Stabilizing linear systems with saturation through optimal control

We construct a continuous feedback for a saturated system x˙(t) = Ax(t) + Bσ(u(t)). The feedback renders the system asymptotically stable on the whole set of states that can be driven to 0 with an open-loop control. Trajectories of the resulting closed-loop system are optimal for an auxiliary optimal control problem with a convex cost and linear dynamics. The value function for the auxiliary problem, which we show to be differentiable, serves as a Lyapunov function for the saturated system. Relating the saturated system, which is nonlinear, to an optimal control problem with linear dynamics is possible thanks to the monotone structure of saturation.