tailieunhanh - Bohl perron theorem for differential algebraic equations

This paper is concerned with the Bohl-Perron theorem for differential algebraic equations. We prove that the system is exponentially stable if and only if for any bounded input q, the equation. | VNU Journal of Science: Mathematics – Physics, Vol. 34, No. 3 (2018) 61-70 Bohl-Perron Theorem for Differential Algebraic Equations Nguyen Thu Ha* Electric Power University, 235 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam Received 11 September 2018 Revised 14 September 2018; Accepted 24 September 2018 Abstract: This paper is concerned with the Bohl-Perron theorem for differential algebraic equations. We prove that the system E (t ) x '(t ) A(t ) x(t ), t t0 is exponentially stable if and only if for any bounded input q, the equation E (t ) x '(t ) A(t ) x(t ) q (t ), x(t0 ) 0, t t0 has a bounded solution. Keywords: Differential algebraic equation, asymptotic stability, input - output bounded function. 1. Introduction In lots of applications there is a frequently arising question, namely how robust is a characteristic qualitative property of a system (., the stability) when the system comes under the effect of uncertain perturbations. The designer wants to have operation systems working stably under small perturbation. Therefore, the investigation which conditions ensures robust stability play an important role both in theory and practice. On the other hand, to measure the robust stability, one proceed a test and expect that with rather good input, the output will satisfy some desired properties. For example, if the bounded input implies the boundedness of output then our system must be stable. The aim of this paper is to answer the above questions. We focus on studying the robust stability of time-varying systems of differential-algebraic equations (DAE-s) of the form E (t ) x '(t ) A(t ) x(t ), t t0 , () where E(·), A(·) are continuous matrix functions defined on [0, ) , valued in d d . The leading term E(t) is supposed to be singular for all t t0 . If the system () is subjected to an outer force q , then it becomes _ Corresponding author. Tel.: 84-903212531. Email: ntha2009@ https// .