SIAM Journal on Control and Optimization, Vol.54, No.2, 894-917, 2016
STABILIZATION OF DISCRETE-TIME SYSTEMS WITH MULTIPLICATIVE NOISE AND MULTIPLE DELAYS IN THE CONTROL VARIABLE
This paper focuses on the stabilization problem of discrete-time systems with multiplicative noise and multiple delays in the control variable. First, a transformation is constructed to convert the original system into a delay-free system with its input matrix containing future noise. Next, the finite-horizon linear quadratic regulation problem for the delay-free system subject to a constraint on the control is investigated. We provide the necessary and sufficient condition for the problem having a unique solution and the optimal feedback control using Riccati-type difference equations. Finally, we show that the original system can be stabilized in the mean-square sense if and only if the set of solutions to the Riccati-type difference equations is convergent. This further yields the fact that the system under consideration can be stabilized if and only if the algebraic Riccati-type equations have a unique solution such that a specific matrix is positive definite.