79,464 research outputs found
On the Number of Zeros of Abelian Integrals: A Constructive Solution of the Infinitesimal Hilbert Sixteenth Problem
We prove that the number of limit cycles generated by a small
non-conservative perturbation of a Hamiltonian polynomial vector field on the
plane, is bounded by a double exponential of the degree of the fields. This
solves the long-standing tangential Hilbert 16th problem. The proof uses only
the fact that Abelian integrals of a given degree are horizontal sections of a
regular flat meromorphic connection (Gauss-Manin connection) with a
quasiunipotent monodromy group.Comment: Final revisio
Polynomial Bounds for Oscillation of Solutions of Fuchsian Systems
We study the problem of placing effective upper bounds for the number of
zeros of solutions of Fuchsian systems on the Riemann sphere. The principal
result is an explicit (non-uniform) upper bound, polynomially growing on the
frontier of the class of Fuchsian systems of given dimension n having m
singular points. As a function of n,m, this bound turns out to be double
exponential in the precise sense explained in the paper. As a corollary, we
obtain a solution of the so called restricted infinitesimal Hilbert 16th
problem, an explicit upper bound for the number of isolated zeros of Abelian
integrals which is polynomially growing as the Hamiltonian tends to the
degeneracy locus. This improves the exponential bounds recently established by
A. Glutsyuk and Yu. Ilyashenko.Comment: Will appear in Annales de l'institut Fourier vol. 60 (2010
Uniform stabilization for linear systems with persistency of excitation. The neutrally stable and the double integrator cases
Consider the controlled system where the pair
is stabilizable and takes values in and is
persistently exciting, i.e., there exist two positive constants such
that, for every , . In particular,
when becomes zero the system dynamics switches to an uncontrollable
system. In this paper, we address the following question: is it possible to
find a linear time-invariant state-feedback , with only depending on
and possibly on , which globally asymptotically stabilizes the
system? We give a positive answer to this question for two cases: when is
neutrally stable and when the system is the double integrator
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