Characterization of well-posedness of piecewise-linear systems

One of the basic issues in the study of hybrid systems is the well-posednes s (existence and uniqueness of solutions) problem of discontinuous dynamica l systems. The paper addresses this problem for a class of piecewise-linear discontinuous systems under the definition of solutions of Caratheodory, T he concepts of jump solutions or of sliding modes are not considered here. In this sense, the problem to be discussed is one of the most basic problem s in the study of well-posedness for discontinuous dynamical systems. First , we derive necessary and sufficient renditions for bimodal systems to be w ell-posed, in terms of an analysis based on lexicographic inequalities and the smooth continuation property of solutions. Next, its extensions to the multi-modal case are discussed. As an application to switching control, in the case that two state feedback gains are switched according to a criterio n depending on the state, we give a characterization of all admissible stat e feedback gains for which the closed loop system remains well-posed.