1. Introduction
Neural networks (NNs) have always possessed a wide range of applications in associative memory, image processing, dynamic optimization, and machine learning [1–3] over the years, which have aroused many scholars' interest and produced a large number of research results [4–6]. In 1986, the inertial neural networks (INNs) model was produced by introducing an inductor into an artificial neural network circuit [7, 8]. Different from the artificial neural networks model described by the first-order differential equation, the INNs model obeys the second-order one and the second-order term is called the inertial term. The introduction of the inertia term can effectively improve the disordered search performance of the network, and it is also the key factor to produce complex behaviors such as bifurcation and chaos. Moreover, the inertia term itself has a strong biological background. For example, the cell membrane of some organisms can be realized by the circuit system containing inductance, and the axons of squid and penguin can be simulated by the inductance in the circuit [9]. Compared with the first-order NNs, INNs show more complex dynamic behaviors. So, it shows the extremely important theoretical significance and practical application value in studying the dynamical behaviors of INNs deeply [10, 11].
The above research results are all on the basis of real-valued neural networks with real-valued activation functions and state variables. With the involvement of complex signals, complex-valued neural networks (CVNNs) models entered the vision of scholars [12–19]. It is not only a simple extension of real-valued NNs but also has the characteristics of fast calculation and strong processing ability, and it can be used to deal with some problems that cannot be solved in the real number field. Naturally, complex-valued inertial neural networks (CVINNs) have also aroused widespread concern, and the development of dynamic behavior has increased rapidly in recent years. When developing dynamic behaviors, most of the work is based on the separation method by separating the real part and imaginary part of the considered systems, such as stability analysis [20–22], stabilization [23, 24], and synchronization [25–29]. Later, driven by related theories in a complex number field, the non-separation method, that is, regarding the considered systems as a whole, is used fully to study various dynamics, including state estimation [30], exponential convergence [22], finite/fixed-time synchronization [28, 29, 31], anti-synchronization [23, 24, 32], and exponential and adaptive synchronization [33]. Compared with the separation method, the non-separation method can reduce conservatism and computational complexity.
Synchronization has been applied to some practical fields such as image encryption, secure communication, and brain-like intelligence, which makes it a very active research topic. Besides those synchronization issues mentioned above, there are others, for example, polynomial synchronization (PS). The concept of PS originated from polynomial stability and was first proposed in [34], which is a kind of dynamic behavior in some special cases of stochastic differential equations and wave equations [35, 36]. As stated in [37, 38], its basic characteristic is that its top Lyapunov exponent equals 0. It is called polynomial stability because its definition contains t−λ(λ > 0), which is similar to the structure of polynomials. It is more general than exponential stability, which is between exponential stability and asymptotic stability. Its convergence speed is slower than that of exponential stability, but both of them can degenerate into asymptotic stability. Thus, based on these outstanding characteristics, polynomial stability is also a very typical dynamic behavior. For PS of the drive-response systems, that is, polynomial stability of the error systems, there has been no relevant result about CVINNs up to now. This fact promotes our research.
It should be pointed out that the time delay cannot be ignored in the whole design process of CVINNs. As we all know, due to the inherent delay time of inertial neurons and the limited speed of information transmission, signal acquisition, and processing between neurons, the time delay of the inertial neural system is inevitable. It is found that time delay has a great impact on the dynamic behaviors of neural networks, which may lead to instability, oscillation, and other poor performance. Therefore, it is very important to study the dynamic characteristics of delayed neural networks. As one of many delay types, the proportional delay is time-varying and unbounded with τ(t) = (1 − q)t(0 < q < 1). Due to its monotony-increasing property, the network running time can be easily controlled according to the allowable delay of the networks [39, 40]. In addition, systems with proportional delay have many interesting applications, such as nonlinear integro-differential equations, web application servers, and wireless LAN [41–43]. Thus, it plays an increasingly important role in many important fields such as physics, biological systems, and control science [44, 45]. These days, there are few achievements in the dynamical behaviors of CVINNs with proportional delay [21, 31]. In [21], memristor-based CVINNs with proportional and distributed delays are studied and new results on exponential input-to-state stability by using the separation method are given. In [31], the stabilization problem for CVINNs with proportional delay is discussed via the non-separation approach and finite-time stability theory.
Considering the clear feature of multi-proportional delays [38, 46], such as more generality and reality in contrast to single-proportional delays, we will focus on the PS problem of CVINNs with multi-proportional delays in this paper. The main contributions are summarized as follows:
Notations: C and Cn stand for the set of complex numbers and all n-dimensional complex-valued vectors, respectively. For any z ∈ C, $| z| =\sqrt{z\overline{z}}$ denotes the module of z, where $\overline{z}$ is the conjugate of z. For any $z={\left({z}_{1},{z}_{2},\ldots ,{z}_{n}\right)}^{{\rm{T}}}\in {C}^{n}$, the norm of z is defined as $\parallel z\parallel =\sqrt{{\sum }_{j=1}^{n}| {z}_{j}{| }^{2}}$.
| (1)Considering the importance of PS and proportional delay, the PS problem of CVINNs with multi-proportional delays is studied for the first time. | |
| (2)To maintain the originality of the considered system and reduce computational pressure, our work is carried out by a non-separation approach instead of decomposition. | |
| (3)To achieve the PS of the addressed CVINNs with multi-proportional delays, by applying the exponential transformation and designing an appropriate controller, a new criterion for PS is proposed based on the Lyapunov function approach and some inequalities techniques. |
2. Preliminaries
In this paper, we consider the complex-valued inertial neural networks described in the following form:
$\begin{eqnarray}\begin{array}{rcl}{\ddot{u}}_{j}(t) & = & -{a}_{j}{\dot{u}}_{j}(t)-{b}_{j}{u}_{j}(t)+\displaystyle \sum _{r=1}^{n}{c}_{{jr}}{f}_{r}({u}_{r}({p}_{r}t))\\ & & +\displaystyle \sum _{r=1}^{n}{m}_{{jr}}{g}_{r}({u}_{r}({q}_{r}t))+{I}_{j}(t),\end{array}\end{eqnarray}$
where j = 1, 2,…,n, t ≥ t0 ≥ 1, uj(t) ∈ C represents the state of the jth neuron at time t; ${\ddot{u}}_{j}(t)$ indicates the inertial term; aj and bj are the positive constants; cjr and mjr are the delayed connection weight coefficients; fr(· ), gr(· ) ∈ C are complex-valued activation functions; pr, qr are proportional delays factors with pr, qr ∈ (0, 1]; prt = t − (1 − pr)t, qrt = t − (1 − qr)t, in which (1 − pr)t and (1 − qr)t are the transmission delay functions; Ij is the external input.On one hand, as a special type of delay, the proportional delay is a time-varying and unbounded delay, which is different from constant delay, distributed delay, and so on. In many fields, such as control science, and physical and biological systems, it can be found that proportional delay plays an important role. On the other hand, as a special neural network, CVINNs have attracted extensive attention because of their rich biological and engineering background. Compared with first-order neural networks, the dynamic behaviors of the second-order CVINNs are more complex and it is more difficult to deal with them. So far, there have been few reports on CVINNs with proportional delays [21, 31]. Here, we will continue to study deeply the CVINNs model with multi-proportional delays and develop the relative achievement for the PS problem.
The initial conditions of system (1 )3 )
$\begin{eqnarray}{u}_{j}(s)={\phi }_{j}(s),{\dot{u}}_{j}(t)={\psi }_{j}(s),\end{eqnarray}$
where $s\in [\tilde{q}{t}_{0},{t}_{0}]$, $\tilde{q}={\min }_{1\leqslant r\leqslant n}\{{p}_{r},{q}_{r}\}$, φj(s) and $Psi$j(s) are continuous functions. The response system is described by $\begin{eqnarray}\begin{array}{rcl}{\ddot{v}}_{j}(t) & = & -{a}_{j}{\dot{v}}_{j}(t)-{b}_{j}{v}_{j}(t)+\displaystyle \sum _{r=1}^{n}{c}_{{jr}}{f}_{r}({v}_{r}({p}_{r}t))\\ & & +\displaystyle \sum _{r=1}^{n}{m}_{{jr}}{g}_{r}({v}_{r}({q}_{r}t))+{I}_{j}(t)+{U}_{j}(t),\,\,\end{array}\end{eqnarray}$
where vj(t) represents the state of the response system and Uj(t) represents the control input. The initial conditions of system ( $\begin{eqnarray}{v}_{j}(s)=\hat{{\phi }_{j}}(s),\dot{{v}_{j}}(t)=\hat{{\psi }_{j}}(s),\end{eqnarray}$
where $s\in [\tilde{q}{t}_{0},{t}_{0}]$, $\hat{{\phi }_{j}}(s)$ and $\hat{{\psi }_{j}}(s)$ are continuous functions.Define the synchronization error yj(t) = vj(t) − uj(t), the error system can be written as
$\begin{eqnarray}\begin{array}{rcl}{\ddot{y}}_{j}(t) & = & -{a}_{j}{\dot{y}}_{j}(t)-{b}_{j}{y}_{j}(t)+\displaystyle \sum _{r=1}^{n}{c}_{{jr}}\hat{{f}_{r}}({y}_{r}({p}_{r}t))\\ & & +\displaystyle \sum _{r=1}^{n}{m}_{{jr}}\hat{{g}_{r}}({y}_{r}({q}_{r}t))+{U}_{j}(t),\end{array}\end{eqnarray}$
where t ≥ t0, ${\hat{f}}_{r}({y}_{r}({p}_{r}t))$ = fr(vr(prt)) − fr(ur(prt)), ${\hat{g}}_{r}({y}_{r}({q}_{r}t)\,={g}_{r}({v}_{r}({q}_{r}t))-{g}_{r}({u}_{r}({q}_{r}t))$.By introducing the following variable transformation:5 ) can be converted into6 ) 1 ).
$\begin{eqnarray*}{z}_{j}(t)={w}_{j}{\dot{y}}_{j}(t)+{\tilde{w}}_{j}{y}_{j}(t),\end{eqnarray*}$
where wj ≠ 0, ${\tilde{w}}_{j}$ is the appropriate constant, system ( $\begin{eqnarray}\left\{\begin{array}{l}{\dot{y}}_{j}(t)=-\displaystyle \frac{{\tilde{w}}_{j}}{{w}_{j}}{y}_{j}(t)+\displaystyle \frac{1}{{w}_{j}}{z}_{j}(t),\\ {\dot{z}}_{j}(t)=-{\gamma }_{j}{z}_{j}(t)-{\zeta }_{j}{y}_{j}(t)+\displaystyle \sum _{r=1}^{n}{w}_{j}{c}_{{jr}}{\hat{f}}_{r}({y}_{r}({p}_{r}t))\\ +\displaystyle \sum _{r=1}^{n}{w}_{j}{m}_{{jr}}{\hat{g}}_{r}({y}_{r}({q}_{r}t))+{w}_{j}{U}_{j}(t),\end{array}\right.\end{eqnarray}$
where ${\gamma }_{j}={a}_{j}-\tfrac{{\tilde{w}}_{j}}{{w}_{j}}$ and ${\zeta }_{j}={b}_{j}{w}_{j}-{\gamma }_{j}{\tilde{w}}_{j}$. Let ${\hat{y}}_{j}(t)={y}_{j}({{\rm{e}}}^{t})$ and ${\hat{z}}_{j}(t)={z}_{j}({{\rm{e}}}^{t})$, we can obtain the following equivalent system for system ( $\begin{eqnarray}\left\{\begin{array}{l}{\hat{\dot{y}}}_{j}(t)={{\rm{e}}}^{t}\left(-\displaystyle \frac{{\tilde{w}}_{j}}{{w}_{j}}{\hat{y}}_{j}(t)+\displaystyle \frac{1}{{w}_{j}}{\hat{z}}_{j}(t)\right),\\ {\hat{\dot{z}}}_{j}(t)={{\rm{e}}}^{t}\left(-{\gamma }_{j}{\hat{z}}_{j}(t)-{\zeta }_{j}{\hat{y}}_{j}(t)+\sum _{r=1}^{n}{w}_{j}{c}_{{jr}}{\hat{f}}_{r}({\hat{y}}_{r}(t-{\mu }_{r}))\right.\\ \left.+\sum _{r=1}^{n}{w}_{j}{m}_{{jr}}{\hat{g}}_{r}({y}_{r}(t-{h}_{r}))+{w}_{j}{\tilde{U}}_{j}(t\right),\qquad t\geqslant \mathrm{log}{t}_{0},\end{array}\right.\end{eqnarray}$
with $\begin{eqnarray}\begin{array}{rcl}{\hat{y}}_{j}(s) & = & {\xi }_{j}(s),\,{\hat{\dot{y}}}_{j}(s)={\chi }_{j}(s),\\ s & \in & [\mathrm{log}\tilde{q}{t}_{0},\mathrm{log}{t}_{0}],\end{array}\end{eqnarray}$
where ${\mu }_{r}=-\mathrm{log}{p}_{r}$, ${h}_{r}=-\mathrm{log}{q}_{r}$, ${\tilde{U}}_{j}(t)={U}_{j}({{\rm{e}}}^{t})$, ${\xi }_{j}(s)={\hat{\phi }}_{j}({{\rm{e}}}^{s})-{\phi }_{j}({{\rm{e}}}^{s})$ and ${\chi }_{j}(s)={\hat{\psi }}_{j}({{\rm{e}}}^{s})-{\psi }_{j}({{\rm{e}}}^{s})$.In realizing the PS of the addressed model, the main difficulty and challenge are how to obtain the power exponential term ${t}^{-\lambda }$ according to the definitions of polynomial stability and synchronization. Here, firstly, the exponential transformation is involved [34]. Then, by designing a suitable controller and constructing an appropriate Lyapunov function, our purpose is achieved.
In this paper, activation functions ${f}_{r}(\cdot )$ and ${g}_{r}(\cdot )$ are assumed to satisfy the following Lipschitz conditions:
$\begin{eqnarray*}\begin{array}{rcl}| {f}_{r}(v) & & -{f}_{r}(u)| \leqslant {F}_{r}| v-u| ,\\ | {g}_{r}(v) & & -{g}_{r}(u)| \leqslant {G}_{r}| v-u| ,\,\forall v,u\in C,\end{array}\end{eqnarray*}$
where $r=1,2,\ldots ,n$, ${F}_{r}\gt 0$ and ${G}_{r}\gt 0$.[38]. Let $v(t)={\left({v}_{1}(t),{v}_{2}(t),\ldots ,{v}_{n}(t)\right)}^{{\rm{T}}}$ and $\hat{\phi }(s)={\left({\hat{\phi }}_{1}(s),{\hat{\phi }}_{2}(s),\ldots ,{\hat{\phi }}_{n}(s)\right)}^{{\rm{T}}}$, system (
$\begin{eqnarray*}\parallel v(t)-\bar{v}{\parallel }^{2}\leqslant K\parallel \hat{\phi }(s)-\bar{v}{\parallel }_{\tilde{q}}^{2}{t}^{-\lambda },\,t\geqslant {t}_{0},\end{eqnarray*}$
where $\parallel \hat{\phi }(s)-\bar{v}{\parallel }_{\tilde{q}}^{2}={\sup }_{\tilde{q}{t}_{0}\leqslant \ s\leqslant {t}_{0}}\parallel \hat{\phi }(s)-\bar{v}{\parallel }^{2}$, $\bar{v}$ is an equilibrium point of system ([34]. Let $u(t)={\left({u}_{1}(t),{u}_{2}(t),\ldots ,{u}_{n}(t)\right)}^{{\rm{T}}}$ and $\phi (s)={\left({\phi }_{1}(s),{\phi }_{2}(s),\ldots ,{\phi }_{n}(s)\right)}^{{\rm{T}}}$, systems (
$\begin{eqnarray*}\parallel v(t)-u(t){\parallel }^{2}\leqslant K\parallel \hat{\phi }(s)-\phi (s){\parallel }_{\tilde{q}}^{2}{t}^{-\lambda },\,t\geqslant {t}_{0},\end{eqnarray*}$
where $\parallel \hat{\phi }(s)-\phi (s){\parallel }_{\tilde{q}}^{2}={\sup }_{\tilde{q}{t}_{0}\leqslant \ s\leqslant {t}_{0}}\parallel \hat{\phi }(s)-\phi (s){\parallel }^{2}$.3. Main result
In this section, we will develop a sufficient condition to ascertain the PS for systems (1) and (3). First, we design the following control input 9 ). 13 )-(15 ) into (12 ), we can obtain10 ), one has
$\begin{eqnarray}{\tilde{U}}_{j}(t)=\left\{\begin{array}{l}-\displaystyle \frac{1}{{w}_{j}\overline{{\hat{z}}_{j}(t)}}\left({k}_{j}{\hat{y}}_{j}(t)\overline{{\hat{y}}_{j}(t)}+{\eta }_{j}{\hat{z}}_{j}(t)\overline{{\hat{z}}_{j}(t)}+\displaystyle \frac{{\rho }_{j}}{2}\sum _{r=1}^{n}{\hat{y}}_{r}(t-{\mu }_{r})\overline{{\hat{y}}_{r}(t-{\mu }_{r})}\right.\\ \left.+\displaystyle \frac{{\sigma }_{j}}{2}\sum _{r=1}^{n}{\hat{y}}_{r}(t-{h}_{r})\overline{{\hat{y}}_{r}(t-{h}_{r})}\right),\qquad \qquad \qquad \qquad | {\hat{z}}_{j}(t)| \ne 0,\\ 0,\qquad \qquad \qquad \qquad \qquad \qquad \qquad | {\hat{z}}_{j}(t)| =0,\end{array}\right.\end{eqnarray}$
where positive numbers kj, ηj, ρj and σj are control gains.Here, the above controller (
Suppose Assumption 1 holds, if there exist positive numbers kj, ${\eta }_{j}$, ${\rho }_{j}$, ${\sigma }_{j}$, and $\varepsilon \gt 0$ such that the following inequalities hold:
$\begin{eqnarray}\begin{array}{l}\varepsilon -2\displaystyle \frac{{\tilde{w}}_{j}}{{w}_{j}}-2{k}_{j}+| \displaystyle \frac{1}{{w}_{j}}-{\zeta }_{j}| \leqslant 0,\\ \quad \varepsilon -2({\gamma }_{j}+{\eta }_{j})+| \displaystyle \frac{1}{{w}_{j}}-{\zeta }_{j}| +\displaystyle \sum _{r=1}^{n}| {w}_{j}{c}_{{jr}}| {F}_{r}\\ \quad +\displaystyle \sum _{r=1}^{n}| {w}_{j}{m}_{{jr}}| {G}_{r}\leqslant 0,\\ \quad \displaystyle \sum _{r=1}^{n}| {w}_{j}{c}_{{jr}}| {F}_{r}-{\rho }_{j}\leqslant 0,\\ \quad \displaystyle \sum _{r=1}^{n}| {w}_{j}{m}_{{jr}}| {G}_{r}-{\sigma }_{j}\leqslant 0,\end{array}\end{eqnarray}$
then systems (1) and (3) are globally polynomially synchronized based on the controller (Construct the Lyapunov function as below
$\begin{eqnarray}V(t)=\sum _{j=1}^{n}\left({\hat{y}}_{j}(t)\overline{{\hat{y}}_{j}(t)}+{\hat{z}}_{j}(t)\overline{{\hat{z}}_{j}(t)}\right){{\rm{e}}}^{\varepsilon t}.\end{eqnarray}$
Calculating the derivative of V(t), we can obtain $\begin{eqnarray}\begin{array}{rcl}\dot{V}(t) & = & {{\rm{e}}}^{\varepsilon t}\displaystyle \sum _{j=1}^{n}\left\{\Space{0ex}{3.65ex}{0ex}\varepsilon \left({\hat{y}}_{j}(t)\overline{{\hat{y}}_{j}(t)}+{\hat{z}}_{j}(t)\overline{{\hat{z}}_{j}(t)}\right)\right.\\ & & +{{\rm{e}}}^{t}\left(-\displaystyle \frac{{\tilde{w}}_{j}}{{w}_{j}}{\hat{y}}_{j}(t)+\displaystyle \frac{1}{{w}_{j}}{\hat{z}}_{j}(t)\right)\overline{{\hat{y}}_{j}(t)}\\ & & +{{\rm{e}}}^{t}{\hat{y}}_{j}(t)\left(-\displaystyle \frac{{\tilde{w}}_{j}}{{w}_{j}}\overline{{\hat{y}}_{j}(t)}+\displaystyle \frac{1}{{w}_{j}}\overline{{\hat{z}}_{j}(t)}\right)\\ & & +{{\rm{e}}}^{t}\left[\Space{0ex}{3.55ex}{0ex}-{\gamma }_{j}{\hat{z}}_{j}(t)-{\zeta }_{j}{\hat{y}}_{j}(t)\right.\\ & & +\displaystyle \sum _{r=1}^{n}{w}_{j}{c}_{{jr}}{\hat{f}}_{r}({\hat{y}}_{r}(t-{\mu }_{r}))\\ & & +\displaystyle \sum _{r=1}^{n}{w}_{j}{m}_{{jr}}{\hat{g}}_{r}({\hat{y}}_{r}(t-{h}_{r}))-\displaystyle \frac{1}{\overline{{\hat{z}}_{j}(t)}}\\ & & \times \left(\Space{0ex}{3.25ex}{0ex}{k}_{j}{\hat{y}}_{j}(t)\overline{{\hat{y}}_{j}(t)}+{\eta }_{j}{\hat{z}}_{j}(t)\overline{{\hat{z}}_{j}(t)}\right.\\ & & +\displaystyle \frac{{\rho }_{j}}{2}\displaystyle \sum _{r=1}^{n}{\hat{y}}_{r}(t-{\mu }_{r})\overline{{\hat{y}}_{r}(t-{\mu }_{r})}\\ & & \left.\left.+\displaystyle \frac{{\sigma }_{j}}{2}\displaystyle \sum _{r=1}^{n}{\hat{y}}_{r}(t-{h}_{r})\overline{{\hat{y}}_{r}(t-{h}_{r})}\right)\right]\overline{{\hat{z}}_{j}(t)}\\ & & +{{\rm{e}}}^{t}{\hat{z}}_{j}(t)\left[\Space{0ex}{3.65ex}{0ex}-{\gamma }_{j}\overline{{\hat{z}}_{j}(t)}-{\zeta }_{j}\overline{{\hat{y}}_{j}(t)}\right.\\ & & +\displaystyle \sum _{r=1}^{n}{w}_{j}\overline{{c}_{{jr}}{\hat{f}}_{r}({\hat{y}}_{r}(t-{\mu }_{r}))}\\ & & +\displaystyle \sum _{r=1}^{n}{w}_{j}\overline{{m}_{{jr}}{\hat{g}}_{r}({y}_{r}(t-{h}_{r}))}-\displaystyle \frac{1}{{\hat{z}}_{j}(t)}\\ & & \times \left({k}_{j}{\hat{y}}_{j}(t)\overline{{\hat{y}}_{j}(t)}+{\eta }_{j}{\hat{z}}_{j}(t)\overline{{\hat{z}}_{j}(t)}\right.\\ & & +\displaystyle \frac{{\rho }_{j}}{2}\displaystyle \sum _{r=1}^{n}{\hat{y}}_{r}(t-{\mu }_{r})\overline{{\hat{y}}_{r}(t-{\mu }_{r})}+\displaystyle \frac{{\sigma }_{j}}{2}\\ & & \left.\left.\left.\times \displaystyle \sum _{r=1}^{n}{\hat{y}}_{r}(t-{h}_{r})\overline{{\hat{y}}_{r}(t-{h}_{r})}\right)\right]\right\}\\ & = & {{\rm{e}}}^{(\varepsilon +1)t}\displaystyle \sum _{j=1}^{n}\left[\left(\displaystyle \frac{\varepsilon }{{{\rm{e}}}^{t}}-2\displaystyle \frac{{\tilde{w}}_{j}}{{w}_{j}}-2{k}_{j}\right){\hat{y}}_{j}(t)\overline{{\hat{y}}_{j}(t)}\right.\\ & & +\left(\displaystyle \frac{\varepsilon }{{{\rm{e}}}^{t}}-2({\gamma }_{j}+{\eta }_{j})\right){\hat{z}}_{j}(t)\overline{{\hat{z}}_{j}(t)}\\ & & +\left(\displaystyle \frac{2}{{w}_{j}}-2{\zeta }_{j}\right){Re}({\hat{z}}_{j}(t)\overline{{\hat{y}}_{j}(t)})\\ & & +2\displaystyle \sum _{r=1}^{n}{w}_{j}{Re}\left({c}_{{jr}}{\hat{f}}_{r}({\hat{y}}_{r}(t-{\mu }_{r}))\overline{{\hat{z}}_{j}(t)}\right)\\ & & +2\displaystyle \sum _{r=1}^{n}{w}_{j}{Re}\left({m}_{{jr}}{\hat{g}}_{r}({\hat{y}}_{r}(t-{h}_{r}))\overline{{\hat{z}}_{j}(t)}\right)\\ & & -{\rho }_{j}\displaystyle \sum _{r=1}^{n}{\hat{y}}_{r}(t-{\mu }_{r})\overline{{\hat{y}}_{r}(t-{\mu }_{r})}\\ & & \left.-{\sigma }_{j}\displaystyle \sum _{r=1}^{n}{\hat{y}}_{r}(t-{h}_{r})\overline{{\hat{y}}_{r}(t-{h}_{r})}\right].\end{array}\end{eqnarray}$
According to Assumption 1, we have $\begin{eqnarray}\begin{array}{l}\displaystyle \sum _{j=1}^{n}\left(\displaystyle \frac{2}{{w}_{j}}-2{\zeta }_{j}\right)\mathrm{Re}({\hat{z}}_{j}(t)\overline{{\hat{y}}_{j}(t)})\\ \quad \leqslant \displaystyle \sum _{j=1}^{n}| \displaystyle \frac{2}{{w}_{j}}-2{\zeta }_{j}\parallel {\hat{z}}_{j}(t)\parallel \overline{{\hat{y}}_{j}(t)}| \\ \quad \leqslant \displaystyle \sum _{j=1}^{n}| \displaystyle \frac{1}{{w}_{j}}-{\zeta }_{j}| \left({\hat{z}}_{j}(t)\overline{{\hat{z}}_{j}(t)}+{\hat{y}}_{j}(t)\overline{{\hat{y}}_{j}(t)}\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}2\displaystyle \sum _{j=1}^{n}\displaystyle \sum _{r=1}^{n}{w}_{j}\mathrm{Re}\left({c}_{{jr}}{\hat{f}}_{r}({\hat{y}}_{r}(t-{\mu }_{r}))\overline{{\hat{z}}_{j}(t)}\right)\\ \quad \leqslant 2\displaystyle \sum _{j=1}^{n}\displaystyle \sum _{r=1}^{n}| {w}_{j}{c}_{{jr}}\parallel {\hat{f}}_{r}({\hat{y}}_{r}(t-{\mu }_{r}))\parallel \overline{{\hat{z}}_{j}(t)}| \\ \quad \leqslant 2\displaystyle \sum _{j=1}^{n}\displaystyle \sum _{r=1}^{n}| {w}_{j}{c}_{{jr}}| {F}_{r}| {\hat{y}}_{r}(t-{\mu }_{r})\parallel \overline{{\hat{z}}_{j}(t)}| \\ \quad \leqslant \displaystyle \sum _{j=1}^{n}\displaystyle \sum _{r=1}^{n}| {w}_{j}{c}_{{jr}}| {F}_{r}\left({\hat{y}}_{r}(t-{\mu }_{r})\overline{{\hat{y}}_{r}(t-{\mu }_{r})}+{\hat{z}}_{j}(t)\overline{{\hat{z}}_{j}(t)}\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}2\displaystyle \sum _{j=1}^{n}\displaystyle \sum _{r=1}^{n}{w}_{j}\mathrm{Re}\left({m}_{{jr}}{\hat{g}}_{r}({\hat{y}}_{r}(t-{h}_{r}))\overline{{\hat{z}}_{j}(t)}\right)\\ \quad \leqslant 2\displaystyle \sum _{j=1}^{n}\displaystyle \sum _{r=1}^{n}| {w}_{j}{m}_{{jr}}\parallel {\hat{g}}_{r}({\hat{y}}_{r}(t-{h}_{r}))\parallel \overline{{\hat{z}}_{j}(t)}| \\ \quad \leqslant 2\displaystyle \sum _{j=1}^{n}\displaystyle \sum _{r=1}^{n}| {w}_{j}{m}_{{jr}}| {G}_{r}| {\hat{y}}_{r}(t-{h}_{r})\parallel \overline{{\hat{z}}_{j}(t)}| \\ \quad \leqslant \displaystyle \sum _{j=1}^{n}\displaystyle \sum _{r=1}^{n}| {w}_{j}{m}_{{jr}}| {G}_{r}\left({\hat{y}}_{r}(t-{h}_{r})\overline{{\hat{y}}_{r}(t-{h}_{r})}+{\hat{z}}_{j}(t)\overline{{\hat{z}}_{j}(t)}\right).\end{array}\end{eqnarray}$
Submitting ( $\begin{eqnarray}\begin{array}{l}\dot{V}(t)\leqslant {{\rm{e}}}^{(\varepsilon +1)t}\displaystyle \sum _{j=1}^{n}\left[\left(\displaystyle \frac{\varepsilon }{{{\rm{e}}}^{t}}-2\displaystyle \frac{{\tilde{w}}_{j}}{{w}_{j}}-2{k}_{j}+| \displaystyle \frac{1}{{w}_{j}}\right.\right.\\ \quad \left.-{\zeta }_{j}| \right){\hat{y}}_{j}(t)\overline{{\hat{y}}_{j}(t)}+\left(\displaystyle \frac{\varepsilon }{{{\rm{e}}}^{t}}-2({\gamma }_{j}+{\eta }_{j})\right.\\ \quad +| \displaystyle \frac{1}{{w}_{j}}-{\zeta }_{j}| +\displaystyle \sum _{r=1}^{n}| {w}_{j}{c}_{{jr}}| {F}_{r}\\ \quad \left.+\sum _{r=1}^{n}| {w}_{j}{m}_{{jr}}| {G}_{r}\right){\hat{z}}_{j}(t)\overline{{\hat{z}}_{j}(t)}\\ \quad +\left(\displaystyle \sum _{r=1}^{n}| {w}_{j}{c}_{{jr}}| {F}_{r}-{\rho }_{j}\right){\hat{y}}_{r}(t-{\mu }_{r})\overline{{\hat{y}}_{r}(t-{\mu }_{r})}\\ \quad +\left(\displaystyle \sum _{r=1}^{n}| {w}_{j}{m}_{{jr}}| {G}_{r}-{\sigma }_{j}\right){\hat{y}}_{r}(t-{h}_{r})\overline{{\hat{y}}_{r}(t-{h}_{r})}\\ \quad \leqslant {{\rm{e}}}^{(\varepsilon +1)t}\displaystyle \sum _{j=1}^{n}\left[\left(\varepsilon -2\displaystyle \frac{{\tilde{w}}_{j}}{{w}_{j}}-2{k}_{j}+| \displaystyle \frac{1}{{w}_{j}}-{\zeta }_{j}| \right){\hat{y}}_{j}(t)\overline{{\hat{y}}_{j}(t)}\right.\\ \quad +\left(\varepsilon -2({\gamma }_{j}+{\eta }_{j})+| \displaystyle \frac{1}{{w}_{j}}-{\zeta }_{j}| +\sum _{r=1}^{n}| {w}_{j}{c}_{{jr}}| {F}_{r}\right.\\ \quad \left.+\sum _{r=1}^{n}| {w}_{j}{m}_{{jr}}| {G}_{r}\right){\hat{z}}_{j}(t)\overline{{\hat{z}}_{j}(t)}\\ \quad +\left(\displaystyle \sum _{r=1}^{n}| {w}_{j}{c}_{{jr}}| {F}_{r}-{\rho }_{j}\right){\hat{y}}_{r}(t-{\mu }_{r})\overline{{\hat{y}}_{r}(t-{\mu }_{r})}\\ \quad +\left(\displaystyle \sum _{r=1}^{n}| {w}_{j}{m}_{{jr}}| {G}_{r}-{\sigma }_{j}\right){\hat{y}}_{r}(t-{h}_{r})\overline{{\hat{y}}_{r}(t-{h}_{r})}.\end{array}\end{eqnarray}$
From ( $\begin{eqnarray}\dot{V}(t)\leqslant 0,\end{eqnarray}$
which implies $V(t)\leqslant V(\mathrm{log}{t}_{0})$, for $t\geqslant \mathrm{log}{t}_{0}$.Moreover, from (11 ), we can obtain
$\begin{eqnarray}{{\rm{e}}}^{\varepsilon t}\parallel \hat{y}(t){\parallel }^{2}\leqslant V(t),\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{l}V(\mathrm{log}{t}_{0})=\displaystyle \sum _{j=1}^{n}\left({\hat{y}}_{j}(\mathrm{log}{t}_{0})\overline{{\hat{y}}_{j}(\mathrm{log}{t}_{0})}\right.\\ \quad \left.+{\hat{z}}_{j}(\mathrm{log}{t}_{0})\overline{{\hat{z}}_{j}(\mathrm{log}{t}_{0})}\right){{\rm{e}}}^{\varepsilon \mathrm{log}{t}_{0}}\\ \quad \leqslant {{\rm{e}}}^{\varepsilon \mathrm{log}{t}_{0}}\displaystyle \sum _{j=1}^{n}\left(1+\displaystyle \frac{{\hat{z}}_{j}(\mathrm{log}{t}_{0})\overline{{\hat{z}}_{j}(\mathrm{log}{t}_{0})}}{\mathop{\sup }\limits_{\mathrm{log}\tilde{q}{t}_{0}\leqslant s\leqslant \mathrm{log}{t}_{0}}\parallel \hat{y}(s){\parallel }^{2}}\right)\\ \quad \times \mathop{\sup }\limits_{\mathrm{log}\tilde{q}{t}_{0}\leqslant s\leqslant \mathrm{log}{t}_{0}}\parallel \hat{y}(s){\parallel }^{2}\\ \quad \leqslant {t}_{0}^{\varepsilon }M\mathop{\sup }\limits_{\mathrm{log}\tilde{q}{t}_{0}\leqslant s\leqslant \mathrm{log}{t}_{0}}\parallel \hat{y}(s){\parallel }^{2},\end{array}\end{eqnarray}$
where $\begin{eqnarray*}\begin{array}{rcl}\hat{y}(t) & = & {\left({\hat{y}}_{1}(t),{\hat{y}}_{2}(t),\ldots ,{\hat{y}}_{n}(t)\right)}^{{\rm{T}}},\\ M & = & \displaystyle \sum _{j=1}^{n}\left(1+\displaystyle \frac{{\hat{z}}_{j}(\mathrm{log}{t}_{0})\overline{{\hat{z}}_{j}(\mathrm{log}{t}_{0})}}{\mathop{\sup }\limits_{\mathrm{log}\tilde{q}{t}_{0}\leqslant s\leqslant \mathrm{log}{t}_{0}}\parallel \hat{y}(s){\parallel }^{2}}\right).\end{array}\end{eqnarray*}$
Thus, we have $\begin{eqnarray}\begin{array}{rcl}\parallel \hat{y}(t){\parallel }^{2} & \leqslant & {t}_{0}^{\varepsilon }M\mathop{\sup }\limits_{\mathrm{log}\tilde{q}{t}_{0}\leqslant s\leqslant \mathrm{log}{t}_{0}}\parallel \xi (s){\parallel }^{2}{{\rm{e}}}^{-\varepsilon t}\\ & = & K\mathop{\sup }\limits_{\mathrm{log}\tilde{q}{t}_{0}\leqslant s\leqslant \mathrm{log}{t}_{0}}\parallel \xi (s){\parallel }^{2}{{\rm{e}}}^{-\varepsilon t},\end{array}\end{eqnarray}$
where $K={t}_{0}^{\varepsilon }M\geqslant 1$.Note that $\hat{y}(t)=y({{\rm{e}}}^{t})=v({{\rm{e}}}^{t})-u({{\rm{e}}}^{t})$, one can have1 , systems (1) and (3) can achieve global polynomial synchronization. The proof is completed.
$\begin{eqnarray}\parallel v(t)-u(t){\parallel }^{2}\leqslant K\parallel \hat{\phi }(s)-\phi (s){\parallel }_{\tilde{q}}^{2}{t}^{-\varepsilon },\end{eqnarray}$
where $\parallel \hat{\phi }(s)-\phi (s){\parallel }_{\tilde{q}}^{2}={\sup }_{\tilde{q}{t}_{0}\leqslant s\leqslant {t}_{0}}\parallel \hat{\phi }(s)-\phi (s){\parallel }^{2}$. Therefore, by definition Sufficient conditions for polynomial synchronization based on CVINNs systems with multi-proportional delays are established in theorem 1. In our work, by constructing Lyapunov functions and making full use of inequality techniques, the expected results are obtained. To the best of our knowledge, this is the first time that the problem of polynomial synchronization has been discussed under CVINNs.
At present, the methods for dealing with CVINNs are mainly divided into two types, namely the separation method [20, 21, 23–27] and the non-separation method [31–33]. The former can increase the number of variables and dimensions of systems whereas the latter can keep the originality of systems and reduce the limitations. Obviously, the advantages of the latter are more typical, which inspires us to fully utilize the non-separation method to achieve the goal.
As we know, it is the first time that the PS problem of CVINNs with multi-proportional delays is studied. Here, we use the non-separation method which can simplify the process and result. Moreover, we also consider the multi-proportional delays which own the easier controllability. Therefore, our work is very important in theory and application value.
There are many types of time delays, such as time-varying delays, proportional delays, distributed delays, and so on. As we know, a large number of achievements involve time-varying delays, for example, proportional-integral synchronization [47] and finite-time synchronization of Markovian coupled neural networks [48]. Different from these results, this paper focuses on multi-proportional delays, which are unbounded delays and widely used in the light absorption of stellar matter, nonlinear dynamic systems, and many other fields.
4. Numerical example
In this section, we will provide an example and some simulations to illustrate the effectiveness of the theoretical result.
Consider the following two-dimension drive system
$\begin{eqnarray}\begin{array}{rcl}{\ddot{u}}_{j}(t) & = & -{a}_{j}{\dot{u}}_{j}(t)-{b}_{j}{u}_{j}(t)+\displaystyle \sum _{r=1}^{2}{c}_{{jr}}{f}_{r}({u}_{r}({p}_{r}t))\\ & & +\displaystyle \sum _{r=1}^{2}{m}_{{jr}}{g}_{r}({u}_{r}({q}_{r}t))+{I}_{j}(t),\end{array}\end{eqnarray}$
and the response system is given by $\begin{eqnarray}\begin{array}{rcl}{\ddot{v}}_{j}(t) & = & -{a}_{j}{\dot{v}}_{j}(t)-{b}_{j}{v}_{j}(t)+\displaystyle \sum _{r=1}^{n}{c}_{{jr}}{f}_{r}({v}_{r}({p}_{r}t))\\ & & +\displaystyle \sum _{r=1}^{n}{m}_{{jr}}{g}_{r}({v}_{r}({q}_{r}t))+{I}_{j}(t)+{U}_{j}(t),\end{array}\end{eqnarray}$
with ${a}_{1}={a}_{2}=0.6$, ${b}_{1}={b}_{2}=6$, ${p}_{1}=0.9$, ${q}_{1}=0.7$, ${p}_{2}\,=0.8$, ${q}_{2}=0.5$, ${c}_{11}=3-{\rm{i}}$, ${c}_{12}=6-5.5{\rm{i}}$, ${c}_{21}=-5\,+2.7{\rm{i}}$, ${c}_{22}=5-8{\rm{i}}$, ${m}_{11}=8-0.3{\rm{i}}$, ${m}_{12}=10-5{\rm{i}}$, ${m}_{21}\,=-5+4{\rm{i}}$, ${m}_{22}=-5.2+5{\rm{i}}$, ${f}_{r}({u}_{r}(t))={g}_{r}({u}_{r}(t))=\tfrac{1-{{\rm{e}}}^{-\overline{{u}_{j}(t)}}}{1+{{\rm{e}}}^{-\overline{{u}_{j}(t)}}}$ $(r=1,2)$, ${I}_{1}(t)={I}_{2}(t)=\sin (0.1t)$.The state trajectories of systems (22 ) and (23 ) with Uj(t) = 0 are shown in figure 1, where the initial conditions are given by φ1(s) = 1.5 − 1.5i, φ2(s) = − 1.5 − 1.5i, $Psi$1(s) = − 6.5 + 13.5i, $Psi$2(s) = 0.5 + 3.5i, $\hat{{\phi }_{1}}(s)=-1.5\,+1.5{\rm{i}}$, $\hat{{\phi }_{2}}(s)=1.5\,+1.4{\rm{i}}$, $\hat{{\psi }_{1}}(s)=11.5-18.5{\rm{i}}$ and $\hat{{\psi }_{2}}(s)=13.5\,+\,3.6{\rm{i}}$ with s ∈ [0.5, 1]. It is obvious that synchronization cannot be realized.
Figure 1. The curves of the variables u1, v1, u2, v2 without the controller. |
Then, we will consider the PS of systems (22 ) and (23 ) under the controller (9 ). It is easy to verify that Assumption 1 holds and ${F}_{j}={G}_{j}=\tfrac{2}{3}$, for j = 1, 2. Through calculating (10 ) of theorem 1, we can obtain that k1 > 1.7, k2 > 1.7, η1 > 10.9060, η2 > 13.0343, ρ1 > 7.5345, ρ2 > 10.776, and σ1 > 12.7906, σ2 > 9.0780. So we choose k1 = 1.71, k2 = 1.71, η1 = 10.91, η2 = 13.04, ρ1 = 7.54, ρ2 = 10.78, and σ1 = 12.80, σ2 = 9.08. From theorem 1, systems (22 ) and (23 ) can reach the PS under the controller (9 ), which can be also seen in figure 2.
Figure 2. The curves of the variables u1, v1, u2, v2 with the controller. |
5. Conclusion
In this paper, we have studied the PS problem of CVINNs with multi-proportional delays by the non-separation method. In order to realize the PS of the addressed system, we have involved the exponential transformation, obtained the power exponential term via fully utilizing some inequality techniques, and solved the main difficulty based on the designed suitable controller and Lyapunov function method. Then, we obtained a simple sufficient condition for PS. Finally, we have provided a numerical simulation example to verify the validity of our work. In the following work, we will focus on other synchronization phenomena of CVINNs, such as module-phase synchronization, projective synchronization, combination synchronization, and so on.