Linear systems with input constraints: stability and optimality

For discrete-time linear systems with inputs constraints, we characterize the generalized stability region and give an algorithm to find the facet of this region. We show that there exists a finite horizon such that the infinite horizon constrained LQR problem can turn to a finite horizon constrained LQR problem and give an upper bound of this horizon for any given initial state. We also give an upper bound of this finite horizon for all initial states inside the stability region.