Opuscula Math. 37, no. 1 (2017), 21-64

Opuscula Mathematica

The LQ/KYP problem for infinite-dimensional systems

Piotr Grabowski

Abstract. Our aim is to present a solution to a general linear-quadratic (LQ) problem as well as to a Kalman-Yacubovich-Popov (KYP) problem for infinite-dimensional systems with bounded operators. The results are then applied, via the reciprocal system approach, to the question of solvability of some Lur'e resolving equations arising in the stability theory of infinite-dimensional systems in factor form with unbounded control and observation operators. To be more precise the Lur'e resolving equations determine a Lyapunov functional candidate for some closed-loop feedback systems on the base of some properties of an uncontrolled (open-loop) system. Our results are illustrated in details by an example of a temperature of a rod stabilization automatic control system.

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 and Biomedical Engineering, al. Mickiewicza 30, 30-059 Krakow, Poland
  • Communicated by A. Shkalikov.
  • Received: 2016-05-19.
  • Revised: 2016-10-24.
  • Accepted: 2016-10-29.
  • Published online: 2016-12-14.
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Cite this article as:
Piotr Grabowski, The LQ/KYP problem for infinite-dimensional systems, Opuscula Math. 37, no. 1 (2017), 21-64, http://dx.doi.org/10.7494/OpMath.2017.37.1.21

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