tailieunhanh - Báo cáo nghiên cứu khoa học: "Stability Radius of Linear Dynamic Equations with Constant Coefficients on Time Scales"

Tuyển tập báo cáo khoa học trường đại học Quốc Gia Hà Nội đề tài: Stability Radius of Linear Dynamic Equations with Constant Coefficients on Time Scale. | VNU Journal of Science Mathematics - Physics 26 2010 163-173 Stability Radius of Linear Dynamic Equations with Constant Coefficients on Time Scales Le Hong Lan1 Nguyen Chi Liem2 Department of Basic Sciences University of Transport and Communication Hanoi Vietnam Department of Mathematics Mechanics and Informatics University of Science VNU 334 Nguyen Trai Hanoi Vietnam Received 10 August 2010 Abstract. This paper considers the exponential stability and stability radius of time-invarying dynamic equations with respect to linear dynamic perturbations on time scales. A formula for the stability radius is given. Keywords and phrases time scales exponential function linear dynamic equation exponentially stable stability radius 1. Introduction In the last decade there have been extensive works on studying of robustness measures where one of the most powerful ideas is the concept of the stability radii introduced by Hinrichsen and Pritchard 1 . The stability radius is defined as the smallest in norm complex or real perturbations destabilizing the system. In 2 if X Ax is the nominal system they assume that the perturbed system can be represented in the form X A BDC x 1 where D is an unknown disturbance matrix and B c are known scaling matrices defining the structure of the perturbation. The complex stability radius is given by -1 max tEi C H-A -1B 2 If the nominal system is the difference equation Xn-Ị-1 Axn in 3 they assume that the perturbed system can be represented in the form xn 1 A BDC xn. 3 Then the complex stability radius is given by max C lM - A 1B . 4 ue @ 1 Corresponding authors. E-mail honglanle229@ This work was supported by the project B2010 - 04. 163 164 . Lan . Liem VNU Journal of Science Mathematics - Physics 26 2010 163-173 Earlier results for time-varying systems can be found . in 4 5 . The most successful attempt for finding a formula of the stability radius was an elegant result given by Jacob 5 . Using this result the notion and .