Residuation of tropical series: Rationality issues

Decidability of existence, rationality of delay controllers and robust delay controllers are investigated for systems with time weights in the tropical and interval semirings. Depending on the (max,+) or (min,+)-rationality of the series specifying the controlled system and the control objective, cases are identified where the controller series defined by residuation is rational, and when it is positive (i.e., when delay control is feasible). When the control objective is specified by a tolerance, i.e. by two bounding rational series, a nice case is identified in which the controller series is of the same rational type as the system specification series.