Rayleigh Beam Equation Research Articles

Lack of exponential stability for Rayleigh Beam equation of only one dynamical boundary control with delay

Lack of exponential stability for Rayleigh Beam equation of only one dynamical boundary control with delay

Exponential stability of Rayleigh beam equation on a star‐shaped network with indefinite damping

In this paper, we study the stability of the Rayleigh beam equation on a star‐shaped network with indefinite sign damping terms. We prove that the system is well posed via the semigroup theory. Then, by a detailed spectral analysis of the system operator, we provide a necessary condition for the distribution of the spectrum, the completeness and the Riesz basis property of the eigenfunctions of the operator as well as the asymptotic stability of the system. Moreover, we discuss how the stability of the system depends on the sign of damping terms. More precisely, we obtain a condition on the negativity degree of the damping coefficients in order that the system always remains exponentially stable.

Numerical analysis and simulation for Rayleigh beam equation with dynamical boundary controls

In this paper, the Rayleigh beam system with two dynamical boundary controls is treated. Theoretically, the well-posedness of the weak solution is obtained. Later, we discretize the system by using the Implicit Euler scheme in time and the P^3 Hermite finite element in space. In addition, we show the decay of the discrete energy and we establish some a priori error estimates. Finally, some numerical simulations are presented.

Optimal energy decay rate of Rayleigh beam equation with only one dynamic boundary control

In \cite{WehbeRayleigh:06}, Wehbe considered the Rayleigh beam equation with two dynamical boundary controls and established the optimal polynomial energy decay rate of type $\dfrac{1}{t}$. The proof exploits in an explicit way the presence of two boundary controls, hence the case of the Rayleigh beam damped by only one dynamical boundary control remained open. In this paper, we fill this gap by considering a clamped Rayleigh beam equation subject to only on dynamical boundary feedback. First, we consider the Rayleigh beam equation subject to only one dynamical boundary control moment. We give the asymptotic expansion of eigenvalues and eigenfunctions of the undamped underlying system and we establish a polynomial energy decay rate of type $\frac{1}{t}$ for smooth initial data via an observability inequality of the corresponding undamped problem combined with the boundedness property of the transfer function of the associated undamped problem. Moreover, using the real part of the asymptotic expansion of eigenvalues of the damped system, we prove that the obtained energy decay rate is optimal. Next, we consider the Rayleigh beam equation subject to only one dynamical boundary control force. We start by giving the asymptotic expansion of the eigenvalues and the eigenfunctions of the damped and undamped systems using an explicit approximation of the characteristic equation determining these eigenvalues. We next show that the system of eigenvectors of the damped problem form a Riesz basis. Finally, we establish the optimal energy decay rate of polynomial type $\frac{1}{\sqrt{t}}$.

Optimal energy decay rate of Rayleigh beam equation with only one boundary control force

We consider a clamped Rayleigh beam equation subject to only one boundary control force. Using an explicit approximation, we first give the asymptotic expansion of eigenvalues and eigenfunctions of the undamped underlying system. We next establish a polynomial energy decay rate for smooth initial data via an observability inequality of the corresponding undamped problem combined with a boundedness property of the transfer function of the associated undamped problem. Finally, by a frequency domain approach, using the real part of the asymptotic expansion of eigenvalues of the infinitesimal generator of the associated semigroup, we prove that the obtained energy decay rate is optimal.

Hydroelastic analysis of VLFS based on a consistent coupled-mode system and FEM

Three models for the interaction of water waves with large floating elastic structures are analysed. The first model, based on the Euler–Bernoulli beam theory, has already been extensively studied. The second is based on the Rayleigh beam equation. The third approach utilises the Timoshenko approximation and is thus capable of incorporating shear deformation and rotary inertia effects. A novelty of the proposed hydroelastic systems is the consistent local mode expansion of the underlying hydrodynamic field interacting with the floating structure, which leads to coupled-mode systems with respect to the modal amplitudes of the wave potential and the surface elevation. This representation is rapidly convergent to the solution of the full hydroelastic problem. The dispersion relations of these models are derived and analysed, supporting at a next stage the efficient development of finite element method solvers of the coupled system.

Exact controllability of a Rayleigh beam with a single boundary control

We prove exact boundary controllability for the Rayleigh beam equation \({\varphi_{tt} -\alpha\varphi_{ttxx} + A\varphi_{xxxx} = 0, 0 < x < l, t > 0}\) with a single boundary control active at one end of the beam. We consider all combinations of clamped and hinged boundary conditions with the control applied to either the moment \({\varphi_{xx}(l, t)}\) or the rotation angle \({\varphi_{x}(l, t)}\) at an end of the beam. In each case, exact controllability is obtained on the space of optimal regularity for L 2(0, T) controls for \({T > 2l\sqrt{\frac{\alpha}{A}}}\). In certain cases, e.g., the clamped case, the optimal regularity space involves a quotient in the velocity component. In other cases, where the regularity for the observed problem is below the energy level, a quotient space may arise in solutions of the observed problem.

Exponential Stabilization of a Rayleigh Beam Using Collocated Control

We consider a hinged elastic beam described by the Rayleigh beam equation on the interval [0,pi]. We assume the presence of two sensors: one measures the angular velocity of the beam at a point xi epsiv [0,pi] and the other measures the bending (curvature) of the beam at the same point. (If xi = 0 or xi = pi, then the second output is not needed.) The corresponding operator semigroup is unitary on a suitable Hilbert state space. These two measurements are advantageous because they make the open-loop system exactly observable, regardless of the point xi. We design the actuators and the feedback law in order to exponentially stabilize this system. Using the theory of collocated static output feedback developed in our recent paper , we design the actuators such that they are collocated, meaning that B = C*, where B is the control operator and C is the observation operator. It turns out that if xi epsiv [0,pi], then the actuators cause a discontinuity of the bending exactly at (this is the price, in this example, of having collocated actuators and sensors). This obliges us to use an extension of to define the output signal in terms of the left and right limit of the bending at xi. We prove that, for all static output feedback gains in a suitable finite range, the closed-loop system is well posed and exponentially stable. This follows from the general theory in our paper, whose main points are recalled here.

Optimal energy decay rate for Rayleigh beam equation with dynamical boundary controls

We consider a Rayleigh beam equation with two dynamical boundary controls. First, by a multiplier method, we show that the smooth solution has a polynomial energy decay rate. Next, using a spectrum method, we justify that the polynomial energy decay rate is optimal.

Decay Rates for a Beam with Pointwise Force and Moment Feedback

We consider the Rayleigh beam equation and the Euler–Bernoulli beam equation with pointwise feedback shear force and bending moment at the position ξ in a bounded domain (0,π) with certain boundary conditions. The energy decay rate in both cases is investigated. In the case of the Rayleigh beam, we show that the decay rate is exponential if and only if ξ/π is a rational number with coprime factorization ξ/π=p/q, where q is odd. Moreover, for any other location of the actuator we give explicit polynomial decay estimates valid for regular initial data. In the case of the Euler–Bernoulli beam, even for a nonhomogeneous material, exponential decay of the energy is proved, independently of the position of the actuator.

Basis Property of a Rayleigh Beam with Boundary Stabilization

A Rayleigh beam equation with boundary stabilization control is considered. Using an abstract result on the Riesz basis generation of discrete operators in Hilbert spaces, we show that the closed-loop system is a Riesz spectral system; that is, there is a sequence of generalized eigenfunctions of the system, which forms a Riesz basis in the state Hilbert space. The spectrum-determined growth condition, distribution of eigenvalues, as well as stability of the system are developed. This paper generalizes the results in Ref. 1.

A Classical Approach to Uniform Null Controllability for Elastic Beams

The Rayleigh beam equation is the formal limit of the Timoshenko beam equation as the shear modulus $K\rightarrow +\infty$. Following a method in W. Littman, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 5 (1978), pp. 567--580 and W. Littman, Proc. 24th IEEE CDC, Fort Lauderdale, FL, 1985, controllability is possible. That is, the evolution systems associated with the Rayleigh and Timoshenko equations can be driven to rest by applying appropriate controls at both ends of the beam. In this work we show that the process is uniform; more precisely, controllability of the Rayleigh system can be achieved by letting $% K\rightarrow +\infty $ in the solution of the Timoshenko controllability problem.

Le taux optimal de décroissance de l'énergie dans l'équation de poutre de Rayleigh

We consider the Rayleigh beam equation subject to a viscous damping. Using a constructive approximation, we first give the asymptotic form of eigenvalues and eigenvectors of the underlying system. We next identify the optimal energy decay rate as the supremum of the real part of the spectrum of the infinitesimal generator of the associated semigroup.

A compact perturbation method for the boundary stabilization of the Rayleigh beam equation

We prove that the Rayleigh beam equation can be uniformly exponentially stabilized by only one control moment. We also prove the strong asymptotic stabilization and the lack of uniform exponential stabilization in the case of only one control force.

A Compact Perturbation Method for the Boundary Stabilization of the Rayleigh Beam Equation

A Compact Perturbation Method for the Boundary Stabilization of the Rayleigh Beam Equation

Causal Theories of Evolution and Wave Propagation in Mathematical Physics

There are still many phenomena, especially in continuum physics, that are described by means of parabolic partial differential equations whose solution are not compatible with the causality principle. Compatibility with this principle is required also by the theory of relativity. A general form of hyperbolic operators for the most frequently occurring linear governing equations in mathematical physics is written down. It is then easy to convert any given parabolic equation to the hyperbolic form without necessarily entering into the cause of the inadequacy of the governing equation. The method is verified on the well-known example of Timoshenko’s correction of the Bernoulli–Euler–Rayleigh beam equation for flexural motion. The “Love–Rayleigh” fourth-order differential equations for the longitudinal and torsional wave propagation in the rod is generalized with this method. The hyperbolic version (not to mention others) of the linear Korteweg–de Vries equation and of the “telegraph” equation governing electromagnetic wave propagation through relaxing material are given. Lagrangians of all the equations studied are listed. For all the reasons given we believe the hyperbolic governing equations to be physically and mathematically more realistic and adequate.