Robust stability of the time varying singular distributed parameter system in hilbert space

Abstract

Perturbation and robust exponential stability of the time varying singular distributed parameter system are discussed via functional analysis and operator theory in Hilbert space. The perturbation principle of generalized evolution operators and the sufficient condition concerning the robust exponential stability of the time varying singular distributed parameter system are obtained, in which the exponential stability for the time varying singular distributed parameter system is not destroyed, if we perturb the equation by linear operator. The perturbation are assumed to have unbounded structure and the smallest perturbation which destroys exponential stability is determined.

GE-evolution operator methods for the stability of time varying singular distributed parameter system

“…GE 0 -evolution operator is very important for studying the time varying singular distributed parameter system (1), and there are essential distinctions between GE-evolution operator and evolution operator [14,[16][17][18][19][20][21][22][23][24][25]. Ge [17] discussed the mild solution and exponential stability of time varying singular distributed parameter system (1) by using GE 0 -evolution operator in Hilbert space; Liu and Shi [18] studied the exponential stability of time varying singular distributed parameter system (1) by using GE 0 -evolution operator in Hilbert space on the base of [17]. In this paper, we discuss the exponential stability of time varying singular distributed parameter system (1) in Hilbert space on the bases of the above papers.…”

“…They appear in the study of the temperature distribution in a composite heat conductor, voltage distribution in electromagnetically coupled superconductive circuits, signal propagation in a system of electrical cables [1][2][3][4][5][6][7][8][9][10][11][12] etc. There is an essential distinction between singular and ordinary distributed parameter systems [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. Under disturbance, not only singular distributed parameter systems lose stability, but also great changes take place in their structure, such as leading to impulsive behavior etc.…”

“…But there are essential distinctions among C 0 -semigroup, degenerate C 0 -semigroup and GE 0 -semigroup [1,[10][11][12][13][14][15][16]. GE 0 -evolution operator is very important for studying the time varying singular distributed parameter system (1), and there are essential distinctions between GE-evolution operator and evolution operator [14,[16][17][18][19][20][21][22][23][24][25]. Ge [17] discussed the mild solution and exponential stability of time varying singular distributed parameter system (1) by using GE 0 -evolution operator in Hilbert space; Liu and Shi [18] studied the exponential stability of time varying singular distributed parameter system (1) by using GE 0 -evolution operator in Hilbert space on the base of [17].…”

Exact Controllability for Time Varying Singular Distributed Parameter System in Hilbert Space

“…They appear in the study of the temperature distribution in a composite heat conductor, voltage distribution in electromagnetically coupled superconductive circuits, signal propagation in a system of electrical cables [1][2][3] etc. There is an essential distinction between singular and ordinary distributed parameter systems [1][2][3][4][5][6][7][8][9][10]. Under disturbance, not only singular distributed parameter systems lose stability, but also great changes take place in their structure, such as leading to impulsive behavior etc.…”

Some results for exponential stability of singular distributed parameter systems

“…In 2009, Ge [18] discussed the exponential stability of a class for the singular distributed parameter system by using the theory of GE 0 -semigroup (i.e., strongly continuously generalized operator semigroup). In 2010, Yuan and Ge [23] discussed the exponential stability of the singular system by using the degenerate semigroup; Li and Wang [24] discussed the exponential stabilizability for a class of singular distributed parameter control system by using the GE 0 -semigroup theory in Hilbert space; Ge [25] , Liu and Shi [26] discussed the exponential stability of the time varying singular distributed parameter system by using the GE 0 -semigroup and GE-mild evolution operator. In 2011, Li * This work is supported by National Natural Science Foundation (NNSF) of China under Grant 61174081. and Ge [29] discussed the asymptotical stability of singular distributed parameter systems by using C 0 and GE 0 -semigroups methods.…”