TY  - JOUR
ID  - Grabowski2018
LB  - Grabowski2018
AU  - Grabowski, Piotr
TI  - Small-gain theorem for a class of abstract parabolic systems
ST  - Small-gain theorem for a class of abstract parabolic systems
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2018
VL  - 38
IS  - 5
SP  - 651
EP  - 680
DO  - https://doi.org/10.7494/OpMath.2018.38.5.651
UR  - https://doi.org/10.7494/OpMath.2018.38.5.651
AB  - We consider a class of abstract control system of parabolic type with observation which the state, input and output spaces are Hilbert spaces. The state space operator is assumed to generate a linear exponentially stable analytic semigroup. An observation and control action are allowed to be described by unbounded operators. It is assumed that the observation operator is admissible but the control operator may be not. Such a system is controlled in a feedback loop by a controller with static characteristic being a globally Lipschitz map from the space of outputs into the space of controls. Our main interest is to obtain a perturbation theorem of the small-gain-type which guarantees that null equilibrium of the closed-loop system will be globally asymptotically stable in Lyapunov's sense.
KW  - control of infinite-dimensional systems
KW  - semigroups
KW  - infinite-time LQ-control problem
KW  - Lur'e feedback systems
SN  - 1232-9274
ER  -