oc.optimization control - Minimizing a function containing an integral

It looks like a mixed-integer dynamic optimization problem. Your problem can be rewritten as follows: (notice the transformation of the integral into a differential equation? It's a standard trick. Also, note that you need an initial condition for $Y$)

$min_{x(t)} L(T)$

s.t.
$ frac{dL(t)}{dt} = AR(t)-x(t)$, with $L(0) = 0$

$ frac{dR(t)}{dt} = ax(t)R(t)Y(t) - bR(t)$, with $R(0) = R_{0}$

$frac{dY(t)}{dt}=−x(t)R(t)Y(t)$, with $Y(0) = Y_{0}$

$x(t) = delta(t) x_{min} + (1 - delta(t)) x_{max}$ where $delta(t) in {0,1}$

To solve this problem numerically, simply discretize the differential equations using backward Euler (easy), or implicit Runge Kutta (harder, but more accurate). Pose this as a Mixed Integer Nonlinear Program (MINLP) and use one of these solvers to find the solution.

Bonmin
https://projects.coin-or.org/Bonmin

Couenne
https://projects.coin-or.org/Couenne

These solvers will traverse the branch-and-bound tree more intelligently and efficiently than your method of enumerating every single case, which will grow with the no. of discretization grid points you have. (e.g. let's say you discretize over 20 points; the no. of cases you have to search is $2^{20} = 1048576$. Not nice.)

With a branch-and-bound|cut|reduce MINLP solver (and a bit of luck), on average you are unlikely to hit the worst case scenario where every single case is enumerated.

There are other ways of solving this problem -- multiple-shooting, sequential dynamic optimization, etc. In my opinion, optimal control methods (Pontryagin's maximum principle) are typically intractable on problems like this.