Value function in maximum hands-off control for linear systems

In this brief paper, we study the value function in maximum hands-off control. Maximum hands-off control, also known as sparse control, is the L0L0-optimal control among the feasible controls. Although the L0L0 measure is discontinuous and non-convex, we prove that the value function, or the minimum L0L0 norm of the control, is a continuous and strictly convex function of the initial state in the reachable set, under an assumption on the controlled plant model. We then extend the finite-horizon maximum hands-off control to model predictive control (MPC), and prove the recursive feasibility and the stability by using the continuity and convexity properties of the value function.