10,469 research outputs found
Continuous symmetries of Lagrangians and exact solutions of discrete equations
One of the difficulties encountered when studying physical theories in
discrete space-time is that of describing the underlying continuous symmetries
(like Lorentz, or Galilei invariance). One of the ways of addressing this
difficulty is to consider point transformations acting simultaneously on
difference equations and lattices. In a previous article we have classified
ordinary difference schemes invariant under Lie groups of point
transformations. The present article is devoted to an invariant Lagrangian
formalism for scalar single-variable difference schemes. The formalism is used
to obtain first integrals and explicit exact solutions of the schemes.
Equations invariant under two- and three- dimensional groups of Lagrangian
symmetries are considered.Comment: 31 pages, submitted to Journal of Mathematical Physic
Lie Symmetries and Exact Solutions of First Order Difference Schemes
We show that any first order ordinary differential equation with a known Lie
point symmetry group can be discretized into a difference scheme with the same
symmetry group. In general, the lattices are not regular ones, but must be
adapted to the symmetries considered. The invariant difference schemes can be
so chosen that their solutions coincide exactly with those of the original
differential equation.Comment: Minor changes and journal-re
Hamilton-Jacobi equations for optimal control on multidimensional junctions
We consider continuous-state and continuous-time control problems where the
admissible trajectories of the system are constrained to remain on a union of
half-planes which share a common straight line. This set will be named a
junction. We define a notion of constrained viscosity solution of
Hamilton-Jacobi equations on the junction and we propose a comparison principle
whose proof is based on arguments from the optimal control theory.Comment: 37 pages and 2 figure
Global stabilization of a Korteweg-de Vries equation with saturating distributed control
This article deals with the design of saturated controls in the context of
partial differential equations. It focuses on a Korteweg-de Vries equation,
which is a nonlinear mathematical model of waves on shallow water surfaces. Two
different types of saturated controls are considered. The well-posedness is
proven applying a Banach fixed point theorem, using some estimates of this
equation and some properties of the saturation function. The proof of the
asymptotic stability of the closed-loop system is separated in two cases: i)
when the control acts on all the domain, a Lyapunov function together with a
sector condition describing the saturating input is used to conclude on the
stability, ii) when the control is localized, we argue by contradiction. Some
numerical simulations illustrate the stability of the closed-loop nonlinear
partial differential equation. 1. Introduction. In recent decades, a great
effort has been made to take into account input saturations in control designs
(see e.g [39], [15] or more recently [17]). In most applications, actuators are
limited due to some physical constraints and the control input has to be
bounded. Neglecting the amplitude actuator limitation can be source of
undesirable and catastrophic behaviors for the closed-loop system. The standard
method to analyze the stability with such nonlinear controls follows a two
steps design. First the design is carried out without taking into account the
saturation. In a second step, a nonlinear analysis of the closed-loop system is
made when adding the saturation. In this way, we often get local stabilization
results. Tackling this particular nonlinearity in the case of finite
dimensional systems is already a difficult problem. However, nowadays, numerous
techniques are available (see e.g. [39, 41, 37]) and such systems can be
analyzed with an appropriate Lyapunov function and a sector condition of the
saturation map, as introduced in [39]. In the literature, there are few papers
studying this topic in the infinite dimensional case. Among them, we can cite
[18], [29], where a wave equation equipped with a saturated distributed
actuator is studied, and [12], where a coupled PDE/ODE system modeling a
switched power converter with a transmission line is considered. Due to some
restrictions on the system, a saturated feedback has to be designed in the
latter paper. There exist also some papers using the nonlinear semigroup theory
and focusing on abstract systems ([20],[34],[36]). Let us note that in [36],
[34] and [20], the study of a priori bounded controller is tackled using
abstract nonlinear theory. To be more specific, for bounded ([36],[34]) and
unbounded ([34]) control operators, some conditions are derived to deduce, from
the asymptotic stability of an infinite-dimensional linear system in abstract
form, the asymptotic stability when closing the loop with saturating
controller. These articles use the nonlinear semigroup theory (see e.g. [24] or
[1]). The Korteweg-de Vries equation (KdV for short)Comment: arXiv admin note: text overlap with arXiv:1609.0144
On the Linearization of Second-Order Differential and Difference Equations
This article complements recent results of the papers [J. Math. Phys. 41
(2000), 480; 45 (2004), 336] on the symmetry classification of second-order
ordinary difference equations and meshes, as well as the Lagrangian formalism
and Noether-type integration technique. It turned out that there exist
nonlinear superposition principles for solutions of special second-order
ordinary difference equations which possess Lie group symmetries. This
superposition springs from the linearization of second-order ordinary
difference equations by means of non-point transformations which act
simultaneously on equations and meshes. These transformations become some sort
of contact transformations in the continuous limit.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Local analytic regularity in the linearized Calder\'on problem
We consider the linearization of the Dirichlet-to-Neumann (DN) map as a
function of the potential. We show that it is injective at a real analytic
potential for measurements made at an open subset of analyticity of the
boundary. More generally, we relate the analyticity up to the boundary of the
variations of the potential to the analyticity of the symbols of the
corresponding variations of the DN-map.Comment: A gap in the proof of Lemma 1.2 in v1 prompted us to remove that
lemma, causing a superficial change in the formulation of the main resul
Analysis of a HMM time-discretization scheme for a system of Stochastic PDE's
We consider the discretization in time of a system of parabolic stochastic
partial differential equations with slow and fast components; the fast equation
is driven by an additive space-time white noise. The numerical method is
inspired by the Averaging Principle satisfied by this system, and fits to the
framework of Heterogeneous Multiscale Methods.The slow and the fast components
are approximated with two coupled numerical semi-implicit Euler schemes
depending on two different timestep sizes. We derive bounds of the
approximation error on the slow component in the strong sense - approximation
of trajectories - and in the weak sense - approximation of the laws. The
estimates generalize the results of \cite{E-L-V} in the case of infinite
dimensional processes
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