1 Introduction

The study of NNs with values in multidimensional domains has increased over the last few years. As such, CVNNs, for which the domain is the 2D complex numbers algebra, and QVNNs, for which the domain is the 4D quaternion algebra, have appeared. Their generalization, ClVNNs, for which the domain can be any \(2^{n}\)-dimensional Clifford algebra, with \(n\ge 1\), have also begun to raise interest lately.

A different generalization of CVNNs and QVNNs has also recently appeared, in the form of OVNNs, for which the domain is the 8D octonion algebra. The octonion algebra does not belong to the Clifford algebras family, which is easy to see because all Clifford algebras are associative, whereas the octonion algebra is not. However, the octonion algebra is the only other normed division algebra that can be defined over the reals, beside the complex and quaternion algebras. This property makes it interesting for applications, and, indeed, the octonions have been successfully applied in electrodynamics (Chanyal 2013), gravitational field equations (Demir 2012), or fluid dynamics (Demir and Tanişli 2016), but also in salient object detection (Gao and Lam 2014a, b), signal processing (Snopek 2015; Wang et al. 2016; Błaszczyk and Snopek 2017), or hyperspectral fluorescence data fusion (Bauer and Leon 2016), for example.

Because of their potential applications in problems related to high-dimensional data processing, it was a natural idea to apply the octonion algebra to the domain of NNs, also, which gave rise to OVNNs. Feedforward OVNNs were first proposed in Popa (2016). Then, the dynamics of recurrent Hopfield OVNNs was studied in Popa (2018a), Popa (2018b). Since then, other papers appeared, which study different dynamic properties of OVNNs, for example (Kandasamy and Rajan 2020; Chouhan et al. 2022; Li and Huang 2022; Chouhan et al. 2023; Gao and Dai 2023; Huang and Li 2023; Li et al. 2023; Popa 2023). The majority of these papers are very recent, showing a growing interest from the scientific community in the domain of OVNNs, indicating the likely publication of additional papers in this area in the near future.

When discussing the dynamics of NN models, it’s essential to put forward models with time delays, because, in real-world implementations of NNs, they naturally appear because of the finite switching velocity of circuit components, and can determine undesired behavior. One common type of delays is the leakage delay, which appears in the self-feedback term of the NNs. It has been added, as part of OVNN models, in Popa (2018a), Gao and Dai (2023). Then, the conduction velocities along the pathways of the circuit implementation of NNs can cause dispersion, which gives rise to distributed delays. When they appear in conjunction with time-varying delays, which is the type of delays that was most often discussed in the context of NNs, they are called mixed delays. Mixed delays have been added to OVNN models in Popa (2018a), Gao and Dai (2023), Popa (2023).

On the other hand, to our best knowledge, all the papers involving OVNNs are done in continuous time. However, when NNs are implemented in circuits, discretization is an inevitable step, resulting in a discrete time system, which can differ very much in its dynamic properties from those of the original continuous time system. It is this realization that led to the definition of NNs in discrete time, for the first time in Mohamad and Gopalsamy (2000). Since then, discrete time NNs have developed into a field of study in their own right, with an increasing number of papers appearing each year. As previously mentioned, there are no papers in the available literature discussing discrete time OVNNs.

An elegant solution to the problem of continuous time vs. discrete time systems was proposed in the form of time scale calculus, for the first time in Hilger (1990). With this type of calculus, the properties of differential and difference systems, or any hybrid combination of them, can be studied in a unified manner. Today, the most important references for time scale calculus are the books (Bohner and Peterson 2001; Martynyuk 2016; Adıvar and Raffoul 2020), in which the domain was developed further and summarized. Time scales have also been added to the study of NNs, for the first time in Chen and Du (2008). Similarly to discrete time NNs, time scale NNs have become an established area of research in their own right, with many papers appearing yearly that discuss this type of networks. Also, like for discrete time OVNNs, there are no papers currently discussing time scales OVNNs, to the best of our knowledge.

For systems defined on time scales, the traditional Lyapunov theory is not suitable, since time scale calculus is more broad and must take into account the properties of both differential and difference systems. As an alternative, in order to study the dynamics of such systems, different variations of Halanay inequalities have been employed over time, see (Mohamad and Gopalsamy 2000; Wen et al. 2008; Wang 2010; Wen et al. 2018; Kassim and Tatar 2021). Halanay inequalities for time scales are the result of extending these inequalities to time scales, see (Adıvar and Bohner 2011; Ou et al. 2015, 2016; Ou 2020). They were then applied to studying the dynamics of time scale NNs, for example in Xiao and Zeng (2017), Xiao and Zeng (2018), Xiao et al. (2018), Xiao et al. (2020), Wan and Zeng (2021), Wan and Zeng (2022a), Wan and Zeng (2022b).

The key contributions of the paper can be summarized as follows:

  1. 1.

    A model of OVNNs with leakage, time-varying, and distributed delays defined on time scales is formulated, and then, to avoid the problems related to the non-associativity and non-commutativity of the octonion algebra, the system is transformed into a real-valued one.

  2. 2.

    Lyapunov-like functions pertaining to two general categories are formulated, and Halanay-type inequalities for time scales are used in order to deduce sufficient conditions, expressed both as scalar inequalities and as LMIs, which guarantee the exponential stability of the proposed model.

  3. 3.

    Then, two similar Lyapunov-like functions are used in conjunction with a state feedback controller in order to obtain sufficient criteria expressed in the same way for the exponential synchronization of the OVNNs put forward.

  4. 4.

    Through the means of one numerical simulation, both in discrete time and in continuous time, each of the four theorems is demonstrated.

  5. 5.

    The results obtained in the paper are general enough that they can be particularized for continuous time or for discrete time OVNNs, or any hybrid combination of the two, or even for models of CVNNs or QVNNs, for which, to our knowledge, no comparable results have been reported in the literature.

The remaining part of the research unfolds in the following way. Section 2 is dedicated to emphasizing the most important aspects of the octonion algebra and of the time scale calculus, along with useful lemmas and an assumption that will be used to obtain the main results. Also, the proposed model is given in the same section, and its transformation into a real-valued system is performed. Then, Sect. 3 is dedicated to presenting the results of the paper: two theorems for the stability and two theorems for the synchronization of the proposed model. One numerical simulation is employed to demonstrate each of the four theorems, both in discrete time and in continuous time, in Sect. 4. The last section, Sect. 5, is dedicated to concluding the paper.

Notations: \(\mathbb {R}\) – reals, \(\mathbb {R}^{+}\) – positive reals, \(\mathbb {O}\) – octonions, \(||\cdot ||_{p}\) – \(L_{p}\) norm, \(p\in \{1,2\}\), \(\mathbb {R}^{N}\) (\(\mathbb {O}^{N}\)) – real (octonion) vectors of dimension N, \(\mathbb {R}^{N\times N}\) (\(\mathbb {O}^{N\times N}\)) – real (octonion) \(N\times N\)-dimensional matrices, \(Y<0\) – Y is negative definite, \(\lambda _{\min }(Y)\) – smallest eigenvalue of Y, \(Y^{T}\) – transpose of Y.

2 Preliminaries

First, we provide an overview of the octonion algebra, mostly based on Popa (2018a), Popa (2018b). “The set of octonions is defined as:

$$\begin{aligned} \mathbb {O}=\left\{ \left. o=\sum _{q=0}^{7}o^{q}e_{q}\right| o^{q}\in \mathbb {R},\;\forall 0\le q\le 7\right\} , \end{aligned}$$

where \(e_{q}\) represent the unit octonions, \(\forall 0\le q\le 7\). On this set, we define the octonion addition by \(o+p:=\sum _{q=0}^{7}(o^{q}+p^{q})e_{q}\) and the scalar multiplication by \(\alpha o:=\sum _{q=0}^{7}(\alpha o^{q})e_{q}\). The multiplication of octonions is defined by the multiplication table of the octonion units:

\(\times \)

\(e_{0}\)

\(e_{1}\)

\(e_{2}\)

\(e_{3}\)

\(e_{4}\)

\(e_{5}\)

\(e_{6}\)

\(e_{7}\)

\(e_{0}\)

\(e_{0}\)

\(e_{1}\)

\(e_{2}\)

\(e_{3}\)

\(e_{4}\)

\(e_{5}\)

\(e_{6}\)

\(e_{7}\)

\(e_{1}\)

\(e_{1}\)

\(-e_{0}\)

\(e_{3}\)

\(-e_{2}\)

\(e_{5}\)

\(-e_{4}\)

\(-e_{7}\)

\(e_{6}\)

\(e_{2}\)

\(e_{2}\)

\(-e_{3}\)

\(-e_{0}\)

\(e_{1}\)

\(e_{6}\)

\(e_{7}\)

\(-e_{4}\)

\(-e_{5}\)

\(e_{3}\)

\(e_{3}\)

\(e_{2}\)

\(-e_{1}\)

\(-e_{0}\)

\(e_{7}\)

\(-e_{6}\)

\(e_{5}\)

\(-e_{4}\)

\(e_{4}\)

\(e_{4}\)

\(-e_{5}\)

\(-e_{6}\)

\(-e_{7}\)

\(-e_{0}\)

\(e_{1}\)

\(e_{2}\)

\(e_{3}\)

\(e_{5}\)

\(e_{5}\)

\(e_{4}\)

\(-e_{7}\)

\(e_{6}\)

\(-e_{1}\)

\(-e_{0}\)

\(-e_{3}\)

\(e_{2}\)

\(e_{6}\)

\(e_{6}\)

\(e_{7}\)

\(e_{4}\)

\(-e_{5}\)

\(-e_{2}\)

\(e_{3}\)

\(-e_{0}\)

\(-e_{1}\)

\(e_{7}\)

\(e_{7}\)

\(-e_{6}\)

\(e_{5}\)

\(e_{4}\)

\(-e_{3}\)

\(-e_{2}\)

\(e_{1}\)

\(-e_{0}\)

For each octonion \(o\in \mathbb {O}\), its conjugate is defined as \(\overline{o}:=o^{0}e_{0}-\sum _{q=1}^{7}o^{q}e_{q}\). Then, we define the norm of octonion \(o\in \mathbb {O}\) as \(|o|:=\sqrt{\overline{o}o}=\sqrt{\sum _{q=0}^{7}(o^{q})^{2}}\) and its inverse as \(o^{-1}:=\frac{\overline{o}}{|o|^{2}}\). With all these operations, it can be proved that \(\mathbb {O}\) is a normed division algebra. Actually, there exists a famous result by Hurwitz, who showed that the real, complex, quaternion, and octonion number sets with their respective operations are the only normed division algebras over the real numbers.

As can be seen from the multiplication table, we have that \(e_{q}e_{l}=-e_{l}e_{q}\ne e_{l}e_{q}\) if \(q\ne l\), \(q\ne 0\), \(l\ne 0\), which means that, like the algebra of quaternions, the above-defined algebra of octonions is not commutative. Moreover, we can observe that \((e_{q}e_{l})e_{m}=-e_{q}(e_{l}e_{m})\ne e_{q}(e_{l}e_{m})\) if q, l, m are non-zero, different, and \(e_{q}e_{l}\ne \pm e_{m}.\) This tells us that, unlike the algebra of quaternions and other Clifford algebras in general, the algebra of octonions is also not associative.”

Next, we present the fundamentals of calculus on time scales, generally based on Bohner and Peterson (2001). “Time scale \(\mathbb {T}\) is defined as an non-empty closed subset of the real number set \(\mathbb {R}\), from which it inherits the ordering and the topology. The following jump operators can be defined \(\forall t\in \mathbb {T}\): forward jump \(\sigma (t):=\inf \{\left. s\in \mathbb {T}\right| s>t\}\) and backward jump \(\rho (t):=\sup \{\left. s\in \mathbb {T}\right| s<t\}\). Function \(\mu :\mathbb {T}\rightarrow [0,+\infty )\), defined as \(\mu (t):=\sigma (t)-t\), \(\forall t\in \mathbb {T}\), represents the forward graininess function. Denote \(\hat{\mu }=\sup \{\left. \mu (t)\right| t\in \mathbb {T}\}\).

Then, a point \(t\in \mathbb {T}\) is left (right)-dense if \(\rho (t)=t\) (\(\sigma (t)=t\)); a point \(t\in \mathbb {T}\) is left (right)-scattered if \(\rho (t)<t\) (\(\sigma (t)>t\)). If there exists a left-scattered maximum for \(\mathbb {T}\) denoted as m, then \(\mathbb {T}^{\kappa }:=\mathbb {T}\backslash \{m\}\), else \(\mathbb {T}^{\kappa }:=\mathbb {T}\). The function \(f:\mathbb {T}\rightarrow \mathbb {R}\) is said to be rd-continuous if, for any right-dense \(t_{1}\in \mathbb {T}\), \(f(t_{1})=\lim _{\zeta \rightarrow t_{1}^{+}}f(\zeta )\), and for any left-dense \(t_{2}\in \mathbb {T}\), \(\lim _{\zeta \rightarrow t_{2}^{-}}f(\zeta )\) exists. The notation \(C_{rd}(\mathbb {T},\mathbb {R})\) is used for the set of rd-continuous functions \(f:\mathbb {T}\rightarrow \mathbb {R}\). The jump operators for a function \(f:\mathbb {T}\rightarrow \mathbb {R}\) are defined as \(f^{\sigma }(t)=f(\sigma (t))\) and \(f^{\rho }(t)=f(\rho (t))\), respectively. A function \(p:\mathbb {T}\rightarrow \mathbb {R}\) is called regressive, and we write \(p\in \mathcal {R}(\mathbb {T},\mathbb {R})\), if \(p\in C_{rd}(\mathbb {T},\mathbb {R})\) and \(1+\mu (t)p(t)\ne 0\), \(\forall t\in \mathbb {T}^{\kappa }\). The function \(p:\mathbb {T}\rightarrow \mathbb {R}\) is called positively regressive, and we write \(p\in \mathcal {R}^{+}(\mathbb {T},\mathbb {R})\), if \(p\in C_{rd}(\mathbb {T},\mathbb {R})\) and \(1+\mu (t)p(t)>0\), \(\forall t\in \mathbb {T}^{\kappa }\). We define, \(\forall p\in \mathcal {R}(\mathbb {T},\mathbb {R})\), \(\ominus p(t):=-p(t)/(1+\mu (t)p(t))\), \(\forall t\in \mathbb {T}\). For any set \(S\subseteq \mathbb {R}\) we define \(S_{\mathbb {T}}=S\cap \mathbb {T}\).

For a function \(f:\mathbb {T}\rightarrow \mathbb {R}\), if \(t\in \mathbb {T}^{\kappa }\), the \(\Delta \)-derivative of f at t is, if it exists, the number denoted by \(f^{\Delta }(t)\) such that \(\forall \varepsilon >0\) there exists a \(\delta >0\) for which the following inequality holds \(\forall s\in (t-\delta ,t+\delta )_{\mathbb {T}}\):

$$\begin{aligned} |f(\sigma (t))-f(s)-f^{\Delta }(t)(\sigma (t)-s)|\le \varepsilon |\sigma (t)-s|. \end{aligned}$$

If the \(\Delta \)-derivative exists \(\forall t\in \mathbb {T}^{\kappa }\), the function f is called \(\Delta \)-differentiable.

\(\Delta \)-integration constitutes the inverse operation of \(\Delta \)-differentiation, i.e., if \(F^{\Delta }(t)=f(t)\), then

$$\begin{aligned} \int _{a}^{b}f(s)\Delta s=F(b)-F(a),\quad \forall a,b\in \mathbb {T}. \end{aligned}$$

Lastly, if \(p\in \mathcal {R}(\mathbb {T},\mathbb {R})\) is a regressive function, then the \(\Delta \)-exponential function \(e_{p}:\mathbb {T}\times \mathbb {T}\rightarrow \mathbb {R}\) is defined by the formula:

$$\begin{aligned} e_{p}(a,b)=e^{\int _{a}^{b}\xi _{\mu (s)}(p(s))\Delta s},\quad \forall a,b\in \mathbb {T}, \end{aligned}$$

where \(\xi _{\mu (s)}\) denotes the cylinder transformation, which is defined as:

$$\begin{aligned} \xi _{h}(z)={\left\{ \begin{array}{ll} \frac{\log (1+zh)}{h}, &{} h\ne 0\\ z, &{} h=0 \end{array}\right. }." \end{aligned}$$

Related to time scales, we will need the following lemmas for the proofs of our theorems:

Lemma 1

(Bohner and Peterson (2001)) “If \(f,g:\mathbb {T}\rightarrow \mathbb {R}\) are \(\Delta \)-differentiable, then

  1. (i)

    \((f(t)+g(t))^{\Delta }=f^{\Delta }(t)+g^{\Delta }(t)\);

  2. (ii)

    \((f(t)g(t))^{\Delta }=f^{\Delta }(t)g(t)+f(\sigma (t))g^{\Delta }(t)=f(t)g^{\Delta }(t)+f^{\Delta }(t)g(\sigma (t)).\)”

Lemma 2

(Xiao and Zeng (2018)) “Let \(g:\mathbb {T}\rightarrow \mathbb {R}^{+}\) be a non-negative rd-continuous function on \(\mathbb {T}\), and

$$\begin{aligned} y^{\Delta }(t)\le -\alpha _{1}y(t)+\alpha _{2}\sup _{s\in [t-\tau ,t]_{\mathbb {T}}}y(s) \end{aligned}$$

holds for \(t\in [t_{0},+\infty )_{\mathbb {T}}\), where \(\alpha _{1}\) and \(\alpha _{2}\) are positive constants. If \(-\alpha _{1}\in \mathcal {R}^{+}\), \(\alpha _{1}-\alpha _{2}>0\) then

$$\begin{aligned} y(t)\le \sup _{s\in [t_{0}-\tau ,t_{0}]_{\mathbb {T}}}y(s)e_{\ominus \lambda }(t,t_{0}), \end{aligned}$$

for any \(t\in [t_{0},+\infty )_{\mathbb {T}}\), where \(\lambda >0\) satisfies the inequality

$$\begin{aligned} -\ominus \lambda \le \alpha _{1}-\alpha _{2}e^{\lambda \tau }." \end{aligned}$$

Lemma 3

(Xiao and Zeng (2018)) “If \(-\lambda \in \mathcal {R}^{+}\), \(y\in C_{rd}(\mathbb {T},\mathbb {R})\), \(z\in C_{rd}(\mathbb {T},\mathbb {R})\), then, \(\forall t\in \mathbb {T}\),

$$\begin{aligned} y^{\Delta }(t)\le -\lambda y(t)+z(t) \end{aligned}$$

implies

$$\begin{aligned} |y(t)|^{\Delta }\le -\lambda |y(t)|+|z(t)|." \end{aligned}$$

We are now prepared to present the model that will be studied. Define on time scale \(\mathbb {T}\) the subsequent OVNN having leakage, time-varying, and distributed delays:

$$\begin{aligned} o_{i}^{\Delta }(t)= & {} -c_{i}o_{i}(t-\delta )+\sum _{j=1}^{N}a_{ij}f_{j}(o_{j}(t))+\sum _{j=1}^{N}b_{ij}f_{j}(o_{j}(t-\tau (t)))\nonumber \\{} & {} +\sum _{j=1}^{N}g_{ij}\int _{t-\eta }^{t}f_{j}(o_{j}(s))\Delta s+I_{i}, \end{aligned}$$
(1)

\(\forall i\in \{1,\ldots ,N\}\), \(\forall t\in [0,+\infty )_{\mathbb {T}}\), where \(o_{i}(t)\in \mathbb {O}\) – states at \(t\in [0,+\infty )_{\mathbb {T}}\), \(c_{i}\) – self-feedback weights, \(a_{ij}\in \mathbb {O}\) – weights without delay, \(b_{ij}\in \mathbb {O}\) – weights with delay, \(g_{ij}\in \mathbb {O}\) – distributed delay weights, \(f_{j}:\mathbb {O}\rightarrow \mathbb {O}\) – activation functions, and \(I_{i}\in \mathbb {O}\) – external inputs, \(\forall i,j\in \{1,\ldots ,N\}\), \(\delta >0\) is the leakage delay, \(\tau :\mathbb {R}^{+}\rightarrow \mathbb {R}^{+}\) are the time-varying delays, and we assume that there exists \(\tau >0\) such that \(\tau (t)\le \tau \), \(\forall t\in [0,+\infty )_{\mathbb {T}}\), and \(\eta >0\) is the distributed delay. The notation \(\omega :=\max \{\delta ,\tau ,\eta \}\) is made. Also, assume that the activation functions \(f_{j}\) have the form \(f_{j}(o)=\sum _{q=0}^{7}f_{j}^{q}(o)e_{q}\), \(\forall o\in \mathbb {O}\), where \(f_{j}^{q}:\mathbb {O}\rightarrow \mathbb {R}\), \(\forall 0\le q\le 7\), \(\forall j\in \{1,\ldots ,N\}\).

The initial conditions of NN (1) are:

$$\begin{aligned} o_{i}(t)=\varphi _{i}(t),\quad \forall t\in [-\omega ,0]_{\mathbb {T}}, \end{aligned}$$

where \(\varphi _{i}\in \mathcal {C}([-\omega ,0]_{\mathbb {T}},\mathbb {O})\), \(\forall i\in \{1,\ldots ,N\}\). On \(\mathcal {C}([-\omega ,0]_{\mathbb {T}},\mathbb {O}^{N})\) we define the norm as \(||\varphi ||=\sum _{i=1}^{N}\sup _{[-\omega ,0]_{\mathbb {T}}}|\varphi _{i}(t)|\).

System (1) is assumed to possess an unique equilibrium point, denoted by \(\tilde{o}=(\tilde{o}_{1},\ldots ,\tilde{o}_{N})^{T}\) in the following. In this setting, if we put \(\mathfrak {o}_{i}(t)=o_{i}(t)-\tilde{o}_{i}\), system (1) becomes:

$$\begin{aligned} \mathfrak {o}_{i}^{\Delta }(t)= & {} -c_{i}\mathfrak {o}_{i}(t-\delta )+\sum _{j=1}^{N}a_{ij}\tilde{f}_{j}(\mathfrak {o}_{j}(t))+\sum _{j=1}^{N}b_{ij}\tilde{f}_{j}(\mathfrak {o}_{j}(t-\tau (t)))\nonumber \\{} & {} +\sum _{j=1}^{N}g_{ij}\int _{t-\eta }^{t}\tilde{f}_{j}(\mathfrak {o}_{j}(s))\Delta s, \end{aligned}$$
(2)

\(\forall i\in \{1,\ldots ,N\}\), \(\forall t\in [0,+\infty )_{\mathbb {T}}\), where \(\tilde{f}_{j}(\mathfrak {o}_{j}(t))=f_{j}(\mathfrak {o}_{j}(t)+\tilde{o}_{j})-f_{j}(\tilde{o}_{j})\), \(\forall t\in [0,+\infty )_{\mathbb {T}}\), \(\forall j\in \{1,\ldots ,N\}\).

Now, NN (2) has the following initial conditions:

$$\begin{aligned} \mathfrak {o}_{i}(t)=\chi _{i}(t)=\varphi _{i}(t)-\tilde{o}_{i},\;\forall t\in [-\omega ,0]_{\mathbb {T}}, \end{aligned}$$

where \(\chi _{i}\in \mathcal {C}([-\omega ,0]_{\mathbb {T}},\mathbb {O})\), \(\forall i\in \{1,\ldots ,N\}\).

The system of equations (2) will now be converted into 8 real-valued systems. This is accomplished by using the following 8 equations to represent each equation in (2):

$$\begin{aligned} \mathfrak {o}_{i}^{q\Delta }(t)= & {} -c_{i}\mathfrak {o}_{i}^{q}(t-\delta )+\sum _{j=1}^{N}\sum _{l=0}^{7}a_{ij}^{ql}\tilde{f}_{j}^{l}(\mathfrak {o}_{j}(t))+\sum _{j=1}^{N}\sum _{l=0}^{7}b_{ij}^{ql}\tilde{f}_{j}^{l}(\mathfrak {o}_{j}(t-\tau (t)))\nonumber \\{} & {} +\sum _{j=1}^{N}\sum _{l=0}^{7}g_{ij}^{ql}\int _{t-\eta }^{t}\tilde{f}_{j}^{l}(\mathfrak {o}_{j}(s))\Delta s, \end{aligned}$$
(3)

\(\forall i\in \{1,\ldots ,N\}\), \(\forall 0\le q\le 7\), where \(o^{ql}\), \(\forall 0\le q,l\le 7\), denotes a component of \(\text {mat}(o)\), which is a matrix defined by:

$$\begin{aligned} \text {mat}(o):=\begin{bmatrix}o^{0} &{} -o^{1} &{} -o^{2} &{} -o^{3} &{} -o^{4} &{} -o^{5} &{} -o^{6} &{} -o^{7}\\ o^{1} &{} o^{0} &{} -o^{3} &{} o^{2} &{} -o^{5} &{} o^{4} &{} o^{7} &{} -o^{6}\\ o^{2} &{} o^{3} &{} o^{0} &{} -o^{1} &{} -o^{6} &{} -o^{7} &{} o^{4} &{} o^{5}\\ o^{3} &{} -o^{2} &{} o^{1} &{} o^{0} &{} -o^{7} &{} o^{6} &{} -o^{5} &{} -o^{4}\\ o^{4} &{} o^{5} &{} o^{6} &{} o^{7} &{} o^{0} &{} -o^{1} &{} -o^{2} &{} -o^{3}\\ o^{5} &{} -o^{4} &{} o^{7} &{} -o^{6} &{} o^{1} &{} o^{0} &{} o^{3} &{} -o^{2}\\ o^{6} &{} -o^{7} &{} -o^{4} &{} o^{5} &{} o^{2} &{} -o^{3} &{} o^{0} &{} o^{1}\\ o^{7} &{} o^{6} &{} -o^{5} &{} -o^{4} &{} o^{3} &{} o^{2} &{} -o^{1} &{} o^{0} \end{bmatrix}. \end{aligned}$$

Now, by denoting \(\text {vec}(o):=(o^{0},\ldots ,o^{7})^{T}\), NN (2) will be transformed as:

$$\begin{aligned} \text {vec}(\mathfrak {o}_{i}^{\Delta }(t))= & {} -c_{i}\text {vec}(\mathfrak {o}_{i}(t-\delta ))+\sum _{j=1}^{N}\text {mat}(a_{ij})\text {vec}(\tilde{f}_{j}(\mathfrak {o}_{j}(t)))\\{} & {} +\sum _{j=1}^{N}\text {mat}(b_{ij})\text {vec}(\tilde{f}_{j}(\mathfrak {o}_{j}(t-\tau (t))))\\{} & {} +\sum _{j=1}^{N}\text {mat}(g_{ij})\int _{t-\eta }^{t}\text {vec}(\tilde{f}_{j}(\mathfrak {o}_{j}(s)))\Delta s, \end{aligned}$$

\(\forall i\in \{1,\ldots ,N\}\), \(\forall t\in [0,+\infty )_{\mathbb {T}}\).

Finally, with the notations \(\check{C}=\text {diag}(c_{1}I_{8},\ldots ,c_{N}I_{8}),\) \(\check{A}=\left( \text {mat}(a_{ij})\right) _{1\le i,j\le N}\), \(\check{B}=\left( \text {mat}(b_{ij})\right) _{1\le i,j\le N}\), \(\check{G}=\left( \text {mat}(g_{ij})\right) _{1\le i,j\le N}\), \(\check{\mathfrak {o}}(t):=(\text {vec}(\mathfrak {o}_{1}(t))^{T},\ldots ,\text {vec}(\mathfrak {o}_{N}(t))^{T})^{T}\), \(\check{f}(\mathfrak {o}(t)):=(\text {vec}(\tilde{f}_{1}(\mathfrak {o}_{1}(t)))^{T},\ldots ,\text {vec}(\tilde{f}_{N}(\mathfrak {o}_{N}(t)))^{T})^{T}\), NN (2) becomes:

$$\begin{aligned} \check{\mathfrak {o}}^{\Delta }(t)= & {} -\check{C}\check{\mathfrak {o}}(t-\delta )+\check{A}\check{f}(\mathfrak {\check{\mathfrak {o}}}(t))+\check{B}\check{f}(\check{\mathfrak {o}}(t-\tau (t)))\nonumber \\{} & {} +\check{G}\int _{t-\eta }^{t}\check{f}(\check{\mathfrak {o}}(s))\Delta s,\quad \forall t\in [0,+\infty )_{\mathbb {T}}. \end{aligned}$$
(4)

On the other hand, we will define the following system as the response NN and regard NN (1) as the drive NN, for the analysis of synchronization:

$$\begin{aligned} p_{i}^{\Delta }(t)= & {} -c_{i}p_{i}(t-\delta )+\sum _{j=1}^{N}a_{ij}f_{j}(p_{j}(t))+\sum _{j=1}^{N}b_{ij}f_{j}(p_{j}(t-\tau (t)))\nonumber \\{} & {} +\sum _{j=1}^{N}g_{ij}\int _{t-\eta }^{t}f_{j}(p_{j}(s))\Delta s+I_{i}-u_{i}(t), \end{aligned}$$
(5)

\(\forall i\in \{1,\ldots ,N\}\), \(\forall t\in [0,+\infty )_{\mathbb {T}}\), and \(p_{i}(t)\in \mathbb {O}\) – states at \(t\in [0,+\infty )_{\mathbb {T}}\) and \(u_{i}(t)\in \mathbb {O}\) – control inputs at \(t\in [0,+\infty )_{\mathbb {T}}\).

The initial conditions of NN (5) are:

$$\begin{aligned} p_{i}(t)=\psi _{i}(t),\quad \forall t\in [-\omega ,0]_{\mathbb {T}}, \end{aligned}$$

where \(\psi _{i}\in \mathcal {C}([-\omega ,0]_{\mathbb {T}},\mathbb {O})\), \(\forall i\in \{1,\ldots ,N\}\).

In this new setting, by considering (1) and (5), and making the notation \(\varpi _{i}(t)=p_{i}(t)-o_{i}(t)\), the error system will have the following expression:

$$\begin{aligned} \varpi _{i}^{\Delta }(t)= & {} -c_{i}\varpi _{i}(t-\delta )+\sum _{j=1}^{N}a_{ij}\tilde{f}_{j}(\varpi _{j}(t))+\sum _{j=1}^{N}b_{ij}\tilde{f}_{j}(\varpi _{j}(t-\tau (t)))\nonumber \\{} & {} +\sum _{j=1}^{N}g_{ij}\int _{t-\eta }^{t}\tilde{f}_{j}(\varpi _{j}(s))\Delta s-u_{i}(t), \end{aligned}$$
(6)

\(\forall i\in \{1,\ldots ,N\}\), \(\forall t\in [0,+\infty )_{\mathbb {T}}\), where \(\tilde{f}_{j}(\varpi _{j}(t))=f_{j}(\varpi _{j}(t)+o_{j}(t))-f_{j}(o_{j}(t))\), \(\forall t\in [0,+\infty )_{\mathbb {T}}\), \(\forall j\in \{1,\ldots ,N\}\).

The initial conditions of NN (6) are now:

$$\begin{aligned} \varpi _{i}(t)=\phi _{i}(t)=\varphi _{i}(t)-\psi _{i}(t),\;\forall t\in [-\omega ,0]_{\mathbb {T}}, \end{aligned}$$

where \(\phi _{i}\in \mathcal {C}([-\omega ,0]_{\mathbb {T}},\mathbb {O})\), \(\forall i\in \{1,\ldots ,N\}\).

Similarly as before, the OVNNs (6) can be converted into a real-valued system of equations, which, in matrix form, can be designated as:

$$\begin{aligned} \check{\varpi }^{\Delta }(t)=-\check{C}\check{\varpi }(t-\delta )+\check{A}\check{f}(\check{\varpi }(t))+\check{B}\check{f}(\check{\varpi }(t-\tau (t)))+\check{G}\int _{t-\eta }^{t}\check{f}(\check{\varpi }(s))\Delta s-\check{u}. \end{aligned}$$
(7)

We also need to make the following assumption regarding the activation functions:

Assumption 1

(Popa (2023)) “The activation functions \(f_{j}\) satisfy, \(\forall o,\ o'\in \mathbb {O}\), the following Lipschitz conditions:

$$\begin{aligned} ||f_{j}^{q}(o)-f_{j}^{q}(o')||\le l_{j}^{q}||o-o'||, \end{aligned}$$

\(\forall j\in \{1,\ldots ,N\}\), \(\forall 0\le q\le 7\), where \(l_{j}^{q}>0\), represent the Lipschitz constants. Furthermore, we denote \(L:=\text {diag}(l_{1}^{0},\ldots ,l_{1}^{7},\ldots ,l_{N}^{0},\ldots ,l_{N}^{7})\in \mathbb {R}^{8N\times 8N}\).”

3 Main results

For the first two theorems, system (1) is considered to not possess leakage delay, i.e., \(\delta =0\):

$$\begin{aligned} \check{\mathfrak {o}}^{\Delta }(t)=-\check{C}\check{\mathfrak {o}}(t)+\check{A}\check{f}(\check{\mathfrak {o}}(t))+\check{B}\check{f}(\check{\mathfrak {o}}(t-\tau (t)))+\check{G}\int _{t-\eta }^{t}\check{f}(\check{\mathfrak {o}}(s))\Delta s,\quad \forall t\in [0,+\infty )_{\mathbb {T}}. \end{aligned}$$

This system can also be written as:

$$\begin{aligned} \check{\mathfrak {o}}_{i}^{\Delta }(t)= & {} -\check{c}_{i}\check{\mathfrak {o}}_{i}(t)+\sum _{j=1}^{8N}\check{a}_{ij}\check{f}_{j}(\check{\mathfrak {o}}_{j}(t))+\sum _{j=1}^{8N}\check{b}_{ij}\check{f}_{j}(\check{\mathfrak {o}}_{j}(t-\tau (t)))\nonumber \\{} & {} +\sum _{j=1}^{8N}\check{g}_{ij}\int _{t-\eta }^{t}\check{f}_{j}(\check{\mathfrak {o}}_{j}(s))\Delta s, \end{aligned}$$
(8)

\(\forall t\in [0,+\infty )_{\mathbb {T}}\), \(\forall i\in \{1,\ldots ,8N\}\).

Theorem 4

In the setting of Assumption 1, if there exist positive definite (PD) matrix \(P\in \mathbb {R}^{8N\times 8N}\), diagonal PD matrices \(R_{1}\), \(R_{2}\in \mathbb {R}^{8N\times 8N}\), any matrices \(N_{1}\), \(N_{2}\), \(N_{3}\), \(N_{4}\), \(N_{5}\in \mathbb {R}^{8N\times 8N}\), positive numbers \(\alpha _{1}\), \(\alpha _{21}\), \(\alpha _{22}\), such that \(\alpha _{1}>\alpha _{21}+\alpha _{22}\), \(-\alpha _{1}\in \mathcal {R}^{+}\), and the subsequent LMI holds:

$$\begin{aligned} \Omega <0, \end{aligned}$$
(9)

where \(\Omega _{1,1}=-PC-C^{T}P+L^{T}R_{1}L-N_{2}C-C^{T}N_{2}^{T}+\alpha _{1}P\), \(\Omega _{1,2}=-C^{T}N_{1}^{T}-N_{2}\), \(\Omega _{1,5}=PA+N_{2}A+C^{T}N_{3}^{T}\), \(\Omega _{1,6}=PB+N_{2}B+C^{T}N_{4}^{T}\), \(\Omega _{1,7}=PG+N_{2}G+C^{T}N_{5}^{T}\), \(\Omega _{2,2}=\hat{\mu }P-N_{1}-N_{1}^{T}\), \(\Omega _{2,5}=N_{1}A+N_{3}^{T}\), \(\Omega _{2,6}=N_{1}B+N_{4}^{T}\), \(\Omega _{2,7}=N_{1}G+N_{5}^{T}\), \(\Omega _{3,3}=L^{T}R_{2}L-\alpha _{21}P\), \(\Omega _{4,4}=-\alpha _{22}P\), \(\Omega _{5,5}=-R_{1}-N_{3}A-A^{T}N_{3}^{T}\), \(\Omega _{5,6}=-N_{3}B-A^{T}N_{4}^{T}\), \(\Omega _{5,7}=-N_{3}G-A^{T}N_{5}^{T}\), \(\Omega _{6,6}=-R_{2}-N_{4}B-B^{T}N_{4}^{T}\), \(\Omega _{6,7}=-N_{4}G-B^{T}N_{5}^{T}\), \(\Omega _{7,7}=-N_{5}G-G^{T}N_{5}^{T}\), then the equilibrium point of system (1) is exponentially stable.

Proof

We begin by constructing the subsequent Lyapunov-type function:

$$\begin{aligned} V(t)=\check{\mathfrak {o}}^{T}(t)P\check{\mathfrak {o}}(t). \end{aligned}$$

Now, we take, along the positive half trajectory of system (8), the \(\Delta \)-derivative of V, and, also taking Lemma 1 into consideration, we have that:

$$\begin{aligned} V^{\Delta }(t)\le & {} \check{\mathfrak {o}}^{T}(t)P\check{\mathfrak {o}}^{\Delta }(t)+\check{\mathfrak {o}}^{\Delta T}(t)P\check{\mathfrak {o}}(t)+\hat{\mu }\check{\mathfrak {o}}^{\Delta T}(t)P\check{\mathfrak {o}}^{\Delta }(t)\nonumber \\= & {} \check{\mathfrak {o}}^{T}(t)P\left( -\check{C}\check{\mathfrak {o}}(t)+\check{A}\check{f}(\check{\mathfrak {o}}(t))+\check{B}\check{f}(\check{\mathfrak {o}}(t-\tau (t)))+\check{G}\int _{t-\eta }^{t}\check{f}(\check{\mathfrak {o}}(s))\Delta s\right) \nonumber \\{} & {} +\left( -\check{C}\check{\mathfrak {o}}(t)+\check{A}\check{f}(\check{\mathfrak {o}}(t))+\check{B}\check{f}(\check{\mathfrak {o}}(t-\tau (t)))+\check{G}\int _{t-\eta }^{t}\check{f}(\check{\mathfrak {o}}(s))\Delta s\right) ^{T}P\check{\mathfrak {o}}^{\Delta }(t)\nonumber \\{} & {} +\hat{\mu }\check{\mathfrak {o}}^{\Delta T}(t)P\check{\mathfrak {o}}^{\Delta }(t). \end{aligned}$$
(10)

From Assumption 1, we have that the subsequent inequalities are true:

$$\begin{aligned} 0\le & {} \check{\mathfrak {o}}^{T}(t)L^{T}R_{1}L\check{\mathfrak {o}}(t)-\check{f}(\check{\mathfrak {o}}(t))^{T}R_{1}\check{f}(\check{\mathfrak {o}}(t)), \end{aligned}$$
(11)
$$\begin{aligned} 0\le & {} \check{\mathfrak {o}}^{T}(t-\tau (t))L^{T}R_{2}L\check{\mathfrak {o}}(t-\tau (t))-\check{f}(\check{\mathfrak {o}}(t-\tau (t)))^{T}R_{2}\check{f}(\check{\mathfrak {o}}(t-\tau (t))), \end{aligned}$$
(12)

for some diagonal PD matrices \(R_{1},R_{2}\in \mathbb {R}^{8N\times 8N}\).

For any matrices \(N_{1},N_{2},N_{3},N_{4},N_{5}\in \mathbb {R}^{8N\times 8N}\), the following equality is also true:

$$\begin{aligned}{} & {} \left[ \check{\mathfrak {o}}^{\Delta T}(t)N_{1}+\check{\mathfrak {o}}^{T}(t)N_{2}-\check{f}(\check{\mathfrak {o}}(t))^{T}N_{3} -\check{f}(\check{\mathfrak {o}}(t-\tau (t)))^{T}N_{4}-\left( \int _{t-\eta }^{t}\check{f}(\check{\mathfrak {o}}(s))\Delta s\right) ^{T}N_{5}\right] \nonumber \\{} & {} \quad \times \left[ -\check{\mathfrak {o}}^{\Delta }(t)-\check{C}\check{\mathfrak {o}}(t) +\check{A}\check{f}(\check{\mathfrak {o}}(t))+\check{B}\check{f}(\check{\mathfrak {o}}(t-\tau (t))) +\check{G}\int _{t-\eta }^{t}\check{f}(\check{\mathfrak {o}}(s))\Delta s\right] =0.\nonumber \\ \end{aligned}$$
(13)

From relations (10)–(13), we obtain that:

$$\begin{aligned} V^{\Delta }(t)\le & {} -\alpha _{1}V(t)+\alpha _{21}V(t-\tau (t))+\alpha _{22}V(t-\eta )+\gamma ^{T}(t)\Omega \gamma (t)\nonumber \\\le & {} -\alpha _{1}V(t)+(\alpha _{21}+\alpha _{22})\sup _{s\in [t-\omega ,t]_{\mathbb {T}}}V(s), \end{aligned}$$
(14)

where, for the last inequality, we used hypothesis (9) and

$$\begin{aligned} \gamma (t)= & {} \left[ \check{\mathfrak {o}}^{T}(t) \check{\mathfrak {o}}^{\Delta T}(t) \check{\mathfrak {o}}(t-\tau (t)) \check{\mathfrak {o}}(t-\eta ) \check{f}(\check{\mathfrak {o}}(t))^{T} \check{f}(\check{\mathfrak {o}}(t-\tau (t)))^{T} \left( \int _{t-\eta }^{t}\check{f}(\check{\mathfrak {o}}(s))\Delta s\right) ^{T}\right] ^{T}. \end{aligned}$$

From Lemma 2 applied to (14), we get that

$$\begin{aligned} V(t)\le \sup _{s\in [-\omega ,0]_{\mathbb {T}}}V(s)e_{\ominus \lambda }(t,0), \end{aligned}$$

which means that

$$\begin{aligned} \lambda _{\min }(P)||\check{\mathfrak {o}}(t)||_{2}^{2}\le & {} \check{\mathfrak {o}}^{T}(t)P\check{\mathfrak {o}}(t)\\\le & {} \sup _{s\in [-\omega ,0]_{\mathbb {T}}}V(s)e_{\ominus \lambda }(t,0), \end{aligned}$$

that is to say that

$$\begin{aligned} ||\check{\mathfrak {o}}(t)||_{2}\le \sqrt{\frac{\sup _{s\in [-\omega ,0]_{\mathbb {T}}}V(s)}{\lambda _{\min }(P)}}\left( e_{\ominus \lambda }(t,0)\right) ^{\frac{1}{2}}. \end{aligned}$$

This proves the exponential stability of the equilibrium point of system (1), which is exactly what we wanted to obtain. \(\square \)

Theorem 5

In the setting of Assumption 1, if there exist positive numbers \(\rho _{i}\), \(0\le i\le 8N\), such that \(\alpha _{1}>\alpha _{21}+\alpha _{22}\), \(-\alpha _{1}\in \mathcal {R}^{+}\), where \(\alpha _{1}=\min _{1\le i\le 8N}\left\{ c_{i}-\sum _{j=1}^{8N}\frac{\rho _{j}}{\rho _{i}}l_{i}|\check{a}_{ji}|\right\} \), \(\alpha _{21}=\max _{1\le i\le 8N}\left\{ \sum _{j=1}^{8N}\frac{\rho _{j}}{\rho _{i}}l_{i}|\check{b}_{ji}|\right\} \), \(\alpha _{22}=\max _{1\le i\le 8N} \left\{ \sum _{i=1}^{8N}\frac{\rho _{j}}{\rho _{i}}l_{i}\eta |\check{g}_{ji}|\right\} \), then the equilibrium point of system (1) is exponentially stable.

Proof

Construct the following Lyapunov-like function:

$$\begin{aligned} V(t)=\sum _{i=1}^{8N}\rho _{i}|\check{\mathfrak {o}}_{i}(t)|. \end{aligned}$$

By taking, along the positive half trajectory of system (8), the \(\Delta \)-derivative of V, and, also taking Lemma 3 and Assumption 1 into consideration, we have that:

$$\begin{aligned} V^{\Delta }(t)\le & {} -\sum _{i=1}^{8N}c_{i}\rho _{i}|\check{\mathfrak {o}}_{i}(t)|+\sum _{i=1}^{8N}\sum _{j=1}^{8N}\rho _{i}|\check{a}_{ij}|\cdot |\check{f}_{j}(\check{\mathfrak {o}}_{j}(t))|\nonumber \\{} & {} +\sum _{i=1}^{8N}\sum _{j=1}^{8N}\rho _{i}|\check{b}_{ij}|\cdot |\check{f}_{j}(\check{\mathfrak {o}}_{j}(t-\tau (t)))|\nonumber \\{} & {} +\sum _{i=1}^{8N}\sum _{j=1}^{8N}\rho _{i}|\check{g}_{ij}|\int _{t-\eta }^{t}|\check{f}_{j}(\check{\mathfrak {o}}_{j}(s))|\Delta s\nonumber \\\le & {} -\sum _{i=1}^{8N}c_{i}\rho _{i}|\check{\mathfrak {o}}_{i}(t)|+\sum _{i=1}^{8N}\sum _{j=1}^{8N}\rho _{j}|\check{a}_{ji}|\cdot l_{i}|\check{\mathfrak {o}}_{i}(t)|+\sum _{i=1}^{8N}\sum _{j=1}^{8N}\rho _{j}|\check{b}_{ji}|\cdot l_{i}|\check{\mathfrak {o}}_{i}(t-\tau (t))|\nonumber \\{} & {} +\sum _{i=1}^{8N}\sum _{j=1}^{8N}\rho _{j}|\check{g}_{ji}|\int _{t-\eta }^{t}l_{i}|\check{\mathfrak {o}}_{i}(s)|\Delta s\nonumber \\\le & {} -\sum _{i=1}^{8N}\left( c_{i}\rho _{i}-\sum _{j=1}^{8N}\rho _{j}l_{i}|\check{a}_{ji}|\right) |\check{\mathfrak {o}}_{i}(t)|+\sum _{i=1}^{8N}\left( \sum _{j=1}^{8N}\rho _{j}l_{i}|\check{b}_{ji}|\right) |\check{\mathfrak {o}}_{i}(t-\tau (t))|\nonumber \\{} & {} +\sum _{i=1}^{8N}\left( \sum _{j=1}^{8N}\rho _{j}l_{i}\eta |\check{g}_{ji}|\right) \sup _{s\in [t-\eta ,t]_{\mathbb {T}}}|\check{\mathfrak {o}}_{i}(s)|\nonumber \\\le & {} -\alpha _{1}V(t)+\alpha _{21}\sup _{s\in [t-\tau ,t]_{\mathbb {T}}}V(s)+\alpha _{22}\sup _{s\in [t-\eta ,t]_{\mathbb {T}}}V(s)\nonumber \\\le & {} -\alpha _{1}V(t)+(\alpha _{21}+\alpha _{22})\sup _{s\in [t-\omega ,t]_{\mathbb {T}}}V(s). \end{aligned}$$
(15)

From Lemma 2 applied to (15), we get that

$$\begin{aligned} V(t)\le \sup _{s\in [-\omega ,0]_{\mathbb {T}}}V(s)e_{\ominus \lambda }(t,0), \end{aligned}$$

which means that

$$\begin{aligned} \min _{1\le i\le 8N}\{\rho _{i}\}||\check{\mathfrak {o}}(t)||_{1}\le & {} \sum _{i=1}^{8N}\rho _{i}|\check{\mathfrak {o}}_{i}(t)|\\\le & {} \sup _{s\in [-\omega ,0]_{\mathbb {T}}}V(s)e_{\ominus \lambda }(t,0), \end{aligned}$$

that is to say that

$$\begin{aligned} ||\check{\mathfrak {o}}(t)||_{1}\le \frac{\sup _{s\in [-\omega ,0]_{\mathbb {T}}}V(s)}{\min _{1\le i\le 8N}\{\rho _{i}\}}e_{\ominus \lambda }(t,0). \end{aligned}$$

This proves the exponential stability of the equilibrium point of system (1), which is exactly what we wanted to obtain. \(\square \)

Remark 1

The exponential stability of OVNNs was studied in the available literature very rarely, for example, in Popa (2018a), Popa (2018b). Theorems 4 and 5 give sufficient conditions expressed as LMIs and algebraic inequalities, respectively, which ensure the exponential stability of OVNNs with time-varying and distributed delays defined on time scales. Such a general model has not been yet considered in the literature, to our awareness, which means that the obtained results are not directly comparable with the ones already available. However, they can be particularized for simpler models, such as for OVNNs defined in continuous time or discrete time, or even for CVNNs or QVNNs defined on time scales, which again proves the generality of our theorems.

Next, we will use the following controller of state feedback type to achieve synchronization between NNs (1) and (5):

$$\begin{aligned} u(t)=K_{1}\check{\varpi }(t)+K_{2}\check{\varpi }(t-\delta )+K_{3}\check{\varpi }(t-\tau (t))+K_{4}\int _{t-\eta }^{t}\check{\varpi }(s)\Delta s, \end{aligned}$$
(16)

in which \(K_{1},\;K_{2},\;K_{3},\;K_{4}\in \mathbb {R}^{8N\times 8N}\) constitute the control gain matrices. Using this controller, NN (7) becomes:

$$\begin{aligned} \check{\varpi }^{\Delta }(t)= & {} -K_{1}\check{\varpi }(t)-(\check{C}+K_{2})\check{\varpi }(t-\delta )-K_{3}\check{\varpi }(t-\tau (t))\nonumber \\{} & {} -K_{4}\int _{t-\eta }^{t}\check{\varpi }(s)\Delta s\nonumber \\{} & {} +\check{A}\check{f}(\check{\varpi }(t))+\check{B}\check{f}(\check{\varpi }(t-\tau (t)))+\check{G}\int _{t-\eta }^{t}\check{f}(\check{\varpi }(s))\Delta s. \end{aligned}$$
(17)

Theorem 6

In the setting of Assumption 1, if there exist PD matrix \(P\in \mathbb {R}^{8N\times 8N}\), diagonal PD matrices \(R_{1}\), \(R_{2}\in \mathbb {R}^{8N\times 8N}\), any matrices \(N_{1}\), \(N_{2}\), \(N_{3}\), \(N_{4}\), \(N_{5}\), \(N_{6}\), \(N_{7}\), \(N_{8}\in \mathbb {R}^{8N\times 8N}\), positive numbers \(\alpha _{1}\), \(\alpha _{21}\), \(\alpha _{22}\), \(\alpha _{23}\), such that \(\alpha _{1}>\alpha _{21}+\alpha _{22}+\alpha _{23}\), \(-\alpha _{1}\in \mathcal {R}^{+}\), and the subsequent LMI holds:

$$\begin{aligned} \Omega <0, \end{aligned}$$
(18)

where \(\Omega _{1,1}=-PK_{1}-K_{1}P+L^{T}R_{1}L-N_{2}K_{1}-K_{1}N_{2}^{T}+\alpha _{1}P\), \(\Omega _{1,2}=-K_{1}N_{1}^{T}-N_{2}\), \(\Omega _{1,3}=-P(C+K_{2})-N_{2}(C+K_{2})-K_{1}N_{3}^{T}\), \(\Omega _{1,4}=-PK_{3}-N_{2}K_{3}-K_{1}N_{4}^{T}\), \(\Omega _{1,6}=PA+N_{2}A+K_{1}N_{6}^{T}\), \(\Omega _{1,7}=PB+N_{2}B+K_{1}N_{7}^{T}\), \(\Omega _{1,8}=-PK_{4}-N_{2}K_{4}-K_{1}N_{5}^{T}\), \(\Omega _{1,9}=PG+N_{2}G+K_{1}N_{8}^{T}\), \(\Omega _{2,2}=\hat{\mu }P-N_{1}-N_{1}^{T}\), \(\Omega _{2,3}=-N_{1}(C+K_{2})-N_{3}^{T}\), \(\Omega _{2,4}=-N_{1}K_{3}-N_{4}^{T}\), \(\Omega _{2,6}=N_{1}A+N_{6}^{T}\), \(\Omega _{2,7}=N_{1}B+N_{7}^{T}\), \(\Omega _{2,8}=-N_{1}K_{4}-N_{5}^{T}\), \(\Omega _{2,9}=N_{1}G+N_{8}^{T}\), \(\Omega _{3,3}=-N_{3}(C+K_{2})-(C+K_{2})N_{3}^{T}-\alpha _{21}P\), \(\Omega _{3,4}=-N_{3}K_{3}-(C+K_{2})N_{4}^{T}\), \(\Omega _{3,6}=N_{3}A+(C+K_{2})N_{6}^{T}\), \(\Omega _{3,7}=N_{3}B+(C+K_{2})N_{7}^{T}\), \(\Omega _{3,8}=-N_{3}K_{4}-(C+K_{2})N_{5}^{T}\), \(\Omega _{3,9}=N_{3}G+(C+K_{2})N_{8}^{T}\), \(\Omega _{4,4}=L^{T}R_{2}L-N_{4}K_{3}-K_{3}N_{4}^{T}-\alpha _{22}P\), \(\Omega _{4,6}=N_{4}A+K_{3}N_{6}^{T}\), \(\Omega _{4,7}=N_{4}B+K_{3}N_{7}^{T}\), \(\Omega _{4,8}=-N_{4}K_{4}-K_{3}N_{5}^{T}\), \(\Omega _{4,9}=N_{4}G+K_{3}N_{8}^{T}\), \(\Omega _{5,5}=-\alpha _{23}P\), \(\Omega _{6,6}=-R_{1}-N_{6}A-A^{T}N_{6}^{T}\), \(\Omega _{6,7}=-N_{6}B-A^{T}N_{7}^{T}\), \(\Omega _{6,8}=N_{6}K_{4}+A^{T}N_{5}^{T}\), \(\Omega _{6,9}=-N_{6}G-A^{T}N_{8}^{T}\), \(\Omega _{7,7}=-R_{2}-N_{7}B-B^{T}N_{7}^{T}\), \(\Omega _{7,8}=N_{7}K_{4}+B^{T}N_{5}^{T}\), \(\Omega _{7,9}=-N_{7}G-B^{T}N_{8}^{T}\), \(\Omega _{8,8}=-N_{5}K_{4}-K_{4}N_{5}^{T}\), \(\Omega _{8,9}=N_{5}G+K_{4}N_{8}^{T}\), \(\Omega _{9,9}=-N_{8}G-G^{T}N_{8}^{T}\), then drive NN (1) is exponentially synchronized with response NN (5) based on state feedback controller (16).

Proof

The subsequent Lyapunov-type function will be constructed firstly:

$$\begin{aligned} V(t)=\check{\varpi }^{T}(t)P\check{\varpi }(t). \end{aligned}$$

By taking, along the positive half trajectory of system (17), the \(\Delta \)-derivative of V, and also taking Lemma 1 into consideration, we have that:

$$\begin{aligned} V^{\Delta }(t)\le & {} \check{\varpi }^{T}(t)P\check{\varpi }^{\Delta }(t)+\check{\varpi }^{\Delta T}(t)P\check{\varpi }(t)+\hat{\mu }\check{\varpi }^{\Delta T}(t)P\check{\varpi }^{\Delta }(t)\nonumber \\= & {} \check{\varpi }^{T}(t)P\left( -K_{1}\check{\varpi }(t)-(\check{C}+K_{2})\check{\varpi }(t-\delta )-K_{3}\check{\varpi }(t-\tau (t))-K_{4}\int _{t-\eta }^{t}\check{\varpi }(s)\Delta s\right. \nonumber \\{} & {} \left. +\check{A}\check{f}(\check{\varpi }(t))+\check{B}\check{f}(\check{\varpi }(t-\tau (t)))+\check{G}\int _{t-\eta }^{t}\check{f}(\check{\varpi }(s))\Delta s\right) \nonumber \\{} & {} +\left( -K_{1}\check{\varpi }(t)-(\check{C}+K_{2})\check{\varpi }(t-\delta )-K_{3}\check{\varpi }(t-\tau (t))-K_{4}\int _{t-\eta }^{t}\check{\varpi }(s)\Delta s\right. \nonumber \\{} & {} \left. +\check{A}\check{f}(\check{\varpi }(t))+\check{B}\check{f}(\check{\varpi }(t-\tau (t)))+\check{G}\int _{t-\eta }^{t}\check{f}(\check{\varpi }(s))\Delta s\right) ^{T}P\check{\varpi }^{\Delta }(t)\nonumber \\{} & {} +\hat{\mu }\check{\varpi }^{\Delta T}(t)P\check{\varpi }^{\Delta }(t). \end{aligned}$$
(19)

From Assumption 1, we have that the subsequent inequalities are true:

$$\begin{aligned} 0\le & {} \check{\varpi }^{T}(t)L^{T}R_{1}L\check{\varpi }(t)-\check{f}(\check{\varpi }(t))^{T}R_{1}\check{f}(\check{\varpi }(t)), \end{aligned}$$
(20)
$$\begin{aligned} 0\le & {} \check{\varpi }^{T}(t-\tau (t))L^{T}R_{2}L\check{\varpi }(t-\tau (t))-\check{f}(\check{\varpi }(t-\tau (t)))^{T}R_{2}\check{f}(\check{\varpi }(t-\tau (t))), \end{aligned}$$
(21)

for some diagonal PD matrices \(R_{1},R_{2}\in \mathbb {R}^{8N\times 8N}\).

For any matrices, \(N_{1},N_{2},N_{3},N_{4},N_{5},N_{6},N_{7},N_{8}\in \mathbb {R}^{8N\times 8N}\) the following equality is also true:

$$\begin{aligned}{} & {} \Biggl [\check{\varpi }^{\Delta T}(t)N_{1}+\check{\varpi }^{T}(t)N_{2}+\check{\varpi }^{T}(t-\delta )N_{3}+\check{\varpi }^{T}(t-\tau (t))N_{4}+\left( \int _{t-\eta }^{t}\check{\varpi }(s)\Delta s\right) ^{T}N_{5}\nonumber \\{} & {} \quad \left. -\check{f}(\check{\varpi }(t))^{T}N_{6}-\check{f}(\check{\varpi }(t-\tau (t)))^{T}N_{7}-\left( \int _{t-\eta }^{t}\check{f}(\check{\varpi }(s))\Delta s\right) ^{T}N_{8}\right] \times \nonumber \\{} & {} \quad \biggl [-\check{\varpi }^{\Delta }(t)-K_{1}\check{\varpi }(t)-(\check{C}+K_{2})\check{\varpi }(t-\delta )-K_{3}\check{\varpi }(t-\tau (t))-K_{4}\int _{t-\eta }^{t}\check{\varpi }(s)\Delta s\nonumber \\{} & {} \quad \left. +\check{A}\check{f}(\check{\varpi }(t))+\check{B}\check{f}(\check{\varpi }(t-\tau (t)))+\check{G}\int _{t-\eta }^{t}\check{f}(\check{\varpi }(s))\Delta s\right] =0. \end{aligned}$$
(22)

From relations (19)–(22), we obtain that:

$$\begin{aligned} V^{\Delta }(t)\le & {} -\alpha _{1}V(t)+\alpha _{21}V(t-\delta )+\alpha _{22}V(t-\tau (t))+\alpha _{23}V(t-\eta )+\gamma ^{T}(t)\Omega \gamma (t)\nonumber \\\le & {} -\alpha _{1}V(t)+(\alpha _{21}+\alpha _{22}+\alpha _{23})\sup _{t\in [-\omega ,0]_{\mathbb {T}}}V(s), \end{aligned}$$
(23)

where, for the last inequality, we used hypothesis (18) and

$$\begin{aligned} \gamma (t)= & {} \left[ \begin{array}{cccccc} \check{\varpi }^{T}(t)&\check{\varpi }^{\Delta T}(t)&\check{\varpi }(t-\delta )&\check{\varpi }(t-\tau (t))&\check{\varpi }(t-\eta )&\left( \int _{t-\eta }^{t}\check{\varpi }(s)\Delta s\right) ^{T}\end{array}\right. \\{} & {} \left. \begin{array}{ccc} \check{f}(\varpi (t))^{T}&\check{f}(\varpi (t-\tau (t)))^{T}&\left( \int _{t-\eta }^{t}\check{f}(\varpi (s))\Delta s\right) ^{T}\end{array}\right] ^{T}. \end{aligned}$$

From Lemma 2 applied to (23), we get that

$$\begin{aligned} V(t)\le \sup _{s\in [-\omega ,0]_{\mathbb {T}}}V(s)e_{\ominus \lambda }(t,0), \end{aligned}$$

which means that

$$\begin{aligned} \lambda _{\text {min}}(P)||\check{\varpi }(t)||_{2}^{2}\le & {} \check{\varpi }^{T}(t)P\check{\varpi }(t)\\\le & {} \sup _{s\in [-\omega ,0]_{\mathbb {T}}}V(s)e_{\ominus \lambda }(t,0), \end{aligned}$$

that is to say that

$$\begin{aligned} ||\check{\varpi }(t)||_{2}\le \sqrt{\frac{\sup _{s\in [-\omega ,0]_{\mathbb {T}}}V(s)}{\lambda _{\text {min}}(P)}}\left( e_{\ominus \lambda }(t,0)\right) ^{\frac{1}{2}}. \end{aligned}$$

This proves the exponential synchronization of drive NN (1) with response NN (5) under state feedback controller (17), which is exactly what we wanted to obtain. \(\square \)

For the next theorem, we will write system (17) as:

$$\begin{aligned} \check{\varpi }_{i}^{\Delta }(t)= & {} -k_{1i}\check{\varpi }_{i}(t)-(\check{c_{i}}+k_{2i})\check{\varpi }_{i}(t-\delta )-k_{3i}\check{\varpi }_{i}(t-\tau (t))-k_{4i}\int _{t-\eta }^{t}\check{\varpi }_{i}(s)ds\nonumber \\{} & {} +\sum _{j=1}^{8N}\check{a}_{ij}\check{f}_{j}(\check{\varpi }_{j}(t))+\sum _{j=1}^{8N}\check{b}_{ij}\check{f}_{j}(\check{\varpi }_{j}(t-\tau (t)))\nonumber \\{} & {} +\sum _{j=1}^{8N}\check{g}_{ij}\int _{t-\eta }^{t}\check{f}(\check{\varpi }_{j}(s))\Delta s. \end{aligned}$$
(24)

\(\forall t\in [0,+\infty )_{\mathbb {T}}\), \(\forall i\in \{1,\ldots ,8N\}\).

Theorem 7

In the setting of Assumption 1, if there exist positive numbers \(\rho _{i}\), \(0\le i\le 8N\), such that \(\alpha _{1}>\alpha _{21}+\alpha _{22}+\alpha _{23}\), \(-\alpha _{1}\in \mathcal {R}^{+}\), where \(\alpha _{1}=\min _{1\le i\le 8N}\left\{ k_{1i}-\sum _{j=1}^{8N}\frac{\rho _{j}}{\rho _{i}}l_{i}|\check{a}_{ji}|\right\} \), \(\alpha _{21}=\max _{1\le i\le 8N}\left\{ \sum _{j=1}^{8N}\frac{\rho _{j}}{\rho _{i}}l_{i}|\check{b}_{ji}|\right\} \), \(\alpha _{22}=\max _{1\le i\le 8N}\left\{ \check{c_{i}}+k_{2i}\right\} \), \(\alpha _{23}=\max _{1\le i\le 8N}\left\{ k_{4i}l_{i}\eta +\sum _{i=1}^{8N}\frac{\rho _{j}}{\rho _{i}}l_{i}\eta |\check{g}_{ji}|\right\} \), then drive NN (1) is exponentially synchronized with response NN (5) based on state feedback controller (16).

Proof

Construct the following Lyapunov-like function:

$$\begin{aligned} V(t)=\sum _{i=1}^{8N}\rho _{i}|\check{\varpi }_{i}(t)|. \end{aligned}$$

By taking, along the positive half trajectory of system (24), the \(\Delta \)-derivative of V, and also taking Lemma 3 and Assumption 1 into consideration, we have that:

$$\begin{aligned} V^{\Delta }(t)\le & {} -\sum _{i=1}^{8N}k_{1i}\rho _{i}|\check{\varpi }_{i}(t)|+\sum _{i=1}^{8N}(\check{c_{i}}+k_{2i})\rho _{i}|\check{\varpi }_{i}(t-\delta )|+\sum _{i=1}^{8N}k_{3i}\rho _{i}|\check{\varpi }_{i}(t-\tau (t))|\nonumber \\{} & {} +\sum _{i=1}^{8N}k_{4i}\rho _{i}\int _{t-\eta }^{t}|\check{\varpi }_{i}(s)|ds+\sum _{i=1}^{8N}\sum _{j=1}^{8N}\rho _{i}|\check{a}_{ij}|\cdot |\check{f}_{j}(\check{\varpi }_{j}(t))|+\nonumber \\{} & {} \sum _{i=1}^{8N}\sum _{j=1}^{8N}\rho _{i}|\check{b}_{ij}|\cdot |\check{f}_{j}(\check{\varpi }_{j}(t-\tau (t)))|+\sum _{i=1}^{8N}\sum _{j=1}^{8N}\rho _{i}|\check{g}_{ij}|\int _{t-\eta }^{t}|\check{f}_{j}(\check{\varpi }_{j}(s))|\Delta s\nonumber \\\le & {} -\sum _{i=1}^{8N}k_{1i}\rho _{i}|\check{\varpi }_{i}(t)|+\sum _{i=1}^{8N}(\check{c_{i}}+k_{2i})\rho _{i}|\check{\varpi }_{i}(t-\delta )|+\sum _{i=1}^{8N}k_{3i}\rho _{i}|\check{\varpi }_{i}(t-\tau (t))|\nonumber \\{} & {} +\sum _{i=1}^{8N}k_{4i}\rho _{i}l_{i}\eta \sup _{s\in [t-\eta ,t]_{\mathbb {T}}}|\check{\varpi }_{i}(s)|+\sum _{i=1}^{8N}\sum _{j=1}^{8N}\rho _{j}|\check{a}_{ji}|\cdot l_{i}|\check{\varpi }_{i}(t)|\nonumber \\{} & {} +\sum _{i=1}^{8N}\sum _{j=1}^{8N}\rho _{j}|\check{b}_{ji}|\cdot l_{i}|\check{\varpi }_{i}(t-\tau (t))|+\sum _{i=1}^{8N}\sum _{j=1}^{8N}\rho _{j}|\check{g}_{ji}|\int _{t-\eta }^{t}l_{i}|\check{\varpi }_{i}(s)|\Delta s\nonumber \\\le & {} -\sum _{i=1}^{8N}\left( k_{1i}\rho _{i}-\sum _{j=1}^{8N}\rho _{j}l_{i}|\check{a}_{ji}|\right) |\check{\varpi }_{i}(t)|+\sum _{i=1}^{8N}\left( (\check{c_{i}}+k_{2i})\rho _{i}\right) |\check{\varpi }_{i}(t-\delta )|\nonumber \\{} & {} +\sum _{i=1}^{8N}\left( k_{3i}\rho _{i}+\sum _{j=1}^{8N}\rho _{j}l_{i}|\check{b}_{ji}|\right) |\check{\varpi }_{i}(t-\tau (t))|\nonumber \\{} & {} +\sum _{i=1}^{8N}\left( k_{4i}\rho _{i}l_{i}\eta +\sum _{i=1}^{8N}\rho _{j}l_{i}\eta |\check{g}_{ji}|\right) \sup _{s\in [t-\eta ,t]_{\mathbb {T}}}|\check{\varpi }_{i}(s)|\nonumber \\\le & {} -\alpha _{1}V(t)+\alpha _{21}\sup _{s\in [t-\delta ,t]_{\mathbb {T}}}V(s)+\alpha _{22}\sup _{s\in [t-\tau ,t]_{\mathbb {T}}}V(s)+\alpha _{23}\sup _{s\in [t-\eta ,t]_{\mathbb {T}}}V(s)\nonumber \\\le & {} -\alpha _{1}V(t)+(\alpha _{21}+\alpha _{22}+\alpha _{23})\sup _{s\in [t-\omega ,t]_{\mathbb {T}}}V(s). \end{aligned}$$
(25)

From Lemma 2 applied to (25), we get that

$$\begin{aligned} V(t)\le \sup _{s\in [-\omega ,0]_{\mathbb {T}}}V(s)e_{\ominus \lambda }(t,0), \end{aligned}$$

which means that

$$\begin{aligned} \min _{1\le i\le 8N}\{\rho _{i}\}||\check{\varpi }(t)||_{1}\le & {} \sum _{i=1}^{8N}\rho _{i}|\check{\varpi }_{i}(t)|\\\le & {} \sup _{s\in [-\omega ,0]_{\mathbb {T}}}V(s)e_{\ominus \lambda }(t,0), \end{aligned}$$

that is to say that

$$\begin{aligned} ||\check{\varpi }(t)||_{1}\le \frac{\sup _{s\in [-\omega ,0]_{\mathbb {T}}}V(s)}{\min _{1\le i\le 8N}\{\rho _{i}\}}e_{\ominus \lambda }(t,0). \end{aligned}$$

This proves the exponential synchronization of drive NN (1) with response NN (5) under state feedback controller (17), which is exactly what we wanted to obtain. \(\square \)

Remark 2

Results concerning the exponential synchronization of OVNNs are even rarer than the ones concerning exponential stability. For example, paper (Popa 2023) discusses the Mittag–Leffler synchronization of fractional-order OVNNs and paper (Chouhan et al. 2023) discusses the fixed-time synchronization of OVNNs. Theorems 6 and 7 put forward sufficient criteria expressed as LMIs and algebraic inequalities, respectively, for the exponential synchronization of OVNNs with leakage, time-varying, and distributed delays defined on time scales. The model is even more general than the one in Theorems 4 and 5, and also there are no corresponding results in the available literature, to our best knowledge, focusing on such a general model. This means that Theorems 6 and 7 are not directly comparable with already existing ones, but can be particularized for OVNNs defined on time scales with fewer types of delays, for OVNNs defined in continous time or discrete time, or even for CVNNs or QVNNs defined on time scales with the same types of delays.

4 Numerical examples

Example 1

In the first example, take time scale \(\mathbb {T}=0.1\mathbb {Z}\), which means that \(\hat{\mu }=0.1\).

Let the following OVNN with two neurons and leakage, time-varying, and distributed delays be defined on time scale \(\mathbb {T}\):

$$\begin{aligned} o_{i}^{\Delta }(t)= & {} -c_{i}o_{i}(t-\delta )+\sum _{j=1}^{2}a_{ij}f_{j}(o_{j}(t))+\sum _{j=1}^{2}b_{ij}f_{j}(o_{j}(t-\tau (t)))\nonumber \\{} & {} +\sum _{j=1}^{2}g_{ij}\int _{t-\eta }^{t}f_{j}(o_{j}(s))\Delta s+I_{i}, \end{aligned}$$
(26)

\(\forall i\in \{1,2\}\), \(\forall t\in [0,+\infty )_{\mathbb {T}}\). Now, suppose system (26) has a unique equilibrium point, designated by \(\tilde{o}=(\tilde{o}_{1},\tilde{o}_{2})^{T}\). In this setting, if we denote \(\mathfrak {o}_{i}(t)=o_{i}(t)-\tilde{o}_{i}\), system (26) becomes:

$$\begin{aligned} \mathfrak {o}_{i}^{\Delta }(t)= & {} -c_{i}\mathfrak {o}_{i}(t-\delta )+\sum _{j=1}^{2}a_{ij}\tilde{f}_{j}(\mathfrak {o}_{j}(t))+\sum _{j=1}^{2}b_{ij}\tilde{f}_{j}(\mathfrak {o}_{j}(t-\tau (t)))\nonumber \\{} & {} +\sum _{j=1}^{2}g_{ij}\int _{t-\eta }^{t}\tilde{f}_{j}(\mathfrak {o}_{j}(s))\Delta s, \end{aligned}$$
(27)

\(\forall i\in \{1,2\}\), \(\forall t\in [0,+\infty )_{\mathbb {T}}\), where \(\tilde{f}_{j}(\mathfrak {o}_{j}(t))=f_{j}(\mathfrak {o}_{j}(t)+\tilde{o}_{j})-f_{j}(\tilde{o}_{j})\), \(\forall t\in [0,+\infty )_{\mathbb {T}}\), \(\forall j\in \{1,2\}\).

The parameters are as follows:

$$\begin{aligned} C= & {} \left[ \begin{array}{cc} 4 &{} 0\\ 0 &{} 5 \end{array}\right] , \\ A= & {} \left[ \begin{array}{cc} a_{11} &{} a_{12}\\ a_{21} &{} a_{22} \end{array}\right] , \\ a_{11}= & {} -0.7e_{0}+0.9e_{1}-0.2e_{2}+0.4e_{3}+0.2e_{4}+0.8e_{5}+0.3e_{6}+0.9e_{7}, \\ a_{12}= & {} 0.3e_{0}+0.9e_{1}-0.2e_{2}-0.2e_{3}+0.5e_{4}+0.8e_{5}+0.8e_{6}-0.9e_{7}, \\ a_{21}= & {} -0.2e_{0}-0.4e_{1}+0.2e_{2}-0.2e_{3}+0.3e_{4}+0.2e_{5}-0.5e_{6}+0.2e_{7},\\ a_{22}= & {} 0.4e_{0}+0.3e_{1}+0.1e_{2}+0.4e_{3}-0.2e_{4}-0.8e_{5}+0.8e_{6}+0.9e_{7}, \\ B= & {} \left[ \begin{array}{cc} b_{11} &{} b_{12}\\ b_{21} &{} b_{22} \end{array}\right] , \\ b_{11}= & {} -0.4e_{0}+0.7e_{1}+0.2e_{2}+0.5e_{3}-0.9e_{4}+0.9e_{5}-0.8e_{6}+0.9e_{7}, \\ b_{12}= & {} 0.8e_{0}+0.5e_{1}+0.3e_{2}-0.5e_{3}+0.8e_{4}+0.9e_{5}-0.9e_{6}+0.8e_{7}, \\ b_{21}= & {} 0.3e_{0}+0.2e_{1}-0.2e_{2}+0.1e_{3}+0.8e_{4}+0.9e_{5}+0.7e_{6}+0.9e_{7}, \\ b_{22}= & {} -0.5e_{0}+0.5e_{1}+0.2e_{2}+0.4e_{3}+0.8e_{4}-0.9e_{5}-0.8e_{6}+0.7e_{7}, \\ G= & {} \left[ \begin{array}{cc} g_{11} &{} g_{12}\\ g_{21} &{} g_{22} \end{array}\right] , \\ g_{11}= & {} -0.4e_{0}+0.7e_{1}+0.2e_{2}+0.5e_{3}-0.9e_{4}+0.9e_{5}-0.8e_{6}+0.9e_{7}, \\ g_{12}= & {} 0.9e_{0}+0.5e_{1}+0.3e_{2}-0.5e_{3}+0.8e_{4}+0.9e_{5}-0.9e_{6}+0.7e_{7}, \\ g_{21}= & {} 0.3e_{0}+0.2e_{1}-0.2e_{2}+0.1e_{3}+0.8e_{4}+0.9e_{5}+0.8e_{6}+0.9e_{7}, \\ g_{22}= & {} -0.5e_{0}+0.5e_{1}+0.2e_{2}+0.4e_{3}+0.9e_{4}-0.9e_{5}-0.8e_{6}+0.7e_{7}, \\ f_{j}(o)= & {} \frac{\sqrt{2}}{40}\sum _{q=0}^{7}f_{j}^{q}(o)e_{q}=\frac{\sqrt{2}}{40}\sum _{q=0}^{7}\frac{1}{1+\exp (-o^{q})}e_{q},\;\forall o\in \mathbb {O},\ \forall j\in \{1,2\}. \end{aligned}$$

Thus, Assumption 1 is satisfied by the activation functions, and \(L=\left[ \begin{array}{cc} 0.025I_{8} &{} 0\\ 0 &{} 0.025I_{8} \end{array}\right] \).

Also, we take \(\delta =0\) (no leakage delay), \(\tau (t)=0.4|\sin t|\), \(\eta =0.5\), hence \(\tau =0.4\) and \(\omega =\max \{\delta ,\tau ,\eta \}=0.5\), and \(\alpha _{1}=4.7\), \(\alpha _{21}=2.2\), \(\alpha _{22}=2.3\), which verify \(\alpha _{1}>\alpha _{21}+\alpha _{22}\). Taking all of the above into consideration, we can conclude that all the conditions of Theorem 4 are met, which means that we can solve the LMI in (9), obtaining \(R_{1}=\text {diag}(1.4962I_{8},1.6684I_{8})\), \(R_{2}=\text {diag}(1.7715I_{8},1.7993I_{8})\), (in order to reduce the length of the paper, we do not provide the values of the other matrices). Thus, the unique equilibrium point of system (26) is exponentially stable.

In Figs. 1, 2 are depicted the state trajectories of octonions \(o_{1}\) and \(o_{2}\) for NN (26), starting from 8 initial values.

Fig. 1
figure 1

State trajectories for the components of octonion \(o_{1}\) in Example 1. The 8 colors in each graph depict the 8 initial values

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Fig. 2
figure 2

State trajectories for the components of octonion \(o_{2}\) in Example 1. The 8 colors in each graph depict the 8 initial values

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Example 2

For this example, consider the same time scale \(\mathbb {T}=0.1\mathbb {Z}\) and the same system (26), which is again assumed to have a unique equilibrium point and can consequently be transformed into system (27), but, this time, with the following parameters:

$$\begin{aligned} C= & {} \left[ \begin{array}{cc} 4 &{} 0\\ 0 &{} 5 \end{array}\right] , \\ A= & {} \left[ \begin{array}{cc} a_{11} &{} a_{12}\\ a_{21} &{} a_{22} \end{array}\right] , \\ a_{11}= & {} -0.7e_{0}+0.8e_{1}-0.2e_{2}+0.4e_{3}+0.2e_{4}+0.8e_{5}+0.3e_{6}+0.9e_{7}, \\ a_{12}= & {} 0.3e_{0}+0.8e_{1}-0.2e_{2}-0.2e_{3}+0.5e_{4}+0.8e_{5}+0.8e_{6}-0.9e_{7}, \\ a_{21}= & {} -0.2e_{0}-0.4e_{1}+0.2e_{2}-0.2e_{3}+0.3e_{4}+0.2e_{5}-0.5e_{6}+0.2e_{7}, \\ a_{22}= & {} 0.4e_{0}+0.3e_{1}+0.1e_{2}+0.4e_{3}-0.2e_{4}-0.8e_{5}+0.8e_{6}+0.9e_{7}, \\ B= & {} \left[ \begin{array}{cc} b_{11} &{} b_{12}\\ b_{21} &{} b_{22} \end{array}\right] , \\ b_{11}= & {} -0.4e_{0}+0.7e_{1}+0.2e_{2}+0.5e_{3}-0.9e_{4}+0.9e_{5}-0.8e_{6}+0.9e_{7}, \\ b_{12}= & {} 0.8e_{0}+0.5e_{1}+0.3e_{2}-0.5e_{3}+0.8e_{4}+0.9e_{5}-0.9e_{6}+0.9e_{7}, \\ b_{21}= & {} 0.3e_{0}+0.2e_{1}-0.2e_{2}+0.1e_{3}+0.8e_{4}+0.9e_{5}+0.9e_{6}+0.9e_{7}, \\ b_{22}= & {} -0.5e_{0}+0.5e_{1}+0.2e_{2}+0.4e_{3}+0.9e_{4}-0.9e_{5}-0.9e_{6}+0.8e_{7}, \\ G= & {} \left[ \begin{array}{cc} g_{11} &{} g_{12}\\ g_{21} &{} g_{22} \end{array}\right] , \\ g_{11}= & {} -0.4e_{0}+0.7e_{1}+0.2e_{2}+0.5e_{3}-0.9e_{4}+0.9e_{5}-0.8e_{6}+0.9e_{7}, \\ g_{12}= & {} 0.8e_{0}+0.5e_{1}+0.3e_{2}-0.5e_{3}+0.8e_{4}+0.9e_{5}-0.9e_{6}+0.9e_{7}, \\ g_{21}= & {} 0.3e_{0}+0.2e_{1}-0.2e_{2}+0.1e_{3}+0.8e_{4}+0.9e_{5}+0.9e_{6}+0.9e_{7}, \\ g_{22}= & {} -0.5e_{0}+0.5e_{1}+0.2e_{2}+0.4e_{3}+0.9e_{4}-0.9e_{5}-0.9e_{6}+0.8e_{7}, \\ f_{j}(o)= & {} \frac{\sqrt{2}}{40}\sum _{q=0}^{7}f_{j}^{q}(o)e_{q}=\frac{\sqrt{2}}{40}\sum _{q=0}^{7}\frac{1}{1+\exp (-o^{q})}e_{q},\;\forall o\in \mathbb {O},\ \forall j\in \{1,2\}. \end{aligned}$$

Thus, Assumption 1 is satisfied by the activation functions, and \(L=\left[ \begin{array}{cc} 0.025I_{8} &{} 0\\ 0 &{} 0.025I_{8} \end{array}\right] \).

We also take \(\delta =0\) (no leakage delay), \(\tau (t)=0.3|\cos t|\), \(\eta =0.5\), hence \(\tau =0.3\) and \(\omega =\max \{\delta ,\tau ,\eta \}=0.5\), and compute \(\alpha _{1}=3.74\), \(\alpha _{21}=0.125\), \(\alpha _{22}=0.2013\), which satisfy \(\alpha _{1}>\alpha _{21}+\alpha _{22}\), and \(\rho _{1}=1\), \(\rho _{2}=2\). All of the above allow us to ascertain that all the hypotheses of Theorem 5 are met, which means that the equilibrium point of system (26) is exponentially stable.

Example 3

For the next example, consider time scale \(\mathbb {T}=\mathbb {R}\), from which we deduce that \(\hat{\mu }=0\). In order to study the synchronization property, we will consider system (26) as being the drive NN, and the following system as being the response NN:

$$\begin{aligned} p_{i}^{\Delta }(t)= & {} -c_{i}p_{i}(t-\delta )+\sum _{j=1}^{2}a_{ij}f_{j}(p_{j}(t))+\sum _{j=1}^{2}b_{ij}f_{j}(p_{j}(t-\tau (t)))\nonumber \\{} & {} +\sum _{j=1}^{2}g_{ij}\int _{t-\eta }^{t}f_{j}(p_{j}(s))\Delta s+I_{i}-u_{i}(t), \end{aligned}$$
(28)

\(\forall i\in \{1,2\}\), \(\forall t\in [0,+\infty )_{\mathbb {T}}\). In this new setting, by considering (26) and (28), and making the notation \(\varpi _{i}(t)=p_{i}(t)-o_{i}(t)\), the error system will have the following expression:

$$\begin{aligned} \varpi _{i}^{\Delta }(t)= & {} -c_{i}\varpi _{i}(t-\delta )+\sum _{j=1}^{2}a_{ij}\tilde{f}_{j}(\varpi _{j}(t))+\sum _{j=1}^{2}b_{ij}\tilde{f}_{j}(\varpi _{j}(t-\tau (t)))\nonumber \\{} & {} +\sum _{j=1}^{2}g_{ij}\int _{t-\eta }^{t}\tilde{f}_{j}(\varpi _{j}(s))\Delta s-u_{i}(t), \end{aligned}$$
(29)

\(\forall i\in \{1,2\}\), \(\forall t\in [0,+\infty )_{\mathbb {T}}\), where \(\tilde{f}_{j}(\varpi _{j}(t))=f_{j}(\varpi _{j}(t)+o_{j}(t))-f_{j}(o_{j}(t))\), \(\forall t\in [0,+\infty )_{\mathbb {T}}\), \(\forall j\in \{1,2\}\).

Next, the subsequent controller of state feedback type will be employed to achieve synchronization between systems (26) and (28):

$$\begin{aligned} u(t)=K_{1}\check{\varpi }(t)+K_{2}\check{\varpi }(t-\delta )+K_{3}\check{\varpi }(t-\tau (t))+K_{4}\int _{t-\eta }^{t}\check{\varpi }(s)\Delta s, \end{aligned}$$
(30)

in which \(K_{1},\;K_{2},\;K_{3},\;K_{4}\in \mathbb {R}^{8N\times 8N}\) constitute the control gain matrices. Using this controller, NN (29) becomes:

$$\begin{aligned} \check{\varpi }^{\Delta }(t)= & {} -K_{1}\check{\varpi }(t)-(\check{C}+K_{2})\check{\varpi }(t-\delta )-K_{3}\check{\varpi }(t-\tau (t))\nonumber \\{} & {} -K_{4}\int _{t-\eta }^{t}\check{\varpi }(s)\Delta s+\check{A}\check{f}(\check{\varpi }(t))\nonumber \\{} & {} +\check{B}\check{f}(\check{\varpi }(t-\tau (t)))+\check{G}\int _{t-\eta }^{t}\check{f}(\check{\varpi }(s))\Delta s. \end{aligned}$$
(31)

The parameters, in this example, are taken as:

$$\begin{aligned} C= & {} \left[ \begin{array}{cc} 4 &{} 0\\ 0 &{} 5 \end{array}\right] , \\ A= & {} \left[ \begin{array}{cc} a_{11} &{} a_{12}\\ a_{21} &{} a_{22} \end{array}\right] , \\ a_{11}= & {} -0.7e_{0}+0.9e_{1}-0.2e_{2}+0.4e_{3}+0.2e_{4}+0.8e_{5}+0.3e_{6}+0.9e_{7}, \\ a_{12}= & {} 0.3e_{0}+0.9e_{1}-0.2e_{2}-0.2e_{3}+0.5e_{4}+0.8e_{5}+0.8e_{6}-0.9e_{7}, \\ a_{21}= & {} -0.2e_{0}-0.4e_{1}+0.2e_{2}-0.2e_{3}+0.3e_{4}+0.2e_{5}-0.5e_{6}+0.2e_{7}, \\ a_{22}= & {} 0.4e_{0}+0.3e_{1}+0.1e_{2}+0.4e_{3}-0.2e_{4}-0.8e_{5}+0.8e_{6}+0.9e_{7}, \\ B= & {} \left[ \begin{array}{cc} b_{11} &{} b_{12}\\ b_{21} &{} b_{22} \end{array}\right] , \\ b_{11}= & {} -0.4e_{0}+0.7e_{1}+0.2e_{2}+0.5e_{3}-0.9e_{4}+0.9e_{5}-0.8e_{6}+0.9e_{7}, \\ b_{12}= & {} 0.8e_{0}+0.5e_{1}+0.3e_{2}-0.5e_{3}+0.8e_{4}+0.9e_{5}-0.9e_{6}+0.8e_{7}, \\ b_{21}= & {} 0.3e_{0}+0.2e_{1}-0.2e_{2}+0.1e_{3}+0.8e_{4}+0.9e_{5}+0.7e_{6}+0.9e_{7}, \\ b_{22}= & {} -0.5e_{0}+0.5e_{1}+0.2e_{2}+0.4e_{3}+0.8e_{4}-0.9e_{5}-0.8e_{6}+0.7e_{7}, \\ G= & {} \left[ \begin{array}{cc} g_{11} &{} g_{12}\\ g_{21} &{} g_{22} \end{array}\right] , \\ g_{11}= & {} -0.4e_{0}+0.7e_{1}+0.2e_{2}+0.5e_{3}-0.9e_{4}+0.9e_{5}-0.8e_{6}+0.9e_{7}, \\ g_{12}= & {} 0.9e_{0}+0.5e_{1}+0.3e_{2}-0.5e_{3}+0.8e_{4}+0.9e_{5}-0.9e_{6}+0.7e_{7}, \\ g_{21}= & {} 0.3e_{0}+0.2e_{1}-0.2e_{2}+0.1e_{3}+0.8e_{4}+0.9e_{5}+0.8e_{6}+0.9e_{7}, \\ g_{22}= & {} -0.5e_{0}+0.5e_{1}+0.2e_{2}+0.4e_{3}+0.9e_{4}-0.9e_{5}-0.8e_{6}+0.7e_{7}, \\ f_{j}(o)= & {} \frac{\sqrt{2}}{40}\sum _{q=0}^{7}f_{j}^{q}(o)e_{q}=\frac{\sqrt{2}}{40}\sum _{q=0}^{7}\frac{1}{1+\exp (-o^{q})}e_{q},\;\forall o\in \mathbb {O},\ \forall j\in \{1,2\}. \end{aligned}$$

Thus, Assumption 1 is satisfied by the activation functions, and \(L=\left[ \begin{array}{cc} 0.025I_{8} &{} 0\\ 0 &{} 0.025I_{8} \end{array}\right] \).

The delays are \(\delta =0.08\), \(\tau (t)=0.3|\sin t|\), so \(\tau =0.3\), and \(\eta =0.4\), so \(\omega =\max \{\delta ,\tau ,\eta \}=0.4\), and also take \(\alpha _{1}=7\), \(\alpha _{21}=2.2\), \(\alpha _{22}=2.3\), \(\alpha _{23}=2.4\), which satisfy \(\alpha _{1}>\alpha _{21}+\alpha _{22}+\alpha _{23}\). The control gain matrices are designed as:

$$\begin{aligned} K_{1}=\left[ \begin{array}{cc} 10.2 &{} 0\\ 0 &{} 10.1 \end{array}\right] ,\;K_{2}=\left[ \begin{array}{cc} 0.2 &{} 0\\ 0 &{} 0.3 \end{array}\right] ,\;K_{3}=\left[ \begin{array}{cc} 0.3 &{} 0\\ 0 &{} 0.1 \end{array}\right] ,\;K_{4}=\left[ \begin{array}{cc} 0.4 &{} 0\\ 0 &{} 0.3 \end{array}\right] . \end{aligned}$$

Solving LMI (18), we obtain \(R_{1}=\text {diag}(0.46336,1.0832)\), \(R_{2}=\text {diag}(0.36233,0.37425)\), \(R_{3}=\text {diag}(1.9368,2.1147)\), \(R_{4}=\text {diag}(1.7712,2.3323)\) (the other matrices’ values are not displayed for brevity’s sake), which means that all the conditions of Theorem 6 are met, reaching to the conclusion that drive NN (26) is exponentially synchronized with response NN (28) based on state feedback controller (30).

Figures 3, 4 depict graphically the trajectories of the components of the states \(\check{\varpi }_{1}\) and \(\check{\varpi }_{2}\), starting from 8 initial points.

Fig. 3
figure 3

State trajectories for the components of octonion \(\check{\varpi }_{1}\) in Example 3. The 8 colors in each graph depict the 8 initial values

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Fig. 4
figure 4

State trajectories for the components of octonion \(\check{\varpi }_{2}\) in Example 3. The 8 colors in each graph depict the 8 initial values

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Example 4

We take the same time scale \(\mathbb {T}=\mathbb {R}\) for our last example, and try to study the same problem of realizing synchronization between drive NN (26) and response NN (28) based on controller of state feedback type (30), but now having the following parameters:

$$\begin{aligned} C= & {} \left[ \begin{array}{cc} 4 &{} 0\\ 0 &{} 5 \end{array}\right] , \\ A= & {} \left[ \begin{array}{cc} a_{11} &{} a_{12}\\ a_{21} &{} a_{22} \end{array}\right] , \\ a_{11}= & {} -0.7e_{0}+0.9e_{1}-0.2e_{2}+0.4e_{3}+0.2e_{4}+0.8e_{5}+0.3e_{6}+0.9e_{7}, \\ a_{12}= & {} 0.3e_{0}+0.9e_{1}-0.2e_{2}-0.2e_{3}+0.5e_{4}+0.8e_{5}+0.8e_{6}-0.9e_{7}, \\ a_{21}= & {} -0.2e_{0}-0.4e_{1}+0.2e_{2}-0.2e_{3}+0.3e_{4}+0.2e_{5}-0.5e_{6}+0.2e_{7}, \\ a_{22}= & {} 0.4e_{0}+0.3e_{1}+0.1e_{2}+0.4e_{3}-0.2e_{4}-0.8e_{5}+0.8e_{6}+0.9e_{7}, \\ B= & {} \left[ \begin{array}{cc} b_{11} &{} b_{12}\\ b_{21} &{} b_{22} \end{array}\right] , \\ b_{11}= & {} -0.4e_{0}+0.7e_{1}+0.2e_{2}+0.5e_{3}-0.9e_{4}+0.9e_{5}-0.8e_{6}+0.9e_{7}, \\ b_{12}= & {} 0.8e_{0}+0.5e_{1}+0.3e_{2}-0.5e_{3}+0.8e_{4}+0.9e_{5}-0.9e_{6}+0.8e_{7}, \\ b_{21}= & {} 0.3e_{0}+0.2e_{1}-0.2e_{2}+0.1e_{3}+0.8e_{4}+0.9e_{5}+0.7e_{6}+0.9e_{7}, \\ b_{22}= & {} -0.5e_{0}+0.5e_{1}+0.2e_{2}+0.4e_{3}+0.8e_{4}-0.9e_{5}-0.8e_{6}+0.7e_{7}, \\ G= & {} \left[ \begin{array}{cc} g_{11} &{} g_{12}\\ g_{21} &{} g_{22} \end{array}\right] , \\ g_{11}= & {} -0.4e_{0}+0.7e_{1}+0.2e_{2}+0.5e_{3}-0.9e_{4}+0.9e_{5}-0.8e_{6}+0.9e_{7}, \\ g_{12}= & {} 0.9e_{0}+0.5e_{1}+0.3e_{2}-0.5e_{3}+0.8e_{4}+0.9e_{5}-0.9e_{6}+0.7e_{7}, \\ g_{21}= & {} 0.3e_{0}+0.2e_{1}-0.2e_{2}+0.1e_{3}+0.8e_{4}+0.9e_{5}+0.8e_{6}+0.9e_{7}, \\ g_{22}= & {} -0.5e_{0}+0.5e_{1}+0.2e_{2}+0.4e_{3}+0.9e_{4}-0.9e_{5}-0.8e_{6}+0.7e_{7}, \\ f_{j}(o)= & {} \frac{\sqrt{2}}{40}\sum _{q=0}^{7}f_{j}^{q}(o)e_{q}=\frac{\sqrt{2}}{40}\sum _{q=0}^{7}\frac{1}{1+\exp (-o^{q})}e_{q},\;\forall o\in \mathbb {O},\ \forall j\in \{1,2\}. \end{aligned}$$

Thus, Assumption 1 is satisfied by the activation functions, and \(L=\left[ \begin{array}{cc} 0.025I_{8} &{} 0\\ 0 &{} 0.025I_{8} \end{array}\right] \).

Now, we take \(\delta =0.01\), \(\tau (t)=0.1|\sin t|\), \(\eta =0.05\), from which we have \(\tau =0.1\) and \(\omega =\max \{\delta ,\tau ,\eta \}=0.1\), and we compute \(\alpha _{1}=9.7625\), \(\alpha _{21}=0.4025\), \(\alpha _{22}=5.3\), \(\alpha _{23}=0.0205\), from which we have \(\alpha _{1}>\alpha _{21}+\alpha _{22}+\alpha _{23}\), and \(\rho _{1}=1\), \(\rho _{2}=2\). The conditions of Theorem 7 are all met, reaching to the conclusion that drive NN (26) is exponentially synchronized with response NN (28) based on state feedback controller (30).

5 Conclusions

In this paper, a general model of OVNNs defined on time scales with leakage, time-varying, and distributed delays was put forward. First of all, to avoid the problems related to the non-commutativity and non-associativity of the octonion algebra, the system with octonion values was transformed into one with real values. Then, a Halanay inequality specially designed for time scales was used in order to obtain sufficient conditions, expressed as scalar inequalities and LMIs, which guarantee the exponential stability of the proposed model, using two general Lyapunov-like functions. Then, using the same Halanay inequality and the same Lyapunov-like functions, but also a state feedback controller, sufficient criteria for the exponential synchronization of the NNs put forward were obtained. Finally, one numerical example is provided to illustrate each of the four theorems deduced in the paper.

The results presented in this research are sufficiently general to allow for their particularization for both discrete time and continuous time OVNNs, as well as any hybrid combination of the two. They can also be particularized for CVNNs or QVNNs, for which, as far as we are aware, no similar results have been published in the literature. Additionally, the techniques utilized in the paper can be applied to other types of models defined on time scales, such as NNs with inertial terms, reaction–diffusion terms, Markov jump parameters, or impulsive effects to obtain sufficient criteria for other dynamic properties, such as multistability, dissipativity, passivity, etc. These show promise for future research.