The Sharp Upper Bound on the Mobility of the Highest Exponent of a Linear System Under Perturbations Whose Weighted Mean Is Small

摘要

Consider the linear differential system x = A(t)x,x∈R,t≥0(1)with bounded piecewise continuous coefficient matrix A such that ||A(t)||≤M≤+∞ for all t≥0 and with Cauchy matrix X(t,T). Along with system (1), consider the perturbed system y = A(t)y + Q(t)y, y∈R~n,t≥0 (2)with piecewise continuous perturbation matrix Q satisfying the integral boundedness condition [1. p. 252], i.e., the inequality ∫_t~t+1||Q(T)||dT≤C_Q<+∞for all t≥0, where CQ is a constant depending on Q. The highest exponent of system (2) will be denoted by λ_n(A+Q).