I am not sure if it is my mistake or it is that this control problem happens to have no solution.
The problem is to $\max _{ \{ c_{t} \}_{t} } \int_{0}^{2} 2s_{t}-3c_{t} dt$ such that $s'_{t} = s_{t}+c_{t}$, $s(0)=5$, and $0 \leq c_{t} \leq 2$ for all $t$. I wrote down the Hamiltonian $H := 2s_{t}-3c_{3} - \mu_{t}(s_{t}+c_{t})$ and know by the maximum principle that $\partial H /\partial c = 0 = -3 - \mu_{t}$ and $\partial H / \partial s = 2 - \mu_{t} = -\mu'_{t}$. (For convenience the symbol "prime" denotes the time derivative, which is the usual dot notation.) But now we have $\mu_{t} = -3$ for all $t$, so $\mu'_{t} = 0$; this contradicts the second condition.
So is it that I am wrong from somewhere on or that the contradiction happens to be the nature of the problem?