On Relationships Between Weierstrass-Erdmannn Corner Condition, Snell´s Law and the Hybrid Minimum Principle

Over the last few years various classes of optimal control problems for hybrid systems, with continuous and discrete states and dynamics, have been formalized. In particular, generalizing the standard minimum principle (MP), Sussmann |1| and Riedinger et al. |2|, among other authors, have given versions of the hybrid minimum principle (HMP). In this paper first we show how the Weierstrass-Erdmann corner conditions of calculus of variations |3| can be recovered from the hybrid maximum principle (HMP) and then we show how a classical result in Physics, namely Snell´s law, can be obtained from HMP.