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A stability result for a degenerate beam equation
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Abstract
We consider a degenerate beam equation in presence of a leading operator which is not in divergence form. We impose clamped conditions where the degeneracy occurs and dissipative conditions at the other endpoint. We provide some conditions for the uniform exponential decay of solutions for the associated Cauchy problem.
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In this article we consider the fourth-order operators and in divergence and non divergence form, where degenerates in an interior point of the interval. Using the semigroup technique, under suitable assumptions on a, we study the generation property of these operators associated to generalized Wentzell boundary conditions. We prove the well posedness of the corresponding parabolic problems.
The paper deals with the controllability of a degenerate beam equation. In particular, we assume that the left end of the beam is fixed, while a suitable control f f acts on the right end of it. As a first step, we prove the existence of a solution for the homogeneous problem, then we prove some estimates on its energy. Thanks to them, we prove an observability inequality, and using the notion of solution by transposition, we prove that the initial problem is null controllable.
This paper contains results about the existence, uniqueness and stability of solutions for the damped nonlinear extensible beam equation
\begin{document} u_{tt}+\Delta ^2u-M(\|\nabla u(t)\|^2)\Delta u+\|\Delta u(t)\|^{2\alpha}\,|u_t|^{\gamma}u_t = 0\ \mbox{ in } \ \Omega \times \mathbb{R}^+, \end{document}
where \begin{document}\end{document}, \begin{document}\end{document}, \begin{document}\end{document} is a bounded domain with smooth boundary \begin{document}\end{document}, and \begin{document} M \end{document} is a nonlocal function that represents beam's extensibility term. The novelty of the work is to consider the damping as a product of a degenerate and nonlocal term with a nonlinear function. This work complements the recent article by Cavalcanti et al. [8] who treated this model with degenerate nonlocal weak (and strong) damping. The main result of the work is to show that for regular initial data the energy associated with the problem proposed goes to zero when \begin{document} t \end{document} goes to infinity.
The Euler-Bernoulli (E-B) beam is the most commonly utilized model in the study of vibrating beams. The exact frequency equations for this problem, subject to energy-conserving boundary conditions, are well-known; however, the corresponding dissipative problem has been solved only approximately, via asymptotic methods. These methods, of course, are not accurate when looking at the low end of the spectrum. Here, we solve for the exact frequency equations for the E-B beam subject to boundary damping. Numerous numerical examples are provided, showing plots of both the complex wave numbers and the exponential damping rates for the first five frequencies in each case. Some of these results are surprising.
While the Euler-Bernoulli beam is the most commonly utilized model in studying vibrating beams, one often requires a model that captures the additional effects of rotary inertia or deformation due to shear. The Rayleigh beam improves upon the Euler-Bernoulli by including the former effect, while the shear beam is an improvement that includes the latter. While all of these problems have been well studied when subject to energy-conserving boundary conditions, none have been solved for the case of boundary damping. We compute the exact frequency equations for the Rayleigh and shear beams, subject to boundary damping and, in the process, we find interesting connections between the two models, despite their being very different.
We study a wave equation in one space dimension with a general diffusion
coefficient which degenerates on part of the boundary. Degeneracy is measured
by a real parameter . We establish observability inequalities for
weakly (when ) as well as strongly (when )
degenerate equations. We also prove a negative result when the diffusion
coefficient degenerates too violently (i.e. when ) and the blow-up of
the observability time when converges to 2 from below. Thus, using
the HUM method we deduce the exact controllability of the corresponding
degenerate control problem when . We conclude the paper by
studying the boundary stabilization of the degenerate linearly damped wave
equation and show that a suitable boundary feedback stabilizes the system
exponentially. We extend this stability analysis to the degenerate nonlinearly
boundary damped wave equation, for an arbitrarily growing nonlinear feedback
close to the origin. This analysis proves that the degeneracy does not affect
the optimal energy decay rates at large time. We apply the optimal-weight
convexity method of \cite{alaamo2005, alajde2010} together with the results of
the previous section, to perform this stability analysis.
We consider a finite difference semi-discrete scheme for the approximation of the boundary controls of
a 1-D equation modelling the transversal vibrations of a hinged beam. It is known that, due to the high
frequency numerical spurious oscillations, the uniform (with respect to the mesh-size) controllability property
of the semi-discrete model fails in the natural setting. Consequently, the convergence of the approximate
controls corresponding to initial data in the finite energy space cannot be guaranteed. We prove that,
by adding a vanishing numerical viscosity, the uniform controllability property and the convergence of the
scheme is ensured.
In this paper, an initial-boundary value problem for a linear-homogeneous axially moving tensioned beam equation is considered. One end of the beam is assumed to be simply-supported and to the other end of the beam a spring and a dashpot are attached, where the damping generated by the dashpot is assumed to be small. In this paper only boundary damping is considered. The problem can be used as a simple model to describe the vertical vibrations of a conveyor belt, for which the velocity is assumed to be constant and relatively small compared to the wave speed. A multiple time-scales perturbation method is used to construct formal asymptotic approximations of the solutions, and it is shown how different oscillation modes are damped.
Many flexible structures consist of a large number of components coupled end to end in the form of a chain. In this paper, we consider the simplest type of such structures which is formed by N serially connected Euler-Bernoulli beams, with N actuators and sensors co-located at nodal points. When these N beams are strongly connected at all intermediate nodes and their material coefficients satisfy certain properties, uniform exponential stabilization can be achieved by stabilizing at one end point of the composite beam. We use finite elements to discretize the partial differential equation and compute the spectra of these boundary damped operators. Numerical results are also illustrated.
We give null controllability results for some degenerate parabolic equations in non divergence form on a bounded interval.
In particular, the coefficient of the second order term degenerates at the extreme points of the domain. For this reason,
we obtain an observability inequality for the adjoint problem. Then we prove Carleman estimates for such a problem. Finally,
in a standard way, we deduce null controllability also for semilinear equations.
We consider a degenerate wave equation in one dimension, with drift and in presence of a leading operator which is not in divergence form. We impose a homogeneous Dirichlet boundary condition where the degeneracy occurs and a boundary damping at the other endpoint. We provide some conditions for the uniform exponential decay of solutions for the associated Cauchy problem.
This paper investigates the well-posedness and stability of the beam model with degenerate nonlocal damping: where is a bounded domain with smooth boundary, and . The main purpose in the present paper is to show that the transition from the case q=0 to the case produces an explicit influence on the stability of energy solutions. More precisely, when q=0, we conclude that the energy goes to zero as t goes to infinity without an explicit decay rate; while when , we present a polynomial decay rate of type that depends only on the exponent of the velocity term, not on and . Furthermore, we prove that the energy cannot be exponentially stable and derive more accurate decay rates of the energy.
In this paper, motivated by recent papers on the stabilization of evolution problems with nonlocal degenerate damping terms, we address an extensible beam model with degenerate nonlocal damping of Balakrishnan-Taylor type. We discuss initially on the well-posedness with respect to weak and regular solutions. Then we show for the first time how hard is to guarantee the stability of the energy solution (related to regular solutions) in the scenarios of constant and non-constant coefficient of extensibility. The degeneracy (in time) of the single nonlocal damping coefficient and the methodology employed in the stability approach are the main novelty for this kind of beam models with degenerate damping.
Many problems in structural dynamics involve stabilizing the elastic
energy of partial differential equations (PDE) such as the
Euler-Bernoulli beam equation by boundary conditions. Exponential
stability is a very desirable property in such elastic systems. The
energy multiplier method has been successfully applied by several people
to establish exponential stability for various PDEs and boundary
conditions. However, it has also been found that for certain boundary
conditions the energy multiplier method is not effective in proving the
exponential stability property. A recent theorem of F. L. Huang
introduces a frequency domain method to study such exponential decay
problems. In this paper, we derive estimates of the resolvent operator
on the imaginary axis and apply Huang's theorem to establish an
exponential decay result for an Euler-Bernoulli beam with rate control
of the bending moment only. We also derive asymptotic limits of
eigenfrequencies, which was also done earlier by P. Rideau. Finally, we
indicate the realizability of these boundary feedback stabilization
schemes by illustrating some mechanical designs of passive damping
devices.
In the modern vibration control of flexible space structures and flexible robots, various boundary feedback schemes have been employed to cause energy dissipation and damping, thereby achieving stabilization. The mathematical analysis of the eigenspectrum of vibration is usually carried out by classical separation of variables and by solving the transcendental equations. This involves rather lengthy and tedious work due to the complexity and the numerous boundary conditions. A different approach, developed by J. B. Keller and S. I. Rubinow, uses ideas from wave propagation to obtain asymptotic estimates of eigenvalues for multidimensional scattering problems. This approach is powerful and yields accurate eigenvalue estimates even at a relatively low frequency range [Ann. Physics 9, 24-75 (1960)]. In this paper, we take advantage of this wave approach to study one-dimensional vibration problems with boundary damping. We decompose vibration waves into incident, reflected (including transmitted) and evanescent waves. Based on their amplitude and reflection coefficients we are able to derive eigenfrequency estimates and compute numerical values. This wave propagation method greatly simplifies the asymptotic estimation procedures and reproduces earlier results [the first author, M. P. Coleman and H. H. West, SIAM J. Appl. Math. 47, 751-780 (1987; Zbl 0641.93047); the first author, S. G. Krantz, D. L. Russel, C. E. Wayne and H. H. West, M. P. Coleman, SIAM J. Appl., Math. 49, 1665-1693 (1989; Zbl 0685.73046)]. It also yields new insights into various structural control problems such as feedback with tipped mass, nonrobustness of feedback delays [R. Datko, M. P. Polis and J. Lagnese, SIAM J. Control Optimization 24, 152-156 (1986; Zbl 0592.93047); R. Datko, ibid. 26, No.3, 697-713 (1988; Zbl 0643.93050)], noncollocated sensors and actuators, and special medium-low frequency structural damping patterns.
Exact controllability is studied for distributed systems, of hyperbolic type or for Petrowsky systems (like plate equations). The control is a boundary control or a local distributed control. Exact controllability consists in trying to drive the system to rest in a given finite time. The solution of the problems depends on the function spaces where the initial data are taken, and also depends on the function space where the control can be chosen. A systematic method (named HUM, for Hilbert Uniqueness Method) is introduced. It is based on uniqueness results (classical or new) and on Hilbert spaces constructed (in infinitely many ways) by using Uniqueness. A number of applications are indicated. Nonlinear Riccati type PDEs are obtained. Finally, we consider how all this behaves for perturbed systems.
This paper considers the Euler-Bernoulli problem with boundary controls g1, g2 in the Dirichlet and Neumann boundary conditions, respectively. Several exact controllability results are shown, including the following. The problem is exactly controllable in an arbitrarily short time T>0 in the space (of maximal regularity) H-1(Ω) × V′, V as specified, (i) with boundary controls g1 member of L2(Σ), g2≡0 under some geometrical conditions on Ω; (ii) with boundary controls g1 member of L2(Σ) and g2 member of L2(0, T; H-1(Γ)) without geometrical conditions on Ω. A direct approach is given, based on an operator model for the problem and on multiplier techniques.
A vibrating plate is here taken to satisfy the model equation:u
tt + 2u = 0 (where
2u:= (u); = Laplacian) with boundary conditions of the form:u
v = 0 and(u)
v = = control. Thus, the state is the pair [u, u
t] and controllability means existence of on := (0,T) transfering any[u, u
t]0 to any[u, u
t]T. The formulation is given by eigenfunction expansion and duality. The substantive results apply to a rectangular plate. For largeT one has such controllability with = O(T
–1/2). More surprising is that (based on a harmonic analysis estimate [11]) one has controllability for arbitrarily short times (in contrast to the wave equation:u
tt = u) with log = O(T
–1) asT0. Some related results on minimum time control are also included.
We propose a finite difference semi-discrete scheme for the approximation of the boundary exact controllability problem of the 1-D beam equation modelling the transversal vibrations of a beam with fixed ends. First of all we show that, due to the high frequency spurious oscillations, the uniform (with respect to the mesh-size) controllability property of the semi-discrete model fails in the natural functional setting. We then prove that there are two ways of restoring the uniform controllability property: a) filtering the high frequencies, i.e. controlling projections on subspaces where the high frequencies have been filtered; b) adding an extra boundary control to kill the spurious high frequency oscillations. In both cases the convergence of controls and controlled solutions is proved in weak and strong topologies, under suitable assumptions on the convergence of the initial data.
We consider the exact controllability of a hybrid
system consisting of an elastic beam, clamped at one end and attached
at the other end to a
rigid antenna. Such a system is governed by one partial
differential equation and two ordinary differential equations. Using the
HUM method, we prove that the hybrid system is exactly
controllable in an arbitrarily short time in the usual energy space.
A degenerate operator in non divergence form
- A Camasta
- G Fragnelli
A. Camasta, G. Fragnelli, A degenerate operator in non divergence form, Recent Advances in Mathematical Analysis, Trends in Mathematics, https://doi.org/10.1007/978-3-031-20021-2.
Degenerate fourth order parabolic equations with Neumann boundary conditions
- A Camasta
- G Fragnelli
A. Camasta, G. Fragnelli, Degenerate fourth order parabolic equations with Neumann
boundary conditions, submitted for publication, arXiv:2203.02739.
- A Camasta
- G Fragnelli
A. Camasta, G. Fragnelli, New results on controllability and stability for degenerate
Euler-Bernoulli type equations, submitted for publication, arXiv: 2306.11851.