Description of the linear Perron effect under parametric perturbations exponentially vanishing at infinity

Let ${\mathcal M}_n$ be the set of linear differential systems of order $n\geqslant 2$ whose coefficients are continuous and bounded on the time semiaxis $\mathbb{R}_+$. Denote by $\lambda_1(A)\leqslant\ldots\leqslant \lambda_n(A)$ the Lyapunov exponents of a system $A\in {\mathcal M}_n,$ by $\Lambda(A)=(\lambda_1(A),\ldots,\lambda_n(A))$ their spectrum, and by $\mathrm{es}(A)$ the exponential stability index of $A$ (the dimension of the linear subspace of solutions with negative characteristic exponents). For a system $A\in {\mathcal M}_n$ and a metric space $M,$ we consider the class ${\mathcal E}_n[A](M)$ of continuous $(n\times n)$ matrix-valued functions $Q\colon \mathbb{R}_+\times M\to \mathbb{R}^{n\times n}$ satisfying the bound $\|Q(t,\mu)\|\leqslant C_Q\exp(-\sigma_Qt)$ for all $(t,\mu)\in\mathbb{R}_+\times M,$ where $C_Q$ and $\sigma_Q$ are positive constants (possibly different for each function $Q$), and such that the Lyapunov exponents of the system $A+Q,$ which are functions of $\mu\in M$ and are denoted by $\lambda_1(\mu;A+Q)\leqslant\ldots\leqslant \lambda_n(\mu;A+Q),$ are not less than the corresponding Lyapunov exponents of the system $A$; i.e., $\lambda_k(\mu;A+Q)\geqslant \lambda_k(A),$ $k=\overline{1,n},$ for all $\mu\in M$. The problem is to obtain a complete description for each $n\in\mathbb{N}$ and each metric space $M$ of the class of pairs $\bigl(\Lambda(A),\Lambda(\cdot\,;A+Q)\bigr)$ composed of the spectrum $\Lambda(A)\in\mathbb{R}^n$ of a system $A\in {\mathcal M}_n$ and the spectrum $\Lambda(\cdot\,;A+Q)\colon M\to \mathbb{R}^n$ of a family $A+Q,$ where $A$ ranges over ${\mathcal M}_n$ and the matrix-valued function $Q$ ranges over the class ${\mathcal E}_n[A](M)$ for each $A,$ i.e., of the class $\Pi {\mathcal E}_n(M)=\{\bigl(\Lambda(A),\Lambda(\cdot\,;A+Q)\bigr)\,\vert\, A\in {\mathcal M}_{n},\,Q\in {\mathcal E}_n[A](M)\}$. The solution of this problem is provided by the following statement: for each integer $n\geqslant 2$ and every metric space $M$, a pair $\bigl(l,F(\cdot)\bigr),$ where $l=(l_1,\ldots,l_n)\in\mathbb{R}^n$ and $F(\cdot)=(f_1(\cdot),\ldots,f_n(\cdot))\colon M\to \mathbb{R}^n,$ belongs to the class $\Pi {\mathcal E}_n(M)$ if and only if the following conditions are met: (1) $l_1\leqslant \ldots \leqslant l_n,$ (2) $f_1(\mu)\leqslant \ldots \leqslant f_n(\mu)$ for all $\mu\in M,$ (3) $f_i(\mu)\geqslant l_i$ for all $i=\overline{1,n}$ and $\mu\in M,$ (4) for each $i=\overline{1,n}$, the function $f_i(\cdot)\colon M\to \mathbb{R}$ is bounded and, for any $r\in\mathbb{R}$, the preimage $f_i^{-1}([r,+\infty))$ of the half-interval $[r,+\infty)$ is a $G_{\delta}$-set. The solution of the similar problem of describing the pairs composed of the exponential stability index $\mathrm{es}(A)\in \{0,\ldots,n\}$ of a system $A$ and the exponential stability index $\mathrm{es}(\cdot\,;A+Q)\colon M\to \{0,\ldots,n\}$ of a family $A+Q,$ i.e., the class ${\mathcal I}{\mathcal E}_n(M)=\{\bigl(\mathrm{es}(A),\mathrm{es}(\cdot\,;A+Q)\bigr)\,\vert\, A\in {\mathcal M}_{n},\,Q\in {\mathcal E}_n[A](M)\}$, is contained in the following statement: for any positive integer $n\geqslant 2$ and every metric space $M$, a pair $\bigl(d,f(\cdot)\bigr),$ where $d\in\{0,\ldots,n\}$ and $f\colon M\to\{0,\ldots,n\},$ belongs to the class ${\mathcal I}{\mathcal E}_n(M)$ if and only if $f(\mu)\leqslant d$ for all $\mu\in M$ and, for any $r\in\mathbb{R}$, the preimage $f^{-1}((-\infty,r])$ of the half-interval $(-\infty,r]$ is a $G_{\delta}$-set.