Exponential stability of an elastic string with local Kelvin–Voigt damping

This paper is devoted to analyzing an elastic string with local Kelvin–Voigt damping. We prove the exponential stability of the system when the material coefficient function near the interface is smooth enough. Our method is based on the frequency method and semigroup theory.     

Exponential stability of an elastic string with local Kelvin–Voigt damping

This paper is devoted to analyzing an elastic string with local Kelvin–Voigt damping. We prove the exponential stability of the system when the material coefficient function near the interface is smooth enough. Our method is based on the frequency method and semigroup theory.     

On the lack of exponential stability for an elastic–viscoelastic waves interaction system

In this paper, we consider an interaction system in which a wave and a viscoelastic wave equation evolve in two bounded domains, with natural transmission conditions at a common interface. We show the lack of uniform decay of solutions in general domains. The method is based on the construction of ray-like solutions by means of geometric optics expansions and a careful analysis of the transfer of the energy at the interface.     

Mindlin–Timoshenko systems with Kelvin–Voigt: analyticity and optimal decay rates

This paper is concerned with asymptotic stability of Mindlin–Timoshenko plates with dissipation of Kelvin–Voigt type on the equations for the rotation angles. We prove that the corresponding evolution semigroup is analytic if a viscoelastic damping is also effective over the equation for the transversal displacements. On the contrary, if the transversal displacement is undamped, we show that the semigroup is neither analytic nor exponentially stable. In addition, in the latter case, we show that the solution decays polynomially and we prove that the decay rate found is optimal.     

Exponential stability for the wave equations with local Kelvin–Voigt damping

We consider the wave equations with local viscoelastic damping distributed around the boundary of a bounded open set We show that the energy of the wave equations goes uniformly and exponentially to zero for all initial data of finite energy. This author supported partially the National Natural Sciences Foundation grant 10271111. Received: February 8, 2005; revised: July 3, 2005     

Global well-posedness and stability of semilinear Mindlin–Timoshenko system

We study long-term behavior of Reissner–Mindlin–Timoshenko (RMT) plate systems, focusing on the interplay between nonlinear viscous damping and source terms. The sources may represent restoring forces, but may also be focusing thus potentially amplifying the total energy which is the primary scenario of interest. This work complements [28] which established local well-posedness of this problem, global well-posedness when damping dominates the sources (in an appropriate sense) and a blow-up in the complementary scenario assuming negative “total” initial energy. The current paper develops the potential well theory for the RMT system: it proves global existence for potential well solutions without restricting the source exponents, derives explicit energy decay rates dependent on the order of the damping exponents, and verifies a blow-up result for positive total initial energy.     

Weak Solvability of Kelvin–Voigt Model of Thermoviscoelasticity

In this paper we study the weak solvability of an initial-boundary value thermoviscoelasticity problem for one Kelvin–Voigt mathematical model which describes the flow of weakly concentrated aqueous polymer solutions.     

Well-posedness and exponential stability of a thermoelastic Joint–Leg–Beam system with Robin boundary conditions

An important class of proposed large space structures features a triangular truss backbone. In this paper we study thermomechanical behavior of a truss component; namely, a triangular frame consisting of two thin-walled circular beams connected through a joint. Transverse and axial mechanical motions of the beams are coupled though a mechanical joint. The nature of the external solar load suggests a decomposition of the temperature fields in the beams leading to two heat equations for each beam. One of these fields models the circumferential average temperature and is coupled to axial motions of the beam, while the second field accounts for a temperature gradient across the beam and is coupled to beam bending. The resulting system of partial and ordinary differential equations formally describes the coupled thermomechanical behavior of the joint–beam system. The main work is in developing an appropriate state-space form and then using semigroup theory to establish well-posedness and exponential stability.     

Asymptotic behavior of the spectrum of one-dimensional vibrations in a layered medium consisting of elastic and Kelvin–Voigt viscoelastic materials

For every algebraically closed field k of characteristic different from 2, we prove the following: (1) Finite-dimensional (not necessarily associative) k-algebras of general type of a fixed dimension, considered up to isomorphism, are parametrized by the values of a tuple of algebraically independent (over k) rational functions of the structure constants. (2) There exists an “algebraic normal form” to which the set of structure constants of every such algebra can be uniquely transformed by means of passing to its new basis—namely, there are two finite systems of nonconstant polynomials on the space of structure constants, {f_i}i∈I and {b_j}j∈J, such that the ideal generated by the set {f_i}i∈I is prime and, for every tuple c of structure constants satisfying the property b_j(c) ≠ 0 for all j ∈ J, there exists a unique new basis of this algebra in which the tuple c′ of its structure constants satisfies the property f_i(c′) = 0 for all i ∈ I.     

Mathematical justification of Kelvin–Voigt beam models by asymptotic methods

The authors derive and justify two models for the bending-stretching of a viscoelastic rod by using the asymptotic expansion method. The material behaviour is modelled by using a general Kelvin?CVoigt constitutive law.     

A constructive method for the stabilization of the wave equation with localized Kelvin–Voigt damping

We consider the wave equation with Kelvin–Voigt damping in a bounded domain. The damping is localized in a suitable open subset of the domain under consideration. The exponential stability result proposed by Liu and Rao for that system assumes that the damping is localized in a neighborhood of the whole boundary, and the damping coefficient is continuously differentiable with a bounded Laplacian. We propose a new solution to the exponential stability problem based on the introduction of a new variable, and a constructive frequency domain approach. The main features of our method are: (i) the damping region need not be a neighborhood of the whole boundary; (ii) the damping coefficient is assumed to be bounded measurable with bounded measurable gradient only; (iii) the introduction of a new variable. These features enable us to improve on the damping coefficient smoothness and more especially on the feedback control region. Further, when combined with a recent result of Borichev and Tomilov on the polynomial decay of bounded semigroups, the new method enables us to prove a polynomial decay estimate of the energy when the damping coefficient is bounded measurable only.     

Mixed initial–boundary value problem for equations of motion of Kelvin–Voigt fluids

The initial–boundary value problem for equations of motion of Kelvin–Voigt fluids with mixed boundary conditions is studied. The no-slip condition is used on some portion of the boundary, while the impermeability condition and the tangential component of the surface force field are specified on the rest of the boundary. The global-in-time existence of a weak solution is proved. It is shown that the solution is unique and depends continuously on the field of external forces, the field of surface forces, and initial data.     

On some properties of Hermite–Padé approximants to an exponential system

Extremal properties and localization of zeros of general (including nondiagonal) type I Hermite–Padé polynomials are studied for the exponential system {e ^λjz } _j=0 ^k with arbitrary different complex numbers λ₀, λ₁,..., λ_k. The theorems proved in the paper complement and generalize the results obtained earlier by other authors.     

Mean square exponential stability of stochastic neural networks with reaction–diffusion terms and delays

In this paper, some sufficient conditions ensuring mean square exponential stability of the equilibrium point of a class of stochastic neural networks with reaction–diffusion terms and time-varying delays are obtained. The conditions involving the effect of diffusion terms reduce the conservatism of the previous results. Finally, we give a numerical example to verify the effectiveness of our results.