Exact Boundary Controllability and Exact Boundary Observability for a Coupled System of Quasilinear Wave Equations

Based on the theory of semi-global classical solutions to quasilinear hyperbolic systems, the authors apply a unified constructive method to establish the local exact boundary(null) controllability and the local boundary(weak) observability for a coupled system of 1-D quasilinear wave equations with various types of boundary conditions.     

Exact Boundary Controllability for a Coupled System of Wave Equations with Neumann Boundary Controls

This paper first shows the exact boundary controllability for a coupled system of wave equations with Neumann boundary controls.In order to establish the corresponding observability inequality,the authors introduce a compact perturbation method which does not depend on the Riesz basis property,but depends only on the continuity of projection with respect to a weaker norm,which is obviously true in many cases of application.Next,in the case of fewer Neumann boundary controls,the non-exact boundary controllability for the initial data with the same level of energy is shown.     

Global Exact Boundary Controllability for General First-Order Quasilinear Hyperbolic Systems

For general first-order quasilinear hyperbolic systems, based on the analysis of simple wave solutions along characteristic trajectories, the global two-sided exact boundary controllability is achieved in a relatively short controlling time.     

一类波动方程耦合系统具Dirichlet边界控制的部分精确边界同步性

作者对具有Dirichlet边界控制的一类波动方程耦合系统提出了部分精确边界同步性的概念,并对其进行了深入的讨论,得到了实现部分精确边界同步性的充要条件.同时对部分同步态进行了讨论,得到了部分同步态不依赖于边界控制的一个充分条件.     

双曲系统的节点状态的精确边界能控性

在此综述性文章中,我们将回顾关于节点状态的精确边界能控性的已有结果,并对此主题之进一步研究给出若干建议     

Generalized Exact Boundary Synchronization for a Coupled System of Wave Equations with Dirichlet Boundary Controls

This paper deals with the generalized exact boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls in the framework of weak solutions. A necessary and sufficient condition for the generalized exact boundary synchronization is obtained, and some results for its generalized exactly synchronizable states are given.     

Semi-linear Wave Equations with Effective Damping

The authors study the Cauchy problem for the semi-linear damped wave equation $$u_{tt} - \Delta u + b\left( t \right)u_t = f\left( u \right), u\left( {0,x} \right) = u_0 \left( x \right), u_t \left( {0,x} \right) = u_1 \left( x \right)$$ in any space dimension n ≥ 1. It is assumed that the time-dependent damping term b(t) > 0 is effective, and in particular tb(t) → ∞ as t → ∞. The global existence of small energy data solutions for |f(u)| ≈ |u| ^p in the supercritical case of $p > \tfrac{2} {n}$ and $p \leqslant \tfrac{n} {{n - 2}}$ for n ≥ 3 is proved.     

Exact Boundary Synchronization by Groups for a Kind of System of Wave Equations Coupled with Velocities*

This paper deals with the exact boundary controllability and the exact boundary synchronization for a 1-D system of wave equations coupled with velocities. These problems can not be solved directly by the usual HUM method for wave equations, however, by transforming the system into a first order hyperbolic system, the HUM method for 1-D first order hyperbolic systems, established by Li-Lu(2022) and Lu-Li(2022), can be applied to get the corresponding results.     

Exact Controllability and Boundary Stabilization of Torsional Vibrations of an Internally Damped Flexible Space Structure

In this paper, we study the exact controllability and boundary stabilization of the torsional vibrations of a flexible space structure (such as a solar cell array) modeled by a rectangular panel, incorporating the material damping of the structure. The panel is hoisted at one end by a rigid hub and the other end is totally free. For the attachment of this hub on one side of the panel, the hub dynamics leads to a nonstandard boundary condition. To incorporate internal damping of the material, we assume Voigt-type viscoelasticity of the structure. Exact controllability theory is established using the Hilbert uniqueness method by means of a control torque applied only on the rigid hub of the panel. At the same time, uniform exponential energy decay rate is obtained directly for the solution of this problem.     

非线性Klein-Gordon方程整体解存在的最佳条件

本文研究一类带竞争势函数的非线性Klein-Gordon方程的柯西问题.根据基 态的特征,运用势井方法和凹方法导出了该问题解爆破和整体存在的最佳条件. 同时 还回答了当初值为多小时,整体解存在这个问题.     

Klein-Gordon方程的定性分析及精确解

本文运用动力系统的定性理论,得到了Klein-Gordon方程的孤立波分支图,并求得了与定性分析结果一致的孤波解.     

Exact Boundary Synchronization for a Coupled System of Wave Equations with Neumann Boundary Controls

In this paper, for a coupled system of wave equations with Neumann boundary controls, the exact boundary synchronization is taken into consideration. Results are then extended to the case of synchronization by groups. Moreover, the determination of the state of synchronization by groups is discussed with details for the synchronization and for the synchronization by 3-groups, respectively.     

一类波动方程组的初边值问题

本文主要讨论Ｒ＾ｎ中区域Ω上的广义ＢＢＭ方程组的初边值问题，通过引入强解的概念和应用不动点原理，得到了该问题强解之存在与唯一性。文中亦讨论了解的正则性。     

Exact Boundary Controllability for the Spatial Vibration of String with Dynamical Boundary Conditions

This paper deals with the spatial vibration of an elastic string with masses at the endpoints. The authors derive the corresponding quasilinear wave equation with dynamical boundary conditions, and prove the exact boundary controllability of this system by means of a constructive method with modular structure.     

浅水波方程和Klein-Gordon方程新的精确行波解

采用了一种新的方法来求解浅水波方程和Klein-Gordon的行波解.在该方法下,Klein-Gordon方程和浅水波方程都得到了其精确的周期孤立波解,从而该方法的有效性得到了验证.     

非线性Klein-Gordon方程柯西问题解的整体存在性与Blow-up

研究非线性Klein-Gordon方程的柯西问题u_(tt)-Δu+u=u|u|~(p-1),x∈R~n,t>0;u(x,0)=u_0(x),u_t(x,0)=u_1(x),x∈R~n.通过引进一族位势井,得到了解的整体存在性与不存在的门槛结果.     

On the Exact Controllability of the Wave Equation with Interior and Boundary Controls

The exact controllability of the wave equation in a bounded domain of Rⁿ, with controls on a part of the boundary and in the interior, is studied. Feedback laws are established.     

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.     

Exact Internal Controllability and Synchronization for a Coupled System of Wave Equations*

In this paper the exact internal controllability for a coupled system of wave equations with arbitrarily given coupling matrix is established. Based on this result, the exact internal synchronization and the exact internal synchronization by p-groups are successfully considered.     

可化约拟线性双曲组带一类非局部边界条件的精确边界能控性

先证明了一类非线性积分方程解的存在唯一性,利用此结果,建立了可化约拟线性双曲组带一类非局部边界条件的单侧精确边界能控性.