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Asymptotic behavior of gradient-like dynamical systems involving inertia and multiscale aspects
In a Hilbert space , we study the asymptotic behaviour, as time
variable goes to , of nonautonomous gradient-like dynamical
systems involving inertia and multiscale features.
Given a general Hilbert space, and two convex
differentiable functions, a positive damping parameter, and a function of which tends to zero as goes to , we
consider the second-order differential equation This
system models the emergence of various collective behaviors in game theory, as
well as the asymptotic control of coupled nonlinear oscillators. Assuming that
tends to zero moderately slowly as goes to infinity, we show
that the trajectories converge weakly in . The limiting equilibria
are solutions of the hierarchical minimization problem which consists in
minimizing over the set of minimizers of . As key assumptions,
we suppose that and that, for
every belonging to a convex cone depending on the data
and where is
the Fenchel conjugate of , and is the support function of
. An application is given to coupled oscillators
A minimization problem for the lapse and the initial-boundary value problem for Einstein's field equations
We discuss the initial-boundary value problem of General Relativity. Previous
considerations for a toy model problem in electrodynamics motivate the
introduction of a variational principle for the lapse with several attractive
properties. In particular, it is argued that the resulting elliptic gauge
condition for the lapse together with a suitable condition for the shift and
constraint-preserving boundary conditions controlling the Weyl scalar Psi_0 are
expected to yield a well posed initial-boundary value problem for metric
formulations of Einstein's field equations which are commonly used in numerical
relativity.
To present a simple and explicit example we consider the 3+1 decomposition
introduced by York of the field equations on a cubic domain with two periodic
directions and prove in the weak field limit that our gauge condition for the
lapse and our boundary conditions lead to a well posed problem. The method
discussed here is quite general and should also yield well posed problems for
different ways of writing the evolution equations, including first order
symmetric hyperbolic or mixed first-order second-order formulations. Well posed
initial-boundary value formulations for the linearization about arbitrary
stationary configurations will be presented elsewhere.Comment: 34 pages, no figure
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