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 L0-optimal control among the admissible controls. Although the L0 measure is discontinuous and non- convex, we prove that the value function, or the minimum L0 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. This property is important, in particular, for discussing the sensitivity of the optimality against uncertainties in the initial state, and also for investigating the stability by using the value function as a Lyapunov function in model predictive control.
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