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Remarks on weak stabilization of semilinear wave equations

Alain Haraux — 2001

ESAIM: Control, Optimisation and Calculus of Variations

If a second order semilinear conservative equation with esssentially oscillatory solutions such as the wave equation is perturbed by a possibly non monotone damping term which is effective in a non negligible sub-region for at least one sign of the velocity, all solutions of the perturbed system converge weakly to 0 as time tends to infinity. We present here a simple and natural method of proof of this kind of property, implying as a consequence some recent very general results of Judith Vancostenoble....

Équations d'évolution non linéaires : solutions bornées et périodiques

Alain Haraux — 1978

Annales de l'institut Fourier

Soit φ un sous-différentiel (non coercif) dans un espace de Hilbert. On étudie l’existence de solutions bornées ou périodiques pour l’équation d u d t + φ ( u ( t ) ) f ( t ) , t 0 . Deux solutions périodiques éventuelles diffèrent d’une constante. Si f est périodique et ( I ˙ + φ ) - 1 compact, toute trajectoire bornée est asymptote pour t + à une trajectoire périodique.

Remarks on weak stabilization of semilinear wave equations

Alain Haraux — 2010

ESAIM: Control, Optimisation and Calculus of Variations

If a second order semilinear conservative equation with esssentially oscillatory solutions such as the wave equation is perturbed by a possibly non monotone damping term which is effective in a non negligible sub-region for at least one sign of the velocity, all solutions of the perturbed system converge weakly to 0 as time tends to infinity. We present here a simple and natural method of proof of this kind of property, implying as a consequence some recent very general results of Judith Vancostenoble. ...

Positively homogeneous functions and the Łojasiewicz gradient inequality

Alain Haraux — 2005

Annales Polonici Mathematici

It is quite natural to conjecture that a positively homogeneous function with degree d ≥ 2 on N satisfies the Łojasiewicz gradient inequality with exponent θ = 1/d without any need for an analyticity assumption. We show that this property is true under some additional hypotheses, but not always, even for N = 2.

Pointwise and spectral control of plate vibrations.

Alain HarauxStéphane Jaffard — 1991

Revista Matemática Iberoamericana

We consider the problem of controlling pointwise (by means of a time dependent Dirac measure supported by a given point) the motion of a vibrating plate Ω. Under general boundary conditions, including the special cases of simply supported or clamped plates, but of course excluding the cases where multiple eigenvalues exist for the biharmonic operator, we show the controlability of finite linear combinations of the eigenfunctions at any point of Ω where no eigenfunction vanishes at any time greater...

Oscillations of anharmonic Fourier series and the wave equation.

Alain HarauxVilmos Komornik — 1985

Revista Matemática Iberoamericana

In this paper we have collected some partial results on the sign of u(t,x) where u is a (sufficiently regular) solution of ⎧     utt + (-1)m Δmu = 0     (t,x) ∈ R x Ω ⎨ ⎩     u = ... = Δm-1 u = 0     t ∈ R. These results rely on the study of a sign of almost periodic functions of a special form restricted...

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