Asymptotic stability of abstract dissipative systems with infinite memory

We consider in this paper the problem of asymptotic behavior of solutions to an abstract linear dissipative integrodifferential equation with infinite memory (past history) modeling linear viscoelasticity. We show that the stability of the system holds for a much larger class of the convolution kernels than the one considered in the literature, and we provide a relation between the decay rate of the solutions and the growth of the kernel at infinity. Some applications are also given.     

The optimal decay rate for a weak dissipative Bresse system

In this work we consider the Bresse system with frictional damping operating only on the angle displacement and we show that under a certain assertion the solution decays polynomially and the decay rate is optimal.     

Asymptotic stability of nonlinear systems with unbounded delays

Some asymptotic stability criteria are derived for systems of nonlinear functional differential equations with unbounded delays. The criteria are described as matrix equations or matrix inequalities, which are computationally flexible and efficient. The theories are then applied to the stabilization of time-delay systems via standard feedback control (SFC) or time-delayed feedback control (DFC). Several examples are given to illustrate the results.     

Asymptotic stability of neutral differential systems with many delays

We are concerned with delay-independent asymptotic stability of linear system of neutral differential equations. We first establish a sufficient and necessary condition for the system to be delay-independently asymptotically stable, and then give some equivalent stability conditions. This paper improves many recent results on the asymptotic stability in the literature. One example is given to show that the sufficient and necessary condition is easy to verify.     

Asymptotic stability of switched linear time-varying systems based on top-floor function

This paper considers the problem of asymptotic stability for switched linear time-varying (SLTV) systems. First, some stability conditions for SLTV systems are given by using infinite integrals. Then, based on the results obtained, two stability conditions are proposed by combining the methods of top-floor function and average dwell time. Moreover, using strict top-floor function, a stability condition is also provided when some subsystems are unstable. With the help of top-floor function, the stability problem of SLTV systems can be simplified and solved by using the technique of linear matrix inequalities (LMIs). Finally, a numerical example is given to illustrate the theoretical results.     

Asymptotic stability in thermoelectromagnetism with memory

A model describing a linear homogeneous dielectric with memory which obeys the Cattaneo–Maxwell law for the heat conduction is presented. The restrictions on the constitutive functionals are found as a direct consequence of the Second Law of Thermodynamics and some free energy potentials exhibited. Such potentials allow to determine a domain of dependence theorem for the first‐order integro‐differential system of equations governing the evolution of the thermoelectromagnetic radiation. The dissipativity due to the memory and to the heat conduction allows to establish some estimates on the asymptotic behaviour and prove the exponential decay of the solution of the system in absence of external sources. Copyright © 1999 John Wiley & Sons, Ltd.     

Asymptotic behavior of the energy for a class of weakly dissipative second-order systems with memory

A class of second-order abstract systems with memory and Dirichlet boundary conditions is investigated. By suitable Liapunov functionals, existence of solutions as well as asymptotic behavior, are determined. In particular, when the memory kernel decays exponentially, the polynomially decay of the solutions is proved.     

About Asymptotic Behavior for a Transmission Problem in Hyperbolic Thermoelasticity

We consider a transmission problem in thermoelasticity with memory. We show the exponential decay of the solution in case of radially symmetric situations, as time goes to infinity.      

Asymptotic stability of differential systems of neutral type

We offer sufficient conditions for the asymptotic stability of the equilibrium point of linear neutral differential systems. An application of our results to a family of artificial neural networks of neutral type is also illustrated.     

Asymptotic property of linear Volterra difference systems

By using the resolvent matrix and the comparison principle, we investigate the asymptotic behavior of linear Volterra difference systems.     

Asymptotic Log-Harnack inequality and applications for stochastic systems of infinite memory

The asymptotic log-Harnack inequality is established for several kinds of models on stochastic differential systems with infinite memory: non-degenerate SDEs, neutral SDEs, semi-linear SPDEs, and stochastic Hamiltonian systems. As applications, the following properties are derived for the associated segment Markov semigroups: asymptotic heat kernel estimate, uniqueness of the invariant probability measure, asymptotic gradient estimate (hence, asymptotically strong Feller property), as well as asymptotic irreducibility.     

Asymptotic behavior of Timoshenko beam with dissipative boundary feedback

This paper is concerned with the stabilization problem of Timoshenko beam in the presence of linear dissipative boundary feedback controls. Using C₀-semigroups theory we establish the existence and the uniqueness of solution of the proposed closed loop system. In order to consider the asymptotic behavior of the closed loop system, we first discuss the existence of nonzero solution of a closely related boundary value problem. Then we derive various necessary and sufficient conditions for the system to be asymptotically stable. Finally, we prove the equivalence between the exponential stability and the asymptotic stability for the closed loop system.     

Influence of anti-diagonals on the asymptotic stability for linear differential systems

Sufficient conditions are given for asymptotic stability of the linear differential system x′  =  B(t)x with B(t) being a 2  ×  2 matrix. All components of B(t) are not assumed to be positive. The matrix B(t) is naturally divisible into a diagonal matrix D(t) and an anti-diagonal matrix A(t). Our concern is to clarify a positive effect of the anti-diagonal part A(t)x on the asymptotic stability for the system x′  =  B(t)x.      

Asymptotic stability and instability of the solutions of systems with impulse action

In this paper, we study the stability of the zero solution of a system of ordinary differential equations subject to impulse action. Using the method of Lyapunov functions, we obtain tests for asymptotic stability or instability of the system. Illustrative examples are given.     

Asymptotic stability of linear multistep methods for nonlinear neutral delay differential equations

This paper is concerned with the numerical solution to initial value problems of nonlinear delay differential equations of neutral type. We use A-stable linear multistep methods to compute the numerical solution. The asymptotic stability of the A-stable linear multistep methods when applied to the nonlinear delay differential equations of neutral type is investigated, and it is shown that the A-stable linear multistep methods with linear interpolation are GAS-stable. We validate our conclusions by numerical experiments.     

Some new simple stability criteria of linear neutral systems with a single delay

This article mainly considers the linear neutral delay-differential systems with a single delay. Using the characteristic equation of the system, new simple delay-independent asymptotic and exponential stability criteria are derived in terms of the matrix measure, the spectral norm and the spectral radius of the corresponding matrices. Numerical examples demonstrate that our criteria are less conservative than those of previous corresponding results [L.M. Li, Stability of linear neutral delay-differential systems, Bull. Aust. Math. Soc. 38 (1988) 339–344; G.D. Hu, G.D. Hu. Some simple criteria for stability of neutral delay-differential systems, Appl. Math. Comput. 80 (1996) 257–271; D.Q. Cao, Ping He, Sufficient conditions for stability of linear neutral systems with a single delay, Appl. Math. Lett. 17 (2004) 139–144; G.D. Hu, G.D. Hu, B. Cahlon, Algebraic criteria for stability of linear neutral systems with a single delay, J. Comput. Appl. Math. 135 (2001) 125–130].     

Asymptotic and total stability for generalized dynamical systems

The existence of solutions to a class of nonlinear problems depending on a parameter is proved using the Gaierkin method The principal difficulty is that when the parameter crosses a critical value the bound edness of the approximate solutions is no longer obvious. Applications are indicated.