Can you point me at an existence theorem for an optimal control problem with binary control set? In particular,
$$ \max_u \int_0^T u(t) e^{-f(A(t))}(v(A(t))-c) \, dt $$ $$\text{s.t. } \dot{A}(t)=u $$
where $u \in \{0,1\}$. The existence results that I found typically required the control set to be convex.
Let us temporarily remove the binary constraint and solve the problem without that. Then, we shall see if we can achieve the same value under the binary constraint.
Fix $T>0$, $c \in \mathbb{R}$, and let $f(x)$ and $v(x)$ be given differentiable functions of $x \in \mathbb{R}$. For $t \in \mathbb{R}$ define: $$ H(t) = \int_0^t e^{-f(x)}(v(x)-c)dx $$ Let $A(t)$ be a given differentiable function of $t \in \mathbb{R}$. Suppose that $A(t)$ is nondecreasing and has $A'(t) \in [0,1]$ for all $t$. So $A(t)$ increases by at most $T$ over any interval of duration $T$. Then the objective function value for this given $A(t)$ is: \begin{align} val &= \int_0^T A'(t) e^{-f(A(t))}(v(A(t))-c)dt \\ &= \int_0^T \frac{d}{dt}[H(A(t))]dt\\ &= H(A(T))-H(A(0)) \\ &\leq \sup_{z \in [0,T]} [H(A(0)+z)-H(A(0))]\\ &\leq \sup_{a \in \mathbb{R}, z \in [0,T]} [H(a+z)-H(a)] \end{align}
Suppose the supremum is finite and is achieved at particular values $a^* \in \mathbb{R}, z^* \in [0,T]$. Then the optimal value is: $$ val^* = H(a^*+z^*) - H(a^*) $$ and we can achieve this value using the function $A(t) = a^* + (z^*/T)t$. So linear functions are optimal for this problem. Of course, the objective value only depends on the $A(t)$ function at its endpoints, so many other kinds of functions are also optimal.
We can also achieve $val^*$ using binary control: Define: $$ u(t) = \left\{ \begin{array}{ll} 0 &\mbox{ if $t <T-z^*$} \\ 1 & \mbox{ if $t \geq T-z^*$} \end{array} \right.$$ Define $A(t) = a^* + \int_0^t u(\tau)d\tau$. Then $A(T-z^*)=a^*$ and $A(T)=a^*+z^*$. We get: \begin{align} val &= \int_0^T u(t)e^{-f(A(t))}(v(A(t))-c)dt \\ &= \int_0^{T-z^*} u(t)e^{-f(A(t))}(v(A(t))-c)dt + \int_{T-z^*}^Tu(t)e^{-f(A(t))}(v(A(t))-c)dt \\ &= \int_{T-z^*}^T A'(t) e^{-f(A(t))}(v(A(t))-c)dt\\ &=H(A(T))-H(A(T-z^*))\\ &=H(a^*+z^*)-H(a^*)\\ &=val^* \end{align}