Quasi-diagonalization of linear impulsive systems and applications

[摘要] This work is concerned with the quasi-diagonalization of the impulsive linear system x' = A(t)x, x(t(k)(+)) = B(k)x(t(k)(-)), where the function A(t) is bounded and piecewise uniformly continuous, and (B-k)(k=1)(infinity) is a bounded sequence of impulse matrices. Let Lambda(t) and D-k be the diagonal matrices of eigenvalues of A(t) and B-k. We prove that there exists a transformation x = T(t)y which reduces this impulsive system to y' = [Lambda(t) + F(t) + Delta(t, sigma) + R(t)]y, y(t(k)) = [D-k + Delta(k)]y(t(k)(-)), where F(t), Delta(t, sigma), and (Delta(k))(k=1)(infinity) are functions with small norms in L-1, L-infinity, and l(infinity), respectively, and R(t) = -T-1(t)T'(t). An estimate for integral(s)(t)R(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 epsilon x' = A(t(k)(+)), x(t(k)(+)) = B(k)x(t(k)(-)). (C) 1997 Academic Press.

[发布日期] 1997-04-15