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Applied Mathematics and Optimization

, Volume 37, Issue 1, pp 99–126 | Cite as

A New Approach to Linearly Perturbed Riccati Equations Arising in Stochastic Control

  • M. D. Fragoso
  • O. L. V. Costa
  • C. E. de Souza

Abstract.

In this paper a linearly perturbed version of the well-known matrix Riccati equations which arise in certain stochastic optimal control problems is studied. Via the concepts of mean square stabilizability and mean square detectability we improve previous results on both the convergence properties of the linearly perturbed Riccati differential equation and the solutions of the linearly perturbed algebraic Riccati equation. Furthermore, our approach unifies, in some way, the study for this class of Riccati equations with the one for classical theory, by eliminating a certain inconvenient assumption used in previous works (e.g., [10] and [26]). The results are derived under relatively weaker assumptions and include, inter alia, the following: (a) An extension of Theorem 4.1 of [26] to handle systems not necessarily observable. (b) The existence of a strong solution, subject only to the mean square stabilizability assumption. (c) Conditions for the existence and uniqueness of stabilizing solutions for systems not necessarily detectable. (d) Conditions for the existence and uniqueness of mean square stabilizing solutions instead of just stabilizing. (e) Relaxing the assumptions for convergence of the solution of the linearly perturbed Riccati differential equation and deriving new convergence results for systems not necessarily observable.

Key words. Linearly perturbed Riccati equation, Strong solution, Mean square stabilizability, Nonobservable systems, Stochastic control. AMS Classification. 93E20, 93C60. 

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Copyright information

© Springer-Verlag New York Inc. 1998

Authors and Affiliations

  • M. D. Fragoso
    • 1
  • O. L. V. Costa
    • 2
  • C. E. de Souza
    • 3
  1. 1.National Laboratory for Scientific Computing - LNCC/CNPq, Department of Research and Development, Rua Lauro Müller 455, 22290-160 Rio de Janeiro, RJ, Brazil frag@server01.lncc.br BR
  2. 2.Escola Politécnica da Universidade de São Paulo, Departamento de Engenharia Eletrônica, 05508-900 São Paulo SP, Brazil oswaldo@ubirajara.lac.usp.br BR
  3. 3.Department of Electrical and Computer Engineering, University of Newcastle, Newcastle, N.S.W. 2308, AustraliaAU

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