Stabilization Of Wave Equation Research Articles

Stabilization for Wave Equation with Localized Kelvin–Voigt Damping on Cuboidal Domain: A Degenerate Case

Stabilization for Wave Equation with Localized Kelvin–Voigt Damping on Cuboidal Domain: A Degenerate Case

Stability of Wave Equation with Variable Coefficients by Boundary Fractional Dissipation Law

Stability of Wave Equation with Variable Coefficients by Boundary Fractional Dissipation Law

Regularity and stability of wave equations with variable coefficients and Wentzell type boundary conditions

We investigate regularity and stability of wave equations with variable coefficients and the frictional damping, where the damping effect is only on Wentzell boundary and there is no interior damping at all. Based on the spectral approach and the Reimannian geometry method, we present a new way of studying the challenging Wentzell-type problem concerning “the lack of interior damping” and the variable coefficients adiv(A(x)∇u). First, we establish a high-order regularity estimate for the solution of variable coefficient Wentzell problem and a spectral theorem for the operator derived from this problem. And then, after a series of specialized analysis and investigation on the special structure and basic properties of some auxiliary systems, the lower-order term and the resolvent of the operator associated to the variable coefficient Wentzell system by virtue of the Riemannian geometry method and other ideas, we prove eventually that the energies of the system decay with the rate 1/t. As can be seen, even for the constant coefficient case, the decay rate 1/t obtained in this paper is essentially better than the decay rate t−4/5 given in the previous work.

Stabilization of the wave equation through nonlinear Dirichlet actuation

In this paper, we consider the problem of nonlinear (in particular, saturated) stabilization of the high-dimensional wave equation with Dirichlet boundary conditions. The wave dynamics are subject to a dissipative nonlinear velocity feedback and generate a strongly continuous semigroup of contractions on the optimal energy space L2(Ω) × H−1(Ω). It is first proved that any solution to the closed-loop equations converges to zero in the aforementioned topology. Secondly, under the condition that the feedback nonlinearity has linear growth around zero, polynomial energy decay rates are established for solutions with smooth initial data. This constitutes new Dirichlet counterparts to well-known results pertaining to nonlinear stabilization in H1(Ω) × L2(Ω) of the wave equation with Neumann boundary conditions.

Stability of Wave Equations on Riemannian Manifolds with Locally Boundary Fractional Feedback Laws Under Geometric Conditions

Stability of Wave Equations on Riemannian Manifolds with Locally Boundary Fractional Feedback Laws Under Geometric Conditions

Exponential stability for wave equation with delay and dynamic boundary conditions

In this paper, we consider a multi-dimensional wave equation with delay and dynamic boundary conditions, related to the Kelvin–Voigt damping. By using the Faedo–Galerkin approximations together with some priori estimates, we prove the local existence of solution. Since the damping may stabilize the system while the delay may destabilize it, we discuss the interaction between the damping and the delay, and obtain that the system is uniformly stable when the effect of damping is greater than that of time delay. Exponential stability result of system is also established by constructing suitable Lyapunov functionals.

Static boundary feedback stabilization of an anti-stable wave equation with both collocated and non-collocated measurements

In this paper, we consider the stabilization of a wave equation with an unknown anti-stable injection on the left boundary and the control input on the right boundary, where there are both collocated and non-collocated measurements. A static output feedback control law is designed to stabilize the wave equation. The value ranges of the feedback gains are given, such that all eigenvalues of the closed-loop system are shown to be inside the left-half complex plane by applying the Nyquist criterion for distributed parameter systems. Then the exponential stability of the closed-loop system is established. Numerical simulations are presented to verify the effectiveness of the proposed feedback control law.

Stabilization of wave equations on the torus with rough dampings

For the damped wave equation on a compact manifold with {\em continuous} dampings, the geometric control condition is necessary and sufficient for {uniform} stabilisation. In this article, on the two dimensional torus, in the special case where $a(x) = \sum\_{j=1}^N a\_j 1\_{x\in R\_j}$ ($R\_j$ are polygons), we give a very simple necessary and sufficient geometric condition for uniform stabilisation. We also propose a natural generalization of the geometric control condition which makes sense for $L^\infty$ dampings. We show that this condition is always necessary for uniform stabilisation (for any compact (smooth) manifold and any $L^\infty$ damping), and we prove that it is sufficient in our particular case on $\mathbb{T}^2$ (and for our particular dampings).

Stability of wave equation with locally viscoelastic damping and nonlinear boundary source

In this paper we consider a nonlinear wave equation with nonlinear boundary damping and source terms, subject to a localized viscoelastic damping and localized frictional damping. We obtain the global existence of solution via potential well method. Furthermore, by introducing suitable Lyapunov functionals, using multiplier method and constructing convex differential inequality, we compare the effect of locally viscoelastic term and frictional damping and establish an explicit and general decay rate result. This stability result allows the localized viscoelastic damping and localized frictional damping acting simultaneously or complementarily in the domain, which gives the competing relationship of the viscoelastic and frictional damping.

Geometrically constrained stabilization of wave equations with Wentzell boundary conditions

Uniform stabilization of wave equation subject to second-order boundary conditions is considered in this article. Both dynamic (Wentzell) and static (with higher derivatives in space only) boundary conditions are discussed. In contrast to the classical wave equation where stabilization can be achieved by applying boundary velocity feedback, for a Wentzell-type problem boundary damping alone does not cause the energy to decay uniformly to zero. This is the case for both dynamic and static second-order conditions. In order to achieve uniform decay rates of the associated energy, it is necessary to dissipate part of the collar near the boundary. It will be shown how a combination of partially localized boundary feedback and partially localized collar feedback leads to uniform decay rates that are described by a nonlinear differential equation. This goal is attained by combining techniques used for stabilization of ‘unobserved’ Neumann conditions with differential geometry techniques effective for stabilization on compact manifolds. These lead to a construction of special non-radial multipliers which are geometry dependent and allow reconstruction of the high-order part of the potential energy from the damping that is supported only in a far-off region of the domain.

Stabilization of wave equation with variable coefficients by nonlinear boundary feedback

A wave equation with variable coefficients and nonlinear boundary feedback is studied. The results of energy decay of the solution are obtained by multiplier method and Riemann geometry method. Previous results obtained in the literatures are generalized in this paper.

Stabilité exponentielle des équations des ondes avec amortissement local de Kelvin–Voigt

We consider the stability of wave equations with local viscoelastic damping distributed around the boundary of domain. We show that the energy of the system goes uniformly and exponentially to zero for all initial data of finite energy. To cite this article: K. Liu, B. Rao, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

On Boundary Stability of Wave Equations with Variable Coefficients

In this paper, we consider the boundary stabilization of the wave equation with variable coefficients by Riemannian geometry method subject to a different geometric condition which is motivated by the geometric multiplier identities. Several (multiplier) identities (inequalities) which have been built for constant wave equation by Kormornik and Zuazua [2] are generalized to the variable coefficient case by some computational techniques in Riemannian geometry, so that the precise estimates on the exponential decay rate are derived from those inequalities. Also, the exponential decay for the solutions of semilinear wave equation with variable coefficients is obtained under natural growth and sign assumptions on the nonlinearity. Our method is rather general and can be adapted to other evolution systems with variable coefficients (e.g. elasticity plates) as well.

Stability of wave equations with dissipative boundary conditions in a bounded domain

Stability of wave equations with dissipative boundary conditions in a bounded domain

Stabilization of wave equations coupled in parallel by viscous damping

In this article, we consider stabilization of two wave equations coupled in parallel by viscous damping. Uniform exponential stabilization can be achieved when damping is presented in two equations. In general only strong stabilization can be achieved when damping is present in one equation, while uniform exponential stabilization is still possible for some special cases.

Stabilization of Wave Equation by Periodical Moving Actuators

The actuator B(t) is a moving object in D and we consider the wave equation in the non cylindrical domain U = tx(D\\B(t)). The boundary S(t) of B(t) is decomposed in three parts : S(t) = S_(t) U S_(t) U S0(t) with respect to the sign of the normal speed on S(t). Homogeneous Neumann and Dirichlet conditions are imposed respectively on S_(t.) and S_(t.) U S0(t). By penalisation and Galerkin technique we prove the existence and the uniqueness of solutions. We discuss only the energy dissipation associated to that, passive actuator.