Differential and Integral Equations

Quadratic Lyapunov functions for linear skew-product flows and weighted composition operators

C. Chicone and Yu. Latushkin

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The Daleckij-Krein method for constructing a quadratic Lyapunov function for the equation $f'=Df(t)$ in Hilbert space is extended to include the case of an unbounded operator $D$ that generates a $C_0$-group. The extension is applied to obtain a quadratic Lyapunov function for the case of a group of weighted composition operators generated by a flow on a compact metric space together with a cocycle over this flow. These results are used to characterize the hyperbolicity of linear skew-product flows in terms of the existence of such a Lyapunov function. Also, the "trajectorial" method for constructing the Lyapunov function is discussed. Interrelations with Schrödinger, Riccati and Hamiltonian equations are discussed and an application to geodesic flows on two-dimensional Riemannian manifolds is given.

Article information

Differential Integral Equations, Volume 8, Number 2 (1995), 289-307.

First available in Project Euclid: 20 May 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30]
Secondary: 34C35 34G10: Linear equations [See also 47D06, 47D09] 47B38: Operators on function spaces (general) 58F17


Chicone, C.; Latushkin, Yu. Quadratic Lyapunov functions for linear skew-product flows and weighted composition operators. Differential Integral Equations 8 (1995), no. 2, 289--307. https://projecteuclid.org/euclid.die/1369083470

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