Necessary and sufficient conditions for the stability of Lyapunov exponents of linear differential systems with exponentially decaying perturbations (Q2459756)

The paper considers the system of linear differential equations \[ \dot z=A(t)z, \quad z\in\mathbb R^n,\quad t\geq 0, \tag{1} \] where the matrix \(A(t)\) is piecewise continuous and uniformly bounded. Let \(\lambda_1(A)\leq\cdots\leq\lambda_n(A)\) be the Lyapunov exponents of system (1). For the perturbed system \[ \dot y=(A(t)+Q(t))y, \quad y\in\mathbb R^n,\quad t\geq 0, \] with \(Q(t)\) piecewise continuous and uniformly bounded, let \(\lambda_1(A+Q)\leq\cdots\leq\lambda_n(A+Q)\) be its Lyapunov exponents. The Lyapunov exponents of system (1) are said to be stable if, for any piecewise continuous matrix \(Q(\cdot)\) satisfying \(Q(t)\to 0\) as \(t\to +\infty\), one has \(\lambda_i(A+Q)=\lambda_i(A)\), \(i=1,\dots,n\). The Lyapunov exponents of system (1) are said to be stable under perturbations of the class \(\mathcal N\) if the relations \(\lambda_i(A+Q)=\lambda_i(A)\), \(i=1,\dots,n\), are valid for any matrix \(Q(\cdot)\in\mathcal{N}\), where \(\mathcal{N}\) is a subset of the set \(\{Q(\cdot): Q(t)\to 0\text{ as }t\to +\infty\}\). The paper presents necessary and sufficient conditions for the stability of Lyapunov exponents of system (1) under exponentially decaying perturbations of the coefficient matrix, i.e. under perturbations of the class \[ \Bigl\{Q(\cdot): \limsup_{t\to +\infty}t^{-1}\ln \| Q(t)\| <0\Bigr\}. \]