39,311 research outputs found
Stabilization by Unbounded-Variation Noises
In this paper, we claim the availability of deterministic noises for
stabilization of the origins of dynamical systems, provided that the noises
have unbounded variations. To achieve the result, we first consider the system
representations based on rough path analysis; then, we provide the notion of
asymptotic stability in roughness to analyze the stability for the systems. In
the procedure, we also confirm that the system representations include
stochastic differential equations; we also found that asymptotic stability in
roughness is the same property as uniform almost sure asymptotic stability
provided by Bardi and Cesaroni. After the discussion, we confirm that there is
a case that deterministic noises are capable of making the origin become
asymptotically stable in roughness while stochastic noises do not achieve the
same stabilization results.Comment: 22 pages, 5 figure
Analytic and Asymptotic Methods for Nonlinear Singularity Analysis: a Review and Extensions of Tests for the Painlev\'e Property
The integrability (solvability via an associated single-valued linear
problem) of a differential equation is closely related to the singularity
structure of its solutions. In particular, there is strong evidence that all
integrable equations have the Painlev\'e property, that is, all solutions are
single-valued around all movable singularities. In this expository article, we
review methods for analysing such singularity structure. In particular, we
describe well known techniques of nonlinear regular-singular-type analysis,
i.e. the Painlev\'e tests for ordinary and partial differential equations. Then
we discuss methods of obtaining sufficiency conditions for the Painlev\'e
property. Recently, extensions of \textit{irregular} singularity analysis to
nonlinear equations have been achieved. Also, new asymptotic limits of
differential equations preserving the Painlev\'e property have been found. We
discuss these also.Comment: 40 pages in LaTeX2e. To appear in the Proceedings of the CIMPA Summer
School on "Nonlinear Systems," Pondicherry, India, January 1996, (eds) B.
Grammaticos and K. Tamizhman
Comparative Study of Homotopy Analysis and Renormalization Group Methods on Rayleigh and Van der Pol Equations
A comparative study of the Homotopy Analysis method and an improved
Renormalization Group method is presented in the context of the Rayleigh and
the Van der Pol equations. Efficient approximate formulae as functions of the
nonlinearity parameter for the amplitudes of the
limit cycles for both these oscillators are derived. The improvement in the
Renormalization group analysis is achieved by invoking the idea of nonlinear
time that should have significance in a nonlinear system. Good approximate
plots of limit cycles of the concerned oscillators are also presented within
this framework.Comment: 25 pages, 7 figures. Revised and upgraded: Differ Equ Dyn Syst, (26
July, 2015
Some Results on the Boundary Control of Systems of Conservation Laws
This note is concerned with the study of the initial boundary value problem
for systems of conservation laws from the point of view of control theory,
where the initial data is fixed and the boundary data are regarded as control
functions. We first consider the problem of controllability at a fixed time for
genuinely nonlinear Temple class systems, and present a description of the set
of attainable configurations of the corresponding solutions in terms of
suitable Oleinik-type estimates. We next present a result concerning the
asymptotic stabilization near a constant state for general systems.
Finally we show with an example that in general one cannot achieve exact
controllability to a constant state in finite time.Comment: 10 pages, 4 figures, conferenc
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