Stabilization analysis for discrete-time systems with time delay

The stabilization problem for a class of discrete-time systems with time-varying delay is investigated. By constructing an augmented Lyapunov function, some sufficient conditions guaranteeing exponential stabilization are established in forms of linear matrix inequality (LMI) technique. When norm-bounded parameter uncertainties appear in the delayed discrete-time system, a delay-dependent robust exponential stabilization criterion is also presented. All of the criteria obtained in this paper are strict linear matrix inequality conditions, which make the controller gain matrix can be solved directly. Three numerical examples are provided to demonstrate the effectiveness and improvement of the derived results.     

Stabilization for hybrid stochastic systems by aperiodically intermittent control

In this paper, the mean-square exponential stabilization for stochastic differential equations with Markovian switching is studied. Specifically, a new set of sufficient conditions is derived to obtain the aperiodically intermittent control design which exponentially stabilizes the addressed hybrid stochastic differential equations. Further, stabilization problem by periodically intermittent control can be deduced as a special case from the developed results. As an application, we consider the Hopfield neutral network model with simulations to illustrate the effectiveness of developed aperiodically intermittent control design.     

On the time polynomial decay in elastic solids with voids

In this paper we investigate the temporal decay behavior of the solutions of the one-dimensional problem in various theories of continua with voids. It has been proved that the coupling of the elastic structure with porous microstructure is weak in the sense that in many situations the temporal decay of solutions is slow. We have considered some theories of porous continua when the deformation-rate tensor or time-rate or porosity function or thermal effects is present. We have proved that the decay cannot be controlled by a negative exponential. The natural question now is whether there exist or not a polynomial rate of decay of the solution in some appropriate norms. In this paper we consider some cases where the decay is slow and we obtain polynomial decay estimates. In concrete we consider the case when only the viscoelastic effect is present, the case when the motion of voids is assumed to be quasi-static and the porous viscosity is present and we finish with the case of the porous-elasticity when thermal effect is coupled.     

On the decay of solutions in nonsimple elastic solids with memory

The decay of solutions in nonsimple elasticity with memory is addressed, analyzing how the decay rate is influenced by the different dissipation mechanisms appearing in the equations. In particular, a first order dissipation is shown to guarantee the asymptotic stability of the related solution semigroup, but is not strong enough to entail exponential stability. The latter occurs for a dissipation mechanism of the second order, that is, the same order as the one of the leading operator.     

Stabilization of non-homogeneous elastic materials with voids

We study the asymptotic behavior of the solution of the non-homogeneous elastic system with voids and a thermal effect. We first prove the well-posedness of this system under some realistic assumptions on the coefficients. Since this system suffers of exponential stability (as shown in dimension 1 in Pamplona, Muñoz Rivera, and Quintanilla (2009) [18]), our main results concern strong and polynomial stabilities again under some assumptions on the coefficients. These stabilities are obtained in a closed subspace of the natural Hilbert space. Hence we characterize its orthogonal and further show that in the whole space the energy tends strongly or polynomially to the energy of the projection of the initial datum on this orthogonal space. In this respect we extend and precise former results obtained in one dimension in Pamplona, Muñoz Rivera, and Quintanilla (2009) [18].     

Material voids in elastic solids with anisotropic surface energies

This work discusses the role of highly anisotropic interfacial energy for problems involving a material void in a linearly elastic solid. Using the calculus of variations it is shown that important qualitative features of the equilibrium shape of the void may be deduced from smoothness and convexity properties of the interfacial energy.     

Material voids in elastic solids with anisotropic surface energies

This work discusses the role of highly anisotropic interfacial energy for problems involving a material void in a linearly elastic solid. Using the calculus of variations it is shown that important qualitative features of the equilibrium shape of the void may be deduced from smoothness and convexity properties of the interfacial energy.     

Decay property of regularity-loss type of solutions in elastic solids with voids

In this article, we consider two porous systems of nonclassical thermoelasticity in the whole real line. We discuss the long-time behaviour of the solutions in the presence of a strong damping acting, together with the heat effect, on the elastic equation and establish several decay results. Those decay results are shown to be very slow and of regularity-loss type. Some improvements of the decay rates have also been given, provided that the initial data belong to some weighted spaces.     

Decay property of regularity-loss type for solutions in elastic solids with voids

In this paper, we consider the Cauchy problem for a system of elastic solids with voids. First, we show that a linear porous dissipation leads to decay rates of regularity-loss type of the solution. We show some decay estimates for initial data in H^s(R)∩L¹(R)$H^s\left(\mathbb{R}\right)\cap L^1\left(\mathbb{R}\right)$. Furthermore, we prove that by restricting the initial data to be in H^s(R)∩L^1,γ(R)$H^s\left(\mathbb{R}\right)\cap L^{1,\gamma }\left(\mathbb{R}\right)$ and γ∈[0,1]$\gamma \in [0,1]$, we can derive faster decay estimates of the solution. Second, we show that by adding a viscoelastic damping term, then we gain the regularity of the solution and obtain the optimal decay rate.     

Stability of Timoshenko systems with past history

We consider vibrating systems of Timoshenko type with past history acting only in one equation. We show that the dissipation given by the history term is strong enough to produce exponential stability if and only if the equations have the same wave speeds. Otherwise the corresponding system does not decay exponentially as time goes to infinity. In the case that the wave speeds of the equations are different, which is more realistic from the physical point of view, we show that the solution decays polynomially to zero, with rates that can be improved depending on the regularity of the initial data.     

On uniqueness and analyticity in thermoviscoelastic solids with voids

In this paper we consider the most general system proposed to describe the thermoviscoelasticity with voids. We study two qualitative properties of the solutions of this theory. First, we obtain a uniqueness result when we do not assume any sign to the internal energy. Second we extend some previous results and prove the analyticity of the solutions. The impossibility of localization in time of the solutions is a consequence. Last result we present corresponds to the analyticity of solutions in case that the dissipation is not very strong, but with suitable coupling terms.     

Thermo-poroacoustic acceleration waves in elastic materials with voids

A model for coupled elasto-acoustic waves, thermal waves, and waves associated with the voids, in a porous medium is investigated. Due to the use of lighter materials in modern buildings and noise concerns in the environment such models for thermo-poroacoustic waves are of much interest to the building industry. Analysis of such waves is also of interest in acoustic microscopy where the identification of material defects is of paramount importance to industry and medicine. We present a model for acoustic wave propagation in a porous material which also allows for propagation of a thermal wave. The thermodynamics is based on an entropy inequality of A.E. Green, F.R.S. and N. Laws and is presented for a modification of the theory of elastic materials with voids due to J.W. Nunziato and S.C. Cowin. A fully nonlinear acceleration wave analysis is initiated.     

Analyticity and exponential stability of semigroups for the elastic systems with structural damping in Banach spaces

In this paper, we consider a second order evolution equation in a Banach space, which can model an elastic system with structural damping. New forms of the corresponding first order evolution equation are introduced, and their well-posed property is proved by means of the operator semigroup theory. We give sufficient conditions for analyticity and exponential stability of the associated semigroups.     

On interface waves in misoriented pre-stressed incompressible elastic solids

Some relationships, fundamental to the resolution of interfacewave problems, are presented. These equations allow for thederivation of explicit secular equations for problems involvingwaves localized near the plane boundary of anisotropic elastichalf-spaces, such as Rayleigh, Scholte, or Stoneley waves. Theyare obtained rapidly, without recourse to the Stroh formalism.As an application, the problems of Stoneley wave propagationand of interface stability for misaligned predeformed incompressiblehalfspaces are treated. The upper and lower half-spaces aremade of the same material, subject to the same prestress, andare rigidly bonded along a common principal plane. The principalaxes in this plane do not, however, coincide, and the wave propagationis studied in the direction of the bisectrix of the angle betweena principal axis of the upper half-space and a principal axisof the lower half-space.     

Propagating waves in elastic material with voids subjected to electro-magnetic interactions

Constitutive relations and field equations are developed for an elastic solid with voids subjected to electro-magnetic field. The linearized form of the relations and equations are presented separately when medium is subjected to a large magnetic field and when it is subjected to a large electric field. The possibility of propagation of time harmonic plane waves in an infinite elastic solid with voids has been explored. It is found that when the medium is subjected to large magnetic field, there exist two coupled longitudinal waves propagating with distinct speeds and a transverse wave mode. However, when the medium is subjected to a large electric field, there may propagate five basic waves comprising of four coupled longitudinal waves propagating with distinct speeds and a lone transverse wave. The effects of magnetic and electric fields are observed on the propagation characteristics of the existing waves. Under the limiting cases of frequency and for different electric conductive materials, the speeds of various waves are investigated. The phase speeds of different waves and their corresponding attenuations have been computed against the frequency parameter and depicted graphically for a specific material.     

Wave propagation in an elastic layer with voids located in a liquid

We obtain the dispersion equations that describe the propagation of waves in an elastic layer with voids locted between two liquid half-spaces. We study certain limiting cases corresponding to the absence of voids or liquid. We obtain the roots of the dispersion equations for both dissipative and nondissipative systems. It is shown that the relation of the real part of the phase velocity to the wave number in a dissipative system is qualitatively similar to the corresponding relation for the real value of the phase velocity in the case when dissipation is absent. Translated fromMatematichni Metodi i Fiziko-mekhanichni Polya, Vol. 40, No. 1, 1997, pp. 90–96.     

Lie,Jordan and proper codimensions of associative algebras

We study the exponential rate of growth of the sequence of proper, Lie and Jordan codimensions of an associative algebra. We show that for any finite dimensional associative algebra, the exponential rates of growth can be explicitly computed and are strictly related to the PI-exponent of the algebra. The first author was partially supported by MIUR of Italy. The second author was partially supported by RFBR grant No 06-01-00485 and SSC-5666.2006.1     

Polynomial Identities of Algebras of Small Dimension

It is well known that given an associative algebra or a Lie algebra A, its codimension sequence c _n (A) is either polynomially bounded or grows at least as fast as 2 ⁿ . In [2 Giambruno , A. , Mishchenko , S. , Zaicev , M. ( 2006 ). Algebras with intermediate growth of the codimensions . Adv. Appl. Math. 37 ( 3 ): 360 – 377 .[Crossref], [Web of Science ®] , [Google Scholar]] we proved that for a finite dimensional (in general nonassociative) algebra A, dim A = d, the sequence c _n (A) is also polynomially bounded or c _n (A) ≥ a ⁿ asymptotically, for some real number a > 1 which might be less than 2. Nevertheless, for d = 2, we may take a = 2. Here we prove that for d = 3 the same conclusion holds. We also construct a five-dimensional algebra A with c _n (A) < 2 ⁿ .     

Uniform decay in a Timoshenko-type system with past history

Fernández Sare and Rivera [H.D. Fernández Sare, J.E. Muñoz Rivera, Stability of Timoshenko systems with past history, J. Math. Anal. Appl. 339 (1) (2008) 482–502] considered the following Timoshenko-type system

ρ₁φ_tt−K(φ_x+ψ)_x=0,