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Stability of periodic stationary solutions of scalar conservation laws with space-periodic flux

Anne-Laure Dalibard (2011)

Journal of the European Mathematical Society

This article investigates the long-time behaviour of parabolic scalar conservation laws of the type ∂ t u + div y A ( y , u ) - Δ y u = 0 , where y ∈ ℝ N and the flux A is periodic in y . More specifically, we consider the case when the initial data is an L 1 disturbance of a stationary periodic solution. We show, under polynomial growth assumptions on the flux, that the difference between u and the stationary solution behaves in L 1 norm like a self-similar profile for large times. The proof uses a time and space change of variables which is...

Stability result for a thermoelastic Bresse system with delay term in the internal feedback

Lamine Bouzettouta, Sabah Baibeche, Manel Abdelli, Amar Guesmia (2023)

Mathematica Bohemica

The studies considered here are concerend with a linear thermoelastic Bresse system with delay term in the feedback. The heat conduction is also given by Cattaneo's law. Under an appropriate assumption between the weight of the delay and the weight of the damping, we prove the well-posedness of the problem using the semigroup method. Furthermore, based on the energy method, we establish an exponential stability result depending of a condition on the constants of the system that was first considered...

Stabilization of a layered piezoelectric 3-D body by boundary dissipation

Boris Kapitonov, Bernadette Miara, Gustavo Perla Menzala (2006)

ESAIM: Control, Optimisation and Calculus of Variations

We consider a linear coupled system of quasi-electrostatic equations which govern the evolution of a 3-D layered piezoelectric body. Assuming that a dissipative effect is effective at the boundary, we study the uniform stabilization problem. We prove that this is indeed the case, provided some geometric conditions on the region and the interfaces hold. We also assume a monotonicity condition on the coefficients. As an application, we deduce exact controllability of the system with boundary control...

Stabilization of Berger–Timoshenko’s equation as limit of the uniform stabilization of the von Kármán system of beams and plates

G. Perla Menzala, Ademir F. Pazoto, Enrique Zuazua (2002)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We consider a dynamical one-dimensional nonlinear von Kármán model for beams depending on a parameter ε > 0 and study its asymptotic behavior for t large, as ε → 0 . Introducing appropriate damping mechanisms we show that the energy of solutions of the corresponding damped models decay exponentially uniformly with respect to the parameter ε . In order for this to be true the damping mechanism has to have the appropriate scale with respect to ε . In the limit as ε → 0 we obtain damped Berger–Timoshenko beam models...

Stabilization of Berger–Timoshenko's equation as limit of the uniform stabilization of the von Kármán system of beams and plates

G. Perla Menzala, Ademir F. Pazoto, Enrique Zuazua (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider a dynamical one-dimensional nonlinear von Kármán model for beams depending on a parameter ε > 0 and study its asymptotic behavior for t large, as ε → 0. Introducing appropriate damping mechanisms we show that the energy of solutions of the corresponding damped models decay exponentially uniformly with respect to the parameter ε. In order for this to be true the damping mechanism has to have the appropriate scale with respect to ε. In the limit as ε → 0 we obtain damped Berger–Timoshenko...

Stabilization of the Kawahara equation with localized damping

Carlos F. Vasconcellos, Patricia N. da Silva (2011)

ESAIM: Control, Optimisation and Calculus of Variations

We study the stabilization of global solutions of the Kawahara (K) equation in a bounded interval, under the effect of a localized damping mechanism. The Kawahara equation is a model for small amplitude long waves. Using multiplier techniques and compactness arguments we prove the exponential decay of the solutions of the (K) model. The proof requires of a unique continuation theorem and the smoothing effect of the (K) equation on the real line, which are proved in this work.

Stabilization of the Kawahara equation with localized damping

Carlos F. Vasconcellos, Patricia N. da Silva (2011)

ESAIM: Control, Optimisation and Calculus of Variations

We study the stabilization of global solutions of the Kawahara (K) equation in a bounded interval, under the effect of a localized damping mechanism. The Kawahara equation is a model for small amplitude long waves. Using multiplier techniques and compactness arguments we prove the exponential decay of the solutions of the (K) model. The proof requires of a unique continuation theorem and the smoothing effect of the (K) equation on the real line, which are proved in this work.

Stabilization of the wave equation by on-off and positive-negative feedbacks

Patrick Martinez, Judith Vancostenoble (2002)

ESAIM: Control, Optimisation and Calculus of Variations

Motivated by several works on the stabilization of the oscillator by on-off feedbacks, we study the related problem for the one-dimensional wave equation, damped by an on-off feedback a ( t ) u t . We obtain results that are radically different from those known in the case of the oscillator. We consider periodic functions a : typically a is equal to 1 on ( 0 , T ) , equal to 0 on ( T , q T ) and is q T -periodic. We study the boundary case and next the locally distributed case, and we give optimal results of stability. In both cases,...

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