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The control of the boundary value problem for linear impulsive integro-differential systems. (English) Zbl 0943.93007

The impulse control system of integro-differential equations defined on the interval [a,b] is of the form:

x ˙(t)=A(t)x(t)+C(t)u(t)+f(t)+ a t K(t,s)x(s)dx,tθ i ,
Δx(θ i )=B i x(θ i )+ a<θ j θ i D i,j x(θ j )+Q i ν i +I i ,
x(a)=α,x(b)=β,α,β n ,

where {θ i ,i=1,2,,p} is a strictly increasing sequence of real numbers belonging to the interval [a,b], Δx(θ j )=x(θ j +)-x(θ j ), A(t), K(t,s), D i,j , B i , i,j{1,,p} are matrices of size n×n, C(t), and Q i , i=1,2,,p are matrices of size n×m, f is a vector function, and I i n , i=1,2,,p. The pair (u,ν) where uL 2 m (a,b), ν={ν i }, ν i m denotes a control.

Under some additional assumptions necessary and sufficient conditions for controllability of the system are given.

34B37Boundary value problems for ODE with impulses