Stochastic linear quadratic regulation for discrete-time systems with single input and multiple delays

This paper is concerned with linear quadratic regulation (LQR) problem for multiplicative-noise systems with multiple delays in single input channel. The problem is solved by applying the stochastic maximum principle. We show that the optimal costate is a linear function of the augmented state, which incorporates the state and a finite window history of input. The corresponding coefficients satisfy some partial Riccati difference equations of the same order of the plant (ignoring the delay). The optimal controller possesses the same form as the optimal costate and it is determined by the solution of the aforementioned partial Riccati difference equations. The presented results are new to the best of our knowledge.