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Maximal solution to algebraic Riccati equations linked to infinite Markov jump linear systems

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Abstract

In this paper, we deal with a perturbed algebraic Riccati equation in an infinite dimensional Banach space. Besides the interest in its own right, this class of equations appears, for instance, in the optimal control problem for infinite Markov jump linear systems (from now on iMJLS). Here, infinite or finite has to do with the state space of the Markov chain being infinite countable or finite (see Fragoso and Baczynski in SIAM J Control Optim 40(1):270–297, 2001). By using a certain concept of stochastic stability (a sort of L 2-stability), we prove the existence (and uniqueness) of maximal solution for this class of equation and provide a tool to compute this solution recursively, based on an initial stabilizing controller. When we recast the problem in the finite setting (finite state space of the Markov chain), we recover the result of de Souza and Fragoso (Syst Control Lett 14:233–239, 1999) set to the Markovian jump scenario, now free from an inconvenient technical hypothesis used there, originally introduced in Wonham in (SIAM J Control 6(4):681–697).

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Authors and Affiliations

  1. National Laboratory for Scientific Computing -LNCC/CNPq, Av. Getulio Vargas 333, Petrópolis, Rio de Janeiro, 25651-070, Brazil

    Jack Baczynski & Marcelo D. Fragoso

Authors
  1. Jack Baczynski
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  2. Marcelo D. Fragoso
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Correspondence to Jack Baczynski.

Additional information

Research supported by grants CNPq 520367-97-9, 300662/2003-3 and 474653/2003-0, FAPERJ 171384/2002, PRONEX and IM-AGIMB.

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Baczynski, J., Fragoso, M.D. Maximal solution to algebraic Riccati equations linked to infinite Markov jump linear systems. Math. Control Signals Syst. 20, 157–172 (2008). https://doi.org/10.1007/s00498-008-0027-y

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