Well-Posedness of a Class of Piecewise Linear Systems with No Jumps

In this paper, a well-posedness (existence and uniqueness of solutions) problem of bimodal systems given by two linear systems is addressed, where the definition of solutions of CarathEodory is used. This problem is a basic problem in the study of well-posedness for discontinuous dynamical systems. We give here a complete answer to this problem. The obtained result shows that the well-posedness of bimodal systems can be characterized by two properties: the preservation property of the lexicographic inequality relation between the two regions specifying the two modes, and the smooth continuation property.