On sets of linear differential systems to which perturbed linear systems cannot be reduced

Abstract

The paper [2] defines the noncoinciding irreducibility sets N ₂(a, σ) and N ₃(a, σ), σ ∈ (0, 2a], of all n-dimensional linear differential systems with piecewise continuous coefficient matrices A(t) such that ‖A(t)‖ ≤ a < + ∞ for t ∈ [0,+∞) and there exists a linear differential system that is not Lyapunov reducible to the original system and has coefficient matrix B(t) satisfying [for the case of N ₂(a, σ)] the condition

$\left\| {B(t) - A(t)} \right\| \leqslant const \times e^{ - \sigma t} ,t \geqslant 0,$

or [for the case of N ₃(a, σ)] the more general condition that the Lyapunov exponent of the difference B(t) − A(t) does not exceed −σ. For these sets, which are related by the obvious inclusions

$N_i (a,\sigma _1 ) \supseteq N_i (a,\sigma _2 ),0 < \sigma _1 < \sigma _2 \leqslant 2a,i = 2,3,$

, we prove that (i) they strictly decrease with increasing parameter σ ∈ (0, 2a], N _i (a, σ ₁) ⊃ N _i (a, σ ₂) for σ ₁ < σ ₂; (ii) there is a strict inclusion N ₂(a, σ) ⊂ N ₃(a, σ) for all σ ∈ (0, 2a].