Generalized Asymptoticity Problem in Max Algebra; the Two-Dimensional Case

摘要

The asymptotic behavior of an algorithm in max algebra, which can be seen as anextension of the concept of periodicity, is discussed. This concept of periodicity means that for every irreducible matrix M, M(sup(n + d) = lambda(sup d)M(sup n) in max algebra sense, for n greater than or equal to n(sub 0). Here lambda is the eigenvalue and d denotes the length of the critical circuit. The main result is that the asymptotic behavior of the algorithm is characterized by one or more critical circuits, and that two cases can be distinguished. In the first case, the generalized critical circuit has length two, and generalized order 2 periodicity is found. In the second case, the two generalized critical circuits are of length one, and generalized order 1 periodicity is stated. Only the case of 2 by 2 matrices is treated. It is expected that the treatment of this specific case will lead to a more general set up in which square matrices of any size can be included. This extension seems to be such that the asymptotic behavior is order d periodic in a generalized way, where d is the least common multiple of the lengths of some of the generalized critical circuits.