Well-posedness of hybrid systems

Well-posedness problems arise in hybrid systems theory as a consequence of the use of implicit descriptions and of solution concepts that are based on relaxations. Examples show that the well-posedness issue is considerably more complex in hybrid systems than in continuous systems, as a result of a number of factors including the possible presence of sliding modes, the interaction of guards and invariants, and the occurrence of left or right accumulations of event times. Description formats that are based on implicit or relaxed specifications are typically connected to particular subclasses of hybrid systems, and so there is no general theory of well-posedness of hybrid systems; however, the questions that need to be answered are similar in each case. This chapter surveys several description formats and solution concepts that are used for hybrid systems. We concentrate on well-posedness in the sense of existence and uniqueness of solution, without requiring continuous dependence on initial conditions. A selection of results available in the literature is presented for the subclasses of multi-modal linear systems, complementarity systems, and differential equations with discontinuous righthand sides.