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By combining parabolic partial differential equation (PDE) theory with Lyapunov technique, the synchronization is studied for a class of coupled distributed parameter systems (DPS) described by PDEs. First, based on Kronecker product and Lyapunov functional, some easy-to-test sufficient condition is given to ensure the synchronization of coupled DPS with time delay. Secondly, in the case that the whole coupled system cannot synchronize by itself, the proportional-spatial derivative (P-sD) state feedback controller is designed and applied to force the network to synchronize. The sufficient condition on the existence of synchronization controller is given in terms of a set of linear matrix inequalities. Finally, the effectiveness of the proposed control design methodology is demonstrated in numerical simulations.

Most practical systems are distributed in space and time, for example, systems related to heat flows, fluid flow, or flexible structure. The systems are called distributed parameter systems (DPSs), which are mathematically modelled by partial differential equations (PDEs) with boundary conditions. The key characteristic of DPSs is that their inputs, outputs, process states, and the relevant parameters may vary temporally as well as spatially. Due to such characteristic and infinite dimensional, the distributed parameter system is difficult to control. Recent research [

The coupled system can be regarded as a large set of interconnected nodes, which can be expressed by the graph. Each node of the graph represents individual, and the edge of the graph represents the connections among them. The dynamical analysis of coupled system has become a focal topic of great interest, particularly the synchronization phenomena. Synchronization in an array of coupled dynamical systems was first investigated in [

In the case that the whole coupled system cannot synchronize by itself, some controllers should be designed and applied to force the network to synchronize. Due to the fact that the outputs, inputs, process states, and the relevant parameters of DPS may vary temporally as well as spatially, a special control strategy called proportional-spatial derivative (P-sD) control [

Motivated by the above discussion, the aim of this paper is to synchronize coupled distributed parameter systems with time delay. First, based on Kronecker product and Lyapunov functional, some easy-to-test sufficient condition is given to ensure the synchronization of coupled DPS with time delay. Secondly, in the case that the whole coupled system cannot synchronize by itself, the proportional-spatial derivative (P-sD) state feedback controller is designed and applied to force the network to synchronize. The sufficient condition on the existence of synchronization controller is given in terms of a set of linear matrix inequalities. Finally, a numerical example is given to show effectiveness of the proposed method.

Consider a dynamical network consisting of

Equivalently, system (

Let

Our objective is to investigate the synchronization of network (

The coupled distributed parameter system in (

Let

For the coupled DPS (

If the coupled distributed parameter system cannot synchronize to the isolate node (

With the control law (

Next, the asymptotical stability will be analyzed for the closed loop system (

For the coupled DPS (

Consider the following Lyapunov functional candidate for system:

Consider the following distributed parameter system:

State of the first distributed parameter system.

State of the second distributed parameter system.

State of the third distributed parameter system.

Let

Under the designed controller, Figures

State of the first distributed parameter system under the controller.

State of the second distributed parameter system under the controller.

State of the third distributed parameter system under the controller.

In this paper, we have addressed the problem of synchronization problem for a class of coupled DPS. First, some sufficient conditions are given to ensure the synchronization of the coupled DPS. Secondly, in the case that the coupled DPS cannot synchronize by itself, the P-sD state feedback control has been designed and applied to force the coupled DPS to synchronize. The controller has been developed in terms of LMIs based directly on the error system. The coupled DPS can synchronize to the isolated node under the controller. Finally, the developed design method is applied to the simulation example, and the achieved simulation results show the effectiveness and benefit of the proposed controller. It is worth pointing out that the proposed design method is easily developed for the coupled DPS with Lipschtiz nonlinear function.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University (KAU), under Grant 3-130/1434/HiCi. The authors, therefore, acknowledge the technical and financial support of KAU.