I want to solve this dynamic program but I am quite unsure about the approach, as there is an open interval for the control space. I would appreciate any suggestions.
$$\max_{\{u_t\}^{T-1}_{t=0}} \ \ \sum_{t = 0}^{T-1}-(x_t^2-(x_t+u_t)^2)^2-x_T^4$$
subject to: $$u_t \in (-x_t, 0),\ \ \ t= 0, \ldots T-1,$$ $$x_{t+1} = x_t + u_t,\ \ \ t = 0, \ldots T-1,$$ $x_0$ given.