Category:Path-constrained arcs

Whenever a path constraint is active, i.e., it holds ${\displaystyle c_i(x(t))=0\mathrm{\forall }t\in [t^{\text{start}},t^{\text{end}}]\subseteq [0,t_f]}$, and no continuous control ${\displaystyle u(\cdot )}$ can be determined to compensate for the changes in ${\displaystyle x(\cdot )}$, naturally ${\displaystyle \alpha (\cdot )}$ 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.

References

[Sager2009]

Sager, S.; Reinelt, G.; Bock, H.G. (2009): Direct Methods With Maximal Lower Bound for Mixed-Integer Optimal Control Problems. Mathematical Programming, 118, 109--149