Optimal Regulator for Linear Stochastic Systems with Multiple State-Delay

We consider an optimal control problem for linear stochastic systems with multiple state delays. By formulating a general quadratic-linear cost functional, we provide an explicit solution involving coupled Riccati and partial differential equations. The derived optimal control law is in an affine feedback form with respect to the current state, the delayed state, and the integral of past state values. Additionally, we demonstrate the application of this solution to an optimal investment problem with logarithmic utility in market with interest rate influenced by a multi-delayed factor process.