Synchronization of Partial Differential Systems via Diffusion Coupling

Abstract

In this paper, we address the synchronization problem of coupled partial differential systems (PDSs). First, the asymptotical synchronization and the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> synchronization of N-coupled PDSs with space-independent coefficients are considered without or with spatio-temporal disturbance, respectively. The sufficient conditions to guarantee the asymptotical synchronization and the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> synchronization are derived. The effect of the spatial domain on the synchronization of the coupled PDSs is also presented. Then the problem of asymptotical synchronization of N-coupled PDSs with space-dependent coefficients is dealt with and the sufficient condition to guarantee the asymptotical synchronization is obtained by using the Lyapunov-Krasovskii method. The condition of the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> synchronization for N-coupled PDSs with space-dependent coefficients is also presented. Both conditions are given by integral inequalities, which are difficult to be verified. In order to avoid solving these integral inequalities, we adopt the semi-discrete difference method to turn the PDSs into an equivalent spatial space state system, then the sufficient condition of the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> synchronization for N-coupled PDSs is given by an LMI, which is easier to be verified. Further, the relationship between the sufficient conditions for the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> synchronization, obtained by the Lyapunov-Krasovskii method and semi-discrete difference method respectively, is investigated. Finally, two examples of coupled PDSs are given to illustrate the correctness of our results obtained in this paper.