Well-posedness and stability of planar conewise linear systems

Abstract

Planar conewise linear systems constitute a subset of piecewise linear systems. The state space of a conewise linear system is a nite number of convex polyhedral cones lling up the space. Each cone is generated by a positive linear combination of a nite set of vectors, not all zero. In each cone the dynamics is that of a linear system and any pair of neighboring cones share the same dynamics at the common border, which is itself a cone of one lower dimension. Each cone with its linear dynamics is called a mode of the conewise system. This thesis focuses on the simplest case of planar systems that is composed of a nite number of cones of dimension two; with borders that are cones of dimension one, that is rays. Stability of such conewise linear systems is well understood and there are a number of necessary and su cient conditions. Somewhat surprisingly, their well-posedness is not so well understood or studied except for the special case where there are two modes only, i.e, the bimodal case. A graphical necessary and su cient condition is here derived for the wellposedness of a planar conewise linear system of arbitrary number of modes and the well-known condition for stability is re-stated on this same graph. This graphical result is expected to provide some guidance to well-posedness studies of conewise systems in a higher dimension.