The question is a reformulation of an incomplete version.
Consider the following dynamic dividing a dollar game where agent 1 claims $x(t)$ of the dollar and agent 2 $y(t)$ (paper).
\begin{align} &\max_{\{u(t)\}^T_{t=0}} e^{-rT}x(T)\\[2mm] &\max_{\{v(t)\}^T_{t=0}} e^{-rT}y(T)\\[2mm] \text{s.t. }& \dot x(t) = -u(t)\in[0,1-x(t)]\\ & \dot y(t) = -v(t)\in[0,1-y(t)]\\ & x(0) + y(0) > 1 \end{align}
where $y(T) = 1 - x(T)$ with $T:=\min\{t \in \mathbb{R}_{++} : x(t) + y(t) = 1\}$ and $r > 0$ is the time preference rate. Optimal feedback control is given by $u(t) = r(1-x(t))$ and $v(t) = r(1-y(t))$. This solution is somehow motivated by guess and verify.
- Can we obtain the solution by some first order conditions (e.g. maximum principle)?
Transformation of objective function
I tried to somehow transform the objective function into a common functional. Can someone confirm the following: \begin{align} &1 + \int^T_0{-re^{-rt}dt} = 1 + \left[e^{-rt}\right]^T_0 = 1 + e^{-rT} - e^0 = e^{-rT}\\ & x(0) - \int^T_0{u(t)dt} = x(0) + \int^T_0{\dot x(t) dt} = x(T)\\ \Longrightarrow \qquad& e^{-rT} x(0) + \int^T_0{e^{-rt}u(t)(r - e^{rt})dt} = e^{-rT}x(T) \end{align}
- What next?
