Idempotent versions of Haar’s Lemma: links between comparison of discrete event systems with different state spaces and control

summary:Haar's Lemma (1918) deals with the algebraic characterization of the inclusion of polyhedral sets. This Lemma has been involved many times in automatic control of linear dynamical systems via positive invariance of polyhedrons. More recently, it has been used to characterize stochastic comparison w.r.t. linear/integral ordering of Markov (reward) chains. In this paper we develop a state space oriented approach to the control of Discrete Event Systems (DES) based on the remark that most of control constraints of practical interest are naturally expressed as the inclusion of two systems of linear (w.r.t. idempotent semiring or semifield operations) inequalities. Thus, we establish tropical version of Haar's Lemma to obtain the algebraic characterization of such inclusion. As in the linear case this Lemma exhibits the links between two apparently different problems: comparison of DES and control via positive invariance. Our approach to the control differs from the ones based on formal series and is a kind of dual approach of the geometric one recently developed. Control oriented applications of the main results of the paper are given. One of these applications concerns the study of transportation networks which evolve according to a time table. Although complexity of calculus is discussed the algorithmic implementation needs further work and is beyond the scope of this paper