Exploring positively invariant sets by linear systems over idempotent semirings

Abstract

It is now almost well‐known that linear systems over idempotent semirings modelize Discrete Event Systems of practical interest. The control of such systems is a recent topic of research which has not yet been as developed control theory in classical algebra. The properties of positively invariant sets are involved in many different problems in classical control theory, such as constrained control, robustness analysis and optimization, and also in aggregation of Markov chains (namely strong lumpability and coherency). In this paper we identify special families of positively invariant sets for linear discrete‐time systems over idempotent semirings. Necessary and sufficient conditions for a given set to be a positively invariant set of a linear system are obtained. The results obtained lead to further works on this subject.