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Controllability of impulsive functional differential systems in Banach spaces. (English) Zbl 1110.34057

In a Banach space X, the impulsive functional-differential equation

x ˙(t)=Ax(t)+Bu(t)+f(t,x t ),tJ=[0,b],tt k ,k=1,,m,u(·)L 2 (J,U),(1)
x(t k +)=x(t k -)+I k (x(t k -)),k=1,,m,x(t)=Φ(t),t[τ,0],Φ(·)D,(2)

is considered, where A is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators T(t),B is a bounded linear operator; and D is a set of piecewise continuous functions φ:[τ,0]X having finitely many discontinuity points of the first kind. A function x(t)X,t[τ,b], is said to be a solution of (1), (2) if x(t) satisfies condition (2);the restriction of x(t) on the interval [t k ,t k+1 ],k=1,,m is continuous and the following integral equation is verified

x(t)=T(t)Φ(t)+ 0 t T(t-s)[(Bu)(s)+f(s,x s )]ds+
0<t<t k T(t-t k )I k (x(t k -)),tJ·

System (1), (2) is said be controllable if for every initial function Φ(·)D and x 1 X, there exists a control u(·)L 2 (J,U) such that the solution x(t) corresponding to the control u(t) satisfies x(b)=x 1 · On the basis of Schaefer’s fixed-point theorem, sufficient conditions for controllability are obtained.

MSC:
34K35Functional-differential equations connected with control problems
34K45Functional-differential equations with impulses
93C23Systems governed by functional-differential equations