Stabilization of linear systems with input constraints

We consider the control of an unstable LTI system with a constraint on the control input. We show that for every compact subset of the null-controllable region of the system, we can constructively design a nonlinear state feedback controller which ensures the internal exponential stability of the closed loop system, i.e. the state and control signals go to zero exponentially, for every initial condition in this subset. Two controllers are explicitly constructed: one has the property of being continuous and homogeneous, the other one can be considered as a one-step-ahead model predictive control (MPC) scheme with a special cost function