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Finite time stability conditions for non autonomous continuous systems
Finite time stability is defined for continuous non autonomous systems. Starting with a result from Haimo Haimo (1986) we then extend this result to n¡dimensional non autonomous systems through the use of smooth and nonsmooth Lyapunov functions as in Perruquetti and Drakunov (2000). One obtains two different sufficient conditions and a necessary one for finite time stability of continuous non autonomous systems
Exponential stability for infinite-dimensional non-autonomous port-Hamiltonian Systems
We study the non-autonomous version of an infinite-dimensional
port-Hamiltonian system on an interval . Employing abstract results on
evolution families, we show -well-posedness of the corresponding Cauchy
problem, and thereby existence and uniqueness of classical solutions for
sufficiently regular initial data. Further, we demonstrate that a dissipation
condition in the style of the dissipation condition sufficient for uniform
exponential stability in the autonomous case also leads to a uniform
exponential decay of the energy in this non-autonomous setting
Finite-time Lagrangian transport analysis: Stable and unstable manifolds of hyperbolic trajectories and finite-time Lyapunov exponents
We consider issues associated with the Lagrangian characterisation of flow
structures arising in aperiodically time-dependent vector fields that are only
known on a finite time interval. A major motivation for the consideration of
this problem arises from the desire to study transport and mixing problems in
geophysical flows where the flow is obtained from a numerical solution, on a
finite space-time grid, of an appropriate partial differential equation model
for the velocity field. Of particular interest is the characterisation,
location, and evolution of "transport barriers" in the flow, i.e. material
curves and surfaces. We argue that a general theory of Lagrangian transport has
to account for the effects of transient flow phenomena which are not captured
by the infinite-time notions of hyperbolicity even for flows defined for all
time. Notions of finite-time hyperbolic trajectories, their finite time stable
and unstable manifolds, as well as finite-time Lyapunov exponent (FTLE) fields
and associated Lagrangian coherent structures have been the main tools for
characterizing transport barriers in the time-aperiodic situation. In this
paper we consider a variety of examples, some with explicit solutions, that
illustrate, in a concrete manner, the issues and phenomena that arise in the
setting of finite-time dynamical systems. Of particular significance for
geophysical applications is the notion of "flow transition" which occurs when
finite-time hyperbolicity is lost, or gained. The phenomena discovered and
analysed in our examples point the way to a variety of directions for rigorous
mathematical research in this rapidly developing, and important, new area of
dynamical systems theory
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