On solution concepts and well-posedness of linear relay systems

In this paper we study the well-posedness (existence and uniqueness of solutions) of linear relay systems with respect to two di5erent solution concepts, Filippov solutions and forward solutions. We derive necessary and su7cient conditions for well-posedness in the sense of Filippov of linear systems of relative degree one and two in closed loop with relay feedback. To be precise, uniqueness of Filippov (and also forward) solutions follows in this case if the 9rst non-zero Markov parameter is positive. By means of an example it is shown that this intuitively clear condition is not true for systems with relative degree larger than two. The in;uence of the Zeno phenomenon (an in9nite number of relay switching times in a 9nite length time interval) on well-posedness is highlighted and although linear relay systems form a rather limited subclass of hybrid dynamical systems, the consequences of the presence of the Zeno behaviour is typical for many other classes of non-smooth and hybrid systems. ? 2003 Elsevier Ltd. All rights reserved.