Localized Damping Term Research Articles

Local indirect stabilization of same coupled evolution systems through resolvent estimates

<p style='text-indent:20px;'>In this paper, we consider same systems of two coupled equations (wave-wave, Schrödinger-Schrödinger) in a bounded domain. Only one of the two equations is directly damped by a localized damping term (indirect stabilization). Under geometric control conditions on both coupling and damping regions (internal or boundary), we establish the energy decay rate by means of a suitable resolvent estimate. The numerical contribution is interpreted to confirm the theoretical result of a wave-wave system.</p>

STABILIZATION OF TWO STRONGLY COUPLED HYPERBOLIC EQUATIONS IN EXTERIOR DOMAINS

In this paper we study the behavior of the total energy and the $L^2$-norm of solutions of two coupled hyperbolic equations by velocities in exterior domains. Only one of the two equations is directly damped by a localized damping term. We show that, when the damping set contains the coupling one and the coupling term is effective at infinity and on captive region, then the total energy decays uniformly and the $L^2$-norm of smooth solutions is bounded. In the case of two Klein-Gordon equations with equal speeds we deduce an exponential decay of the energy.

Well-posedness and asymptotic behavior of a generalized higher order nonlinear Schrödinger equation with localized dissipation

In this work, we study at the L2 – level global well-posedness as well as long-time stability of an initial-boundary value problem, posed on a bounded interval, for a generalized higher order nonlinear Schrödinger equation, modeling the propagation of pulses in optical fiber, with a localized damping term. In addition, we implement a precise and efficient code to study the energy decay of the higher order nonlinear Schrödinger equation and we prove its convergence and exponential stability of the discrete energy.

Global existence and decay of solutions of a singular nonlocal viscoelastic system with a nonlinear source term, nonlocal boundary condition, and localized damping term

The paper deals with the existence of a global solution of a singular one‐dimensional viscoelastic system with a nonlinear source term, nonlocal boundary condition, and localized frictional damping a(x)ut using the potential well theory. Furthermore, the general decay result is proved. We construct a suitable Lyapunov functional and make use of the perturbed energy method.

Stabilization of a Boussinesq system with localized damping

A family of Boussinesq systems was proposed by J. L. Bona, M. Chen, and J.-C. Saut to describe the two-way propagation of small amplitude gravity waves on the surface of water in a canal. Our work considers a class of these Boussinesq systems which couples two Benjamin-Bona-Mahony type equations posed on a bounded interval. We study the stabilization of the resulting system when a localized damping term acts in one equation only. By means of spectral analysis and eigenvectors expansion of solutions, we prove that the energy associated to the model converges to zero as time tends to infinity. Also, we address the problem of unique continuation property for the corresponding conservative system.

Control of Bilateral Teleoperation System Consisting of Heterogeneous Manipulators with Communication Delay

This paper proposes a new control method for bilateral teleoperation systems with communication delay. Under the proposed control law, the passivity from external forces to velocities and the position synchronization in the working spaces are attained. The master and slave arms can have different mechanical structures and have redundant degree-of-freedom. The proposed control can work even on singular postures, because it uses no inverses of Jacobian matrices. The bad effect of the transmission delay on stability is canceled by local damping terms, since there exists a dissipativity for a supply rate on the actions of the position-signal’s delay and the local damping term. We show the property of the storage functional for the supply rate, which makes it possible to show the synchronization via Barbalat’s lemma. When both the arms are stopped, the force from the environment in the working space is properly transmitted to the other arm as the force on the operation space balancing with the human’s manipulation force.

On the well-posedness and asymptotic behaviour of the generalized Korteweg–de Vries–Burgers equation

In this paper we are concerned with the well-posedness and the exponential stabilization of the generalized Korteweg–de Vries–Burgers equation, posed on the whole real line, under the effect of a damping term. Both problems are investigated when the exponent p in the nonlinear term ranges over the interval [1, 5). We first prove the global well-posedness in Hs(ℝ) for 0 ≤ s ≤ 3 and 1 ≤ p < 2, and in H3(ℝ) when p ≥ 2. For 2 ≤ p < 5, we prove the existence of global solutions in the L2-setting. Then, by using multiplier techniques and interpolation theory, the exponential stabilization is obtained with an indefinite damping term and 1 ≤ p < 2. Under the effect of a localized damping term the result is obtained when 2 ≤ p < 5. Combining multiplier techniques and compactness arguments, we show that the problem of exponential decay is reduced to proving the unique continuation property of weak solutions. Here, the unique continuation is obtained via the usual Carleman estimate.

Analysis of localized damping effects in channel flows with arbitrary rough boundary

We justify a modified Navier–Stokes system that includes the damping effect for a channel flow with an arbitrary irregular boundary. The effects of rough walls are modeled by a localized damping term in the momentum equation. We prove the existence and uniqueness of a solution to the modified Navier–Stokes system in a smooth extended domain using the fixed-point theorem and Saint-Venant technique. The proposed damping term yields an quadratic approximation of real flow for a sufficiently small flux, under the assumption of the almost periodicity of the roughness profile . We further obtain quadratic approximation of real flows by the damping model with a quasi-periodic function and the diophantine condition. Consequently, we confirm that the localized damping effect can provide an effective model to predict channel flows with an arbitrary irregular surface.

Stabilization of two coupled wave equations on a compact manifold with boundary

In this paper we study the indirect stabilization of coupled wave equations by an order one term on a compact manifold with boundary. Only one of the two equations is directly damped by a localized damping term. We prove that the energy of smooth solutions of the system decays polynomially under geometric conditions on both the coupling and the damping regions.

Exponential decay for a locally damped fifth-order equation posed on the line

We prove the exponential decay of the energy related to a locally damped fifth-order equation posed on the whole real line with the initial datum from a bounded set of L2. A local smoothing effect in H2 is established, which is essential to obtain the necessary a priory estimates. Moreover, it is shown that arguments used in the article can be applied to prove the exponential decay rate of solutions for the Korteweg–de Vries equation with a similar localized damping term provided the initial data are uniformly bounded in L2. This last fact improves some previous results.

Uniform stabilization of a nonlinear dispersive system

The purpose of this work is to study the internal stabilization of a coupled system of two generalized Korteweg-de Vries equations under the effect of a localized damping term. To obtain the decay we use multiplier techniques combined with compactness arguments and reduce the problem to prove a unique continuation property for weak solutions. A locally exponential decay result is derived.

Well-posedness and stabilization of a model system for long waves posed on a quarter plane

In this paper we are concerned with an initial--boundary-value problem for a coupled system of two KdV equations, posed on the positive half line, under the effect of a localized damping term. The model arises when modeling the propagation of long waves generated by a wave maker in a channel. It is shown that the solutions of the system are exponentially stable and globally well-posed in the weighted space $L^2(e^{2bx}\,dx)$ for $b>0$. The stabilization problem is studied using a Lyapunov approach, while the well-posedness result is obtained combining fixed-point arguments and energy-type estimates.

On the Well-posedness and Asymptotic Behavior of a Nonlinear Dispersive System in Weighted Spaces

This paper is concerned with a model for propagation of long waves in a channel generated by a wave maker mounted at one end. The mathematical structure consists in a coupled system of two nonlinear Korteweg-de Vries equations posed on the positive half line. Under the effect of a localized damping term it is shown that the solutions of the system are exponentially stable and globally well-posed in the weighted space L2((x+1)mdx) for m≥1. The stabilization problem is studied constructing a Lyapunov function by induction on m and the well-posedness is obtained by passing to the limit in a sequence of solutions in L2(e2bxdx) for b>0.

Stability properties of a higher order scheme for a GKdV-4 equation modelling surface water waves

This work is devoted to the study of a higher order numerical scheme for the critical generalized Korteweg-de Vries equation (GKdV with p=4) in a bounded domain. The KdV equation and some of its generalizations as the GKdV type equations appear in Physics, for example in the study of waves on shallow water. Based on the analysis of stability of the first order scheme introduced by Pazoto et al. (Numer. Math. 116:317–356, 2010), we add a vanishing numerical viscosity term to a semi-discrete scheme so as to preserve similar properties of stability, and thus able to prove the convergence in L4-strong. The semi-discretization of the spatial structure via central finite difference method yields a stiff system of ODE. Hence, for the temporal discretization, we resort to the two-stage implicit Runge-Kutta scheme of the Gauss-Legendre type. The resulting system is unconditionally stable and possesses favorable nonlinear properties. On the other hand, despite the formation of blow up for the critical case of GKdV, it is known that a localized damping term added to the GKdV-4 equation leads to the exponential decay of the energy for small enough initial conditions, which is interesting from the standpoint of the Control Theory. Then, combining the result of convergence in L4-strong with discrete multipliers and a contradiction argument, we show that the presence of the vanishing numerical viscosity term allows the uniform (with respect to the mesh size) exponential decay of the total energy associated to the semi-discrete scheme of higher-order in space with the localized damping term. Numerical experiments are provided to illustrate the performance of the method and to confirm the theoretical results.

Energy decay for the modified Kawahara equation posed in a bounded domain

Studied here is the eventual dissipation of solutions to initial–boundary value problems for the modified Kawahara equation with and without a localized damping term included. It is shown that solutions of undamped problems posed on a bounded interval may not decay if the length of the interval is critical. In contrast, the energy associated to the locally damped problems is shown to be exponentially decreased independently of the interval length.

Global stabilization of a coupled system of two generalized Korteweg-de Vries type equations posed on a finite domain

The purpose of this work is to study the internal stabilizationof a coupled system of two generalized Korteweg-de Vries equations under theeffect of a localized damping term. The exponential stability, as well as,the global existence of weak solutions are investigated when the exponent inthe nonlinear term ranges over the interval $[1, 4)$. To obtain the decay we use multiplier techniquescombined with compactness arguments and reduce the problem to prove a unique continuation propertyfor weak solutions. Here, the unique continuation is obtained via the usual Carleman estimate.

Uniform stabilization of numerical schemes for the critical generalized Korteweg-de Vries equation with damping

This work is devoted to the analysis of a fully-implicit numerical scheme for the critical generalized Korteweg–de Vries equation (GKdV with p = 4) in a bounded domain with a localized damping term. The damping is supported in a subset of the domain, so that the solutions of the continuous model issuing from small data are globally defined and exponentially decreasing in the energy space. Based in this asymptotic behavior of the solution, we introduce a finite difference scheme, which despite being one of the first order, has the good property to converge in L 4-strong. Combining this strong convergence with discrete multipliers and a contradiction argument, we show that the smallness of the initial condition leads to the uniform (with respect to the mesh size) exponential decay of the energy associated to the scheme. Numerical experiments are provided to illustrate the performance of the method and to confirm the theoretical results.

Indirect linear locally distributed damping of coupled systems

The aim of this paper is to prove indirect internal stabilization results for different coupled systems with linear locally distributed damping (coupled wave equations, wave equations with different speeds of propagation). In our case, a linear local damping term appears only in the first equation whereas no damping term is applied to the second one (this is indirect stabilization, see [11]). Using the piecewise multiplier method we prove that the full system is stabilized and that the total energy of the solution of this system decays polynomially.

Asymptotic behavior of the Korteweg–de Vries equation posed in a quarter plane

The purpose of this work is to study the exponential stabilization of the Korteweg–de Vries equation in the right half-line under the effect of a localized damping term. We follow the methods in [G.P. Menzala, C.F. Vasconcellos, E. Zuazua, Stabilization of the Korteweg–de Vries equation with localized damping, Quart. Appl. Math. LX (1) (2002) 111–129] which combine multiplier techniques and compactness arguments and reduce the problem to prove the unique continuation property of weak solutions. Here, the unique continuation is obtained in two steps: we first prove that solutions vanishing on the support of the damping function are necessarily smooth and then we apply the unique continuation results proved in [J.C. Saut, B. Scheurer, Unique continuation for some evolution equations, J. Differential Equations 66 (1987) 118–139]. In particular, we show that the exponential rate of decay is uniform in bounded sets of initial data.

Stabilization of the Kirchhoff type wave equation with locally distributed damping

We derive energy decay estimates of the Kirchhoff type wave equation with a localized damping term in a bounded domain. The damping coefficient function may act alive only on a neighborhood of some part of the boundary.