Stochastic LQR of linear systems with multiple input channels

In this paper, we study the stochastic linear quadratic regulation problem for linear discrete-time systems with multiple input channels. A necessary and sufficient condition is obtained by applying the leader-follower game approach. Different from the augmentation technique, the complete solution is given in terms of one difference Riccati equation and one Riccati-type equation. The key is to introduce a new costate and completely solve the forward-backward stochastic difference equation by establishing a homogeneous relationship between the new costate which is backward and the original state which is forward. It is noted that the problem is more complicated than the deterministic case because of the state- and control-dependent noises.