Quasi-Diagonalization of Linear Impulsive Systems and Applications

This work is concerned with the quasi-diagonalization of the impulsive linear system xXsA t.x, x tqk.sBkx tyk., where the function A t. is bounded and piecewise uniformly continuous, and Bk.`ks1 is a bounded sequence of impulse matrices. Let L t. and Dk be the diagonal matrices of eigenvalues of A t. and Bk . We prove that there exists a transformation xsT t.y which reduces this impul- sive system to yXswL t.qF t.qD t,s.qR t.xy, y tk.swDkqDkxy tyk ., where F t., D t,s., and Dk.`ks1 are functions with small norms in L1, L`, and l`, respectively, and R t.syTy1 t.TX t.. An estimate for HstR u. du is 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 « xXsA t.x, x tqk.sBkx tyk..