On maximal solution to infinite dimensional perturbed Riccati differential equations arising in stochastic control

Finding the maximal solution for a certain class of infinite dimensional perturbed Riccati algebraic equations is the main concern of this paper. In addition, we provide a sufficient and necessary condition for stochastic stability. Also, we obtain necessary conditions which unveil some structural properties. Besides the interest in its own right, this class of equations turns out to be essential, for instance, when dealing with linear systems with infinite countable Markov jump parameters or infinite dimensional linear time-invariant systems with state-dependent noise