# Category:Outer convexification

For time-dependent and space- independent integer controls often another formulation is beneficial, e.g., [Kirches2010]Author: C. Kirches; S. Sager; H.G. Bock; J.P. Schl\"oder
Journal: Optimal Control Applications and Methods
Month: March/April
Number: 2
Pages: 137--153
Title: Time-optimal control of automobile test drives with gear shifts
Url: http://mathopt.de/PUBLICATIONS/Kirches2010.pdf
Volume: 31
Year: 2010 . For every element $v^i$ of $\Omega$ a binary control function $\omega_i(\cdot)$ is introduced.

The general equation $0 = F[x,u,v(t)]$

can then be written as $0 = \sum_{i=1}^{n_{\omega}} F[x,u,v^i] \; \omega_i (t), \;\;\;\; t \in [0, t_f].$

If we impose the special ordered set type one condition $\sum_{i=1}^{n_{\omega}} \omega_i (t) = 1, \;\;\;\; t \in [0, t_f],$

there is a bijection between every feasible integer function $v(\cdot) \in \Omega$ and an appropriately chosen binary function $\omega(\cdot) \in \{0,1\}^{n_{\omega}}$, compare [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 . The relaxation of $\omega(t) \in \{0,1\}^{n_{\omega}}$ is given by $\omega(t) \in [0,1]^{n_{\omega}}$. We will refer to the two constraints as outer convexification [Sager2005]Address: Tönning, Lübeck, Marburg
Author: S. Sager
Editor: ISBN 3-89959-416-9
Publisher: Der andere Verlag
Title: Numerical methods for mixed--integer optimal control problems
Url: http://mathopt.de/PUBLICATIONS/Sager2005.pdf
Year: 2005 of the original model.