Non-dissipative boundary feedback for Rayleigh and Timoshenko beams

We show that a non-dissipative feedback that has been shown in the literature to exponentially stabilize an EulerBernoulli beam makes a Rayleigh beam and a Timoshenko beam unstable

Stability analysis of laminated beam systems with delay using lyapunov functional

This work is concerned with systems of laminated beams model subject to linear and nonlinear delay feedback. In a dynamic laminated beam, time delay manifests in the form of lags in restoring the desired system stability after perturbations. Four prevalent categories of time delay are considered. For laminated beams with relatively high adhesive stiffness, a constant delay feedback is considered for systems made up of individual beams with same elasticity, and neutral delay otherwise. In systems where delay is significantly due to adhesive softening, distributed delay is considered. Lastly, in structures where the mechanism of dissipating energy is nonlinear, a corresponding nonlinear delay effect is investigated. The mechanism of stabilization mainly relies on the intrinsic structural damping, unlike in previous works where researchers introduced additional dampings such as boundary feedback and material damping. The objective of this work is to establish the asymptotic behavior of a vibrating Timoshenko laminated beam using structural or utmost a single frictional damping in presence of different forms of time delay. The energy method for partial differential equations is the main tool used to establish wellposedness results and asymptotic behavior. The existence and uniqueness of the solution is proved using the linear semi group theory, whereas for energy decay properties, the multiplier technique involving constructing a suitable Lyapunov functional equivalent to the energy is utilized. With appropriate assumptions on the delay weight and wave speeds, it is established that the energy of the system at least decays exponentially due to structural damping. Furthermore, a single additional frictional damping guarantees polynomial decay despite the presence of constant or distributed delay feedback. For nonlinear structural damping, with help of some convexity arguments, general decay result is achieved. In summary, depending on the damping mechanism(s), exponential, polynomial, or general decay results of a laminated beam system subject to different forms of delay feedback are established

Étude de la stabilité de quelques systèmes d'équations des ondes couplées sur des domaines bornés et non bornés

The thesis is driven mainly on indirect stabilization system of two coupled wave equations and the boundary stabilization of Rayleigh beam equation. In the case of stabilization of a coupled wave equations, the Control is introduced into the system directly on the edge of the field of a single equation in the case of a bounded domain or inside a single equation but in the case of an unbounded domain. The nature of thus coupled system depends on the coupling equations and arithmetic Nature of speeds of propagation, and this gives different results for the polynomial stability and the instability. In the case of stabilization of Rayleigh beam equation, we consider an equation with one control force acting on the edge of the area. First, using the asymptotic expansion of the eigenvalues and vectors of the uncontrolled system an observability result and a result of boundedness of the transfer function are obtained. Then a polynomial decay rate of the energy of the system is established. Then through a spectral study combined with a frequency method, optimality of the rate obtained is assured.La thèse est portée essentiellement sur la stabilisation indirecte d’un système de deux équations des ondes couplées et sur la stabilisation frontière de poutre de Rayleigh.Dans le cas de la stabilisation d’un système d’équations d’onde couplées, le contrôle est introduit dans le système directement sur le bord du domaine d’une seule équation dans le cas d’un domaine borne ou à l’intérieur d’une seule équation mais dans le cas d’un domaine non borné. La nature du système ainsi couplé dépend du couplage des équations et de la nature arithmétique des vitesses de propagations, et ceci donne divers résultats pour la stabilisation polynomiale ainsi la non stabilité.Dans le cas de la stabilisation de poutre de Rayleigh, l’équation est considérée avec un seul contrôle force agissant sur bord du domaine. D’abord, moyennant le développement asymptotique des valeurs propres et des vecteurs propres du système non contrôlé, un résultat d’observabilité ainsi qu’un résultat de bornétude de la fonction de transfert correspondant sont obtenus. Alors, un taux de décroissance polynomial de l’énergie du système est établi. Ensuite, moyennant une étude spectrale combinée avec une méthode fréquentielle, l’optimalité du taux obtenu est assurée

Well-Posedness and Polynomial energy decay rate of a transmission problem for Rayleigh beam model with heat conduction

In this paper, we investigate the stability of the transmission problem for Rayleigh beam model with heat conduction. First, we reformulate our system into an evolution equation and prove our problem's well-posedness. Next, we demonstrate the resolvent of the operator is compact in the energy space, then by using the general criteria of Arendt-Batty, we prove that the thermal dissipation is enough to stabilize our model. Finally, a polynomial energy decay rate has been obtained which depends on the mass densities and the moments of inertia of the Rayleigh beams

BOUNDARY FEEDBACK STABILIZATION FOR THE INTRINSIC GEOMETRICALLY EXACT BEAM MODEL

The geometrically exact beam (GEB) model is a 1-D second-order non-linear system of six equations which gives the position of a beam in R 3. The beam may undergo large deflections and rotations, as well as shear deformation. A closely related model, the intrinsic formulation of GEB (IGEB), is a 1-D first-order semilin-ear hyperbolic system of twelve equations which has for states velocities and strains. Here, we consider a freely vibrating slender beam made of an isotropic linear-elastic material. Applying a feedback boundary control at one end of the beam, while the other end is clamped, we show that the steady state 0 of IGEB is locally exponentially stable for the H 1 and H 2 norms. The strategy employed is to choose the control so that the energy of the beam is nonincreasing and find appropriate quadratic Lyapunov functions, relying on the energy of the beam, the relationship between GEB and IGEB, and properties of the system's coefficients