**3. Stabilization of complex networks with couplings exchanging randomly**

Based on the proposed model, this section focuses on the design of stochastic pinning controller. By investigating the existing references, it is found that most of the stabilization results of complex networks are achieved by either non-delay or delay controllers. However, from the above explanations, it is said that two such controllers may not describe the actual systems very well. Here, a kind of partially delay-dependent pinning controller containing both non-delay and delay states that take place with a certain probability is proposed to deal with the general case. Without loss of generality, it is assumed that the first *l* nodes are selected to be added the desired pinning controller *ui* (*t*), which are described as

$$\begin{cases} u\_i(t) = -c\alpha(t)k\_i \mathbf{x}\_i(t) - c\left(1 - \alpha(t)\right)k\_{di} \mathbf{x}\_i(t-\tau), \; i \in \mathbb{S}\_l\\ u\_i(t) = 0, & i \in \overline{\mathbb{S}}\_l \end{cases} \tag{5}$$

where *ki* and *kdi* are the non-delayed and delayed coupling control gains, respectively. *α*(*t*) is the Bernoulli stochastic variable and is described as follows:

$$\alpha(t) = \begin{cases} 1, & \text{if } \ge(t) \text{ is valid} \\ 0, & \text{if } \ge(t-\tau) \text{ is valid} \end{cases} \tag{6}$$

whose probabilities are expressed by

where *xi*

28 Robust Control - Theoretical Models and Case Studies

also satisfy

network:

(*t*) = (*xi*1(*t*), *xi*2(*t*), …, *xin*(*t*))*<sup>T</sup>* ∈ ▯*<sup>n</sup>* is the state vector of the *i*th node. *f* : ▯*<sup>n</sup>* → ▯*<sup>n</sup>* is a

continuously differentiable function that describes the activity of an individual system.

*c* > 0 is the coupling strength among the nodes. *τ* > 0 is the coupling delay. *A* = (*aij*) ∈ ▯*<sup>N</sup>* × *<sup>N</sup>* and *B* = (*bij*) ∈ ▯*<sup>N</sup>* × *<sup>N</sup>* stand for the configuration matrices of the complex dynamical network with the non-delayed and delayed couplings, respectively. *A* and *B* can be defined as follows: for *i* ≠ *j*, if there exist non-delayed and delayed couplings between nodes *i* and *j*, then *aij* > 0 and *bij* > 0; Otherwise, *aij* = 0 and *bij* = 0, respectively. Assuming both *A* and *B* are symmetric and

1, 1,

Here, the topologies of the complex network are more general, whose related coupling matrices exchange each other randomly. That is, *A* changes into *B*, while *B* changes into *A* simultane‐ ously. In other words, matrices *A* and *B* exchange. In this case, we have the following complex

> 1 1 ( ) ( ( )) ( ) ( ), S *N N*

From these demonstrations, it is seen that the above two complex networks occur separately and randomly. To describe the above random switching between coupling matrices *A* and *B*,

when *ΔA* = (*Δaij*) ∈ ▯*<sup>N</sup>* × *<sup>N</sup>* and *ΔB* = (*Δbij*) ∈ ▯*<sup>N</sup>* × *<sup>N</sup>*. Especially, such uncertainties are selected to

**Definition 1.** The complex network (1) is asymptotically stable over topologies exchanging

\* *B A* - £ d

randomly, if the complex network (3) with condition (4) is asymptotically stable.

= =

*j j x t f x t c bx t c ax t i*

1 1 ( ) ( ( )) ( ) ( ) ( ) ( ), S

= =

*N N i i ij ij j ij ij j j j xt fxt c a a xt c b b xt i*

*i i ij j ij j*

a robust method will be exploited. That is

where *δ*\* is a given positive scalar.

be *ΔA* = *B* − *A* and *ΔB* = *A* − *B*, which is assumed to be

Before giving the main results, a definition is needed.

*N N ii ij ii ij j ji j ji a ab bi* = ¹ = ¹ =- =- Î å å <sup>S</sup>

, ,

t

> t

(4)

& = + + -Î å å (2)

& = + +D + +D - Î å å (3)

$$P\_r\{\alpha(t) = \mathbf{l}\} = \mathbb{E}\{\alpha(t) = \alpha^\*, P\_r\{\alpha(t) = 0\} = \mathbf{l} - \alpha^\*. \tag{7}$$

where *α*\* ∈ [0, 1]. In addition, it is obtained that

$$\mathbb{E}\left(\boldsymbol{\alpha}(t) - \boldsymbol{\alpha}^\*\right) = 0\tag{8}$$

Substituting *ui* (*t*) into complex network (3), one has

$$\begin{cases} \dot{\mathbf{x}}\_{i}(t) = f(\mathbf{x}\_{i}(t)) + c \sum\_{j=1}^{N} (a\_{\bar{y}} + \Delta a\_{\bar{y}}) \mathbf{x}\_{j}(t) \\ \qquad + c \sum\_{j=1}^{N} (b\_{\bar{y}} + \Delta b\_{\bar{y}}) \mathbf{x}\_{j}(t-\tau) \\ \qquad - c\alpha(t) k\_{i} \mathbf{x}\_{i}(t) - c(\mathbf{l} - \alpha(t)) k\_{\alpha} \mathbf{x}\_{i}(t-\tau), i \in S\_{\ell} \\ \dot{\mathbf{x}}\_{i}(t) = f(\mathbf{x}\_{i}(t)) \\ \qquad + c \sum\_{j=1}^{N} (a\_{\bar{y}} + \Delta a\_{\bar{y}}) \mathbf{x}\_{j}(t) \\ \qquad + c \sum\_{j=1}^{N} (b\_{\bar{y}} + \Delta b\_{\bar{y}}) \mathbf{x}\_{j}(t-\tau), i \in \overline{S}\_{\ell} \end{cases} \tag{9}$$

which is equivalent to

$$\begin{aligned} \dot{\mathbf{x}}\_{i}(t) &= f(\mathbf{x}\_{i}(t)) \\ &+ c \sum\_{j=1}^{N} (a\_{y} + \Delta a\_{y}) \mathbf{x}\_{j}(t) \\ &+ c \sum\_{j=1}^{N} (b\_{y} + \Delta b\_{y}) \mathbf{x}\_{j}(t-\tau) \\ &- c(\alpha(t) - \alpha^{\*}) k\_{i} \mathbf{x}\_{i}(t) \\ &+ c(\alpha(t) - \alpha^{\*}) k\_{\alpha^{\*}} \mathbf{x}\_{i}(t-\tau) \\ &- c\alpha^{\*} k\_{i} \mathbf{x}\_{i}(t) - c(\mathbf{l} - \alpha^{\*}) k\_{\alpha^{\*}} \mathbf{x}\_{i}(t-\tau), i \in S\_{\ell} \\ \dot{\mathbf{x}}\_{i}(t) &= f(\mathbf{x}\_{i}(t)) \\ &+ c \sum\_{j=1}^{N} (a\_{y} + \Delta a\_{y}) \mathbf{x}\_{j}(t) \\ &+ c \sum\_{j=1}^{N} (b\_{y} + \Delta b\_{y}) \mathbf{x}\_{j}(t-\tau), i \in \overline{S}\_{\ell} \end{aligned} \tag{10}$$

Assumption 1. Supposing that there exists a positive definite diagonal matrix P = diag{p1, p2, ..., p,,} and η > 0, such that

$$\mathbf{x}^{\top}(t)Pf(\mathbf{x}(t)) \le \eta \mathbf{x}^{\top}(t)\mathbf{x}\_{l}(t), \forall \mathbf{x}\_{l}(t) \in \mathbb{R}^{\kappa}, t \ge 0 \tag{11}$$

### 3.1 Stabilization realized by a partially delay-dependent pinning controller

THEOREM 1. Let Assumption 1 hold, for given scalars a\* and o\*, there exists a pinning controller (5) such that the complex network (9) is asymptotically stable over topology exchange (4), if there exist Q > 0, k; > 0, and k;a > 0, V i ESg; such that the following condition

$$
\begin{bmatrix}
2\varrho I\_N + 2c\tilde{A} + 2c\delta^\* I\_N + Q & c(\tilde{B} + \delta^\* I\_N) \\
\* & -Q
\end{bmatrix} < 0
\tag{12}
$$

is satisfied, where

$$\varphi = \frac{\eta}{\min\_{1 \le i \le n} (p\_i)}$$

$$
\tilde{A} = A - \operatorname{diag} \left( \underbrace{\alpha^\* k\_1, \alpha^\* k\_2, \dots, \alpha^\* k\_l}\_{\uparrow}, \underbrace{0, \dots, 0}\_{N-l} \right),
$$

$$\tilde{B} = B - \operatorname{diag} \underbrace{(\operatorname{l} - \alpha^\*)k\_{d1}, (\operatorname{l} - \alpha^\*)k\_{d2}, \dots, (\operatorname{l} - \alpha^\*)k\_{d\ell}}\_{l}, \underbrace{0, \dots, 0}\_{N-l} \dots$$

Proof. For complex network (9), we choose a Lyapunov function as follows:

$$V(\mathbf{x}(t)) = \frac{1}{2} \sum\_{i=1}^{N} \mathbf{x}\_i^{\tau}(t) P \mathbf{x}\_i(t) + \frac{1}{2} \sum\_{j=1}^{n} p\_j \int\_{t-\tau}^{t} \tilde{\mathbf{x}}\_j^{\tau}(s) Q \tilde{\mathbf{x}}\_j(s) ds \tag{13}$$

where x ;(t)=(xi,(t), x2;(t), ... , xx;(t))" ∈ u", j = 1, 2, ..., n, and Q is a positive definite of suitable dimensions matrix. Let L be the weak infinitesimal generator of stochastic process, it is defined as

$$\operatorname{L}V(\mathbf{x}(t)) = \lim\_{\Delta \to 0^{+}} \frac{\operatorname{E} \left\langle V(\mathbf{x}(t+\Delta)) \right\rangle - V(\mathbf{x}(t))}{\Delta} \tag{14}$$

Then, one has

LV j=1 j=1

New Stabilization of Complex Networks with Non-delayed Couplings over Random Exchanges 33 http://dx.doi.org/10.5772/62504

$$\tilde{\mathbf{x}} = \frac{1}{2} \sum\_{j=1}^{n} p\_j \left[ \tilde{\mathbf{x}}\_j^T(t) \quad \tilde{\mathbf{x}}\_j^T(t-\tau) \right] \Pi\_1 \left[ \begin{matrix} \tilde{\mathbf{x}}\_j(t) \\ \tilde{\mathbf{x}}\_j(t-\tau) \end{matrix} \right] < 0$$

where

$$
\Pi\_1 = \begin{bmatrix}
2\varrho I\_N + 2c\tilde{\mathcal{A}} + 2c\mathcal{S}^\* I\_N + \mathcal{Q} & c(\tilde{\mathcal{B}} + \mathcal{S}^\* I\_N) \\
\* & -\mathcal{Q}
\end{bmatrix},
$$

It is guaranteed by Π < 0. By condition (12), it is known that LV(x(t)) < 0. This completes the proof.

REMARK 1. It is worth mentioning that for any given function f(x;(t)), it is necessary to find suitable parameters P and n. There, P is related to f(x;(t)), where n can be obtained by the given matrix P. Moreover, Theorem 1 is also extended to other general cases that the coupling matrices A and B change to the other ones independently. Here, we only consider the special case that A and B exchanges each other.

Based on Theorem 1, it is claimed that Q is selected with a general case. However, it may be selected to be some special cases. When Q is chosen as the special case that Q=comax(B + o \*IN)IN, we will have the following corollary.

COROLLARY 1. Let Assumption 1 hold, for given scalars a\* and o\* > 0, there exists a pinning controller (5) such that the complex network (9) is asymptotically stable over topology exchange (4), if there exist k; > 0, and k;; > 0, V i ∈Sy, such that the following condition

$$c\varphi I\_N + c\tilde{A} + c\delta^\* I\_N + c\sigma\_{\text{mov}} (\tilde{B} + \delta^\* I\_N) I\_N < 0 \tag{16}$$

is satisfied, where the other symbols are defined in Theorem 1.

Proof. Based on Theorem 1 and using the Schur complement lemma, one has

$$2\sigma I\_N + 2c\bar{A} + 2c\delta^"I\_N + Q + c^2(\tilde{B} + \delta^"I\_N)Q^{-1}(\tilde{B} + \delta^"I\_N)^\top < 0 \tag{17}$$

implying /7 < 0. By choosing Q=cc max(8 + 0 \* x)/x, it is concluded that (17) is guaranteed by

$$2\mathfrak{q}I\_{\times} + 2c\tilde{A} + 2c\mathcal{S}^{\prime}I\_{\times} + 2c\sigma\_{mn}(\tilde{B} + \mathcal{S}^{\prime}I\_{\times})I\_{\times} < 0\tag{18}$$

This completes the proof.

When there is no topology exchange, we will have the following corollary directly.

**COROLLARY 2.** Let Assumption 1 hold, for given scalar *α*\*, there exists a pinning controller (5) such that the complex network (9) is asymptotically stable over topology exchange (4), if there exist *Q* > 0, *ki* > 0, and *kdi* > 0, ∀*i* ∈Sℓ, such that the following condition holds:

$$
\begin{bmatrix}
2\varrho I\_N + 2c\tilde{A} + Q & c\tilde{B} \\
\* & -Q
\end{bmatrix} < 0\tag{19}
$$

where *φ*, *Ã*, and *B*˜ are defined as those in (12).

It is seen that the expectation of *α*(*t*) in Theorem 1 plays a vital role in the control of the complex network, which needs to be given exactly. However, in practice, it may be very hard to get *α*\* exactly, and only its estimation *α*˜ is available. For an uncertain *α*\* with its estimation *α*˜, its admissible uncertainty *Δα* is defined as

$$
\Delta \mathcal{a} = \stackrel{\ast}{\alpha}^\* - \tilde{\alpha}, \tilde{\alpha} \in [0, 1] \tag{20}
$$

where *Δα* ∈ [−*μ*, *μ*] with *μ* ∈ [0, 1]. Then, we have the following theorem.

**THEOREM 2.** Let Assumption 1 hold, for given scalars *α*˜ and *δ*\* > 0, there exists a pinning controller (5) satisfying condition (20) such that the complex network (9) is asymptotically stable over topology exchange (4), if there exist *Q* > 0, *W* > 0, *ki* > 0, and *kdi* > 0, ∀*i* ∈Sℓ, such that the following conditions

$$
\begin{bmatrix}
2\rho I\_N + 2c\overline{A} + 2c\mu K\_1 + 2c\mu W\_{11} + 2c\delta^\* I\_N + Q & c(\overline{B} - \mu K\_2 + 2\mu W\_{12} + \delta^\* I\_N) \\
\ast & 2c\mu W\_{22} - Q
\end{bmatrix} < 0\tag{21}
$$

$$
\begin{bmatrix}
\* & -W\_{22}
\end{bmatrix} < 0\tag{22}
$$

hold, where

$$W = \begin{bmatrix} W\_{11} & W\_{12} \\ W\_{21} & W\_{22} \end{bmatrix},$$

$$K\_1 = \operatorname{diag} \left\{ \underbrace{k\_1, k\_2, \dots, k\_l}\_{l}, \underbrace{0, \dots, 0}\_{N-l} \right\},$$

New Stabilization of Complex Networks with Non-delayed Couplings over Random Exchanges 35 http://dx.doi.org/10.5772/62504

$$\begin{aligned} K\_2 &= \operatorname{diag} \left\{ \underbrace{k\_{d1}, k\_{d2}, \dots, k\_{dl}}\_{l}, \underbrace{0, \dots, 0}\_{N-l} \right\}, \\\\ \overline{A} &= A - \operatorname{diag} \left\{ \underbrace{\tilde{a}k\_1, \tilde{a}k\_2, \dots, \tilde{a}k\_l}\_{l}, \underbrace{0, \dots, 0}\_{N-l} \right\}, \\\\ \overline{B} &= B - \operatorname{diag} \left\{ \underbrace{(1-\tilde{a})k\_{d1}, (1-\tilde{a})k\_{d2}, \dots, (1-\tilde{a})k\_{dl}}\_{l}, \underbrace{0, \dots, 0}\_{N-l} \right\} \end{aligned}$$

Proof. Based on the proof of Theorem 1, it is known that the stabilization of complex network (9) over random exchanges with (20) is guaranteed by (12), which is equivalent to

$$
\begin{bmatrix}
2\varrho I\_N + 2c\overline{A} - 2c\Delta a K\_1 + 2c\delta^" I\_N + Q & c(\overline{B} + \Delta a K\_2 + \delta^" I\_N) \\
\* & -Q
\end{bmatrix} < 0\tag{23}
$$

It could be rewritten as

$$
\begin{bmatrix}
2\varrho I\_N + 2c\overline{A} + 2c\delta^\* I\_N + \underline{Q} & c(\overline{B} + \delta^\* I\_N) \\
\* & -\underline{Q}
\end{bmatrix}
$$

$$
+c\Delta a \begin{bmatrix}
\* & 0
\end{bmatrix} < 0
$$

That is

$$\begin{bmatrix} 2\varrho I\_N + 2c\overline{A} + 2c\delta'' I\_N + Q & c(\overline{B} + \delta'' I\_N) \\ \* & -Q \end{bmatrix} + c(\Delta a + \mu) \begin{bmatrix} -2K\_1 & K\_2 \\ \* & 0 \end{bmatrix} \tag{25}$$
 
$$-c(\Delta a + \mu)W - c\mu \begin{bmatrix} -2K\_1 & K\_2 \\ \* & 0 \end{bmatrix} + c(\Delta a + \mu)W < 0$$

which is implied by

$$
\begin{bmatrix}
2\rho I\_N + 2c\overline{A} + 2c\delta^\* I\_N + Q & c(\overline{B} + \delta^\* I\_N) \\
\* & -Q
\end{bmatrix}
$$

$$
+c(\Delta\alpha + \mu)\begin{bmatrix}
\* & -W\_{22}
\end{bmatrix} - c\mu\begin{bmatrix}
\* & 0
\end{bmatrix}
\tag{26}
$$

$$
+2c\mu\begin{bmatrix}
W\_{11} & W\_{12} \\
\* & W\_{22}
\end{bmatrix} < 0
$$

Taking into account condition (22), it is further guaranteed by

$$\begin{bmatrix} 2\varrho I\_N + 2c\overline{A} + 2c\delta^\* I\_N + Q & c(\overline{B} + \delta^\* I\_N) \\ \* & -Q \end{bmatrix} - c\mu \begin{bmatrix} -2K\_1 & K\_2 \\ \* & 0 \end{bmatrix} \tag{27}$$
  $+2c\mu \begin{bmatrix} W\_{11} & W\_{12} \\ \* & W\_{22} \end{bmatrix} < 0$ 

which is (21) actually. This completes the proof.

### **3.2 Stabilization realized by adaptive pinning controller**

When *α*\* is unknown, how to stabilize a complex network through a pinning controller should also be taken into consideration. In this section, we will exploit the adaptive pinning control method to deal with this general case.

**THEOREM 3.** Let Assumption 1 hold, for given scalar *δ*\*, if there exist *Q* > 0, *ki* > 0, and *kdi* > 0, ∀*i* ∈Sℓ, such that the following condition

$$
\begin{bmatrix}
2\varrho I\_N + 2c\hat{A} + 2c\boldsymbol{\delta}^\* I\_N + \underline{Q} & \mathfrak{c}(\hat{B} + \boldsymbol{\delta}^\* I\_N) \\
\* & -\underline{Q}
\end{bmatrix} < 0\tag{28}
$$

holds with *Â* = *A* − *K*1 and *B* ^ <sup>=</sup> *<sup>B</sup>* <sup>−</sup> *<sup>K</sup>*2, then the complex network (9) is asymptotically stable over topology exchange (4) under the adaptive pinning controller

$$\begin{cases} u\_i(t) = -ck\_i \mathbf{x}\_i(t) - ck\_{di} \mathbf{x}\_i(t-\tau) + \mathbf{v}\_i(t), i \in \mathbf{s}\_\ell\\ u\_i(t) = 0, & i \in \overline{\mathbf{s}}\_\ell \end{cases} \tag{29}$$

where

() () () ˆ *i i vt c txt* = a

New Stabilization of Complex Networks with Non-delayed Couplings over Random Exchanges 37 http://dx.doi.org/10.5772/62504

and the updating law

$$\dot{\hat{\alpha}}(t) = \begin{cases} 0, & \text{if } \hat{\alpha}(t) = \text{l} \\ \dot{\alpha} \sum\_{i=1}^{l} \mathbf{x}\_{i}^{\top}(t) P \mathbf{x}\_{i}(t), & \text{others} \end{cases} \tag{30}$$

where ∀ό>0 and âº∈[0, 1].

Proof. Here, the Lyapunov function is defined as

$$\begin{split} V(\mathbf{x}(t)) &= \frac{1}{2} \sum\_{i=1}^{N} \mathbf{x}\_{i}^{\top}(t) P \mathbf{x}\_{i}(t) \\ &+ \frac{1}{2} \sum\_{j=1}^{n} p\_{j} \int\_{t-\tau}^{t} \tilde{\mathbf{x}}\_{j}^{\top}(s) Q \tilde{\mathbf{x}}\_{j}(s) ds \\ &+ \frac{1}{2\mathsf{\tilde{G}}} \tilde{\alpha}(t) \tilde{\alpha}(t) \end{split} \tag{31}$$

where a(t)=a(t)=a \*, x ;(t), and Q are same as the ones in (13). Then, it is obtained

I

1 1 \* 1 1 1 \* \* 1 1 1 1 () () ( ) ( ) 2 2 <sup>1</sup> ( () ) () ˆ ˆ ˆ () () () () ˆ () ( ) () () () ( ) = = = = = = = + - -- + - £ + + - + +- + å å å å å å å %% % % & %% % % % % %% %% *n n T T jj j j j j j j n n T T jj j jj j j j <sup>n</sup> <sup>T</sup> jj j j n n T T jj j jj j j j p x t Qx t p x t Qx t t t p x t x t c p x t Ax t c p x t Bx t c px tx t c px tx t x* t t a aa j t d dt *ó* 1 1 1 \* 1 1 1 \* 1 \* 1 () () 1 1 () () ( ) ( ) 2 2 <sup>1</sup> ( () ) () ˆ ˆ ˆ () () () () ˆ () ( ) () () = = = = = = = = + - -- + - £ + + - + + å å å å å å å å %% % % & %% % % % % % % % *<sup>l</sup> <sup>T</sup> i i i n n T T jj j j j j j j n n T T jj j jj j j j <sup>n</sup> <sup>T</sup> jj j j <sup>n</sup> <sup>T</sup> jj j j n j j t Pv t p x t Qx t p x t Qx t t t p x t x t c p x t Ax t c p x t Bx t c px tx t c px* t t a aa j t d d *ó* 1 1 2 1 () ( ) 1 () () <sup>2</sup> <sup>1</sup> ( )( ) <sup>2</sup> <sup>1</sup> ( ) () ( ) <sup>0</sup> 2 ( ) = = = - + - - é ù <sup>=</sup> é ù ë û -P < ê ú - ë û å å å % % % % % % % % % *T j j <sup>n</sup> <sup>T</sup> jj j j <sup>n</sup> <sup>T</sup> j j j j <sup>n</sup> T T <sup>j</sup> jj j j j tx t p x t Qx t p x t Qx t x t p xt xt x t* t t t t t

where

$$
\Pi\_2 = \begin{bmatrix}
2\varrho I\_N + 2c\hat{A} + 2c\mathcal{S}^\* I\_N + \mathcal{Q} & c(\hat{B} + \mathcal{S}^\* I\_N) \\
\ast & -\mathcal{Q}
\end{bmatrix},
$$

This completes the proof.

On the other hand, it is obtained that *δ*\* is also important to the control of the complex network. When it is unavailable, how to get the sufficient condition for the stabilization of complex network is an interesting problem to be discussed. In the next, such a problem will be solved by the following theorem.

**THEOREM 4.** Let Assumption 1 hold, for given scalar *α*\*, if there exist *Q* > 0, *ki* > 0, and *kdi* > 0, ∀*i* ∈Sℓ, such that the following condition

$$
\begin{bmatrix}
2\varrho I\_N + 2c\tilde{A} + \mathcal{Q} & c\tilde{B} \\
\* & -\mathcal{Q}
\end{bmatrix} < 0
\tag{33}
$$

holds, then the complex network (9) is asymptotically stable over topology exchange (4) under the adaptive pinning controller

$$\begin{cases} u\_i(t) = -c\alpha(t)k\_i\mathbf{x}\_i(t) - c(\mathbf{l} - \alpha(t))k\_{\mathrm{ad}}\mathbf{x}\_i(t - \tau) + \varpi\_i(t), & i \in \mathrm{s}\_\ell \\ u\_i(t) = 0, & i \in \mathbb{N} \end{cases} \tag{34}$$

where

1 1

*n n T T*

= =

*j j*

å å

ˆ () () () ()

%% % %

*p x t x t c p x t Ax t*

*jj j jj j*

1 1 () () ( ) ( ) 2 2

%% % %

*p x t Qx t p x t Qx t*

t

dt

t

 t

 t

*jj j j j j*

+ - --

\*

*t t*

&

<sup>1</sup> ( () ) () ˆ ˆ

 aa

£ +

+ -

*c p x t Bx t*

% %

*jj j*

() ()

*t Pv t*

1 1

= =

*j j*

å å

\* \* 1 1

= =

*j j*

1 1

*n n T T*

= =

*j j*

å å

\*

*t t*

<sup>1</sup> ( () ) () ˆ ˆ

 aa

£ +

+ -

*c p x t Bx t*

% %

*jj j*

() ()

*c px tx t*

*<sup>n</sup> <sup>T</sup> jj j*

%

*T j j*

1 () () <sup>2</sup>

*<sup>n</sup> <sup>T</sup>*

<sup>1</sup> ( )( ) <sup>2</sup>

*j j j*

*jj j*

% %

*p x t Qx t*

t

% %

*p x t Qx t*

*jj j*


=

*j n j j*

å

1 1

&

= =

*j j*

å å

ˆ () ( )

*n n T T*

% %

%

*tx t*

() ( )


t

å å

() () () ( )

%% %%

1 1 () () ( ) ( ) 2 2

%% % %

*p x t Qx t p x t Qx t*

*jj j j j j*

+ - --

ˆ () () () ()

2

*x t*

\* \*

*Q*

 d

t

<sup>1</sup> ( ) () ( ) <sup>0</sup> 2 ( )

t

% % % %

ë û -P < ê ú - ë û

ˆ ˆ 2 22 ( )

 ++ + + P = ê ú ë û -

é ù

*NN N I cA c I Q c B I*

 t

é ù <sup>=</sup> é ù

*p xt xt x t*

\*

jd

*<sup>n</sup> T T <sup>j</sup>*

*j j*

%% % %

*p x t x t c p x t Ax t*

*jj j jj j*

t

+ +-

*c px tx t c px tx t*

*n n T T jj j jj j*

t

ˆ () ( )

*n n T T*

1

d

*x*

*<sup>l</sup> <sup>T</sup> i i*

+ -

a

1 \* 1

d

*<sup>n</sup> <sup>T</sup>*

=

*j*

j

*ó*

å

+

+

\* 1

d

+

=

1

=

*j <sup>n</sup> <sup>T</sup>*

å

1

=

*j*

å

1

=

2

where

This completes the proof.

å

å

*c px*

1

=

*i*

å

*<sup>n</sup> <sup>T</sup>*

+ -

a

=

*j*

j

*ó*

38 Robust Control - Theoretical Models and Case Studies

å

+

$$\sigma\_i(t) = \begin{cases} 0, & \text{if } \sum\_{i=1}^{l} \mathbf{x}\_i^T(t) P \mathbf{x}\_i(t) = \mathbf{0} \\\\ -c\hat{\boldsymbol{\delta}} \mathbf{x}\_i(t) [2\sum\_{i=1}^{N} \mathbf{x}\_i^T(t) P \mathbf{x}\_i(t) + \sum\_{i=1}^{N} \mathbf{x}\_i^T(t-\tau) P \mathbf{x}\_i(t-\tau)]}{\sum\_{i=1}^{l} \mathbf{x}\_i^T(t) P \mathbf{x}\_i(t)}, & \text{others} \end{cases}$$

and the updating law

$$\dot{\hat{\delta}} = 2\xi \mathbf{c} \sum\_{i=1}^{N} \mathbf{x}\_{i}^{\top}(t) P \mathbf{x}\_{i}(t) + \xi \mathbf{c} \sum\_{i=1}^{N} \mathbf{x}\_{i}^{\top}(t-\tau) P \mathbf{x}\_{i}(t-\tau) \tag{35}$$

where *ξ* is a positive constant and *δ*0 ≥ 0.

**Proof.** For this case, we choose the Lyapunov function as

$$V(t) = \frac{1}{2} \sum\_{i=1}^{N} \mathbf{x}\_i^{\top}(t) P \mathbf{x}\_i(t) + \frac{1}{2} \sum\_{j=1}^{n} p\_j \int\_{t-\varepsilon}^{t} \tilde{\mathbf{x}}\_j^{\top}(s) Q \tilde{\mathbf{x}}\_j(s) ds + \frac{1}{2\xi} \tilde{\delta}^2 \tag{36}$$

where *δ*˜ =*δ* ^ −*δ* \* . Then, it is obtained

$$\begin{split} & \mathcal{E}(\boldsymbol{\beta},t) \\ &= \mathcal{E}\_{\boldsymbol{\beta}}^{(t)}(t) P\_{\boldsymbol{\beta}}(\boldsymbol{\beta}(t),\boldsymbol{\zeta}(t)) + \mathcal{E}\_{\sum}^{\bar{\gamma}}(\boldsymbol{\beta}\_{\bar{\ell}} + \boldsymbol{\Delta}\boldsymbol{\beta}\_{\bar{\ell}}) \boldsymbol{\chi}\_{\boldsymbol{\ell}}(t) \\ &+ \mathcal{E}\_{\sum}^{\bar{\ell}\_{\bar{\ell}}}(\boldsymbol{\beta}\_{\bar{\ell}} + \boldsymbol{\Delta}\boldsymbol{\beta}\_{\bar{\ell}}) \boldsymbol{\chi}\_{\boldsymbol{\ell}}(t-\tau) \\ &- \alpha\xi^{\ell} + \sum\_{i=1}^{s} \boldsymbol{\chi}\_{\boldsymbol{\ell}}^{\ell} \boldsymbol{\chi}\_{\boldsymbol{\ell}}^{\ell}(t) P\_{\bar{\ell}}(\boldsymbol{\beta}(t-\tau)) - \alpha\{1-\alpha\} \sum\_{i=1}^{s} \boldsymbol{\delta}\_{\bar{\ell}} \boldsymbol{\chi}\_{\boldsymbol{\ell}}^{\ell}(t) P\_{\bar{\ell}}(\boldsymbol{\eta}(t-\tau)) \\ &+ \sum\_{i=1}^{s} \boldsymbol{\chi}\_{\boldsymbol{\ell}}^{\ell}(t) P\_{\bar{\ell}}(\boldsymbol{\beta}) \\ &+ \sum\_{i=1}^{s} \boldsymbol{\chi}\_{\bar{\ell}}^{\ell}(t) P\_{\bar{\ell}}(\boldsymbol{\beta}(t) - \boldsymbol{\delta}\_{i}^{\ell} - \boldsymbol{\delta}\_{i}^{\ell}(t-\tau)) \boldsymbol{\Delta}\_{i}^{\ell}(t-\tau) \\ &+ \sum\_{i=1}^{s} \boldsymbol{\delta}\_{\bar{\ell}}^{\ell}(t) P\_{\bar{\ell}}(\boldsymbol{\beta}(t) + \boldsymbol{\epsilon}) \sum\_{i=1}^{s} \boldsymbol{\chi}\_{\bar{\ell}}^{\ell}(t) P\_{\bar{\ell}}(\boldsymbol{\$$

New Stabilization of Complex Networks with Non-delayed and Delayed Couplings over Random Exchanges http://dx.doi.org/10.5772/62504 41

1 1 1 \* 1 \* 1 \* 1 1 1 () () () () () ( ) () () () ( ) ( )( ) 1 1 () () ( ) 2 2 *n n T T jj j jj j j j <sup>n</sup> <sup>T</sup> jj j j <sup>n</sup> <sup>T</sup> jj j j <sup>n</sup> <sup>T</sup> jj j j <sup>n</sup> <sup>T</sup> jj j j n n T T jj j j j j j j p x t x t c p x t Ax t c p x t Bx t c px tx t c px tx t c px t x t p x t Qx t p x t Qx* j t d d t d tt t = = = = = = = = £ + + - - + - - -- + - å å å å å å å å % %% % % % % % % % % % % % %% % % 1 1 1 \* 1 2 1 ( ) <sup>1</sup> ( )( ) () <sup>2</sup> () ( ) <sup>1</sup> ( )( ) <sup>2</sup> [ () () () ( ) ( ) ( )] <sup>1</sup> ( ) () ( ) <sup>2</sup> ( *<sup>n</sup> <sup>T</sup> jj N j j <sup>n</sup> <sup>T</sup> jj j j <sup>n</sup> <sup>T</sup> j j j j <sup>n</sup> T T jj j j j j T j j <sup>n</sup> T T <sup>j</sup> jj j j j t p x t I cA Q x t p x t cBx t p x t Qx t c p x tx t x tx t xt xt x t p xt xt x t* t j t t t d t t t t t = = = = = - £ + + + - - -- +- +- -- - <sup>=</sup> é ù - P ë û å å å å å % % % % % % % % %% %% % % % % % % \* 1 3 1 ) 1 ( ) 2 () ( ) <sup>1</sup> ( ) 2 <sup>1</sup> ( ) () ( ) <sup>0</sup> 2 ( ) *n N N T T j jj j j j N N <sup>n</sup> T T <sup>j</sup> jj j j j I I x t c p xt xt x t I I x t p xt xt x t* d t t t t = = é ù ê ú ë û é ù -ê úé ù + - é ù ë ûê úê ú - ê úë û - ê ú ë û é ù £ é ù ë û -P < ê ú - ë û å å % % % % % % % %

where

1 1

= + +D

= =

*i j*

*ij ij j*

*c b bxt*

+ +D -

*N N <sup>T</sup>*

å å

( ) ( )]

*<sup>n</sup> T T*

() ()

v

*i i*

*x tP t*

\*

d d

ˆ )

&

( ) [ ( ( )) ( ) ( )

*i i ij ij j*

*x tPf xt c a a x t*

*l l T T ii i di i i*

t

*c k x t Px t c k x t Px t*


( ) ( ) (1 ) ( ) ( )

t

() () () [ ( ) ()

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( ) ( ) (1 ) ( ) ( )

t

at

 t

at

(37)

 t

\* \* 1 1

*i i*

= =

å å

<sup>1</sup> [ ( ) ( ) ( ) ( )] <sup>2</sup>

*p x t Qx t x t Qx t*

+ -- -

å %% % %

1 11

£ + + D

= ==

*i ij*

*N NN T T*

å åå

\* \* 1 1

*i i*

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å å

<sup>1</sup> [ ( ) ( ) ( ) ( )] <sup>2</sup>

%% % %

() () () ()

*jj j jj j*

*p x t x t c p x t Ax t*

% %% % %

() () () ( )

dt

t

( )


*t* t

*p x t Qx t x t Qx t*

+ -- -

*jj j j j*

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t

*c k x t Px t c k x t Px t*


( ) ( )]

*ij ij j*

+ +D -

*b bxt*

() ()

v

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1 1

= =

å å

*j j*

*jj j*

% % %

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+ -

() ( )

*n n T T*

\* \* 1 1

*j j*

= =

å å

() ()

v

*i i*

*x tP t*

\*

<sup>1</sup> ˆ ˆ ( )

+ - &

d dd

1 1

= =

å å

*j j*

1 1 () () ( ) 2 2

*jj j j j j*

%% % %

*p x t Qx t p x t Qx*

*n n T T*

+ --

*n n T T jj j jj j*

*c px tx t c px tx t*

+ +-

%% %%

t

*i i*

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\*

&

£ +

*jj j j j*

1

=

*j*

a

*N*

å

L ( ( ))

40 Robust Control - Theoretical Models and Case Studies

*V xt*

1

=

å

*i*

+

*<sup>l</sup> <sup>T</sup>*

1

=

d


1 ˆ (

1

=

a

*j*

*N*

å

1

=

å

*i*

+

*<sup>l</sup> <sup>T</sup>*

1

<sup>1</sup> ˆ ˆ ( )

=

d dd

1

=

å

*j*

d

*<sup>n</sup> <sup>T</sup>*

1

=

x

å

*i*

+

*<sup>l</sup> <sup>T</sup>*

å

*j*

+ -

x

j

x

h

+

*j*

$$
\Pi\_3 = \begin{bmatrix}
2\varrho I\_N + 2c\tilde{A} + Q & c\tilde{B} \\
 c\tilde{B}^T & -Q
\end{bmatrix}
$$

It is guaranteed by *Π*3 < 0 which is equivalent to (33). This completes the proof.
