580 research outputs found
Lyapunov operator inequalities for exponential stability of Banach space semigroups of operators
AbstractWe obtain continuous-time and discrete-time Lyapunov operator inequalities for the exponential stability of strongly continuous, one-parameter semigroups acting on Banach spaces. Thus we extend the classic result of Datko (1970) [2] from Hilbert spaces to Banach spaces
Converse Lyapunov Theorems for Switched Systems in Banach and Hilbert Spaces
We consider switched systems on Banach and Hilbert spaces governed by
strongly continuous one-parameter semigroups of linear evolution operators. We
provide necessary and sufficient conditions for their global exponential
stability, uniform with respect to the switching signal, in terms of the
existence of a Lyapunov function common to all modes
Stable foliations near a traveling front for reaction diffusion systems
We establish the existence of a stable foliation in the vicinity of a
traveling front solution for systems of reaction diffusion equations in one
space dimension that arise in the study of chemical reactions models and solid
fuel combustion. In this way we complement the orbital stability results from
earlier papers by A. Ghazaryan, S. Schecter and Y. Latushkin. The essential
spectrum of the differential operator obtained by linearization at the front
touches the imaginary axis. In spaces with exponential weights, one can shift
the spectrum to the left. We study the nonlinear equation on the intersection
of the unweighted and weighted spaces. Small translations of the front form a
center unstable manifold. For each small translation we prove the existence of
a stable manifold containing the translated front and show that the stable
manifolds foliate a small ball centered at the front
Exponential dichotomies of evolution operators in Banach spaces
This paper considers three dichotomy concepts (exponential dichotomy, uniform
exponential dichotomy and strong exponential dichotomy) in the general context
of non-invertible evolution operators in Banach spaces. Connections between
these concepts are illustrated. Using the notion of Green function, we give
necessary conditions and sufficient ones for strong exponential dichotomy. Some
illustrative examples are presented to prove that the converse of some
implication type theorems are not valid
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