Lyapunov functions for trichotomies with growth rates
Let \(T(t, \tau)\) be the linear evolution operator associated to the linear equation \[ x' = A(t)x \] in the finite-dimensional space \({\mathbb R}^n\). We say that \(x' = A(t)x\) admits a \(\rho\)-nonuniform exponential contraction if there exist constants \(d\), \(D > 0\) and \(\varepsilon \geq 0\) such that \(\| T(t, \tau) \| \leq D e^{-d(\rho(t) - \rho(\tau)) + \varepsilon|\rho(\tau)|}\) for \(t \geq \tau > 0\). When the equation admits a \(\rho\)-nonuniform exponential contraction with \(\varepsilon = 0\), we say that it admits a \(\rho\)-uniform exponential contraction. In this paper, in strong contrast with the usual stable, unstable, and central behavior, the authors consider asymptotic rates of the form \(e^{c \rho(t)}\) determined by an arbitrary function \(\rho(t)\). The usual notion of nonuniform exponential contraction corresponds to the particular case when \(\rho(t) = t\), that is, \(\| T(t, \tau) \| \leq D e^{-d(t - \tau) + \varepsilon \tau}\) for \(t \geq \tau > 0\) [see \textit{L. Barreira} and \textit{C. Valls}, J. Differ. Equations 217, No.~1, 204--248 (2005; Zbl 1088.34053)]. The notion of nonuniform polynomial contraction is recovered when \(\rho(t) = \log (1 + t)\), that is, \[ \| T(t, \tau) \| \leq D \left( \frac{1 + t}{1 + \tau} \right)^{-d} (1 + \tau)^\varepsilon \] for \(t \geq \tau > 0\) [see \textit{L. Barreira} and \textit{C. Valls}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No.~11, 5208--5219 (2009; Zbl 1181.34046)]. Let \(V : {\mathbb R}_0^+ \times {\mathbb R}^n \to {\mathbb R}_0^-\) be a function, and assume that there exist \(C > 0\) and \(\delta \geq 0\) such that \(|V(t, x)| \leq C e^{\delta |\rho(t)|} \| x \|\) for every \(t \geq 0\) and \(x \in {\mathbb R}^n\). Given \(\alpha > 0\) and \(\gamma \geq 0\), we say that \(V\) is a strict Lyapunov function for the equation \(x' = A(t)x\) if \[ V(t, T(t, \tau) x) \leq \alpha^{\rho(t) - \rho(\tau)} V(\tau, x) \] for \(t \geq \tau\), and \[ V(\tau, x) \geq \frac1C e^{-\gamma \rho(\tau)} \| x \| \] for every \(\tau \geq 0\) and \(x \in {\mathbb R}^n\). The main objective is to give a complete characterization in terms of strict Lyapunov functions of the linear equations admitting a \(\rho\)-nonuniform exponential trichotomy. This includes criteria for the existence of a \(\rho\)-nonuniform exponential trichotomy, as well as inverse theorems providing explicit strict Lyapunov functions for each given exponential trichotomy. In the particular case of quadratic Lyapunov functions, it is shown that the existence of strict Lyapunov sequences can be deduced from more algebraic relations between the quadratic forms defining the Lyapunov functions. As an application of the characterization of nonuniform exponential trichotomy in terms of strict Lyapunov functions, the authors establish the robustness of \(\rho\)-nonuniform exponential trichotomy under sufficiently small linear perturbations. It is emphasized that, in comparison with former works, the proof of the robustness is much simpler even when \(\rho(t) = t\).
scientific article; zbMATH DE number 5651186
scientific article