Uniform asymptotic stability of linear time-varying systems

Abstract

Let ẋ(t) = A(t)x(t), with x : R → R ⁿ and A : R → R ^{n ×n}, be a linear time-varying system. We consider the case where the eigenvalues of the system matrix as functions of the time t ∈ R may have positive real parts. Consider the following examples: the matrix A₁(t) := diag(−l + 2cost, −1 −2cost) and the matrix A ₂(t) := diag(−l + max{2cost, −2cost}, −1 − max{2cost, − 2cost}). Notice that A ₁(t) and A ₂(t) have the same eigenvalues ∀t ∈ R. The system ẋ(t) = A ₁(t)x(t) is uniformly asymptotically stable and the system ẋ(t) = A ₂(t)x(t) is unstable. This illustrates that the stability properties do not only depend on the evolution of the eigenvalues. The evolution of the corresponding eigenvectors will also play a crucial role.