tailieunhanh - Báo cáo hóa học: "STABILITY OF A DELAY DIFFERENCE SYSTEM"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: STABILITY OF A DELAY DIFFERENCE SYSTEM | STABILITY OF A DELAY DIFFERENCE SYSTEM MIKHAIL KIPNIS AND DARYA KOMISSAROVA Received 28 January 2006 Revised 22 May 2006 Accepted 1 June 2006 We consider the stability problem for the difference system xn Axn-1 Bxn-k where A B are real matrixes and the delay k is a positive integer. In the case A -I the equation is asymptotically stable if and only if all eigenvalues of the matrix B lie inside a special stability oval in the complex plane. If k is odd then the oval is in the right half-plane otherwise in the left half-plane. If HAH BH 1 then the equation is asymptotically stable. We derive explicit sufficient stability conditions for A I and A -I. Copyright 2006 M. Kipnis and D. Komissarova. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction Our purpose is to investigate the stability of the system xn Axn-1 Bxn-k 1 1 where A B are m X m real matrixes xn N Rm a positive integer k is a delay. The structure of the solution of equation xn Axn-1 Bxn-k fn with commutative matrixes A B is considered in 6 . A similar scalar equation xn axn-1 bxn-k where a b are real numbers was studied by Kuruklis 8 and Papanicolaou 15 . The following assertion describes the boundaries of the stability domain for in the a b plane see also Figure . Theorem see 7 Theorem 2 . The zero solution of with k 1 is asymptotically stable if and only if the pair a b is the internal point of the finite domain bounded by the Hindawi Publishing Corporation Advances in Difference Equations Volume 2006 Article ID 31409 Pages 1-9 DOI ADE 2006 31409 2 Stability of a delay difference system Figure . The stability domains of . a k is odd k 1. b k is even. The Cohn domain boundary a b 1 are shown both in a b . following lines I a b 1 II a - sinkw sin k - 1 w b -1 k 1 sin w sin k - 1 w III -1 kb - a 1 IV

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