Quasi-diagonalization of linear impulsive systems and applications

Abstract

This work is concerned with the quasi-diagonalization of the impulsive linear systemx′=A(t)x,x(t+k)=Bkx(t-k), where the functionA(t) is bounded and piecewise uniformly continuous, and (Bk)∞k=1is a bounded sequence of impulse matrices. Let Λ(t) andDkbe the diagonal matrices of eigenvalues ofA(t) andBk. We prove that there exists a transformationx=T(t)ywhich reduces this impulsive system toy′=[Λ(t)+F(t)+Δ(t,σ)+R(t)]y,y(tk)=[Dk+Δk]y(t-k), whereF(t), Δ(t,σ), and (Δk)∞k=1are functions with small norms inL1,L∞, andl∞, respectively, andR(t)=-T-1(t)T′(t). An estimate for ∫tsR(u)duis given. We apply these results to the problem of the existence of periodic solutions of impulsive systems and to the problem of stability of the singularly perturbed linear impulsive system εx′=A(t)x,x(t+k)=Bkx(t-k). © 1997 Academic Press.