I'm looking for an existential result in optimal control for the following class of problems:
Given $T > 0$, $\bar x, \hat x\in\mathbb R^d$, an instantaneous cost function $c:\mathbb R^d\times \mathbb R^m\to\mathbb R$ that is bounded and lower semicontinuous, and a continuously differentiable map $f:\mathbb R^d\times\mathbb R^m\to \mathbb R^d$, consider the optimal control problem minimize $\int_0^T c(x(t), u(t))\,dt $ over controls $u$ subject to $\dot x(t) = f(x(t), u(t))$, measurable control maps $u:[0, T]\to \mathbb U\subset\mathbb R^m$ compact, with $x(0) = \bar x$ and $x(T) = \hat x$. We assume that there exists a control that satisfies the end-point constraints.
All the traditional results that I could lay my hands on, e.g., in Fleming and Rishel, require continuity of the cost function $c$, which we don't have.
Any pointer(s) will be appreciated.