Category:Path-constrained arcs

Whenever a path constraint is active, i.e., it holds $c_i(x(t)) = 0 \; \forall \; t \in [t^\text{start}, t^\text{end}] \subseteq [0, t_f]$ , and no continuous control $u(\cdot)$ can be determined to compensate for the changes in $x(\cdot)$ , naturally $\alpha(\cdot)$ needs to do so by taking values in the interior of its feasible domain. An illustrating example has been given in [Sager2009]Author: Sager, S.; Reinelt, G.; Bock, H.G.
Journal: Mathematical Programming
Number: 1
Pages: 109--149
Title: Direct Methods With Maximal Lower Bound for Mixed-Integer Optimal Control Problems
Url: http://mathopt.de/PUBLICATIONS/Sager2009.pdf
Volume: 118
Year: 2009
, where velocity limitations for the energy-optimal operation of New York subway trains are taken into account. The optimal integer solution does only exist in the limit case of infinite switching (Zeno behavior), or when a tolerance is given.