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Growth conditions for the stability of a class of time-varying perturbed singular systems

Faten Ezzine, Mohamed Ali Hammami (2022)

Kybernetika

In this paper, we investigate the problem of stability of linear time-varying singular systems, which are transferable into a standard canonical form. Sufficient conditions on exponential stability and practical exponential stability of solutions of linear perturbed singular systems are obtained based on generalized Gronwall inequalities and Lyapunov techniques. Moreover, we study the problem of stability and stabilization for some classes of singular systems. Finally, we present a numerical example...

Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation

Marcel Guardia, Vadim Kaloshin (2015)

Journal of the European Mathematical Society

We consider the cubic defocusing nonlinear Schrödinger equation in the two dimensional torus. Fix s > 1 . Recently Colliander, Keel, Staffilani, Tao and Takaoka proved the existence of solutions with s -Sobolev norm growing in time. We establish the existence of solutions with polynomial time estimates. More exactly, there is c > 0 such that for any 𝒦 ≫ 1 we find a solution u and a time T such that ∥ u ( T ) ∥ H s ≥ 𝒦 ∥ u ( 0 ) ∥ H s . Moreover, the time T satisfies the polynomial bound 0 < T < 𝒦 C .

Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold

Nalini Anantharaman, Stéphane Nonnenmacher (2007)

Annales de l’institut Fourier

We study the high-energy eigenfunctions of the Laplacian on a compact Riemannian manifold with Anosov geodesic flow. The localization of a semiclassical measure associated with a sequence of eigenfunctions is characterized by the Kolmogorov-Sinai entropy of this measure. We show that this entropy is necessarily bounded from below by a constant which, in the case of constant negative curvature, equals half the maximal entropy. In this sense, high-energy eigenfunctions are at least half-delocalized....

Hamiltonian loops from the ergodic point of view

Leonid Polterovich (1999)

Journal of the European Mathematical Society

Let G be the group of Hamiltonian diffeomorphisms of a closed symplectic manifold Y . A loop h : S 1 → G is called strictly ergodic if for some irrational number the associated skew product map T : S 1 × Y → S 1 × Y defined by T ( t , y ) = ( t + α ; h ( t ) y ) is strictly ergodic. In the present paper we address the following question. Which elements of the fundamental group of G can be represented by strictly ergodic loops? We prove existence of contractible strictly ergodic loops for a wide class of symplectic manifolds (for instance for simply connected...

Hamiltonian stability and subanalytic geometry

Laurent Niederman (2006)

Annales de l’institut Fourier

In the 70’s, Nekhorochev proved that for an analytic nearly integrable Hamiltonian system, the action variables of the unperturbed Hamiltonian remain nearly constant over an exponentially long time with respect to the size of the perturbation, provided that the unperturbed Hamiltonian satisfies some generic transversality condition known as steepness. Using theorems of real subanalytic geometry, we derive a geometric criterion for steepness: a numerical function h which is real analytic around a...

Currently displaying 1821 – 1840 of 4754