On exact controllability of Itô stochastic systems with input delay

This paper considers the exact controllability of Itô stochastic systems with input delay. In particular, one delay-free controller and one delayed controller are involved in the systems which complicates the study due to the inconsistency of adaptiveness caused by input delay. The main contribution of this paper is to provide the necessary and sufficient Gramian matrix condition and the necessary Rank condition for the exact controllability. The key is to solve the backward stochastic differential equations (BSDEs) with input delay.