On the generalized exact boundary synchronization for a coupled system of wave equations

In this paper, we deliver a normalized synchronization transformation to study the generalized exact boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls. The clear relationship among the generalized exact boundary synchronization, the exact boundary null controllability, and the generalized exactly synchronizable states is precisely obtained. This approach gives further a forthright decomposition for the generalized exact boundary synchronization problem, whereby, we gain directly the determination of generalized exactly synchronizable states.     

Determination of generalized exact boundary synchronization matrix for a coupled system of wave equations

For a coupled system of wave equations with Dirichlet boundary controls, this paper deals with the possible choice of its generalized synchronization matrices so that the admissible generalized exact boundary synchronizations for this system are obtained.     

On exact boundary controllability for linearly coupled wave equations

In this paper we study exact boundary controllability for a system of two linear wave equations coupled by lower order terms. We obtain square integrable control of Neuman type for initial state with finite energy, in nonsmooth domains of the plane.     

Partial approximate boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls

In this paper, we propose the concept of partial approximate boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls, and make a deep discussion on it. We analyze the relation-ship between the partial approximate boundary synchronization and the partial exact boundary synchronization, and obtain sufficient conditions to realize the partial approximate boundary synchronization and necessary conditions of Kalman's criterion. In addition, with the help of partial synchronization decomposition, a condition that the approximately synchronizable state does not depend on the sequence of boundary controls is also given.     

A note on the exact synchronization by groups for a coupled system of wave equations

By means of a non‐exact controllability result, we show the necessity of the conditions of compatibility for the exact synchronization by two groups for a coupled system of wave equations with Dirichlet boundary controls. Copyright © 2013 John Wiley & Sons, Ltd.     

A note on the exact synchronization by groups for a coupled system of wave equations

By means of a non‐exact controllability result, we show the necessity of the conditions of compatibility for the exact synchronization by two groups for a coupled system of wave equations with Dirichlet boundary controls. Copyright © 2014 John Wiley & Sons, Ltd.     

Abstract wave equations with generalized Wentzell boundary conditions

We introduce a general framework which allows to verify if abstract wave equations with generalized Wentzell boundary conditions are well-posed, i.e., are governed by a cosine family. As an example we study wave equations for second order differential operators on C[0,1] with non-local Wentzell-type boundary conditions. Moreover, in Appendix A we give a perturbation result for sine and cosine families.     

Initial boundary value problem for a system of generalized IMBq equations

The existence and uniqueness of the global generalized solution and the global classical solution to the initial boundary value problem for a system of generalized IMBq equations are proved. This paper also arrives at some sufficient conditions of blow up of the solution in finite time. Copyright © 2004 John Wiley & Sons, Ltd.     

Global exact boundary controllability for 1‐D quasilinear wave equations

Based on the local exact boundary controllability for 1‐D quasilinear wave equations, the global exact boundary controllability for 1‐D quasilinear wave equations in a neighborbood of any connected set of constant equilibria is obtained by an extension method. Similar results are also given for a kind of general 1‐D quasilinear hyperbolic equations. Copyright © 2010 John Wiley & Sons, Ltd.     

Positive solutions for a system of difference equations with coupled multi-point boundary conditions

We investigate the existence and nonexistence of positive solutions for a system of nonlinear second-order difference equations with parameters subject to coupled multi-point boundary conditions.     

Numerical quenching of a system of equations coupled at the boundary

We study numerical approximations of solutions of the following system of heat equations, coupled at the boundary through a nonlinear flux: where p and q are parameters. We prove that the solutions of a semidiscretization in space quench in finite time. Moreover, we describe in terms of p and q the simultaneous versus non‐simultaneous quenching phenomena. We also find the numerical extinction sets. Finally, in order to obtain the correct quenching rates in the non‐simultaneous case we present some adaptive methods. Copyright © 2009 John Wiley & Sons, Ltd.     

The approximately synchronizable state for a kind of coupled system of wave equations

This paper deals with the approximately synchronizable state by groups for a kind of coupled system of wave equations with Dirichlet boundary controls. So far, the approximate boundary synchronization for a kind of coupled system of wave equations has already been deeply studied; however, the study on the approximately synchronizable state still needs to be done in details. In this paper, we will give some results on the determination of approximately synchronizable state by groups and the attainable set of them.     

New exact Jacobi elliptic functions solutions for the generalized coupled Hirota-Satsuma KdV system

In this paper, an generalized Jacobi elliptic functions expansion method with computerized symbolic computation is used for constructing more new exact Jacobi elliptic functions solutions of the generalized coupled Hirota-Satsuma KdV system. As a result, eight families of new doubly periodic solutions are obtained by using this method, some of these solutions are degenerated to solitary wave solutions and triangular functions solutions in the limit cases when the modulus of the Jacobi elliptic functions m → 1 or 0, which shows that the applied method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear partial differential equations arising in mathematical physics.     

Polynomial stability of a coupled system of wave equations weakly dissipative

In this work, we consider a coupled system of wave equation. We show that the solution of this system has a polynomial rate of decay as time tends to infinity, but does not have exponential decay. We presented a class of examples of application of the main result.     

Strong asymptotic stability of a compactly coupled system of wave equations

We prove the well-posedness and study the strong asymptotic stability of a compactly coupled system of wave equations with a nonlinear feedback acting on one end only.     

Blow-up for a semilinear reaction-diffusion system coupled in both equations and boundary conditions

We study the blow-up behavior for a semilinear reaction-diffusion system coupled in both equations and boundary conditions. The main purpose is to understand how the reaction terms and the absorption terms affect the blow-up properties. We obtain a necessary and sufficient condition for blow-up, derive the upper bound and lower bound for the blow-up rate, and find the blow-up set under certain assumptions.     

Positive solutions for singular boundary value problems of a coupled system of differential equations

By using fixed point theorem of cone expansion and compression, this paper investigates the existence of multiple positive solutions for singular boundary value problems of a coupled system of nonlinear ordinary differential equations.     

Asymptotic stability of the exact boundary controllability of nodal profile for 1D quasi-linear wave equations

In this paper, we consider the asymptotic stability of the exact boundary controllability of nodal profile for 1D quasi-linear wave equations. First, for 1D quasi-linear hyperbolic systems with zero eigenvalues, we establish the existence and uniqueness of semiglobal classical solution to the one-sided mixed initial-boundary value problem on a semibounded initial axis and discuss the asymptotic behavior of the corresponding solutions under different hypotheses on the initial data. Based on these results, we obtain the asymptotic stability of the exact boundary controllability of nodal profile for 1D quasi-linear wave equations on a semibounded time interval.