Eigenvalues of convex processes and convergence properties of differential inclusions

Abstract

In this paper, we study (real) eigenvalues and eigenvectors of convex processes, and provide conditions for the existence of eigenvectors in a given convex coneK⊂ℝⁿ. It is established that the maximal eigenvalue ofG(·) inK is expressed by

(whereK ⁰ is the polar cone ofK) provided that the minimum is attained in intK ⁰. This result is applied to study the asymptotic behaviour of certain differential inclusions{∈G(x(t)). We extend some known results for the von Neumann-Gale model to our more general framework. We prove that ifx ₀ is the unique eigenvector corresponding to the maximal eigenvalue λ₀ ofG(·) inK, then the nonexistence of solutions of a certain special trigonometric form is necessary and sufficient for every viable solutionx(·) to satisfy^-λ ₀ ^t x(t)→cx ₀ ast←∞ for somec≥0. Our method is to study the family of convex conesW _λ=cl{v−λx :x∈K,v∈G(x) where λ is any real number. We characterize the maximal eigenvalue λ₀ as the minimal λ for whichW _λ can be separated fromK.