On the Geometry and Deformation of Switching Manifolds for Autonomous Hybrid Systems

This paper provides a geometrical analysis of autonomous hybrid optimal control systems (HOCS) by studying the properties of switching manifolds in Euclidean space and their associated optimal hybrid trajectories. Motivational examples are to be found in the speed dependent operation of automatic gear shift systems of heavy trucks [1]. In this paper the mathematical formulation of a hybrid system is presented and then the Hybrid Maximum Principle (HMP) necessary conditions for the optimality of a hybrid system trajectory are given, (see [6],[7]). Second order optimality conditions are given in terms of the Hessian matrix of the value function and geometrical data involving the curvature of the switching manifold at its intersection with an optimal trajectory. At the end, the energy of a switching manifold deformation mapping is defined and the hybrid cost optimization is performed with respect to such deformation mappings.