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

Patrick Martinez, Judith Vancostenoble (2010)

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, qT) and is qT-periodic. We study the boundary case and next the locally distributed case, and we give optimal results of stability....

Steady plane flow of viscoelastic fluid past an obstacle

Antonín Novotný, Milan Pokorný (2002)

Applications of Mathematics

We consider the steady plane flow of certain classes of viscoelastic fluids in exterior domains with a non-zero velocity prescribed at infinity. We study existence as well as asymptotic behaviour of solutions near infinity and show that for sufficiently small data the solution decays near infinity as fast as the fundamental solution to the Oseen problem.

Stochastic homogenization of a class of monotone eigenvalue problems

Nils Svanstedt (2010)

Applications of Mathematics

Stochastic homogenization (with multiple fine scales) is studied for a class of nonlinear monotone eigenvalue problems. More specifically, we are interested in the asymptotic behaviour of a sequence of realizations of the form - div a T 1 x ε 1 ω 1 , T 2 x ε 2 ω 2 , ∇ u ε ω = λ ε ω 𝒞 ( u ε ω ) . It is shown, under certain structure assumptions on the random map a ( ω 1 , ω 2 , ξ ) , that the sequence { λ ε ω , k , u ε ω , k } of k th eigenpairs converges to the k th eigenpair { λ k , u k } of the homogenized eigenvalue problem - div ( b ( ∇ u ) ) = λ 𝒞 ¯ ( u ) . For the case of p -Laplacian type maps we characterize b explicitly.

Strong asymptotic stability for n-dimensional thermoelasticity systems

Mohammed Aassila (1998)

Colloquium Mathematicae

We use a new approach to prove the strong asymptotic stability for n-dimensional thermoelasticity systems. Unlike the earlier works, our method can be applied in the case of feedbacks with no growth assumption at the origin, and when LaSalle's invariance principle cannot be applied due to the lack of compactness.

Supersolutions and stabilization of the solutions of the equation∂u/∂t - div(|∇p|p-2 ∇u) = h(x,u), Part II.

Abderrahmane El Hachimi, François De Thélin (1991)

Publicacions Matemàtiques

In this paper we consider a nonlinear parabolic equation of the following type:(P)      ∂u/∂t - div(|∇p|p-2 ∇u) = h(x,u)with Dirichlet boundary conditions and initial data in the case when 1 < p < 2.We construct supersolutions of (P), and by use of them, we prove that for tn → +∞, the solution of (P) converges to some solution of the elliptic equation associated with (P).

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