# 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 (2005)

- 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 (2005): 169-197. <http://eudml.org/doc/90787>.

@article{Grabowski2005,

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 [Automatica34 (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.; infinite-dimensional control systems; Lyapunov functionals; circle criterion},

language = {eng},

month = {12},

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/90787},

volume = {12},

year = {2005},

}

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

DA - 2005/12//

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 [Automatica34 (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.; infinite-dimensional control systems; Lyapunov functionals; circle criterion

UR - http://eudml.org/doc/90787

ER -

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