In this paper, we examine regularity and stability issues for two damped abstract elastic systems. The damping involves the average velocity and a fractional power θ , with θ in [−1, 1], of the principal operator. The matrix operator defining the damping mechanism for the coupled system is degenerate. First, we prove that for θ in (1/2, 1], the underlying semigroup is not analytic, but is differentiable for θ in (0, 1); this is in sharp contrast with known results for a single similarly damped elastic system, where the semigroup is analytic for θ in [1/2, 1]; this shows that the degeneracy dominates the dynamics of the interacting systems, preventing analyticity in that range. Next, we show that for θ in (0, 1/2], the semigroup is of certain Gevrey classes. Finally, we show that the semigroup decays exponentially for θ in [0, 1], and polynomially for θ in [−1, 0). To prove our results, we use the frequency domain method, which relies on resolvent estimates. Optimality of our resolvent estimates is also established. Two examples of application are provided. Contents1. Problem formulation and statements of main results 4973 2. Some technical Lemmas 4977 3. Proof of Theorem 1.1 4978 4. Proof of Theorem 1.2 4986 5. Proof of Theorem 1.3 4995 6. Examples of application 4999 Acknowledgements 5000 REFERENCES 5000 Problem formulation and statements of main resultsLet H be a Hilbert space with inner product (., .) and norm |.|. Let A be a positive unbounded self-adjoint operator, with domain D(A) dense in the Hilbert space H . Set V = D(A 1 2 ). We assume that V → H → V , each injection being dense and compact, where V denotes the topological dual of V .

In this paper we study the stability problem of a tree of elastic strings with local Kelvin-Voigt damping on some of the edges. Under the compatibility condition of displacement and strain and continuity condition of damping coefficients at the vertices of the tree, exponential/polynomial stability are proved. Our results generalizes the cases of single elastic string with local Kelvin-Voigt damping in [21,24,5].2010 Mathematics Subject Classification. 35B35, 35B40, 93D20.

No abstract

The wave equation on an elastic body is conservative; to make the system stable, several authors have introduced different types of dissipative mechanisms, for example, a frictional damping [1] or frictional boundary conditions [2,3]. For the stabilization of a network governed by wave equations, we refer to [4,5], where the authors considered a star-shaped and tree-shaped networks of elastic strings, and they proved that when a feedback is applied on particular nodes, then the system will be polynomially stable but not exponentially stable. We can see, in [6], that the authors considered a network with delay term in the nodal feedbacks. In [7], the authors studied, in particular, the stabilization of a chain of beams and strings. See also [8][9][10][11].Another type of stabilization of an elastic material is to add thermoelastic materials to it. In [12][13][14], the authors proved that the system is then exponentially stable; see also [15] where the authors considered the case of beams and proved that the whole system is also exponentially stable. We want to know if this result holds true for a network of elastic and thermoelastic materials. To our knowledge, the asymptotic behavior of such a system has not been studied yet. In this paper, we consider particular cases of such network that can be partially generalized. In the first case, we suppose that two elastic edges cannot be adjacent (Figure 1). In the second one, we consider a tree of elastic materials, the leaves of which thermoelastic materials are added as follows: the thermoelastic body is related to only one leaf by an end, and the second is free or connects two leaves, with the condition that each leaf is connected to only one thermoelastic body (Figure 2). With the continuity condition for the displacement and the Neumann condition for the temperature at the internal nodes, we prove that the thermal effect is strong enough to stabilize the system. Similar to [16], we will use a frequency method as described in [17,18] (see also [19]).Before going on to the next discussion, let us first recall some definitions and notations about a 1-D network that will be used in this paper. We refer to [10,16,20] for more details.Let G be a network in the Euclidean space R m , with n vertices V D fa 1

In this paper, we consider a network of elastic and thermoelastic Euler-Bernoulli beams. Our main result is to show that under certain conditions, the thermoelastic dissipation over some edges is sufficient to stabilize the whole system. Precisely, we establish the exponential stability of the network.Keywords Exponential stability · Frequency domain method · Network · Euler-Bernoulli beams · Elasticity and thermoelasticity Mathematics Subject Classifications (2010) 35B40 · 35M33 · 93D20.