I am trying to solve the following optimization program and I am hoping for some help as I am stuck:
$\min_{R_{t}\in[0,1]}\,\,\int_{0}^{\tau}e^{-t}R_{t}\,dt$
s.t. $e^{-t}R_{t}-\int_{t}^{\tau}e^{-s}R_{s}ds\geq c$ for all $t$ (with equality at $t=\tau$)
for some fixed $\tau >0$ and $c>0$.
I conjecture that the optimal solution has $R_t = c e^\tau$ for all t. But I cannot prove it.
This problem looks somewhat like an optimal control problem, but it's in an odd form. For example, I tried converting it into an optimal control problem by replacing the constraint with $e^{-t}R_{t}-\int_{t}^{\tau}e^{-s}R_{s}ds + u_t = c$, where $u_t \geq 0$ is the control, and $u_\tau = 0$, but I don't know how to deal with the integral.
If I assume that $\dot{R}_t$ exists (which I would rather not), I can use integration by parts, and rewrite the constraint as $\int_{t}^{\tau}e^{-s}\dot{R}_{s}ds\leq0$ for all t and $R_\tau = c e^\tau$. It's still unclear, however, how I might go about solving this problem.
Even if I consider the relaxed problem where I consider only the constraint corresponding to t=0, so that the problem can be rewritten as
$\min_{R_{t}\in[0,1]}\,\,\int_{0}^{\tau}e^{-t}R_{t}\,dt$
s.t. $\int_{0}^{\tau}e^{-t}\dot{R}_{t}dt\leq0$,
I still don't know how to solve this program.
If I can show that $\dot{R}_t = 0$ for all $t$ in this relaxed problem, because the constraints for all $t \in (0,\tau)$ are satisfied by this solution, I'll be done.
Any help would be very much appreciated! Thank you!
Happy Holidays!
Figured out how to solve the problem. First, define the state variable $X_{t}=\int_{\tau}^{t}e^{-s}R_{s}ds$. Now the problem can be rewritten as
$\max_{u_{t}\geq0}X_{0}$
s.t. $\dot{X}_{t}=u_{t}+c-X_{t}\\X_{\tau}=0$
This problem is non-standard, because the terminal value of $X$ is fixed, while the initial value is free. By changing variables $r = \tau - t$, the problem can be transformed into a canonical optimal control problem with a free endpoint and solved using standard techniques.
The optimal solution has $R_t = ce^{\tau}$.