Continuity of the value function in sparse optimal control

The purpose of this article is to show the continuity of the value function of the sparse optimal (or L
⁰
-optimal) control problem. The sparse optimal control is a control whose support is minimum among all admissible controls. Under the normality assumption, it is known that a sparse optimal control is given by L
¹
optimal control. Furthermore, the value function of the sparse optimal control problem is identical with that of the L
¹
-optimal control problem. From these properties, we prove the continuity of the value function of the sparse optimal control problem by verifying that of the L
¹
-optimal control problem.