A necessary optimality condition for constrained optimal control of hybrid systems

Using the fact that continuous piecewise affine systems can be written as special inverse optimization models, we present necessary optimality conditions for constrained optimal control problems for hybrid dynamical systems. The modeling approach is based on the fact that piecewise affine functions can be written as the difference of two convex functions and has been described in previous publications. The inverse optimization model resulting from this approach can be replaced by its Karush-Kuhn-Tucker conditions to yield a linear complementarity model. An optimal control problem for this model class is an instance of a mathematical program with complementarity constraints for which classical Karush-Kuhn-Tucker optimality conditions may not hold. Exploiting the regularity properties of the inverse optimization model, we show why for the class of control problems under consideration this is not the case and the classical optimality conditions also characterize optimal input trajectories.