Nonlinear Delay Term Research Articles

A general stability result for swelling porous elastic media with nonlinear damping and nonlinear delay term

A general stability result for swelling porous elastic media with nonlinear damping and nonlinear delay term

Exponential stability of a geometric nonlinear beam with a nonlinear delay term in boundary feedbacks

Exponential stability of a geometric nonlinear beam with a nonlinear delay term in boundary feedbacks

Energy decay for a weakly nonlinear damped piezoelectric beams with magnetic effects and a nonlinear delay term

Energy decay for a weakly nonlinear damped piezoelectric beams with magnetic effects and a nonlinear delay term

General decay for a wave equation with Wentzell boundary conditions and nonlinear delay terms

ABSTRACT In this paper, we consider a wave equation with Wentzell dynamical boundary conditions and internal and boundary delay terms. We have obtained a global existence and energy decay estimates of the solutions to our problem for nonlinear dampings and nonlinear time delay terms. We would like to see the influence of frictional damping on the rate of decay of solutions in the presence of nonlinear delay term. We study the asymptotic behaviour of solutions using a perturbed energy method under some conditions between the weight of delay terms in the feedbacks and the weight of the terms without delay.

Stabilization of the wave equation with a nonlinear delay term in the boundary conditions

Abstract A wave equation in a bounded and smooth domain of ℝ n {\mathbb{R}^{n}} with a delay term in the nonlinear boundary feedback is considered. Under suitable assumptions, global existence and uniform decay rates for the solutions are established. The proof of existence of solutions relies on a construction of suitable approximating problems for which the existence of the unique solution will be established using nonlinear semigroup theory and then passage to the limit gives the existence of solutions to the original problem. The uniform decay rates for the solutions are obtained by proving certain integral inequalities for the energy function and by establishing a comparison theorem which relates the asymptotic behavior of the energy and of the solutions to an appropriate dissipative ordinary differential equation.

Global existence and energy decay of solutions to a viscoelastic Bresse-type system with a nonlinear delay term

ABSTRACT In this paper, we consider a viscoelastic Bresse-type system subjects to nonlinear damping, nonlinear time delay term, a finite memory and with homogeneous Dirichlet-Neumann-Neumann boundary conditions. Under a condition between the weight of delay term in the feedback and the weight of the term without delay, we prove, in case of equal speeds of wave propagation, new decay results that generalise and improve the recent results in the literature.

Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback

In the present paper, we consider a suspension bridge problem with a nonlinear delay term in the internal feedback. Namely, we investigate the following equation: \begin{document}$ \begin{equation*} u_{tt}+ \Delta^2 u + \delta_1 g_1 (u_t (x,y,t))+ \delta_2 g_2 (u_t (x,y, t-\tau))+ h(u(x,y,t)) = f(x,y), \end{equation*} $\end{document} together with some suitable initial data and boundary conditions. We prove the global existence of solutions by means of the energy method combined with the Faedo-Galerkin procedure under a certain relation between the weight of the delay term in the feedback and the weight of the nonlinear frictional damping term without delay. Moreover, we establish the existence of a global attractor for the above-mentioned system by proving the existence of an absorbing set and the asymptotic smoothness of the semigroup \begin{document}$ S(t) $\end{document} .

Chaos Analysis of Fractional-order System with Delay Based on Adomian Decomposition

In this paper, fractional-order Lü chaotic system with nonlinear delay term is proposed, and the fractional-order Lü chaotic system with time-delay is numerically decomposed by Adomian decomposition. At the same time, the influence of fractional-order number on the system is verified. Then the bifurcation diagram, complexity and attractor phase diagram of the system are analyzed by Matlab software. The numerical simulation results show that the 0.9order system has rich dynamic characteristics, which provides a theoretical basis for parameter selection when fractional order delay system is applied to multimedia encryption.

Asymptotic behavior for a past history viscoelastic problem with acoustic boundary conditions

ABSTRACTIn this paper, we consider a viscoelastic equation of variable coefficients in the presence of infinite memory (past history) with nonlinear damping term and nonlinear delay term in the boundary feedback and acoustic boundary conditions. Under suitable assumptions, two arbitrary decay results of the energy solution are established via suitable Lyapunov functionals and some properties of the convex functions. The first stability result is given with a relation between the damping term and relaxation function and the second result is given without imposing any restrictive growth assumption on the damping term and the kernel function g. Our result extends the decay result obtained for problems with finite history to those with infinite history.

Energy decay rate for a von Karman system with a boundary nonlinear delay term

In this paper, we show the energy decay rate for a von Karman system with a boundary nonlinear delay term. This work is devoted to investigate the influence of kernel function g and the effect of the boundary nonlinear term μ1|ut(t)|m−1ut(t), a boundary nonlinear time delay term μ2|ut(t−τ)|m−1ut(t−τ) and prove energy decay rates of solutions when g do not necessarily decay exponentially and the boundary condition has a time delay.

General Decay for a Viscoelastic Equation of Variable Coefficients in the Presence of Past History with Delay Term in the Boundary Feedback and Acoustic Boundary Conditions

In this paper, we consider a viscoelastic equation of variable coefficients in the presence of infinite memory (past history) with nonlinear damping term and nonlinear delay term in the boundary feedback and acoustic boundary conditions. Under suitable assumptions, two arbitrary decay results of the energy solution are established via suitable Lyapunov functionals and some properties of the convex functions. The first stability result is given with relation between the damping term and relaxation function. The second result is given without imposing any restrictive growth assumption on the damping term and the kernel function g$g$. Our result extends the decay result obtained for problems with finite history to those with infinite history.

Stabilization of the variable-coefficient structural acoustic model with curved middle surface and delay effects in the structural component

This paper is concerned with the uniform energy decay rates of a structural acoustic model which describes the interactions between the acoustic medium and the elastic structural in an acoustic chamber where one “wall” is flexible and curved. The coupled system considered consists of a variable-coefficient wave equation and a variable-coefficient plate equation defined on the Riemannian manifold with the coupling on the interface between the acoustic medium and the flexible wall. Both the components of the dynamics are subject to nonlinear boundary damping. Furthermore, a nonlinear delay term acting in the boundary feedbacks of the structure component is considered. The Riemannian geometry method is applied to derive the energy estimates of the nonlinear coupled system with variable coefficients. The uniform energy decay rates of the nonlinear variable-coefficient structural acoustic system with delay effects on the Riemannian manifold are quantified by a solution to a constructed nonlinear ODE.

Existence and energy decay rates of solutions to the variable-coefficient Euler–Bernoulli plate with a delay in localized nonlinear internal feedback

This paper is concerned with the existence and energy decay estimates of solutions to the variable-coefficient Euler–Bernoulli plate with a delay in localized nonlinear internal feedback. The existence of solutions to the nonlinear system is obtained by using Galerkin method combined with some energy estimates. Furthermore, applying Riemannian geometry method, we establish uniform decay rates of the plate equation with variable coefficients and a nonlinear delay term.

Global existence and energy decay result for a weak viscoelastic wave equations with a dynamic boundary and nonlinear delay term

In this paper, we consider the weak viscoelastic wave equation utt−Δu+δΔut−σ(t)∫0tg(t−s)Δu(s)ds=|u|p−2u with dynamic boundary conditions, and nonlinear delay term. First, we prove a local existence theorem by using the Faedo–Galerkin approximations combined with a contraction mapping theorem. Secondly, we show that, under suitable conditions on the initial data and the relaxation function, the solution exists globally in time, in using the concept of stable sets. Finally, by exploiting the perturbed Lyapunov functionals, we extend and improve the previous result from Gerbi and Said-Houari (2011).

Global existence and energy decay of solutions to a viscoelastic Timoshenko beam system with a nonlinear delay term

In this paper, we consider the viscoelastic Timoshenko beam system with a delay term in the weakly nonlinear internal feedback in a bounded domain:We prove a global existence result using the energy method combined with the Faedo- Galerkin procedure under a condition between the weight of delay term in the feedback and the weight of the term without delay. Furthermore, we study the asymptotic behavior of solutions using a perturbed energy method.

Local Controllability of Functional Integrodifferential Equations in Banach Space

In this paper, we establish a set of sufficient conditions for the local controllability of functional integrodifferential equations in Banach space. The results are obtained by using the methods of analytic semigroups, fractional powers of operators, and a fixed-point argument. These results generalize previous results on bounded linear operators to unbounded linear operators in which the equation involves a nonlinear delay term. An application to a partial integrodifferential equation is given.

Suboptimal control of neutral systems

Abstract The usual approach for obtaining the optimal control for a neutral system with quadratic cost consists of first deriving a set of necessary conditions for optimality and then using conventional iterative numerical methods to find a control satisfying those conditions. The burden of computation in this approach is enormous. The suboptimum iterative scheme proposed in this paper does not proceed along those lines. Instead, the neutral system is first converted into a non-linear time-delay system. The non-linear delay term is then treated like an extra perturbing input. A linear non-delay system is optimized at each stage and the resulting sequence of control functions converges rapidly to a suboptimal control for the original problem. One numerical example illustrates the technique