Discrete control dependent constraints in multiple shooting optimal control problems

Discrete controls can only take values from a fixed set and appear in numerous real world applications (e.g. car gear shift). On an aircraft, discrete controls normally appear in control surfaces such as flaps and slats or the landing gear. These discrete controls can be taken into account in gradient based optimizations via different reformulation methods. Additionally discrete controls introduce discrete constraints. For instance changing the flaps on an aircraft introduces discrete changes in the speed flight envelope. These discrete control dependent constraints have to be taken into account in the optimization. Three formulation methods are implemented for discrete controls and the corresponding constraints: the use of Multiple Phases, the Hyperbolic Tangent function and the Variable Time Transformation. The reformulations are compared with respect to similarity, physical constraint violation as well as performance using an example landing problem.