Optimal time control for the system of two non-linear ODE

[Guest]

I have the following system of two non-linear ODE with one control variable (modified model of Lotka-Volterra):

[tex]\left\{\begin{matrix}
\frac{dx}{dt}=(\alpha-\beta y)x
\\
\frac{dy}{dt}=(-\gamma + \delta x)y + u
\end{matrix}\right.[/tex]

Here is [tex]\alpha, \beta, \gamma, \delta[/tex] - some constants, [tex]u[/tex] - control variable.

Also I have initial conditions:
[tex]y(0) = y_0,
x(0) = x_0[/tex]
and the final conditions:
[tex]y(t^*) = y^*, x(t^*) =0[/tex]
Another variant of the final conditions is:
[tex]y(t^*) = 0, x(t^*) = x^*[/tex]
So, I want to find some function [tex]u[/tex], that minimize [tex]t^*[/tex].

Is it possible for this system and how I can do it?