Hybrid Memristor-Based Impulsive Neural Networks

Abstract

This chapter introduces a class of heterogeneous delayed impulsive neural networks with memristors and focuses on their collective evolution for multisynchronization. The multisynchronization represents a diversified collective behavior that is inspired by multitasking as well as observations of heterogeneity and hybridity arising from system models. In view of memristor, the memristor-based impulsive neural network is first represented by an impulsive differential inclusion. According to the memristive and impulsive mechanism, a fuzzy logic rule is introduced, and then a new fuzzy hybrid impulsive and switching control method is presented correspondingly. It is shown that using the proposed fuzzy hybrid control scheme, multisynchronization of interconnected memristor-based impulsive neural networks can be guaranteed with a positive exponential convergence rate. The heterogeneity and hybridity in system models thus can be indicated by the obtained error thresholds that contribute to the multisynchronization. Numerical examples are presented and compared to demonstrate the effectiveness of the developed theoretical results.

Appendix

Proof of Theorem 6.1

Proof

Construct a collection of Lyapunov functions as

$$\displaystyle \begin{aligned} {W_i}\left( t \right) = \big\|e_i(t)\big\|, \quad i\in\digamma_q,~ q=1,\cdots,\mathcal{Q}.\end{aligned} $$

(6.20)

First, consider the flow part of (6.11). Let $\widetilde {\mathcal {D}} W_i(t)$ be the Lie derivative of W _i, and ∂W _i(t) the generalized gradient of W _i at e _i, as given by Definition 6.1. Based on (6.11), for any $\mu _i\in \widetilde {\mathcal {D}}W_i(t)$, t ∈ (t _k−1, t _k], there exists $\varrho _i\in \mathcal {K}[\dot {e}_i]$ such that $\mu _i = \nu _i^\top \varrho _i$, with ν _i = ∂W _i(t). Note that, for any $\varrho _i\in \mathcal {K}[\dot {e}_i]$, there also exist △B ∈co[−B ⁽¹⁾, B ⁽¹⁾], △C ∈co[−C ⁽¹⁾, C ⁽¹⁾], and $\triangle \tilde {C}\in \mathrm {co}[-\tilde {C}^{(1)},\tilde {C}^{(1)}]$ such that

where $\tilde {\varXi }(z_q(t),z_q(t-\tau (t)))=-\big (B^{(0)}-B+\triangle B\big ) z_q(t) +\big (C^{(0)}-C+\triangle C\big ) g(z_q(t)) +\big (\tilde {C}^{(0)}-\tilde {C}+\triangle \tilde {C} \big ) g(z_q(t-\tau (t)))$.

Taking $\nu _i=\dot {e}_i^\top \varPsi (e_i,\dot {e}_i)$, equation $\mu _i = \nu _i^\top \varrho _i$ gives

(6.21)

Next, the right-hand side of Eq. (6.21) is discussed.

By Assumption 6.1, one has

With $\sum _{i=1}^N l_{ij}=0$, the coupling term satisfies

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle -\Big[\sum_{j=1}^N l_{ij}\varGamma x_{j}(t)\Big]^\top\varPsi(e_i,\dot{e}_i) \\ &\displaystyle = &\displaystyle -\Big[\sum_{q=1}^{\mathcal{Q}}\sum_{j\in \digamma_q} l_{ij}\varGamma \big(x_{j}(t)-z_q(t)+z_q(t)\big)\Big]^\top\varPsi(e_i,\dot{e}_i)\\ &\displaystyle = &\displaystyle -\Big[\sum_{q=1}^{\mathcal{Q}}\sum_{j\in \digamma_q} l_{ij}\varGamma \big(x_{j}(t)-z_q(t)\big)\Big]^\top\varPsi(e_i,\dot{e}_i) \\ &\displaystyle &\displaystyle \quad +\Big[\sum_{q=1}^{\mathcal{Q}}\sum_{j\in \digamma_q} l_{ij}\varGamma z_q(t)\Big]^\top \varPsi(e_i,\dot{e}_i) \\ &\displaystyle = &\displaystyle -\Big[\sum_{j=1}^N l_{ij}\varGamma e_{j}(t)\Big]^\top\varPsi(e_i,\dot{e}_i).\vspace{-3pt} \end{array} \end{aligned} $$

This gives

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle -\Big[\sum_{j=1}^N l_{ij}\varGamma e_{j}(t)\Big]^\top\varPsi(e_i,\dot{e}_i)\\ &\displaystyle = &\displaystyle -l_{ii}\|e_i(t)\|-\sum_{j=1,j\neq i}^N l_{ij} \big(\varPsi(e_i,\dot{e}_i)^\top\varPsi(e_j,\dot{e}_j)\big)\|e_j(t)\| \\ &\displaystyle \leq &\displaystyle 0.\vspace{-3pt} \end{array} \end{aligned} $$

Contrarily, if

$$\displaystyle \begin{aligned}-l_{ii}\|e_i(t)\|-\sum_{j=1,j\neq i}^N l_{ij} \big(\varPsi(e_i,\dot{e}_i)^\top\varPsi(e_j,\dot{e}_j)\big) \|e_j(t)\|>0,\end{aligned} $$

then

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \sum_{i=1}^N l_{ii}\|e_i(t)\|+\sum_{i=1}^N \sum_{j=1,j\neq i}^N l_{ij} \big(\varPsi(e_i,\dot{e}_i)^\top \varPsi(e_j,\dot{e}_j)\big)\|e_j(t)\| \\ {} &\displaystyle = &\displaystyle \sum_{i=1}^N l_{ii}\|e_i(t)\|+\sum_{i=1}^N \sum_{j=1,j\neq i}^N l_{ji} \big(\varPsi(e_j,\dot{e}_j)^\top \varPsi(e_i,\dot{e}_i)\big)\|e_i(t)\| \\ {} &\displaystyle = &\displaystyle \sum_{i=1}^N \Big[l_{ii} +\sum_{j=1,j\neq i}^N l_{ji} \big(\varPsi(e_j,\dot{e}_j)^\top\varPsi(e_i,\dot{e}_i)\big)\Big] \|e_i(t)\| \\ {} &\displaystyle < &\displaystyle 0.\vspace{-3pt} \end{array} \end{aligned} $$

The above hypothesis implies

$$\displaystyle \begin{aligned}\sum_{i=1}^N \Big[l_{ii} +\sum_{j=1,j\neq i}^N l_{ji} \big(\varPsi(e_j,\dot{e}_j)^\top \varPsi(e_i,\dot{e}_i)\big)\Big]<0,\end{aligned} $$

which contradicts with the fact that $\sum _{j=1}^N l_{ij}=0$ with l _ii > 0 and l _ij = l _ji ≤ 0 (j ≠ i), i, j = 1, 2, ⋯ , N.

Under Assumptions 6.1 and 6.2, it also follows that

Moreover, one has

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle -\Big\{\sum_{\iota =1}^2 \theta_{i\iota}(t) \big[p_{i_k}^\iota {e_i}(t)+ h_{i_k}^\iota \mathrm{sgn}\big({e_i}(t)\big)\big]\Big\}^\top\varPsi(e_i,\dot{e}_i) \\ &\displaystyle = &\displaystyle -\sum_{\iota =1}^2 \theta_{i\iota}(t) \Big(p_{i_k}^\iota \|e_i(t)\|+ h_{i_k}^\iota \Big) \\ &\displaystyle \leq &\displaystyle -\min_{r}\min\{p_{r}^1,p_{r}^2\}\|e_i(t)\|-\min_{r}\min\{h_{r}^1,h_{r}^2\}.\vspace{-3pt} \end{array} \end{aligned} $$

Hence, Eq. (6.21) together with the preceding relationships gives

$$\displaystyle \begin{aligned} \max \widetilde{\mathcal{D}}{{W}_i}\left( t \right)\big|{}_{\text{(6.11)}} \leq -\alpha_i {W_i}\left( t \right) +\beta_i {W_i}\left( t-\tau(t) \right)+d_i^q,\end{aligned} $$

(6.22)

where t ∈ (t _k−1, t _k], α _i, β _i and $d_i^q$ are as given by (6.13).

In the following, the impulse effect in (6.11) is analyzed. Similarly, there also exists △C ∈co[−C ⁽¹⁾, C ⁽¹⁾] such that

Then with Assumption 6.2, for any $\varOmega _i\in \mathcal {K}[W_i(t_k^+)]$, one can write

where $f_i^q=\big \|C^{(0)}+\triangle C \big \| \varepsilon _q$.

With $\sigma _i^q=1-\min \limits _r\min \{\nu _{r}^1,\nu _{r}^2\}+l_g \big \|C^{(0)}+\triangle C \big \|$, it follows

$$\displaystyle \begin{aligned} \max W_i(t_k^+)\big|{}_{\text{(6.11)}}\leq \sigma_i^q W_i(t_k)+f_i^q. \end{aligned} $$

(6.23)

Take the delayed impulsive differential equation

$$\displaystyle \begin{aligned} \left\{ \begin{array}{lll} \dot{V}_i(t) \;\; = -\alpha_i V_i(t) + \beta_i V_i(t-\tau(t)) +d_i^q, \\ \qquad \qquad \qquad t\in (t_{k-1},t_k],\\ V_i(t_k^+) = \sigma_i^q V_i(t_k)+f_i^q, \\ V_i(\vartheta) \;\,= \|\psi_i(\vartheta)-z_q(\vartheta)\|, \;\; \vartheta\in [t_0-\tau,t_0], \end{array} \right. \end{aligned} $$

(6.24)

as a comparison equation, k = 1, 2, ⋯, i = 1, 2, ⋯ , N. According to the comparison principle for delayed impulsive differential equations [5, 12, 38], it follows that W _i(t) ≤ V _i(t) for all t ≥ t ₀.

Then, the solution V _i(t) of dynamical system (6.24) is discussed. For t ∈ (t _k−1, t _k], one has

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} V_i(t) = e^{-\alpha_i (t-t_{k-1})}V_i(t_{k-1}^+) \qquad \qquad \qquad \qquad \;\, \\ + \int_{t_{k-1}}^t e^{-\alpha_i (t-s)} \big[\beta_i V_i(s-\tau(s))+d_i^q \big] \mathrm{d}s \\ = \sigma_i^q e^{-\alpha_i (t-t_{k-1})}V_i(t_{k-1}) +e^{-\alpha_i (t-t_{k-1})} f_i^q \\ + \int_{t_{k-1}}^t e^{-\alpha_i (t-s)} \big[\beta_i V_i(s-\tau(s))+d_i^q \big] \mathrm{d}s. \end{array} \end{aligned} $$

(6.25)

That is,

$$\displaystyle \begin{aligned} \begin{array}{rcl} V_i(t_k) = \sigma_i^q e^{-\alpha_i (t_k-t_{k-1})}V_i(t_{k-1}) +e^{-\alpha_i (t_k-t_{k-1})} f_i^q \\ + \int_{t_{k-1}}^{t_k} e^{-\alpha_i (t_k-s)} \big[\beta_i V_i(s-\tau(s))+d_i^q \big] \mathrm{d}s. \end{array} \end{aligned} $$

For k = 1, with $V_i(t_0^+)=V_i(t_0)$, it follows

$$\displaystyle \begin{aligned} \begin{array}{rcl} V_i(t_1) &\displaystyle =&\displaystyle \sigma_i^q e^{-\alpha_i (t_1-t_{0})}V_i(t_{0})+e^{-\alpha_i (t_1-t_{0})} f_i^q \\ &\displaystyle &\displaystyle + \int_{t_{0}}^{t_1} e^{-\alpha_i (t_1-s)} \big[\beta_i V_i(s-\tau(s))+d_i^q \big] \mathrm{d}s. \end{array} \end{aligned} $$

For k = 2, one has

$$\displaystyle \begin{aligned} \begin{array}{rcl} V_i(t_2)= \sigma_i^q e^{-\alpha_i (t_2-t_{1})}\Big[\sigma_i^q e^{-\alpha_i (t_1-t_{0})}V_i(t_{0}) \qquad \qquad \quad \\ +e^{-\alpha_i (t_1-t_{0})} f_i^q + \int_{t_{0}}^{t_1} e^{-\alpha_i (t_1-s)}\qquad \qquad \quad \; \\ \cdot\big(\beta_i V_i(s-\tau(s))+d_i^q \big) \mathrm{d}s\Big]+e^{-\alpha_i (t_2-t_{1})} f_i^q\quad \\ + \int_{t_{1}}^{t_2} e^{-\alpha_i (t_2-s)} \big[\beta_i V_i(s-\tau(s))+d_i^q \big] \mathrm{d}s\quad \;\;\, \\ =(\sigma_i^q)^2 e^{-\alpha_i (t_2-t_0)}V_i(t_0) +(\sigma_i^q)^2\Big[(\sigma_i^q)^{-1}e^{-\alpha_i (t_2-t_0)} \\ +(\sigma_i^q)^{-2}e^{-\alpha_i (t_2-t_{1})} \Big]f_i^q+\sigma_i^q e^{-\alpha_i (t_2-t_{1})} \quad \; \\ \cdot\int_{t_{0}}^{t_1} e^{-\alpha_i (t_1-s)}\big[\beta_i V_i(s-\tau(s))+d_i^q \big] \mathrm{d}s\quad \\ + \int_{t_{1}}^{t_2} e^{-\alpha_i (t_2-s)} \big[\beta_i V_i(s-\tau(s))+d_i^q \big] \mathrm{d}s. \;\,\vspace{-3pt} \end{array} \end{aligned} $$

For k = 3, it follows

$$\displaystyle \begin{aligned} \begin{array}{rcl} V_i(t_3) = (\sigma_i^q)^3 e^{-\alpha_i (t_3-t_0)}V_i(t_0) +(\sigma_i^q)^3 \Big[(\sigma_i^q)^{-1}e^{-\alpha_i (t_3-t_0)} \\ +(\sigma_i^q)^{-2} e^{-\alpha_i (t_3-t_{1})}+(\sigma_i^q)^{-3} e^{-\alpha_i (t_3-t_{2})} \Big]f_i^q \\ +\sigma_i^q e^{-\alpha_i (t_3-t_2)} \Big\{\sigma_i^q e^{-\alpha_i (t_2-t_{1})}\qquad \qquad \quad \;\;\, \\ \cdot\int_{t_{0}}^{t_1} e^{-\alpha_i (t_1-s)} \big[\beta_i V_i(s-\tau(s))+d_i^q \big] \mathrm{d}s \quad \;\\ + \int_{t_{1}}^{t_2} e^{-\alpha_i (t_2-s)} \big[\beta_i V_i(s-\tau(s))+d_i^q \big] \mathrm{d}s\Big\}\;\; \\ + \int_{t_{2}}^{t_3} e^{-\alpha_i (t_3-s)} \big[\beta_i V_i(s-\tau(s))+d_i^q \big] \mathrm{d}s. \;\;\;\vspace{-3pt} \end{array} \end{aligned} $$

Overall, it can be verified that for any t ∈ (t _k−1, t _k],

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} V_i(t) &\displaystyle = &\displaystyle \varPhi(t,t_0)V_i(t_0)+(\sigma_i^q)^{\mathbb{N}(t,t_0)} \\ &\displaystyle &\displaystyle \cdot\Big[\sum_{r=1}^{\mathbb{N}(t,t_0)}(\sigma_i^q)^{-r} e^{-\alpha_i(t-t_{r-1})}\Big]f_i^q \\ &\displaystyle &\displaystyle +\int_{t_0}^t \varPhi(t,s)\Big[\beta_i V_i(s-\tau(s))+d_i^q \Big]\mathrm{d}s, \end{array} \end{aligned} $$

(6.26)

where $\varPhi (t,t_{0})=(\sigma _i^q)^{\mathbb {N}(t,t_{0})} e^{-\alpha _i (t-t_{0})}$, $\mathbb {N}(t,t_0)$ denotes the number of impulses during time interval (t ₀, t].

According to Definition 6.2, with the average impulsive intermittence τ _a > 0, α _i > 0, and $0<\sigma _i^q<1$, the following three relationships hold.

$$\displaystyle \begin{aligned}(\sigma_i^q)^{\mathbb{N}(t,t_{0})}\leq (\sigma_i^q)^{-\mathbb{N}_0}e^{\frac{\ln \sigma_i^q}{\tau_a} (t-t_0)},\end{aligned}$$

$$\displaystyle \begin{aligned} \begin{array}{rcl} \varPhi(t,t_0) &\displaystyle \leq &\displaystyle e^{-\alpha_i (t-t_0)} (\sigma_i^q)^{\frac{(t-t_0)}{\tau_a}-\mathbb{N}_0} \\ &\displaystyle \leq &\displaystyle (\sigma_i^q)^{-\mathbb{N}_0}e^{-\big(\alpha_i-\frac{\ln \sigma_i^q}{\tau_a}\big) (t-t_0)}, \end{array} \end{aligned} $$

and

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle (\sigma_i^q)^{\mathbb{N}(t,t_0)}\Big[\sum_{r=1}^{\mathbb{N}(t,t_0)}(\sigma_i^q)^{-r} e^{-\alpha_i(t-t_{r-1})}\Big] \\ &\displaystyle \leq &\displaystyle (\sigma_i^q)^{\mathbb{N}(t,t_0)} \sum_{r=1}^{\mathbb{N}(t,t_0)}(\sigma_i^q)^{-r} \\ &\displaystyle = &\displaystyle (\sigma_i^q)^{\mathbb{N}(t,t_0)} (\sigma_i^q)^{-1}\frac{1-(\sigma_i^q)^{-\mathbb{N}(t,t_0)} }{1-(\sigma_i^q)^{-1}} \\ &\displaystyle \leq &\displaystyle \frac{1}{1-\sigma_i^q}-(1-\sigma_i^q)^{-1}(\sigma_i^q)^{\mathbb{N}_1}e^{\frac{\ln \sigma_i^q}{\tau_a} (t-t_0)} \\ &\displaystyle \leq &\displaystyle \frac{1-(\sigma_i^q)^{\mathbb{N}_1}}{1-\sigma_i^q}, \end{array} \end{aligned} $$

where the last inequality is due to $\ln \sigma _i^q<0$.

Thus for any t ∈ (t _k−1, t _k], Eq. (6.26) gives

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} V_i(t) &\displaystyle \leq &\displaystyle (\sigma_i^q)^{-\mathbb{N}_0}e^{-\big(\alpha_i-\frac{\ln \sigma_i^q}{\tau_a}\big) (t-t_0)}V_i(t_0) \\ &\displaystyle &\displaystyle +\tilde{f}_i^q +\int_{t_0}^t (\sigma_i^q)^{-\mathbb{N}_0}e^{-\big(\alpha_i-\frac{\ln \sigma_i^q}{\tau_a}\big) (t-s)} \\ &\displaystyle &\displaystyle \cdot\big[\beta_i V_i(s-\tau(s))\big]\mathrm{d}s, \end{array} \end{aligned} $$

(6.27)

where $\tilde {f}_i^q=\frac {1-(\sigma _i^q)^{\mathbb {N}_1}}{1-\sigma _i^q} f_i^q+\frac {(\sigma _i^q)^{-\mathbb {N}_0} }{\alpha _i}d_i^q$.

Hence, for all t ≥ t ₀, from (6.27) it follows that

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} V_i(t) \leq \tilde{\psi}_i (\sigma_i^q)^{-\mathbb{N}_0}e^{-\tilde{\alpha}_i (t-t_0)}+\frac{\tilde{f}_i^q}{\alpha_i(\sigma_i^q)^{\mathbb{N}_0}-\beta_i}, \end{array} \end{aligned} $$

(6.28)

where $\tilde {\psi }_i=\sup _{t_0-\bar {\tau }\leq \vartheta \leq t_0} \|\psi _i(\vartheta )-z_q(\vartheta )\|$, and $\tilde {\alpha }_i$ is the unique solution of algebra equation

$$\displaystyle \begin{aligned} \tilde{\alpha}_i-\Big(\alpha_i-\frac{\ln \sigma_i^q}{\tau_a}\Big) + (\sigma_i^q)^{-\mathbb{N}_0} \beta_i e^{\tilde{\alpha}_i\bar{\tau}}=0. \end{aligned} $$

(6.29)

It can be verified that $\tilde {\alpha }_i>0$ since one has $\big (\alpha _i-\frac {\ln \sigma _i^q}{\tau _a}\big )- (\sigma _i^q)^{-\mathbb {N}_0} \beta _i>0$ based on condition (6.13).

It is now to verify the relationship (6.28) by a contradiction [12, 38].

If (6.28) does not hold, then there exists T ₁ > 0 such that

$$\displaystyle \begin{aligned}V_i(T_1) \geq \tilde{\psi}_i (\sigma_i^q)^{-\mathbb{N}_0}e^{-\tilde{\alpha}_i (T_1-t_0)} +\frac{\tilde{f}_i^q}{\big(\alpha_i-\frac{\ln \sigma_i^q}{\tau_a}\big)(\sigma_i^q)^{\mathbb{N}_0}-\beta_i}, \end{aligned}$$

$$\displaystyle \begin{aligned}V_i(t) < \tilde{\psi}_i (\sigma_i^q)^{-\mathbb{N}_0}e^{-\tilde{\alpha}_i (t-t_0)}+\frac{\tilde{f}_i^q}{\big(\alpha_i-\frac{\ln \sigma_i^q}{\tau_a}\big)(\sigma_i^q)^{\mathbb{N}_0}-\beta_i}, \end{aligned}$$

for all t < T ₁.

Based on inequality (6.27), one has

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} V_i(T_1) \leq \tilde{\psi}_i(\sigma_i^q)^{-\mathbb{N}_0}e^{-\big(\alpha_i-\frac{\ln \sigma_i^q}{\tau_a}\big) (T_1-t_0)} +\tilde{f}_i^q \\ +\int_{t_0}^{T_1} (\sigma_i^q)^{-\mathbb{N}_0}e^{-\big(\alpha_i-\frac{\ln \sigma_i^q}{\tau_a}\big) (T_1-s)}\quad \\ \cdot\big[\beta_i V_i(s-\tau(s))\big]\mathrm{d}s \\ \leq e^{-\big(\alpha_i-\frac{\ln \sigma_i^q}{\tau_a}\big) (T_1-t_0)}\Big[\tilde{\psi}_i(\sigma_i^q)^{-\mathbb{N}_0}\quad \quad \\ +\frac{\tilde{f}_i^q}{\big(\alpha_i-\frac{\ln \sigma_i^q}{\tau_a}\big)(\sigma_i^q)^{\mathbb{N}_0}-\beta_i} \Big]\qquad \quad \;\; \\ + e^{-\big(\alpha_i-\frac{\ln \sigma_i^q}{\tau_a}\big) (T_1-t_0)}\qquad \qquad \qquad \, \\ \cdot\int_{t_0}^{T_1} (\sigma_i^q)^{-\mathbb{N}_0}e^{\big(\alpha_i-\frac{\ln \sigma_i^q}{\tau_a}\big)(s-t_0)}\quad \;\;\, \\ \cdot\Big[\beta_i \tilde{\psi}_i (\sigma_i^q)^{-\mathbb{N}_0}e^{-\tilde{\alpha}_i (s-\tau(s)-t_0)}\qquad \\ +\frac{\beta_i\tilde{f}_i^q}{\big(\alpha_i-\frac{\ln \sigma_i^q}{\tau_a}\big)(\sigma_i^q)^{\mathbb{N}_0}-\beta_i} \Big]\mathrm{d}s. \qquad \, \end{array} \end{aligned} $$

(6.30)

Due to the fact that $\alpha _i-\frac {\ln \sigma _i^q}{\tau _a}= \tilde {\alpha }_i+ (\sigma _i^q)^{-\mathbb {N}_0} \beta _i e^{\tilde {\alpha }_i\kappa }>(\sigma _i^q)^{-\mathbb {N}_0} \beta _i$, one gets

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \int_{t_0}^{T_1} (\sigma_i^q)^{-\mathbb{N}_0}e^{\big(\alpha_i-\frac{\ln \sigma_i^q}{\tau_a}\big)(s-t_0)}\Big[\beta_i \tilde{\psi}_i (\sigma_i^q)^{-\mathbb{N}_0} \\ &\displaystyle &\displaystyle \quad \cdot e^{-\tilde{\alpha}_i (s-\tau(s)-t_0)}+\frac{\beta_i\tilde{f}_i^q}{\big(\alpha_i-\frac{\ln \sigma_i^q}{\tau_a}\big)(\sigma_i^q)^{\mathbb{N}_0}-\beta_i} \Big]\mathrm{d}s \\ &\displaystyle < &\displaystyle \tilde{\psi}_i (\sigma_i^q)^{-\mathbb{N}_0} \Big[e^{\big(\alpha_i-\frac{\ln \sigma_i^q}{\tau_a}-\tilde{\alpha}_i\big)(T_1-t_0)} -1\Big] \\ &\displaystyle &\displaystyle +(\sigma_i^q)^{-\mathbb{N}_0}\beta_i \frac{\tilde{f}_i^q}{\big(\alpha_i-\frac{\ln \sigma_i^q}{\tau_a}\big)(\sigma_i^q)^{\mathbb{N}_0}-\beta_i} \\ &\displaystyle &\displaystyle \quad \cdot\frac{e^{\big(\alpha_i-\frac{\ln \sigma_i^q}{\tau_a}\big) (T_1-t_0)} -1}{\alpha_i-\frac{\ln \sigma_i^q}{\tau_a}}\\ &\displaystyle < &\displaystyle \tilde{\psi}_i (\sigma_i^q)^{-\mathbb{N}_0} \Big[e^{\big(\alpha_i-\frac{\ln \sigma_i^q}{\tau_a}-\tilde{\alpha}_i\big)(T_1-t_0)} -1\Big] \\ &\displaystyle &\displaystyle + \Big[e^{\big(\alpha_i-\frac{\ln \sigma_i^q}{\tau_a}\big) (T_1-t_0)} -1\Big] \\ &\displaystyle &\displaystyle \quad \cdot\frac{\tilde{f}_i^q}{\big(\alpha_i-\frac{\ln \sigma_i^q}{\tau_a}\big)(\sigma_i^q)^{\mathbb{N}_0}-\beta_i}\,. \end{array} \end{aligned} $$

Substituting the above inequality into inequality (6.30) gives

$$\displaystyle \begin{aligned}V_i(T_1) < \tilde{\psi}_i (\sigma_i^q)^{-\mathbb{N}_0}e^{-\tilde{\alpha}_i (t-t_0)}+\frac{\tilde{f}_i^q}{\big(\alpha_i-\frac{\ln \sigma_i^q}{\tau_a}\big)(\sigma_i^q)^{\mathbb{N}_0}-\beta_i},\end{aligned}$$

which leads to the contradiction.

Therefore, with $\tilde {\alpha }_i >0$ and W _i(t) ≤ V _i(t) (t ≥ t ₀), for any ε > 0, there exists a positive constant $\mathcal {T}_i$ such that W _i(t) ≤ ε + ξ _q, $\forall t>\mathcal {T}_i$, where ξ _q is given by (6.12), i ∈ г_q, $q=1,\cdots ,\mathcal {Q}$. That is, multisynchronization of MINNs (6.3) is achieved asymptotically in regard to master system (6.6). This completes the proof. □