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New results on controllability and stability for degenerate Euler-Bernoulli type equations
... It is only recently that the control issues for degenerate beam equations are considered (see [6,7,11]). In particular, the authors in [6] study beam equations of the form u tt + (a(x)u xx ) xx = 0 in (0, +∞) × (0, 1), (1.1) where a is positive on (0, 1] but vanishes at zero. More precisely, assuming that the left end of the beam is fixed, and applying a suitable control at the right end, they arrived to prove null controllability for the above model in a sufficiently large time. ...
... It is only recently that the control issues for degenerate beam equations are considered (see [6,7,11]). In particular, the authors in [6] study beam equations of the form u tt + (a(x)u xx ) xx = 0 in (0, +∞) × (0, 1), (1.1) where a is positive on (0, 1] but vanishes at zero. More precisely, assuming that the left end of the beam is fixed, and applying a suitable control at the right end, they arrived to prove null controllability for the above model in a sufficiently large time. ...
... which is equivalent to the norms ∥ · ∥ 2,a and ∥ · ∥ 2 (see [6,Proposition 3.1]). In particular, if a is (WD) or (SD) at 0, then ...
... For parabolic degenerate problems the pioneering papers are [2], [16], [17], [18], [26], [36], [37] (see also [27] and the references therein); for hyperbolic degenerate problems the most important paper is [4] (see also the arxiv version of 2015), where a general degenerate function is considered (see also [29], [58], and the references mentioned within), and [9] for the non divergence case (see also [28]). On the other hand, for degenerate beam problems the first results can be found in [13], [14] and [15]. However, it is important to underline that in all the previous papers there is not a delay term and the equations are linear, except for [17] where there is a semilinear term. ...
... In this section we study the stability for a non linear problem governed by a fourth order degenerate operator in divergence form or by a second order operator in divergence or in non divergence form. In every case the function a is (WD) or (SD) and, as in Section 2, the assumption K < 2 is only a technical hypothesis (see [4], [15] and [28]). ...
... for all (u, v) ∈ W 0 (0, 1)× K 2 a,0 (0, 1) (see [15]), one can prove that (A d , D(A d )) is non positive with dense domain and generates a contraction semigroup (R(t)) t≥0 assuming that a is (WD) or (SD). Therefore, the following existence theorem holds. ...
... For parabolic degenerate problems the pioneering papers are [2], [16], [17], [18], [26], [36], [37] (see also [27] and the references therein); for hyperbolic degenerate problems the most important paper is [4] (see also the arxiv version of 2015), where a general degenerate function is considered (see also [29], [58], and the references mentioned within), and [9] for the non divergence case (see also [28]). On the other hand, for degenerate beam problems the first results can be found in [13], [14] and [15]. However, it is important to underline that in all the previous papers there is not a delay term and the equations are linear, except for [17] where there is a semilinear term. ...
... In this section we study the stability for a non linear problem governed by a fourth order degenerate operator in divergence form or by a second order operator in divergence or in non divergence form. In every case the function a is (WD) or (SD) and, as in Section 2, the assumption K < 2 is only a technical hypothesis (see [4], [15] and [28]). ...
... for all (u, v) ∈ W 0 (0, 1)× K 2 a,0 (0, 1) (see [15]), one can prove that (A d , D(A d )) is non positive with dense domain and generates a contraction semigroup (R(t)) t≥0 assuming that a is (WD) or (SD). Therefore, the following existence theorem holds. ...
We consider several classes of degenerate hyperbolic equations involving delay terms and suitable nonlinearities. The idea is to rewrite the problems in an abstract way and, using semigroup theory and energy method, we study well posedness and stability. Moreover, some illustrative examples are given. Keywords: fourth order degenerate operator, second order degenerate operator, operator in divergence or in non divergence form, exponential stability, nonlinear equation, time delay. 2000AMS Subject Classification: 35L80, 93D23, 93D15, 93B05, 93B07 1
... The study of controllability and stabilization of the one-dimensional Petrovsky equation has a long story, starting from works such as [1][2][3][4][5][6][7] and it has continued in recent years in more general contexts, for degenerate equations, see for instance [8][9][10][11]. See also [12][13][14][15][16][17][18][19][20][21][22][23][24] for many significant results on the higher-dimensional Petrovsky equation. ...
... Proof For the well-posedness of problem (9), according to the Hille-Yosida-Phillips Theorem, we need to prove that the operator A is m-dissipative. To this aim, thanks to [41, Proposition 2.2.6], it is sufficient to prove that A ∶ D(A) → H 0 is dissipative and that I − A is surjective. ...
This paper deals with the Neumann boundary controllability of a system of Petrovsky-Petrovsky type which is coupled through the velocities. We are interested in controlling the corresponding state which has two components by exerting only one boundary control force on the system. We prove that, under the usual multiplier geometric control condition and for small coupling coefficient b, there exists a time such that the system is null controllable at any time . Our approach is based on transposition solutions for coupled systems and both the Hilbert Uniqueness Method (HUM) and the energy multiplier method.
... • Degenerate hyperbolic models: [GL16,CF24]. Among these, we focus on the works most closely related to our study, as they provide essential context and motivation for our contributions. ...
This article deals with the boundary null controllability of some degenerate parabolic equations posed on a square domain, presenting the first study of boundary controllability for such equations in multidimensional settings. The proof combines two classical techniques: the method of moments and the Lebeau-Robbiano strategy. A key novelty of this work lies in the analysis of boundary control localized on a subset of the boundary where the degeneracy occurs. Furthermore, we establish the Kalman rank condition as a full characterization of boundary controllability for coupled degenerate systems. The results are extended to N-dimensional domains, and potential extensions and open problems are discussed to motivate further research in this area.
... Boutaayamou, Fragnelli and Mugnai [2] considered the boundary controllability and the last two authors [11] also studied the linear stabilization for a degenerate wave equation in non-divergence form with drift. For degenerate beam problems, the controllability and stabilization results can be found in [3][4][5]. It is worth noting that there is no delay term in all the previous papers. ...
A degenerate wave equation with time-varying delay in the boundary control input is considered. The well-posedness of the system is established by applying the semigroup theory. The boundary stabilization of the degenerate wave equation is concerned and the uniform exponential decay of solutions is obtained by combining the energy estimates with suitable Lyapunov functionals and an integral inequality under suitable conditions.
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.
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.
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 prove an estimate of Carleman type for the one dimensional heat equation
ut - ( a( x )ux )x + c( t,x )u = h( t,x ), ( t,x ) Î ( 0,T ) ( 0,1 ), u_t - \left( {a\left( x \right)u_x } \right)_x + c\left( {t,x} \right)u = h\left( {t,x} \right),\quad \left( {t,x} \right)
\in \left( {0,T} \right) \times \left( {0,1} \right), where a(·) is degenerate at 0. Such an estimate is derived for a
special pseudo-convex weight function related to the degeneracy rate of a(·). Then, we study the null controllability on [0,
1] of the semilinear degenerate parabolic equation
ut - ( a( x )ux )x + f( t,x,u ) = h( t,x )cw ( x ), u_t - \left( {a\left( x \right)u_x } \right)_x + f\left( {t,x,u} \right) = h\left( {t,x} \right)\chi _\omega \left( x \right),
where (t, x) ∈(0, T) × (0, 1), ω=(α, β) ⊂⊂ [0, 1], and f is locally Lipschitz with respect to u.
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 we consider a fourth order operator in non divergence form Au := au′′′′, where is a function that degenerates somewhere in the interval. We prove that the operator generates an analytic semigroup, under suitable assumptions on the function a. We extend these results to a general operator Anu := au(2n).KeywordsDegenerate operators in non divergence formLinear differential operators of order 2nInterior and boundary degeneracyAnalytic semigroups
The distribution of natural frequencies of the Euler–Bernoulli beam resting on elastic foundation and subject to an axial force in the presence of several damping mechanisms is investigated. The damping mechanisms are: ( i ) an external or viscous damping with damping coefficient ( − a 0 ( x )), ( ii ) a damping proportional to the bending rate with the damping coefficient a 1 ( x ). The beam is clamped at the left end and equipped with a four-parameter (α, β, κ 1 , κ 2 ) linear boundary feedback law at the right end. The 2 × 2 boundary feedback matrix relates the control input (a vector of velocity and its spacial derivative at the right end) to the output (a vector of shear and moment at the right end). The initial boundary value problem describing the dynamics of the beam has been reduced to the first order in time evolution equation in the state Hilbert space of the system. The dynamics generator has a purely discrete spectrum (the vibrational modes). Explicit asymptotic formula for the eigenvalues as the number of an eigenvalue tends to infinity have been obtained. It is shown that the boundary control parameters and the distributed damping play different roles in the asymptotical formulas for the eigenvalues of the dynamics generator. Namely, the damping coefficient a 1 and the boundary controls κ 1 and κ 2 enter the leading asymptotical term explicitly, while damping coefficient a 0 appears in the lower order terms.
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.
Preface to the Second Edition Preface to Volume I of the First Edition Preface to Volume II of the First Edition List of Figures Introduction Part I. Finite Dimensional Linear Control of Dynamical Systems Control of Linear Finite Dimensional Differential Systems Linear Quadratic Two-Person Zero-Sum Differential Games Part II. Representation of Infinite Dimensional Linear Control Dynamical Systems Semi-groups of Operators and Interpolation Variational Theory of Parabolic Systems Semi-group Methods for Systems with Unbounded Control and Observation Operators Differential Systems with Delays Part III. Qualitative Properties of Linear Control Dynamical Systems Controllability and Observability for a Class of Infinite Dimensional Systems Part IV. Quadratic Optimal Control: Finite Time Horizon Systems with Bounded Control Operators: Control Inside the Domain Systems with Unbounded Control Operators: Parabolic Equations with Control on the Boundary Systems with Unbounded Control Operators: Hyperbolic Equations with Control on the Boundary Part V. Quadratic Optimal Control: Infinite Time Horizon Systems with Bounded Control Operators: Control Inside the Domain Systems with Unbounded Control Operators: Parabolic Equations with Control on the Boundary Systems with Unbounded Control Operators: Hyperbolic Equations with Control on the Boundary Appendix A. An Isomorphism Result References Index
Synopsis
It is shown that a cantilevered beam with weak viscoelastic damping of Boltzmann-type can be uniformly stabilised by velocity feedback applied as a shearing force at the free end of the beam. Estimates for the viscoelastic energy are derived using the energy multiplier method. The energy decay is related to the decay of the relaxation modulus associated with the viscoelastic material.
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, to appear in Analysis and Numerics of Design, Control and Inverse
Problems, INdAM-Springer volume, arXiv:2203.02739.