Hybrid Memristor-Based Impulsive Neural Networks

Zhi-Hong Guan; Bin Hu; Xuemin (Sherman) Shen

doi:10.1007/978-3-030-02161-0_6

Introduction to Hybrid Intelligent Networks pp 155-193 | Cite as

Hybrid Memristor-Based Impulsive Neural Networks

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Zhi-Hong Guan
Bin Hu
Xuemin (Sherman) Shen

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First Online: 02 February 2019

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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.

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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

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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

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(6.21)

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

By Assumption 6.1, one has

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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

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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

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

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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 ≥ t₀.

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 (t₀, 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 ≥ t₀, 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 T₁ > 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 < T₁.

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 ≥ t₀), 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. □

References

1.
J. Pu, H. Gong, X. Li, and Q. Luo, “Developing neuronal networks: Self-organized criticality predicts the future,” Sci. Reports, vol. 3, no. 1081, pp. 1–6, 2013.Google Scholar
2.
F. Varela, J. P. Lachaux, E. Rodriguez, and J. Martinerie, “The brainweb: phase synchronization and large-scale integration,” Nature Reviews Neuroscience, vol. 2, pp. 229–239, 2001.CrossRefGoogle Scholar
3.
H. Modares, I. Ranatunga, F. L. Lewis, and D. Popa, “Optimized assistive human-robot interaction using reinforcement learning,” IEEE Trans. Cybern., vol. 46, no. 3, pp. 655–667, 2016.CrossRefGoogle Scholar
4.
J. J. Hopfield, “Neural networks and physical systems with emergent collective computational abilities,” Proc. Natl Acad. Sci., vol. 79, no. 8, pp. 2554–2558, 1982.MathSciNetCrossRefGoogle Scholar
5.
Z.-H. Guan and G. Chen, “On delayed impulsive Hopfield neural networks,” Neural Netw., vol. 12, pp. 273–280, 1999.CrossRefGoogle Scholar
6.
A. Thomas, “Memristor-based neural networks,” J. Phys. D: Appl. Phys., vol. 46, no. 9, pp. 093001(1–12), 2013.CrossRefGoogle Scholar
7.
L. Chua, “Memristor-The missing circuit element,” IEEE Trans. Circuit Theory, vol. 18, no. 5, pp. 507–519, 1971.CrossRefGoogle Scholar
8.
D. B. Strukov, G. S. Snider, D. R. Stewart, and R. S. Williams, “The missing memristor found,” Nature, vol. 453, no. 7191, pp. 80–83, 2008.CrossRefGoogle Scholar
9.
X. Liu, T. Chen, J. Cao, and W. Lu, “Dissipativity and quasi-synchronization for neural networks with discontinuous activations and parameter mismatches,” Neural Netw., vol. 24, no. 10, pp. 1013–1021, 2011.CrossRefGoogle Scholar
10.
B. Hu, D.-X. He, Z.-H. Guan, D.-X. Zhang, and X.-H. Zhang, “Hybrid subgroup coordination of multi-agent systems via nonidentical information exchange,” Neurocomputing, vol. 168, pp. 646–654, 2015.CrossRefGoogle Scholar
11.
D.-X. He, G. Ling, Z.-H. Guan, B. Hu, and R.-Q. Liao, “Multisynchronization of coupled heterogeneous genetic oscillator networks via partial impulsive control,” IEEE Trans. Neural Netw. Learn. Syst., vol. 29, no. 2, pp. 335–342, 2018.MathSciNetCrossRefGoogle Scholar
12.
Z.-H. Guan, Z.-W. Liu, G. Feng, and Y.-W. Wang, “Synchronization of complex dynamical networks with time-varying delays via impulsive distributed control,” IEEE Trans. Circuits Syst. I, vol. 57, no. 8, pp. 2182–2195, 2010.MathSciNetCrossRefGoogle Scholar
13.
W. Zhang, C. Li, T. Huang, and X. He, “Synchronization of memristor-based coupling recurrent neural networks with time-varying delays and impulses,” IEEE Trans. Neural Netw. Learn. Syst., vol. 26, no. 12, pp. 3308–3313, 2015.MathSciNetCrossRefGoogle Scholar
14.
X. Yang and D. W. C. Ho, “Synchronization of delayed memristive neural networks: robust analysis approach,” IEEE Trans. Cybern., vol. 46, no. 12, pp. 3377–3387, 2016.CrossRefGoogle Scholar
15.
Z. Guo, S. Yang, and J. Wang, “Global exponential synchronization of multiple memristive neural networks with time delay via nonlinear coupling,” IEEE Trans. Neural Netw. Learn. Syst., vol. 26, no. 6, pp. 1300–1311, 2015.MathSciNetCrossRefGoogle Scholar
16.
S. Wen, Z. Zeng, T. Huang, and Y. Zhang, “Exponential adaptive lag synchronization of memristive neural networks via fuzzy method and applications in pseudorandom number generators,” IEEE Trans. Fuzzy Syst., vol. 22, no, 6, pp. 1704–1713, 2014.CrossRefGoogle Scholar
17.
A. Wu, S. Wen, and Z. Zeng, “Synchronization control of a class of memristor-based recurrent neural networks,” Inform. Sci., vol. 183, no. 1, pp. 106–116, 2012.MathSciNetCrossRefGoogle Scholar
18.
F. Sorrentino, L. M. Pecora, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Complete characterization of the stability of cluster synchronization in complex dynamical networks,” Sci. Advances, vol. 2, no. 4, pp. e1501737, 2016.CrossRefGoogle Scholar
19.
J. Cao and L. Li, “Cluster synchronization in an array of hybrid coupled neural networks with delay,” Neural Netw., vol. 22, no. 4, pp. 335–342, 2009.CrossRefGoogle Scholar
20.
L. Li, D. W. C. Ho, J. Cao, and J. Lu, “Pinning cluster synchronization in an array of coupled neural networks under event-based mechanism,” Neural Netw., vol. 76, pp. 1–12, 2016.CrossRefGoogle Scholar
21.
W. Wu, W. Zhou, and T. Chen, “Cluster synchronization of linearly coupled complex networks under pinning control,” IEEE Trans. Circuits Syst. I, vol. 56, no. 4, pp. 829–839, 2009.MathSciNetCrossRefGoogle Scholar
22.
Q. Gao, G. Feng, D. Dong, and L. Liu, “Universal fuzzy models and universal fuzzy controllers for discrete-time nonlinear systems,” IEEE Trans. Cybern., vol. 45, no. 5, pp. 880–887, 2015.CrossRefGoogle Scholar
23.
Y. Li and S. Tong, “Hybrid adaptive fuzzy control for uncertain MIMO nonlinear systems with unknown dead-zone,” Inform. Sci., vol. 328, pp. 97–114, 2016.CrossRefGoogle Scholar
24.
X. Yang, D. W. C. Ho, J. Lu, and Q. Song, “Finite-time cluster synchronization of T-S fuzzy complex networks with discontinuous subsystems and random coupling delays,” IEEE Trans. Fuzzy Syst., vol. 23, no. 6, pp. 2302–2316, 2015.CrossRefGoogle Scholar
25.
Y. Liu, S. Tong, D. Li, and Y. Gao, “Fuzzy adaptive control with state observer for a class of nonlinear discrete-time systems with input constraint,” IEEE Trans. Fuzzy Syst., vol. 24, no. 5, pp. 1147–1158, 2016.CrossRefGoogle Scholar
26.
S. Zhang, Z. Wang, D. Ding, H. Dong, F. E. Alsaadi, and T. Hayat, “Nonfragile H_∞ fuzzy filtering with randomly occurring gain variations and channel fadings,” IEEE Trans. Fuzzy Syst., vol. 24, no. 3, pp. 505–518, 2016.CrossRefGoogle Scholar
27.
T. Wang, Y. Zhang, J. Qiu, and H. Gao, “Adaptive fuzzy backstepping control for a class of nonlinear systems with sampled and delayed measurements,” IEEE Trans. Fuzzy Syst., vol. 23, no. 2, pp. 302–312, 2015.CrossRefGoogle Scholar
28.
W.-H. Chen, D. Wei, and W. X. Zheng, “Delayed impulsive control of Takagi-Sugeno fuzzy delay systems,” IEEE Trans. Fuzzy Syst., vol. 21, no. 3, pp. 516–526, 2013.CrossRefGoogle Scholar
29.
H. Zhang, H. Yan, T. Liu, and Q. Chen, “Fuzzy controller design for nonlinear impulsive fuzzy systems with time delay,” IEEE Trans. Fuzzy Syst., vol. 19, no. 5, pp. 844–856, 2011.CrossRefGoogle Scholar
30.
H. Dong, Z. Wang, D. W. C. Ho, and H. Gao, “Robust H_∞ fuzzy output feedback control with multiple probabilistic delays and multiple missing measurements,” IEEE Trans. Fuzzy Syst., vol. 18, no. 4, pp. 712–725, 2010.CrossRefGoogle Scholar
31.
G. Feng, “A survey on analysis and design of model-based fuzzy control systems,” IEEE Trans. Fuzzy Syst., vol. 14, no. 5, pp. 676–697, 2006.CrossRefGoogle Scholar
32.
T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. SMC-15, no. 1, pp. 116–132, 1985.CrossRefGoogle Scholar
33.
I. Saboori and K. Khorasani, “H_∞ consensus achievement of multi-agent systems with directed and switching topology networks,” IEEE Trans. Autom. Contr., vol. 59, no. 11, pp. 3104–3109, 2014.MathSciNetCrossRefGoogle Scholar
34.
Z.-H. Guan, B. Hu, M. Chi, D.-X. He, and X.-M. Cheng, “Guaranteed performance consensus in second-order multi-agent systems with hybrid impulsive control,” Automatica, vol. 50, no. 9, pp. 2415–2418, 2014.MathSciNetCrossRefGoogle Scholar
35.
Z.-H. Guan, D. J. Hill, and J. Yao, “A hybrid impulsive and switching control strategy for synchronizaiton of nonlinear systems and application to Chua’s chaotic circuit,” Int. J. Bifurcation and Chaos, vol. 16, no. 1, pp. 229–238, 2006.CrossRefGoogle Scholar
36.
Z.-H. Guan, D. J. Hill, and X. Shen, “On hybrid impulsive and switching systems and application to nonlinear control,” IEEE Trans. Autom. Contr., vol. 50, no. 7, pp. 1058–1062, 2005.MathSciNetCrossRefGoogle Scholar
37.
J. P. Hespanha and A. S. Morse, “Switching between stabilizing controllers,” Automatica, vol. 38, no. 11, pp. 1905–1917, 2002.MathSciNetCrossRefGoogle Scholar
38.
Z. Yang and D. Xu, “Stability analysis and design of impulsive control systems with time delay,” IEEE Trans. Autom. Contr., vol. 52, no. 8, pp. 1448–1454, 2007.MathSciNetCrossRefGoogle Scholar
39.
M. Forti and P. Nistri, “Global convergence of neural networks with discontinuous neuron activations,” IEEE Trans. Circuits Syst. I, vol. 50, no. 11, pp. 1421–1435, 2003.MathSciNetCrossRefGoogle Scholar
40.
B. Hu, Z.-H. Guan, X.-W. Jiang, M. Chi, R.-Q. Liao, “Event-driven multi-consensus of multi-agent networks with repulsive links,” Inform. Sci., vol. 373, pp. 110–123, 2016.CrossRefGoogle Scholar
41.
F. Yaghmaie, R. Su, F. Lewis, and L. Xie, “Multi-party consensus of linear heterogeneous multi-agent systems,” IEEE Trans. Autom. Contr., vol. 62, no. 11, pp. 5578–5589, 2017.CrossRefGoogle Scholar
42.
S. Jafarzadeh, M. Fadali, and A. Sonbol, “Stability analysis and control of discrete type-1 and type-2 TSK fuzzy systems: Part I. Stability analysis,” IEEE Trans. Fuzzy Syst., vol. 19, no. 6, pp. 989–1000, 2011.CrossRefGoogle Scholar
43.
S. Jafarzadeh, M. Fadali, and A. Sonbol, “Stability analysis and control of discrete type-1 and type-2 TSK fuzzy systems: Part II. Control design,” IEEE Trans. Fuzzy Syst., vol. 19, no. 6, pp. 1001–1013, 2011.CrossRefGoogle Scholar
44.
T. M. Guerra and L. Vermeiren, “LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno’s form,” Automatica, vol. 40, no. 5, pp. 823–829, 2004.MathSciNetCrossRefGoogle Scholar
45.
H. Ying, “Structure and stability analysis of general Mamdani fuzzy dynamic models,” Int. J. Intelligent Systems, vol. 20, no. 1, pp. 103–125, 2005.MathSciNetCrossRefGoogle Scholar
46.
R.-E. Precup, M.-L. Tomescu, and St. Preitl, “Fuzzy logic control system stability analysis based on Lyapunov’s direct method,” Int. J. Computers, Communication & Contr., vol. 4, no. 4, pp. 415–426, 2009.CrossRefGoogle Scholar
47.
A. F. Filippov, “Differential equations with discontinuous right-hand side,” Norwell, MA, USA: Kluwer, 1988.Google Scholar
48.
M. Benchohra, J. Henderson, and S. K. Ntouyas, “Impulsive differential equations and inclusions,” vol. 2, New York: Hindawi Publishing Corporation, Chap. 3, 2006.Google Scholar

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Authors and Affiliations

Zhi-Hong Guan
- 1
Bin Hu
- 2
Xuemin (Sherman) Shen
- 3

1.College of AutomationHuazhong University of Science and TechnologyWuhanChina
2.Wuhan National Laboratory For OptoelectronicsHuazhong University of Science and TechnologyWuhanChina
3.Electrical and Computer Engineering DepartmentUniversity of WaterlooWaterlooCanada

Hybrid Memristor-Based Impulsive Neural Networks

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