On an infinite dimensional perturbed Riccati differential equation arising in stochastic control

We set out a preliminary study of a certain class of infinite dimensional perturbed Riccati differential equation which arises, for instance, when dealing with control problems such as those involving linear systems with infinite Markov jump parameters or infinite dimensional linear time-invariant systems with state-dependent noise, both with quadratic cost. We are here, mainly, concerned with questions of existence and unicity of solution.