Opuscula Math. 38, no. 5 (2018), 651-680
https://doi.org/10.7494/OpMath.2018.38.5.651

Opuscula Mathematica

# Small-gain theorem for a class of abstract parabolic systems

Piotr Grabowski

Abstract. 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.

Keywords: control of infinite-dimensional systems, semigroups, infinite-time LQ-control problem, Lur'e feedback systems.

Mathematics Subject Classification: 49N10, 93B05, 93C25.

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• Piotr Grabowski
• AGH University of Science and Technology, Institute of Control Engineering and Robotics, al. Mickiewicza 30, 30-059 Krakow, Poland
• Communicated by Marek Galewski.
• Revised: 2018-04-04.
• Accepted: 2018-04-04.
• Published online: 2018-06-12.