Numerical direct methods for optimal control problems usually first discretize the controller u(t) piecewise from t_0 to t_f. Then the optimal control problem has been cast into the nonlinear programming problem. After that, various optimization approaches can be employed to solve such an optimization problem. But this is usually based on the terminal time t_f is fixed. So if t_f is not fixed, how to discretize u(t)? For intuition, a situation can be the missile-target engagement problem where we do not know when the interceptor will happen.
You could add one more variable z in the differential equation (in the vector space): dz/dt=1. And after that you could include this variable in the objective function defined over a (n+1) domain (sufficiently wide ) for example. Other solutions could come from the semantic of the problem under solution.
Generally you should introduce an extra parameter for your final time tf. Your decision vector is then augmented with t_f.
For instance, discretize your trajectory over [0, 1] with a new variable tau replacing t, and with the jacobian dt/dtau = t_f-t_0.
So instead of solving: min J(x(t_f); t_f) s.t. dx/dt=f(x(t), u(t); t),
solve: min J(x(t_f); t_f) s.t. dx/dtau=f(x(tau), u(tau); tau); dt/dtau=t_f-t_0; t_f-t_0 > 0
For initial convergence, it is indeed better if the mesh does not change much from one iteration to the next. The details of the implementation depend on your method (transcription, collocation, ...).
As keywords: collocation, direct transcription, pseudospectral methods, free final time.
I also suggest you looking at the excellent solver PSOPT, which handles free terminal time problems.
Joris
Hi Joris,
Do you happen to have a reference? I couldn't find a good reference after a long time search.
Thanks.
Peng Yang Joris Olympio
Hi Peng,
Did you get any luck finding a reference? I would really appreciate it if you could share it with me.
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