:2005027}, zbl = {1105.93044}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv:2005027/} }

TY - JOUR AU - Grabowski, Piotr AU - Callier, Frank M. TI - On the circle criterion for boundary control systems in factor form : Lyapunov stability and Lur'e equations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2006 DA - 2006/// SP - 169 EP - 197 VL - 12 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2005027/ UR - https://zbmath.org/?q=an%3A1105.93044 UR - https://doi.org/10.1051/cocv:2005027 DO - 10.1051/cocv:2005027 LA - en ID - COCV_2006__12_1_169_0 ER -

%0 Journal Article %A Grabowski, Piotr %A Callier, Frank M. %T On the circle criterion for boundary control systems in factor form : Lyapunov stability and Lur'e equations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2006 %P 169-197 %V 12 %N 1 %I EDP-Sciences %U https://doi.org/10.1051/cocv:2005027 %R 10.1051/cocv:2005027 %G en %F COCV_2006__12_1_169_0

Grabowski, Piotr; Callier, Frank M. On the circle criterion for boundary control systems in factor form : Lyapunov stability and Lur'e equations. ESAIM: Control, Optimisation and Calculus of Variations, Volume 12 (2006) no. 1, pp. 169-197. doi : 10.1051/cocv:2005027. http://www.numdam.org/articles/10.1051/cocv:2005027/

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