I have the following problem:
$$ maximize \int_{0}^{T} [\rho \alpha s(t)x(t) - c(x(t))]dt $$
subject to $\dot{s}(t) = -\alpha s(t)x(t), s(0) = s_{0}, s(T) = s_{f}$
where $\rho, \alpha \in \Re$ and $c(\cdot)$ is a strictly convex function.
I tried to solve it by setting up the Hamiltonian
$$ H(x(t),s(t),\lambda(t)) = \rho \alpha s(t)x(t) - c(x(t)) -\lambda(t)[\alpha s(t) x(t)]$$ With FOCs:
$$\frac{\partial H}{\partial x} = \rho \alpha s(t) - c'(x(t)) -\lambda(t)\alpha s(t) = 0$$
$$\frac{\partial H}{\partial s} = -\dot{\lambda}(t) \rightarrow \rho \alpha x(t) - \lambda(t)\alpha x(t) = - \dot{\lambda}(t) $$
In order to solve this I need to show that $x(\cdot)$ is constant and I simply cannot get this from this system. If anyone sees this more clearly than me and could show me how to first show this or provide a hint I would greatly appreciate it. I tried solving the differential equations, and also using the system to get to a differential equation for x(t) but I simply cannot get to showing it is a constant.