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

Piotr Grabowski; Frank M. Callier

ESAIM: Control, Optimisation and Calculus of Variations (2006)

- Volume: 12, Issue: 1, page 169-197
- ISSN: 1292-8119

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topGrabowski, Piotr, and 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 12.1 (2006): 169-197. <http://eudml.org/doc/244909>.

@article{Grabowski2006,

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].},

author = {Grabowski, Piotr, Callier, Frank M.},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {infinite-dimensional control systems; semigroups; Lyapunov functionals; circle criterion},

language = {eng},

number = {1},

pages = {169-197},

publisher = {EDP-Sciences},

title = {On the circle criterion for boundary control systems in factor form : Lyapunov stability and Lur’e equations},

url = {http://eudml.org/doc/244909},

volume = {12},

year = {2006},

}

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

PB - EDP-Sciences

VL - 12

IS - 1

SP - 169

EP - 197

AB - 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].

LA - eng

KW - infinite-dimensional control systems; semigroups; Lyapunov functionals; circle criterion

UR - http://eudml.org/doc/244909

ER -

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