Solvability of a time-varying singular distributed parameter system in Banach space

Singular distributed parameter systems are encountered much more often than distributed parameter systems.There is an essential distinction between singular and ordinary distributed parameter systems.A disturbance not only reduces the stability of singular distributed parameter systems but also strongly affects the structures of such systems,leading to,for example,impulsive behavior.Solvability is one of the most importantproblems in the study of singular distributed parameter systems.The main purpose of this paper is to study the solvability of a time-varying singular distributed parameter system in Banach space.First,the generalized evolution operator induced by a bounded linear operator is introduced in Banach space,the properties of the generalized evolution operator are discussed,the generator of the generalized evolution operator is defined,and the existence of a generalized evolution operator is proved;then the solvability of the time-varying singular distributed parameter system is discussed using the generalized evolution operator,the existence and uniqueness of the strong solution are proved,and a constructive expression of the strong solution of the time-varying singular distributed parameter system is obtained.This research is theoretically and practically important to the study of the stability,controllability,and optimal control problem of the time-varying singular distributed parameter system.