Summary: This paper studies delay effects on the mean-square stabilization of stochastic systems with a single input delay. Both continuous- and discrete-time systems are investigated. A continuous-time system is described by a stochastic differential equation with high-dimensional Brownian motion. Two conclusions are proved: (1) Once a system is unstabilizable for some value of delay, it will be unstabilizable for any larger delay; (2) If a system is stabilizable for some value of delay, it will be stabilizable in a small neighborhood of that delay. Hence, by excluding the trivial cases that the system is unstabilizable with zero delay and is stabilizable for any delay, there always exists a finite and positive number CM such that the system is stabilizable if and only if the delay is less than CM. Upper bounds of CM are presented. In a special case, an upper bound is given analytically in terms of eigenvalues of the matrix in the drift coefficient before the state. In general cases, upper bounds are given via LMI optimization problems. Two cases in which CM can be computed are studied. In the case that the system has no state-dependent noise, CM is presented via an LMI optimization problem. In the case that both the state and the control of the system are scalars, CM is expressed analytically. All the above results have counterparts in discrete-time system.
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