Complementarity modeling of hybrid systems

A complementarity framework is described for the modeling of certain classes of mixed continuous/discrete dynamical systems. The use of such a framework is well-known for mechanical systems with inequality constraints, but we give a more general formulation which applies for instance also to systems with relays in a feedback loop. The main theoretical results in the paper are concerned with uniqueness of smooth continuations; the solution of this problem requires the construction of a map from the continuous state to the discrete state. A crucial technical tool is the so-called linear complementarity problem (LCP); we introduce various generalizations of this problem. Specific results are obtained for Hamiltonian systems, passive systems, and linear systems.