Exact controllability for singular distributed parameter system in Hilbert space

Abstract

Exact controllability of singular distributed parameter control system is discussed via functional analysis and the theory of generalized operator semi-group in Hilbert space. Necessary and sufficient conditions concerning the exact controllability are given. Relations between exact controllability and stability of singular distributed parameter system are specified. exact controllability, singular distributed parameter system, Hilbert space

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

“…This set is called the E -resolvent set of operator A and the operator ( , ) R E A λ is called the E -resolvent of operator A . Definition 1 [10,12,13] A one parameter family of…”

“…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.…”

“…For example, the first group of results concerning the stability of autonomous singular distributed parameter system (2) are obtained by using C 0 -semigroup [2][3][4][5][6][7]; the second group of results concerning the stability of autonomous singular distributed parameter system (2) are obtained by using degenerate C 0 -semigroup [1,8,9,11]; the third group of results concerning the stability of autonomous singular distributed parameter system (2) are obtained by using GE 0 -semigroup (i.e., strongly continuously generalized operator semigroup [10]. 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].…”

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.…”

“…In [8], the classically theoretic concepts of controllability and observability for time-invariant linear infinite dimensional descriptor systems are discussed in Hilbert space. In [9] ∈ . This paper deals with the exact controllability of the following time varying linear singular distributed parameter system by using the functional analysis and theory of GE-evolution operator on the bases of the above papers:…”