The weak robustness of interval matrices in maxmax–plus algebra

A maxmax–plus matrix AA (operations maxmax and plus are denoted by ⊕⊕ and ⊗⊗, respectively) is called weakly robust if the only possibility to arrive at an eigenvector is to start the sequence (orbit) x,A⊗x,A2⊗x,…x,A⊗x,A2⊗x,… by a vector that is itself an eigenvector. The weak robustness of a matrix is extended to interval max–plus matrices distinguishing two possibilities, that at least one matrix or all matrices from a given interval are weakly robust. Weakly robust interval matrices in max–plus algebra are studied and equivalent conditions for interval matrices to be weakly robust are presented. The recognition version of the problem of the existence of at least one weakly robust matrix from a given interval is proved to be NP-complete.