Why optimal solution lies at the corner/extreme point?
Answers
Answer:
The optimal solution of an optimization or optimal control problem must not lie in the interior of the feasible region in geeral.
Take the simple problem:
min f(x) with f(x) = - (x-1)^2 +1 on the interval [-3,4]. The global minimum is in x=4.
In addition there is a local minimum in x=-3. The same situation may be present for finite and infinite dimensional optimization problems.
If you have a time optimal control problems in mind, then indeed the optimal control may lie an the boundary of the admissible set, for example, if the control function
enters objective and state equations linearly. However, this is also not true in general.
Depending on the other side conditions the optimal control may be singular
intervalwise, i.e. it lies intervalwise in the interior of the set of admissible controls
while other subarcs are bang-bang, i.e. lying on the boundary of the set of admissible controls. In case, the control enters objective and side constraints nonlinearly, everything may happen.
Hence, this has nothing to do with time or convergence, it's just the optimal solution
of the problem; see the simple problem above.
Explanation: