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 <bibref>Sager2009</bibref>, 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.