1,234 research outputs found
Non-Smooth Stochastic Lyapunov Functions With Weak Extension of Viscosity Solutions
This paper proposes a notion of viscosity weak supersolutions to build a
bridge between stochastic Lyapunov stability theory and viscosity solution
theory. Different from ordinary differential equations, stochastic differential
equations can have the origins being stable despite having no smooth stochastic
Lyapunov functions (SLFs). The feature naturally requires that the related
Lyapunov equations are illustrated via viscosity solution theory, which deals
with non-smooth solutions to partial differential equations. This paper claims
that stochastic Lyapunov stability theory needs a weak extension of viscosity
supersolutions, and the proposed viscosity weak supersolutions describe
non-smooth SLFs ensuring a large class of the origins being noisily
(asymptotically) stable and (asymptotically) stable in probability. The
contribution of the non-smooth SLFs are confirmed by a few examples;
especially, they ensure that all the linear-quadratic-Gaussian (LQG) controlled
systems have the origins being noisily asymptotically stable for any additive
noises
Loss of regularity for Kolmogorov equations
The celebrated H\"{o}rmander condition is a sufficient (and nearly necessary)
condition for a second-order linear Kolmogorov partial differential equation
(PDE) with smooth coefficients to be hypoelliptic. As a consequence, the
solutions of Kolmogorov PDEs are smooth at all positive times if the
coefficients of the PDE are smooth and satisfy H\"{o}rmander's condition even
if the initial function is only continuous but not differentiable. First-order
linear Kolmogorov PDEs with smooth coefficients do not have this smoothing
effect but at least preserve regularity in the sense that solutions are smooth
if their initial functions are smooth. In this article, we consider the
intermediate regime of nonhypoelliptic second-order Kolmogorov PDEs with smooth
coefficients. The main observation of this article is that there exist
counterexamples to regularity preservation in that case. More precisely, we
give an example of a second-order linear Kolmogorov PDE with globally bounded
and smooth coefficients and a smooth initial function with compact support such
that the unique globally bounded viscosity solution of the PDE is not even
locally H\"{o}lder continuous. From the perspective of probability theory, the
existence of this example PDE has the consequence that there exists a
stochastic differential equation (SDE) with globally bounded and smooth
coefficients and a smooth function with compact support which is mapped by the
corresponding transition semigroup to a function which is not locally
H\"{o}lder continuous. In other words, degenerate noise can have a roughening
effect. A further implication of this loss of regularity phenomenon is that
numerical approximations may converge without any arbitrarily small polynomial
rate of convergence to the true solution of the SDE. More precisely, we prove
for an example SDE with globally bounded and smooth coefficients that the
standard Euler approximations converge to the exact solution of the SDE in the
strong and numerically weak sense, but at a rate that is slower then any power
law.Comment: Published in at http://dx.doi.org/10.1214/13-AOP838 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Nonsquare Spectral Factorization for Nonlinear Control Systems
This paper considers nonsquare spectral factorization of nonlinear input affine state space systems in continuous time. More specifically, we obtain a parametrization of nonsquare spectral factors in terms of invariant Lagrangian submanifolds and associated solutions of Hamilton–Jacobi inequalities. This inequality is a nonlinear analogue of the bounded real lemma and the control algebraic Riccati inequality. By way of an application, we discuss an alternative characterization of minimum and maximum phase spectral factors and introduce the notion of a rigid nonlinear system.
Hamiltonian dynamics, nanosystems, and nonequilibrium statistical mechanics
An overview is given of recent advances in nonequilibrium statistical
mechanics on the basis of the theory of Hamiltonian dynamical systems and in
the perspective provided by the nanosciences. It is shown how the properties of
relaxation toward a state of equilibrium can be derived from Liouville's
equation for Hamiltonian dynamical systems. The relaxation rates can be
conceived in terms of the so-called Pollicott-Ruelle resonances. In spatially
extended systems, the transport coefficients can also be obtained from the
Pollicott-Ruelle resonances. The Liouvillian eigenstates associated with these
resonances are in general singular and present fractal properties. The singular
character of the nonequilibrium states is shown to be at the origin of the
positive entropy production of nonequilibrium thermodynamics. Furthermore,
large-deviation dynamical relationships are obtained which relate the transport
properties to the characteristic quantities of the microscopic dynamics such as
the Lyapunov exponents, the Kolmogorov-Sinai entropy per unit time, and the
fractal dimensions. We show that these large-deviation dynamical relationships
belong to the same family of formulas as the fluctuation theorem, as well as a
new formula relating the entropy production to the difference between an
entropy per unit time of Kolmogorov-Sinai type and a time-reversed entropy per
unit time. The connections to the nonequilibrium work theorem and the transient
fluctuation theorem are also discussed. Applications to nanosystems are
described.Comment: Lecture notes for the International Summer School Fundamental
Problems in Statistical Physics XI (Leuven, Belgium, September 4-17, 2005
Statistics of finite scale local Lyapunov exponents in fully developed homogeneous isotropic turbulence
The present work analyzes the statistics of finite scale local Lyapunov
exponents of pairs of fluid particles trajectories in fully developed
incompressible homogeneous isotropic turbulence. According to the hypothesis of
fully developed chaos, this statistics is here analyzed assuming that the
entropy associated to the fluid kinematic state is maximum. The distribution of
the local Lyapunov exponents results to be an unsymmetrical uniform function in
a proper interval of variation. From this PDF, we determine the relationship
between average and maximum Lyapunov exponents, and the longitudinal velocity
correlation function. This link, which in turn leads to the closure of von
K\`arm\`an-Howarth and Corrsin equations, agrees with results of previous
works, supporting the proposed PDF calculation, at least for the purposes of
the energy cascade main effect estimation. Furthermore, through the property
that the Lyapunov vectors tend to align the direction of the maximum growth
rate of trajectories distance, we obtain the link between maximum and average
Lyapunov exponents in line with the previous results. To validate the proposed
theoretical results, we present different numerical simulations whose results
justify the hypotheses of the present analysis.Comment: Research article. arXiv admin note: text overlap with
arXiv:1706.0097
Statistical Lyapunov theory based on bifurcation analysis of energy cascade in isotropic homogeneous turbulence: a physical -- mathematical review
This work presents a review of previous articles dealing with an original
turbulence theory proposed by the author, and provides new theoretical insights
into some related issues. The new theoretical procedures and methodological
approaches confirm and corroborate the previous results. These articles study
the regime of homogeneous isotropic turbulence for incompressible fluids and
propose theoretical approaches based on a specific Lyapunov theory for
determining the closures of the von K\'arm\'an-Howarth and Corrsin equations,
and the statistics of velocity and temperature difference. Furthermore, novel
theoretical issues are here presented among which we can mention the following
ones. The bifurcation rate of the velocity gradient, calculated along fluid
particles trajectories, is shown to be much larger than the corresponding
maximal Lyapunov exponent. On that basis, an interpretation of the energy
cascade phenomenon is given and the statistics of finite time Lyapunov exponent
of the velocity gradient is shown to be represented by normal distribution
functions. Next, the self--similarity produced by the proposed closures is
analyzed, and a proper bifurcation analysis of the closed von
K\'arm\'an--Howarth equation is performed. This latter investigates the route
from developed turbulence toward the non--chaotic regimes, leading to an
estimate of the critical Taylor scale Reynolds number. A proper statistical
decomposition based on extended distribution functions and on the
Navier--Stokes equations is presented, which leads to the statistics of
velocity and temperature difference.Comment: physical--mathematical review of previous works and new theoretical
insights into some relates issue
A detectability criterion and data assimilation for non-linear differential equations
In this paper we propose a new sequential data assimilation method for
non-linear ordinary differential equations with compact state space. The method
is designed so that the Lyapunov exponents of the corresponding estimation
error dynamics are negative, i.e. the estimation error decays exponentially
fast. The latter is shown to be the case for generic regular flow maps if and
only if the observation matrix H satisfies detectability conditions: the rank
of H must be at least as great as the number of nonnegative Lyapunov exponents
of the underlying attractor. Numerical experiments illustrate the exponential
convergence of the method and the sharpness of the theory for the case of
Lorenz96 and Burgers equations with incomplete and noisy observations
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