I have the following system of two non-linear ODE with one control variable (modified model of Lotka-Volterra):
Here is $\alpha, \beta, \gamma, \delta$ - some constants, $u$ - control variable.
Also I have initial conditions: $$ y(0) = y_0, x(0) = x_0 $$ and the final conditions: $$ y(t^*) = y^*, x(t^*) =0 $$ Another variant of the final conditions is: $$ y(t^*) = 0, x(t^*) = x^* $$ So, I want to find some function $u$, that minimize $t^*$.
Is it possible for this system and how I can do it?