LAUSR

In this paper, we investigate the continuity of the value function for a stochastic sparse optimal control. The most common method to solve stochastic optimal control problems is the dynamic programming. Specifically, if the value function is smooth, it satisfies the associated Hamilton-Jacobi-Bellman (HJB) equation. However, in general, the value function for our problem is not differentiable because of the nonsmoothness of the L cost functional. Instead, we can expect that the value function is a viscosity solution to the HJB equation. This paper shows the continuity of our value function as a first step for showing that the value function is a viscosity solution.