Practical synchronization on complex dynamical networks via optimal pinning control

We consider practical synchronization on complex dynamical networks under linear feedback control designed by optimal control theory. The control goal is to minimize global synchronization error and control strength over a given finite time interval, and synchronization error at terminal time. By utilizing the Pontryagin's minimum principle, and based on a general complex dynamical network, we obtain an optimal system to achieve the control goal. The result is verified by performing some numerical simulations on Star networks, Watts-Strogatz networks, and Barabási-Albert networks. Moreover, by combining optimal control and traditional pinning control, we propose an optimal pinning control strategy which depends on the network's topological structure. Obtained results show that optimal pinning control is very effective for synchronization control in real applications.

Figure 1

The synchronization errors $e\left(t\right)$ on different complex networks with different controls, where $e\left(t\right)={\sum }_{i=1}^n{\parallel e_i\left(t\right)\parallel }_2$. With the parameters (i), (a)–(c) are obtained on the Star network, Watts-Strogatz network, and BA network, respectively.

Figure 2

The synchronization errors $e\left(t\right)$ on different complex networks with different controls. With the parameters (ii), (a)–(c) are obtained on the Star network, Watts-Strogatz network, and Barabási-Albert (BA) network, respectively.

Figure 3

The values of the objective function (3) on different complex networks with different controls. With the parameters (i), (a)–(c) are obtained on the Star network, Watts-Strogatz network, and BA network, respectively; with the parameters (ii), (d)–(f) are obtained on the Star network, Watts-Strogatz network, and BA network, respectively.

Figure 4

The dynamical change processes of the optimal controls $u_1^{*}\left(t\right)$ on different complex networks with the parameters (i), where $u_i^{*}\left(t\right),i=2,3,\cdots ,50$ have similar dynamical change processes, which are omitted for clarity.

Figure 5

The average control strengths $v_i$ of the optimal control for each node in the network with the parameters (i).

Figure 6

The synchronization errors $e\left(t\right)$ on the Star network with the optimal pinning controls and different constant pinning controls on the first node.

Figure 7

The values of objective function (3) on the Star network with the optimal pinning control, and different constant pinning controls on the first node with control strengths from $0.2b$ to $b$. “Opt1” and “Opt2” denote $u_1^{*}$ and $u_2^{*}$, respectively.

Figure 8

The dynamical change processes of the optimal controls on the Star network with size $n=10$. The average control strengths are $v_1=1.6494, v_2=1.3547$.