Even when the integration time is not an unknown parameter, it is questionable whether fmincon is applicable. I assume your ODE is with constant coefficients and is higher than first order. In this case, the solution of the ODE is not a differentiable function of the unknown coefficients (or at least the characteristic roots are not). That doesn't rule out empirical success of course, i.e., you could just try it and hope the iterations don't stray into any bad regions.
But since you're already gambling with fmincon, there's no longer much reason to worry about rigour when you include integration time as an additional parameter. The objective as a function of integration time would have to be well-behaved, though. For example, if you are simply doing a least squares fit of the ODE solution to some given noisy data, I suspect that there might be lots of local minima as a function of the integration time, even when it is the only unknown parameter. Also, you would need appropriate constraints.Otherwise an integration time of zero should trivially give you the global minimum squared error.
