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Controllability of a Nonhomogeneous String and Ring under Time Dependent Tension

S. A. Avdonin, B. P. Belinskiy, L. Pandolfi (2010)

Mathematical Modelling of Natural Phenomena

We study controllability for a nonhomogeneous string and ring under an axial stretching tension that varies with time. We consider the boundary control for a string and distributed control for a ring. For a string, we are looking for a control f(t) ∈ L2(0, T) that drives the state solution to rest. We show that for a ring, two forces are required to achieve controllability. The controllability problem is reduced to a moment problem...

Convergence of a two-grid algorithm for the control of the wave equation

Liviu Ignat, Enrique Zuazua (2009)

Journal of the European Mathematical Society

We analyze the problem of boundary observability of the finite-difference space semidiscretizations of the 2-d wave equation in the square. We prove the uniform (with respect to the meshsize) boundary observability for the solutions obtained by the two-grid preconditioner introduced by Glowinski [9]. Our method uses previously known uniform observability inequalities for low frequency solutions and a dyadic spectral time decomposition. As a consequence we prove the convergence of the two-grid algorithm...

Counterexamples to the Strichartz inequalities for the wave equation in general domains with boundary

Oana Ivanovici (2012)

Journal of the European Mathematical Society

In this paper we consider a smooth and bounded domain Ω ⊂ ℝ d of dimension d ≥ 2 with boundary and we construct sequences of solutions to the wave equation with Dirichlet boundary condition which contradict the Strichartz estimates of the free space, providing losses of derivatives at least for a subset of the usual range of indices. This is due to microlocal phenomena such as caustics generated in arbitrarily small time near the boundary. Moreover, the result holds for microlocally strictly convex domains...

Decay estimates of solutions of a nonlinearly damped semilinear wave equation

Aissa Guesmia, Salim A. Messaoudi (2005)

Annales Polonici Mathematici

We consider an initial boundary value problem for the equation u t t - Δ u - ∇ ϕ · ∇ u + f ( u ) + g ( u t ) = 0 . We first prove local and global existence results under suitable conditions on f and g. Then we show that weak solutions decay either algebraically or exponentially depending on the rate of growth of g. This result improves and includes earlier decay results established by the authors.

Decay of solutions of some degenerate hyperbolic equations of Kirchhoff type

Barbara Szomolay (2003)

Commentationes Mathematicae Universitatis Carolinae

In this paper we study the asymptotic behavior of solutions to the damped, nonlinear vibration equation with self-interaction u ¨ = - γ u ˙ + m ( ∥ ∇ u ∥ 2 ) Δ u - δ | u | α u + f , which is known as degenerate if m ( · ) ≥ 0 , and non-degenerate if m ( · ) ≥ m 0 > 0 . We would like to point out that, to the author’s knowledge, exponential decay for this type of equations has been studied just for the special cases of α . Our aim is to extend the validity of previous results in [5] to α ≥ 0 both to the degenerate and non-degenerate cases of m . We extend our results to equations with...

Dispersive and Strichartz estimates for the wave equation in domains with boundary

Oana Ivanovici (2010)

Journées Équations aux dérivées partielles

In this note we consider a strictly convex domain Ω ⊂ ℝ d of dimension d ≥ 2 with smooth boundary ∂ Ω ≠ ∅ and we describe the dispersive and Strichartz estimates for the wave equation with the Dirichlet boundary condition. We obtain counterexamples to the optimal Strichartz estimates of the flat case; we also discuss the some results concerning the dispersive estimates.

Currently displaying 61 – 80 of 333