Synchronization for a class of coupled linear partial differential systems via boundary control☆
Introduction
As a kind of collective behavior of complex systems, synchronization exists in a widespread field, ranging from natural systems to manmade networks and it has received substantial attention in recent years. The study on the synchronization not only can uncover the mechanism of the synchronization but also may provide the theoretical bases to utilize this special character to benefit our human beings. As a result, many studies on the synchronization have been reported, see [1], [2], [3], [4], [5], [6] and the references therein for a survey.
In the real world, there are many phenomena, such as in chemical engineering, neurophysiology and biodynamics, etc., in which state variables depend on not only the time but also the spatial position. These phenomena are generally modeled in spatial-temporal domain by partial differential systems (PDSs). A great deal of concern has been raised on the study of PDSs [7], [8], [9], [10], [11], [12], [13], [14] and the boundary control problem has caught much attention being the easiness for applications [10], [15], [11], [12], [16], [17], [18]. For example, in [10], the authors considered the boundary stabilization and boundary control problems for semilinear parabolic and hyperbolic systems, respectively. In [16], the authors consider the boundary control problem for nonlinear parabolic PDSs. Using the fuzzy method, they designed the boundary controller for the parabolic PDSs with Neumann boundary conditions. In [17], [18], the authors dealt with the stabilization of the PDS with boundary disturbances. Moreover, the backsetpping method, developed in [19], [15], has attracted attentions [15], [11], [20], [9]. In [9], the authors studied the stabilization problem for coupled PDSs based on the backsteping boundary control.
Recently, synchronization of coupled PDSs has also been one of the research focus [21], [22], [23], [24], [25], [26], [27]. Based on the inequality techniques and Lyapunov functional method, the synchronization and/or synchronization were considered in these papers for different kinds of PDSs with or without delays. In [26], the author considered the synchronization for a class of parabolic PDSs and the cases for full state availability and partial state availability are studied and the observer-based controller was designed. However, the boundary control synchronization problem for the coupled PDSs still gets off the ground. In [28], the authors studied the exponential synchronization of coupled PDSs with Neumann boundary conditions. They designed the boundary controller by employing the Lyapunov׳s direct method, the vector-valued Wirtinger׳s inequality and the technique of integration by parts. However, the techniques used in that paper can not be generalized to the case of coupled PDSs with Dirichlet boundary conditions, which motivates this study.
The contributions of this paper lie in the following aspects. We present the boundary controller of synchronization for the coupled PDSs with the Dirichlet boundary conditions. The same problem with the Neumann boundary conditions or mixed boundary conditions is considered in [28] . However, the methods used in that paper can not deal with the case with Dirichlet boundary conditions. To overcome this problem, we turn to the backstepping method, developed in [20]. Another problem appears immediately. As well known, the backstepping method is suitable to handle the scalar PDS or some very special coupled systems [9] and loses its efficacy for the vector PDSs. Nonsingular matrix transformation method [2] can decouple the coupled systems, i.e, it can turn the vector PDSs into the scalar PDS. In this paper, we combined these two methods, i.e., the nonsingular matrix transformation method and the backstepping method, to study the boundary control of synchronization for the coupled PDSs with Dirichlet boundary conditions. The combined method builds the bridge to handle the boundary control of coupled PDSs with Dirichlet boundary conditions. Finally, a numerical example is given to illustrate the effectiveness of the proposed design method.
Section snippets
Model description and preliminaries
In this note, we consider the following N-coupled linear PDSswhere is the state variable of the i-th subsystem, and are the spatial variable and time variable, respectively. The system coefficients a, b are constants and . The constant c stands for the coupling strength. The coupling matrix is defined as follows: if there exists a connection between the i-th subsystem and the j-th subsystem (),
Boundary control for the synchronization of coupled PDSs
In this section, we present the boundary control strategy which ensures the coupled PDSs (2.1) achieving synchronization. Since the transformation is a nonsingular matrix transformation, it is obvious that the boundary stabilization of (2.5) means the boundary stabilization of (2.4) which implies the boundary synchronization of the coupled PDSs (2.1). Therefore, in this section, we consider the boundary control of (2.5). The backstepping method is used in the analysis.
For the convenience
Examples
In this section, we present a numerical example to show the effectiveness of our result obtained in this note.
The following coupled partial differential systems are considered
The following boundary conditions and initial value are posed The coupling matrix G is chosen asthen we getand
Take the
Conclusion
This paper considers the problem of synchronization via boundary control for coupled linear PDSs. Based on the synchronization error dynamics and in light of the nonsingular matrix transformation method, we decouple the coupled synchronization error systems and turn the synchronization problem of the coupled PDSs into the asymptotical stabilization problem. Making use of the backstepping transformation, we transform the decoupled systems into an exponentially stable systems. By virtue of some
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This work was supported by the Program for IBRSEM in Harbin Institute of Technology under grant HIT.IBRSEM.A.201415, and by the Foundation of Supporting Technology for Aerospace under Grant 2014-HIT-HGD7 and by National Natural Science Foundations of Shandong Province under Grant ZR2014FQ014.