Monotonicity and positive invariance of linear systems over dioids

Abstract

In this paper, we make connections between two apparently different concepts. The first concept is the (linear) monotonicity of a given matrix which is usually used in order to compare Markov chains. This concept is involved in the simplification of complex stochastic systems in order to control the approximation error made. The second concept is the positive invariance of sets by a (linear) map. The properties of positively invariant sets are involved in many different problems in classical control theory, such as constrained control, robustness analysis, optimisation, and also in aggregation of Markov chains (namely strong lumpability and coherency). In the context of linear dynamical systems over semirings which play an important role in the study of discrete event systems, we establish links between monotone (or isotone) linear maps and linear maps which admit some special families of positively invariant sets.