Maximal versus strong solution to algebraic Riccati equations arising in infinite Markov jump linear systems

We deal with a perturbed algebraic Riccati equation in an infinite dimensional Banach space which appears, for instance, in the optimal control problem for infinite Markov jump linear systems (from now on iMJLS). Infinite or finite here has to do with the state space of the Markov chain being infinite countable or finite (see, e.g., [M.D. Fragoso, J. Baczynski, Optimal control for continuous time LQ—problems with infinite Markov jump parameters, SIAM J. Control Optim. 40(1) (2001) 270–297]). By using a certain concept of stochastic stability (a sort of L2L2-stability), we have proved in [J. Baczynski, M.D. Fragoso, Maximal solution to algebraic Riccati equations linked to infinite Markov jump linear systems, Internal Report LNCC, no. 6, 2006] existence (and uniqueness) of maximal solution for this class of equations. As it is noticed in this paper, unlike the finite case (including the linear case), we cannot guarantee anymore that maximal solution is a strong solution in this setting. Via a discussion on the main mathematical hindrance behind this issue, we devise some mild conditions for this implication to hold. Specifically, our main result here is that, under stochastic stability, along with a condition related with convergence in the infinite dimensional scenario, and another one related to spectrum—weaker than spectral continuity—we ensure the maximal solution to be also a strong solution. These conditions hold trivially in the finite case, allowing us to recover the result of strong solution of [C.E. de Souza, M.D. Fragoso, On the existence of maximal solution for generalized algebraic Riccati equations arising in stochastic control, Systems Control Lett. 14 (1990) 233–239] set for MJLS. The issue of whether the convergence condition is restrictive or not is brought to light and, together with some counterexamples, unveil further differences between the finite and the infinite countable case.