Controllable and tolerable generalized eigenvectors of interval max-plus matrices

Abstract

By max-plus algebra we mean the set of reals R equipped with the operations a ⊕ b = max { a, b } and a ⊗ b = a + b for a, b ∈ R . A vector x is said to be a generalized eigenvector of max-plus matrices A, B ∈ R ( m, n ) if A ⊗ x = λ ⊗ B ⊗ x for some λ ∈ R . The investigation of properties of generalized eigenvectors is important for the applications. The values of vector or matrix inputs in practice are usually not exact numbers and they can be rather considered as values in some intervals. In this paper the properties of matrices and vectors with inexact (interval) entries are studied and complete solutions of the controllable, the tolerable and the strong generalized eigenproblem in max-plus algebra are presented. As a consequence of the obtained results, efficient algorithms for checking equivalent conditions are introduced.