Shaft A., Characterization of Well-Posedness of Piecewise-Linear Systems

Abstract—One of the basic issues in the study of hybrid sys-tems is the well-posedness (existence and uniqueness of solutions) problem of discontinuous dynamical systems. The paper addresses this problem for a class of piecewise-linear discontinuous systems under the definition of solutions of Carathéodory. The 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 prob-lems in the study of well-posedness for discontinuous dynamical systems. First, we derive necessary and sufficient conditions for bimodal systems to be well-posed, in terms of an analysis based on lexicographic inequalities and the smooth continuation prop-erty 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 criterion de-pending on the state, we give a characterization of all admissible state feedback gains for which the closed loop system remains well-posed. Index Terms—Discontinuous systems, hybrid systems, lexico-graphic inequalities, piecewise-linear systems, well-posedness. I