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Nonlinear feedback stabilization of a rotating body-beam without damping

Boumediène CHENTOUF, Jean-François COUCHOURON (2010)

ESAIM: Control, Optimisation and Calculus of Variations

This paper deals with nonlinear feedback stabilization problem of a flexible beam clamped at a rigid body and free at the other end. We assume that there is no damping and the feedback law proposed here consists of a nonlinear control torque applied to the rigid body and either a boundary control moment or a nonlinear boundary control force or both of them applied to the free end of the beam. This nonlinear feedback, which insures the exponential decay of the beam vibrations, extends the linear...

Nonlinear Heat Equation with a Fractional Laplacian in a Disk

Vladimir Varlamov (1999)

Colloquium Mathematicae

For the nonlinear heat equation with a fractional Laplacian u t + ( - Δ ) α / 2 u = u 2 , 1 < α ≤ 2, the first initial-boundary value problem in a disk is considered. Small initial conditions, homogeneous boundary conditions, and periodicity conditions in the angular coordinate are imposed. Existence and uniqueness of a global-in-time solution is proved, and the solution is constructed in the form of a series of eigenfunctions of the Laplace operator in the disk. First-order long-time asymptotics of the solution is obtained....

Nonlinear models of suspension bridges: discussion of the results

Pavel Drábek, Gabriela Holubová, Aleš Matas, Petr Nečesal (2003)

Applications of Mathematics

In this paper we present several nonlinear models of suspension bridges; most of them have been introduced by Lazer and McKenna. We discuss some results which were obtained by the authors and other mathematicians for the boundary value problems and initial boundary value problems. Our intention is to point out the character of these results and to show which mathematical methods were used to prove them instead of giving precise proofs and statements.

Nonlinear nonlocal evolution problems.

N.-H. Chang, M. Chipot (2003)

RACSAM

We consider a class of nonlinear parabolic problems where the coefficients are depending on a weighted integral of the solution. We address the issues of existence, uniqueness, stationary solutions and in some cases asymptotic behaviour.

Nonlinear Pulse Propagation

Jeffrey Rauch (2001)

Journées équations aux dérivées partielles

This talk gives a brief review of some recent progress in the asymptotic analysis of short pulse solutions of nonlinear hyperbolic partial differential equations. This includes descriptions on the scales of geometric optics and diffractive geometric optics, and also studies of special situations where pulses passing through focal points can be analysed.

Nonlocal quadratic evolution problems

Piotr Biler, Wojbor Woyczyński (2000)

Banach Center Publications

Nonlinear nonlocal parabolic equations modeling the evolution of density of mutually interacting particles are considered. The inertial type nonlinearity is quadratic and nonlocal while the diffusive term, also nonlocal, is anomalous and fractal, i.e., represented by a fractional power of the Laplacian. Conditions for global in time existence versus finite time blow-up are studied. Self-similar solutions are constructed for certain homogeneous initial data. Monte Carlo approximation schemes by interacting...

Note on the internal stabilization of stochastic parabolic equations with linearly multiplicative gaussian noise

Viorel Barbu (2013)

ESAIM: Control, Optimisation and Calculus of Variations

The parabolic equations driven by linearly multiplicative Gaussian noise are stabilizable in probability by linear feedback controllers with support in a suitably chosen open subset of the domain. This procedure extends to Navier − Stokes equations with multiplicative noise. The exact controllability is also discussed.

Novel method for generalized stability analysis of nonlinear impulsive evolution equations

JinRong Wang, Yong Zhou, Wei Wei (2012)

Kybernetika

In this paper, we discuss some generalized stability of solutions to a class of nonlinear impulsive evolution equations in the certain piecewise essentially bounded functions space. Firstly, stabilization of solutions to nonlinear impulsive evolution equations are studied by means of fixed point methods at an appropriate decay rate. Secondly, stable manifolds for the associated singular perturbation problems with impulses are compared with each other. Finally, an example on initial boundary value...

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