Linear-quadratic optimal control under non-Markovian switching

We study an infinite-dimensional continuous-time optimal control problem on finite horizon for a controlled diffusion driven by Brownian motion, in the linear-quadratic case. We admit stochastic coecients, possibly depending on an underlying independent marked point process, so that our model is general enough to include controlled switching systems where the switching mechanism is not required to be Markovian. The problem is solved by means of a Riccati equation, which a backward stochastic differential equation driven by the Bronwian motion and by the random measure associated to the marked point process.