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Optimal control of stable weakly regular linear systems

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Abstract

The paper extends quadratic optimal control theory to weakly regular linear systems, a rather broad class of infinite-dimensional systems with unbounded control and observation operators. We assume that the system is stable (in a sense to be defined) and that the associated Popov function is bounded from below. We study the properties of the optimally controlled system, of the optimal cost operatorX, and the various Riccati equations which are satisfied byX. We introduce the concept of an optimal state feedback operator, which is an observation operator for the open-loop system, and which produces the optimal feedback system when its output is connected to the input of the system. We show that if the spectral factors of the Popov function are regular, then a (unique) optimal state feedback operator exists, and we give its formula in terms ofX. Most of the formulas are quite reminiscent of the classical formulas from the finite-dimensional theory. However, an unexpected factor appears both in the formula of the optimal state feedback operator as well as in the main Riccati equation. We apply our theory to an extensive example.

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Authors and Affiliations

  1. Faculty of Applied Physics, Techn. Univ. of Eindhoven, 5600, MB Eindhoven, The Netherlands

    Martin Weiss

  2. School of Engineering, University of Exeter, EX4 4QF, Exeter, England

    George Weiss

Authors
  1. Martin Weiss
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  2. George Weiss
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Additional information

Part of the results reported here were obtained while the second author was visiting FUNDP Namur, under the Belgian Program on Inter-University Poles of Attraction initiated by the Belgian state, Prime Minister's Office, Science Policy Programming. The scientific responsibility is assumed by the authors.

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Weiss, M., Weiss, G. Optimal control of stable weakly regular linear systems. Math. Control Signal Systems 10, 287–330 (1997). https://doi.org/10.1007/BF01211550

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  • DOI: https://doi.org/10.1007/BF01211550

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