On the differentiability of the value function of switched linear systems under arbitrary and controlled switching

This paper studies the differentiability of the value function of switched linear systems under arbitrary switching and controlled switching, referred to as worst-case and optimal value functions respectively. First, we show that the value functions are Lipschitz continuous, when the cost function is Lipschitz continuous. Then, as the central contribution of this work, we show with examples that each of these functions can be non-differentiable on dense subsets of the state space, even if the cost function is smooth and Lipschitz continuous. This has implications for optimal control and reinforcement learning since it implies that the exact computation of these value functions requires templates involving functions that are non-differentiable on dense subsets.