A new stability number of the Bresse‐Cattaneo system

In this paper, we consider the Bresse‐Cattaneo system with a frictional damping term and prove some optimal decay results for the L²‐norm of the solution and its higher order derivatives. In fact, we show that there is a completely new stability number δ that controls the decay rate of the solution. To prove our results, we use the energy method in the Fourier space to build some very delicate Lyapunov functionals that give the desired results. We also prove the optimality of the results by using the eigenvalues expansion method. In addition, we show that for the absence of the frictional damping term, the solution of our problem does not decay at all. This result improves some early results     

Boundary stabilization for a non‐linear beam on elastic bearings

In this paper, we study the equation under non‐linear boundary conditions which model the vibrations of a beam clamped at x=0 and supported by a non‐linear bearing at x=L. By adding only one damping mechanism at x=L, we prove the existence of a global solution and exponential decay of the energy. Copyright © 2001 John Wiley & Sons, Ltd.     

Boundary stabilization of a microbeam model

In this paper, we study the boundary stabilization of the deflection of a clamped-free microbeam, which is modeled by a sixth-order hyperbolic equation. We design a boundary feedback control, simpler than the one designed in Vatankhah et al,² that forces the energy associated to the deflection to decay exponentially to zero as the time goes to infinity. The rate in which the energy exponentially decays is explicitly given.     

On the exponential and polynomial stability for a linear Bresse system

In this paper, we consider a linear one-dimensional Bresse system consisting of three hyperbolic equations coupled in a certain manner under mixed homogeneous Dirichlet-Neumann boundary conditions. Here, we consider that only the longitudinal displacement is damped, and the vertical displacement and shear angle displacement are free. We prove the well-posedness of the system and some exponential, lack of exponential and polynomial stability results depending on the coefficients of the equations and the smoothness of initial data. At the end, we use some numerical approximations based on finite difference techniques to validate the theoretical results. The proof is based on the semigroup theory and a combination of the energy method and the frequency domain approach.     

Boundedness and stabilization in a two‐species chemotaxis‐competition system of parabolic‐parabolic–elliptic type

This paper is concerned with the two‐species chemotaxis‐competition system where Ω is a bounded domain in with smooth boundary ∂Ω, n≥2; χ_i and μ_i are constants satisfying some conditions. The above system was studied in the cases that a₁,a₂∈(0,1) and a₁>1>a₂, and it was proved that global existence and asymptotic stability hold when are small. However, the conditions in the above 2 cases strongly depend on a₁,a₂, and have not been obtained in the case that a₁,a₂≥1. Moreover, convergence rates in the cases that a₁,a₂∈(0,1) and a₁>1>a₂ have not been studied. The purpose of this work is to construct conditions which derive global existence of classical bounded solutions for all a₁,a₂>0 which covers the case that a₁,a₂≥1, and lead to convergence rates for solutions of the above system in the cases that a₁,a₂∈(0,1) and a₁≥1>a₂.     

Boundary feedback stabilization of Timoshenko beam with boundary dissipation

Nonlinear boundary feedback control to a Timoshenko beam is studied. Under some nonlinear boundary feedback controls, the nonlinear semi-group theory is used to show the well-posedness for the correspnding closed loop system. Then by using the energy perturbed method, it is proved that the vibration of the beam under the proposed control actions decays asymptotically or exponentially as t→∞. Project supported by the National Natural Science Foundation of China.     

Boundary feedback stabilization of coupled sine-Gordon equations

^** Email: koba{at}cntl.kyutech.ac.jp^*** Email: sakamoto{at}cntl.kyutech.ac.jp This paper is concerned with global stabilization of the systemgoverned by coupled sine-Gordon equations without damping. Astabilizer is constructed by boundary velocity feedback. Theclosed-loop system is shown to be well posed by the non-linearsemigroup approach. Moreover, using a multiplier method, globalexponential stabilization of the closed-loop system is proved.     

非均质Euler-Bernoulli梁的边界反馈镇定问题

基于频域乘子方法，讨论非均质Euler-Bernoulli梁边界反馈镇定问题，利用黄发伦关于C0-半群指数稳定的判据和指数乘子技巧，证明了只用力或力矩反馈，由Euler-Bernoulli梁所决定的闭环系统可以指数稳定。     

Ground state solutions for a coupled Kirchhoff‐type system

In this paper, we consider the coupled Kirchhoff‐type system where ε is a small positive parameter and a_i>0, b_i≥0 are constants for i = 1,2, P,Q are positive continuous potentials satisfying some conditions. Using minimax theorem and the Ljusternik‐Schnirelmann category theory, we obtain the existence and multiplicity results of ground state solutions for the aforementioned system as ε > 0 small enough. Moreover, the concentration phenomena of solutions is also explored. Copyright © 2015 John Wiley & Sons, Ltd.     

Ore‐type degree conditions for a graph to be H‐linked

Given a fixed multigraph H with V(H) = {h₁,…, h_m}, we say that a graph G is H‐linked if for every choice of m vertices v₁, …, _vm in G, there exists a subdivision of H in G such that for every i, v_i is the branch vertex representing h_i. This generalizes the notion of k‐linked graphs (as well as some other notions). For a family of graphs, a graph G is ‐linked if G is H‐linked for every . In this article, we estimate the minimum integer r = r(n, k, d) such that each n‐vertex graph with is ‐linked, where is the family of simple graphs with k edges and minimum degree at least . © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 14–26, 2008     

Ambarzumyan‐type theorem with polynomially dependent eigenparameter

In this paper, we are concerned with the inverse Sturm–Liouville problem with polynomially dependent eigenparameter in discontinuity and boundary conditions. By using a self‐adjoint operator‐theoretic interpretation for this sort of problem, Ambarzumyan theorem is provided for the mentioned Sturm–Liouville operator. Copyright © 2014 John Wiley & Sons, Ltd.     

Adaptive stabilization of the sine‐Gordon equation by boundary control

This paper is concerned with adaptive global stabilization of the sine‐Gordon equation without damping by boundary control. An adaptive stabilizer is constructed by the concept of high‐gain output feedback. The closed‐loop system is shown to be locally well‐posed by the Banach fixed point theorem and then to be globally well‐posed by the Lyapunov method. Moreover, using a multiplier method global exponential stabilization of the system is proved. Copyright © 2004 John Wiley & Sons, Ltd.     

Exponential stabilization and L2‐gain for uncertain switched nonlinear systems with interval time‐varying delay

This paper is concerned with the exponential stabilization and L₂‐gain for a class of uncertain switched nonlinear systems with interval time‐varying delay. Based on Lyapunov–Krasovskii functional method, novel delay‐dependent sufficient conditions of exponential stabilization for a class of uncertain switched nonlinear delay systems are developed under an average dwell time scheme. Then, novel criteria to ensure the exponential stabilization with weighted L₂‐gain performance for a class of uncertain switched nonlinear delay systems are established. Furthermore, an effective method is proposed for the designing of a stabilizing feedback controller with L₂‐gain performance. Finally, some numerical examples are given to illustrate the effectiveness of the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.     

Stability and optimality of decay rate for a weakly dissipative Bresse system

This paper is concerned with asymptotic stability of a Bresse system with two frictional dissipations. Under mathematical condition of equal speed of wave propagation, we prove that the system is exponentially stable. Otherwise, we show that Bresse system is not exponentially stable. Then, in the latter case, by using a recent result in linear operator theory, we prove the solution decays polynomially to zero with optimal decay rate. Better rates of polynomial decay depending on the regularity of initial data are also achieved. Copyright © 2014 John Wiley & Sons, Ltd.     

Semigroup Domination on a Riemannian Manifold with Boundary

We discuss the semigroup domination on a Riemannian manifold with boundary. Our main interest is the Hodge–Kodaira Laplacian for differential forms. We consider two kinds of boundary conditions; the absolutely boundary condition and the relative boundary condition. Our main tool is the square field operator. We also develop a general theory of semigroup commutation.     

Exponential stability of pseudo almost periodic delayed Nicholson‐type system with patch structure

In this paper, we consider the existence and global exponential stability of positive pseudo almost periodic solutions for a delayed Nicholson‐type system with patch structure. With the aid of ergodic theory and differential inequality technique, we prove the existence and global exponential stability of positive pseudo almost periodic solutions for the addressed system, which generalize and improve some known results. Specially, an example and its numerical simulations are provided to manifest the effectiveness of theoretical results.     

On Boundary Stability of Wave Equations with Variable Coefficients

In this paper, we consider the boundary stabilization of the wave equation with variable coefficients by Riemmannian geometry method subject to a different geometric condition which is motivated by the geometric multiplier identities. Several (multiplier) identities (inequalities) which have been built for constant wave equation by Kormornik and Zuazua are generalized to the variable coefficient case by some computational techniques in Riemmannian geometry, so that the precise estimates on the exponential decay rate are derived from those inequalitities. Also, the exponential decay for the solutions of semilinear wave equation with variable coefficients is obtained under natural growth and sign assumptions on the nonlinearity. Our method is rather general and can be adapted to other evolution systems with variable coefficients (e.g. elasticity plates) as well.     

On the solutions of a max‐type difference equation system

In this paper, we study behavior of the solution of the following max‐type difference equation system: where , the parameter A is positive real number, and the initial values x₀,y₀ are positive real numbers. Copyright © 2014 John Wiley & Sons, Ltd.     

带耗散项的浅水波方程的整体边界稳定性

在[0,1]区间上研究带耗散项的浅水波方程由边界反馈引起的整体指数稳定性.在控制边界条件u(0)=u_x(1)=u_(xx)(0)=δu_(xxx)(1)-εu(1)u_(xx)(1)-(?)(u(1))=0之下,证明了解的H1整体指数稳定性和H4整体渐进稳定性.     

Boundary stabilization of wave equations with variable coefficients

The aim of this paper is to obtain the exponential energy decay of the solution of the wave equation with variable coefficients under suitable linear boundary feedback. Multiplier method and Riemannian geometry method are used.