Optimal regulator for linear stochastic systems with discrete and distributed state-delays and random time-horizon

We consider an optimal control problem for linear stochastic systems with discrete and distributed state-delay, and random time-horizon. We derive an explicit closed-form solution for this problem under a general quadratic-linear cost functional, in terms of solutions to a system of coupled Riccati and partial differential equations. The resulting optimal control law takes an affine state-feedback form, depending on the current state, the delayed state, and the integral of past state values.