Sliding Mode Control-Based Synchronization of Complex-Valued Neural Networks

Synchronization is a typical dynamical behavior of interconnected systems that is being extensively studied in neural networks. However, most of the research considers real-valued neural networks, and fewer results have been obtained on their complex-valued counterparts. This article presents two sliding mode control strategies to achieve synchronization in a complex-valued neural network (CVNN). The former simplifies an already existing technique that splits the control into real and imaginary parts. The latter extends a fully complex-valued sliding approach for generic complex-valued dynamical systems to the multi-input–multi-output case, and shows its efficiency and higher performance in terms of finite reaching time in the synchronization of CVNNs. The approach is validated via numerical simulations.

computation, wind prediction, radars, and electroencephalography.For a thorough background on CVNN see [1] and [2] for an extended survey of recent results.
Synchronization is an important dynamical phenomenon of complex systems that also arises in neural networks, as it has been shown that the rhythmic activities of living organisms (breathing, walking, running, flying, swimming, etc.) are connected with the synchronization of the interacting neurons that form the neuronal circuits that control them, i.e., the so-called central pattern generators [3].Hence, on one hand, a number of theoretical works studying this behavior in artificial neural networks from different perspectives, scenarios, and control techniques are available in the literature.In particular, synchronization in coupled neural networks with delays [4], [5]; exponential synchronization under impulsive disturbances [6]; lagsynchronization in switched neural networks [7]; and antisynchronization in the face of delays via impulsive controllers [8].On the other hand, applications have been reported in fields such as pattern recognition [9], associative memory [10], secure exchange of information [11], and image encryption [7].Most of the existing literature deals with real-valued neural networks, although attention has been lately focused on the synchronization in CVNNs with special emphasis on drive-response systems, which are the target of this article.
Module-phase synchronization of CVNNs using adaptive feedback control was reported in [12].Exponential synchronization was analyzed in [13] also using an adaptive strategy, and in [14] via a hybrid impulsive control.In turn, a pinning strategy was considered in [15] for memristive CVNNs, while event-triggered impulsive pinning was studied in [16].It is worth noting that a common feature when dealing with CVNNs is to face the problem using a real-imaginary decomposition of the initially complex-valued system.This encompasses all these works but the latter, where a fully complex approach is conducted.However, the complex domain provides a natural frame where to analyze and control systems described by complex-valued models, as it allows to work with a reduced order representation with respect to the corresponding realimaginary one, and specific complex-valued tools can be used.Besides, in the CVNNs case, the generalization characteristics of the neural network, i.e., its ability to respond not only to situations it learned but to unlearned ones, differs between the complex-valued model and its real-imaginary counterpart.Specifically, CVNNs exhibit smaller generalization errors than the real-valued ones in wave-related processing [1].Hence, a fully complex approach is also adopted in this article.
Sliding mode control (SMC) is a well-known control technique with excellent robustness properties in the face of unmodeled dynamics, parametric variation, and external disturbances [17].In turn, its inherent switched action allows finite time control designs.Despite these interesting performance features, it has seldom been used in the synchronization of neural networks.Finite-time synchronization is achieved in [18], [19], [20], and [21] using real-imaginary separation and different discontinuous controllers based on the sign of the error mismatch between drive and response systems.Interestingly, Feng et al. [22] proposed a discontinuous, fully complex approach that uses a complex error sign.Integral SMC controllers were proposed in [23] for a real-valued neural network, while [24], [25], [26], [27] deal with the CVNNs case.The first and the second use real-imaginary separation, while the latter develops a fully complex approach.However, in all the abovementioned cases where a fully complex approach has been proposed, the complex error sign has been defined as the sign of the real part plus j times the sign of the imaginary part, yielding four possible values: ±1 ± j.Alternatively, a more versatile complex sign stems from a direct extension of the real sign definition [see (1) in Section II], which has been reported to render shorter transients for equivalent control efforts [28].
This article proposes SMC controllers to induce outer (or drive-response) synchronization in two coupled CVNNs, as well as inner synchronization in a single CVNN.The first algorithm simplifies the real-imaginary integral SMC policy introduced in [24] by considering the synchronization error as switching manifold.As in [24], the newly proposed strategy yields synchronization, and no performance losses are observed.In turn, the second algorithm proposes a fully complex SMC control law that stems from a multi-input-multi-output (MIMO) counterpart of a complex SMC algorithm presented in [28].The fully complex SMC uses the complex sign definition in [28], and keeps the finite reaching time feature already reported for the single-input-single-output (SISO) case [28].Numerical simulations confirm this superior performance with respect to the real-imaginary proposal [24] and its simplified version.
The rest of this article is organized as follows.Some basic notation is introduced in Section II, and the CVNN model is presented in Section III.Section IV contains the definitions of the types of synchronization analyzed in the article and a study of network synchronizability in CVNNs using the master stability equations (MSEs) for complex-valued systems.Then, the simplified real-imaginary and the fully complex SMC approaches to CVNNs synchronization are presented, respectively, in Sections V and VI.The numerical validations are conducted in Section VII.Finally, Section VIII concludes this article.

II. NOTATION
In what follows, R ≥0 denotes the set of nonnegative real numbers; C n and R n denote, respectively, the complex and real nth-dimensional spaces; Ω ⊆ C n denotes an open subset of C n ; and indicates the set of continuous maps from Ω to R ≥0 , I m denotes the m-dimensional identity matrix, and Let α = α R + α I j ∈ C: j denotes the imaginary unit, while α R = Re(α) ∈ R and α I = Im(α) ∈ R stand for its real and imaginary parts, respectively; α * denotes the conjugate of α ∈ C; |α| = √ α * α and φ α denote the magnitude and phase, respectively, of α; sign(α) denotes the sign function of a complex value α ∈ C\{0}, which is computed as follows [29]: In turn, given a vector a = (a i ) ∈ C n , with a i ∈ C, a * = (a * i ) denotes its conjugate transpose; Re(a), Im(a), |a| = (|a i |), φ a = (φ a i ), and cos φ a = (cos φ a i ) denote, respectively, the R n vectors gathering the real and imaginary parts of its components, its magnitudes and phases, and the cosine of the phases; e jφ a = (e jφ a i ) and sign(a) = (sign(a i )) stand, respectively, for the C n vector containing the complex exponential of the phase, and the complex sign of each component, in the latter when none of them are 0; √ a * a, denote its 1-and 2-norms; inf{|a|} = inf{|a i |; i = 1, . . ., n}, and D a = diag(a) indicates the C n×n diagonal matrix with the components of a as the diagonal elements.
Finally, for a matrix A = (a ij ) ∈ C n×n , a ij ∈ C, we denote by A the norm of A induced by the vector 2-norm.

III. COMPLEX-VALUED NEURAL NETWORKS
A neural network is a multiagent system where the agents are the neurons, and the edges interconnecting them are the synaptic connections.The model of each agent of the network can be seen as a generic nonlinear control system with state space dynamics where x i ∈ C n is the state variable of the ith neuron, ing the jth state-weighted input channel, and is the coupling law governing the interaction between neurons, and N is the total number of neurons in the neural network.
In this work, we consider networks of nodes with internal dynamics composed of a self-feedback term plus a nonlinear function of the state, i.e., In turn, n = m, G i is taken as the n-dimensional identity matrix, namely, G i = I n , and a linear diffusive protocol plus an outer input is assumed for the coupling law, set as where σ ∈ C is the coupling gain, a ij ∈ {0, 1} is the ij element of the corresponding adjacency matrix A = (a ij ), and Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
I i : R ≥0 → R n are smooth, time-dependent maps.Finally, denoting by L the corresponding Laplacian matrix, obtained as L = D − A, with D, A standing for the degree and adjacency matrices, respectively, the overall system (2)-( 4) can be written as follows [24]: (5) This model is a generalization of the natural complex-valued extension of the so-called continuous-time Hopfield-type neural networks introduced in [30], a pioneering work that very much contributed to triggering the research interest in neural networks [31], [32].The Hopfield neural networks model associative memories, which are able to relate a certain input with a previously memorized pattern [1].See [33] for an update on current research on associative memories.

IV. SYNCHRONIZATION IN COMPLEX-VALUED NEURAL NETWORKS
In this section, we first recall the inner and outer synchronization definitions, which have been adapted from [34] and [35] to a neural network with state dynamics given by (5).Then, the problem of synchronizability is discussed using a complex-valued MSEs approach [34].

A. Definitions
Inner synchronization in complex networks is a phenomenon in which each node of the network converges toward a common time-varying solution (e.g., a common limit cycle or chaotic solution).Synchronization can be obtained under a suitable coupling configuration, or by applying an external forcing.
Definition 1: (see [34]) The neural network ( 5) is said to achieve: i) inner synchronization iff ii) finite-time inner synchronization iff there exists T > 0 such that, for all i, j = 1, . . ., N lim t→T x i − x j = 0, and Inner synchronization is said to be achieved globally iff ( 6) or ( 7) is verified for all x i (0), x j (0) ∈ C n , otherwise it is said to be achieved locally.
Synchronization between two coupled networks, or outer synchronization, emerges when the states of a response (or slave) network asymptotically converge toward those of a drive (or master) network, again by appropriate coupling or suitable external actuation.
Let then (5) be a drive network, and let be the response network, where σ y ∈ C is the coupling gain and u i : Ω → C n , u i ∈ H n , denotes a control action.Definition 2: (see [35]) The neural networks ( 5) and ( 8) are said to achieve the following: i) outer synchronization iff ii) finite-time outer synchronization iff there exists T > such that, for all i, j = 1, . . ., N lim t→T y i − x i = 0, and Outer synchronization is achieved globally iff ( 9) or ( 10) is verified for all x i (0), y i (0) ∈ C n , otherwise it is achieved locally.

B. Network Synchronizability: Master Stability Equations
The network synchronizability problem consists of finding conditions on the coupling gain, the network structure, and the node vector fields that guarantee the emergence of synchronization.In this section, we address this issue for networks of identical nodes with the MSEs technique for complex-valued dynamical systems [34], which stems from the master stability function approach first proposed in [36].
The next result follows immediately from [34, Th. 2 and Remark 5].
Assumption 1: Let (5) be a CVNN with identical nodes, i.e., b i = b, f i = f , I i = I, i = 1, . . ., N. Let also v = v(t) ∈ C n be a holomorphic periodic solution of the individual node dynamics plus the external input, namely Finally, assume that the Laplacian matrix of (5), L, has rank Theorem 1: Consider the CVNN (5) and let Assumption be fulfilled.If ξ = 0 is an asymptotically stable solution of each of the following time-varying linear systems: with Jf denoting the Jacobian matrix of f , then inner synchronization is locally achieved, and the synchronization manifold The N dynamical systems (12) are known as the MSE of ( 5) on v = v(t).
Example: We illustrate the MSE method by applying it to a directed CVNN of the form (5) with identical nodes borrowed from [24, Example I]: and I(t) = 0, and the Laplacian matrix ) the synchronous solution candidate being x i = v(t) = 0, i = 1, . . ., 10, which is the unique asymptotically stable equilibrium of the corresponding internal dynamics (11).The goal is to find the set of values of the complex coupling gain σ ∈ C, which we denote by Σ, that guarantee synchronization of the network.
The expected behavior is confirmed by the plots in Figs. 2  and 3 of the real and imaginary parts of the states of the network for the limiting values of the complex coupling gain highlighted in Fig. 1.As expected, synchronization is observed for σ A , σ C , while no synchronization arises for σ B , σ D .

V. REAL/IMAGINARY SLIDING MODE CONTROL OF CVNNS
In [24], a sliding mode controller was proposed for the outer synchronization of the CVNNs ( 5) and ( 8) with 1-D nodes, i.e., n = 1, and separating real and imaginary parts.Essentially, let ) with e i = y i − x i denoting the synchronization errors, σ = σ y − σ, and let the real and imaginary parts of the switching   manifolds be chosen as The main result in [24] reads as follows.Theorem 2: Let the coupled CVNNs ( 5) and (8) with n = 1.
If, for all i = 1, . . ., N, the control actions u i are set as where θ R and θ I are constants larger than 1, and Re(s i ) and Im(s i ) are defined as in (16), then ( 5) and ( 8) achieve outer synchronization globally.
Notice that the sliding mode controller introduced in Theorem 2 has a number of parameters and that the corresponding sliding surfaces ( 16) are of integral type.In what follows, we propose an alternative version with less parameters and a simplified sliding surface definition.This reduces the implementation effort and the associated computational burden, and it is shown in Section VII that it does not entail any performance decay.
Proof: Letting e i = y i − x i and recalling P i defined in (15), the error dynamics of ( 5) and (8) with n = 1 is given by ėi = −b i e i + P i (e i , x) + u i , i = 1, . . ., N. ( Consider the autonomous, positive definite auxiliary function where s = (s i ) ∈ C N .Notice that V ∈ C N is locally Lipschitz on s.In turn, its time derivative over the trajectories of (19) for all s ∈ Ω N \{0} is where, for the sake of brevity, the argument of P i has been omitted.Then, for all i = 1, . . ., N, and recalling that s i = e i , we have that Hence, a sliding regime exists on s = 0 [37], which guarantees that ( 5) and ( 8) achieve outer synchronization.Finally, as ( 18) is verified for all x(0), y(0) ∈ C N , the result is global.
It is straightforward that this strategy can also induce inner synchronization in (8).
Corollary 1: Let v = v(t) ∈ C be a common, holomorphic solution of the individual node dynamics plus the external input of the neural network (8) Then, the SMC law of Theorem 3 with e i = y i − v and l ij e j (22) induces inner synchronization globally in the CVNN (8).

VI. COMPLEX SLIDING MODE CONTROL OF CVNNS
A fully complex-valued SMC is proposed in [28] for SISO systems, which leads to a reduction of the reaching time with respect to a real/imaginary SMC.Here, we extend the result, particularly that in Proposition 2, to MIMO systems.
Consider the complex-valued state-space system with dynamics where Let the complex switching manifold be defined as where s : Ω −→ C M is the complex switching function, which is assumed holomorphic as well, i.e., s ∈ H M .Finally, letting s(x) = (s 1 (x), . . ., s M (x)), with s i : Ω −→ C, i = 1, . . ., M, and G(x) = (G 1 (x), . . ., G M (x)), we denote as D(x) the matrix product of the Jacobian of s(x) times G(x), i.e., with ∂s i /∂x standing for the gradient of s i , i = 1, . . ., M.
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Assumption 2: Let s ∈ H M be a complex switching function, and let S denote the corresponding switching manifold.Assume that D(x) is diagonal, i.e. ( ∂s i ∂x ) • G j (x) = 0, for all i = j; hence, letting s G(x) = (( Finally, assume also that there exist i = 1, . . ., M, with U eq ∈ C M denoting the equivalent control Theorem 4: Let Assumption 2 be fulfilled.Then, the switched control action induces sliding motion of system (23) on Ω 1 ∩ S.Moreover, for all x(0) close enough to Ω 1 ∩ S, the sliding manifold is reached in a finite time T ≤ √ 2( 1 2 ) −1 s(0) .Proof: Let us consider again the autonomous, positive definite, C M function locally Lipschitz on s as well.Its time derivative ∀x ∈ Ω 1 \S is By definition, U eq is the control that makes S an invariant manifold with respect to (23).Hence, using (26) V can be written as Now, recalling the control law ( 27) where we have used the fact that D K sign(s) = D sign(s) K, and also that square diagonal matrices commute.Moreover, Notice that so the first term of (29) becomes As regards the second term, Consequently, (25) yields
Remark 2: The demand in Assumption 2 of having D s G (x) diagonal is a generalization of the CVNN's case involving (5) and (8).It is also met in systems with a diagonal control matrix G(x) and a set of decoupled switching surfaces, i.e., of the form s i (x) = s i (x i ).In turn, (25) are technical assumptions that become essential to guarantee a finite reaching time of the switching manifold S. On the one hand, (25a) is deeply related to the so-called transversality condition of SMC, which ensures the existence of the equivalent control (26).On the other hand, (25b) contains the design parameter 2 , which is selected according to the desired reaching time, the value of 1 , and a known bound of |U eq i |, by tuning the control gains K i .Besides, (25b) indicates that the better the argument of K i compensates that of (s G) i , the lower control gain effort |K i | is needed for (25b) to be fulfilled.
The application of Theorem 4 to finite-time synchronization of CVNNs is established in the next results.
Let x(t, x 0 ) denote the solution of (5) with x(0, x 0 ) = x 0 , and, given μ ∈ R + , let Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Proof: The proof can be obtained as a straightforward application of Theorem 4 with M = N to the error system (19).Indeed, as the switching function vector is s = (e i ) ∈ C N , and the control vector is G = I N , condition (25a) is verified with 1 = 1.Moreover, using (26), it follows that: and, according to the definition of P i in (15), it results that Therefore, (31) and, by continuity, there exists an open neighborhood Ω 1 ⊆ Ω N of the origin where inf i=1,...,N Hence, (25b) is also satisfied, and by Theorem 4, the sliding manifold is reached in finite time.Finally, as (32) is fulfilled for all t ≥ 0, the sliding motion is never lost, i.e., Ω 1 ∩ S is forward invariant, so ( 5) and ( 8) synchronize in finite time.
It is worth highlighting that the basin of attraction of the sliding manifold depends on the initial conditions of both the drive and response system, on the control gain parameters, and also on the norm of the bounded solution of the drive system and, therefore, it is difficult to estimate analytically.
Finite-time inner synchronization is obtained in an analogous manner to Corollary 1.
Corollary 2: Let v = v(t) ∈ C be a common, bounded, holomorphic solution of the individual node dynamics plus the external input of the neural network (8) with n = 1, i.e., such that it verifies (21).Then, the SMC of Proposition 1, with e i = y i − v and P i defined, as in (22), induces finite-time inner synchronization locally in the CVNN (8).
Remark 3: The synchronization results established in Section V are global, while those in Section VI are local.However, the "globality" of such control strategies is in practice limited by the fact that the control effort in Theorems 2 and 3 lies on a control effort that is proportional to the error.When initial conditions are too far from each other the actuators might saturate in a real-world implementation.Hence, globality cannot be experimentally achieved.Instead, local results that rely on Fig. 4. Example I. Open-loop evolution (I(t) = sin t + j cos t, σ = 1 + j, σ y = −1 − j, and u i = 0) of the real and imaginary states of the drive (left), and response (right) CVNNs ( 5) and (8), respectively.a bounded control action are more realistic from a practical viewpoint.
Finally, it is well known in SMC literature that, when the control action is discontinuous, as is the case in Theorems 2 and 3, the switching manifold is effectively crossed in an unknown, but finite, time.Instead, in finite-time controllers, as the one stemming from Theorem 4, this settling time is fixed by design.

VII. NUMERICAL VALIDATION
The results presented in Sections V and VI regarding outer synchronization are validated through the CVNN introduced as an example in Section IV-B, while inner synchronization is illustrated via a complex-valued Hopfield neural network [32].
Example I: Following [24], the external input is selected as I(t) = sin t + j cos t; as for the coupling gain, σ = σ y = 1 was selected therein, which already yielded inner synchronization for both the drive and the response network in open loop.In order to better test the performance of the algorithms, we choose σ = 1 + j, σ y = −1 − j, so that only the drive network exhibits inner synchronization, as illustrated in Fig. 4.
The real/imaginary outer synchronization strategy introduced in Theorem 3 is run with control parameters k R i = k I i = 14, i = 1, . . ., N. Fig. 5 gathers the evolution of the individual node errors and control inputs: notice that outer synchronization is effectively achieved with bounded control effort.
In order to numerically validate the fully complex-valued SMC algorithm presented in Proposition 1, we select the components of the control gain vector as K i = 18 ∈ R + , i = 1, . . ., 10.This guarantees cos(φ K ) = 1 N ; hence, a minimum control effort is required for constraint (31) to be fulfilled.The evolution of the individual node errors and control inputs, portrayed in Fig. 6, reveals the effectiveness of the strategy to achieve finite-time Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE I KEY PERFORMANCE INDICATORS
outer synchronization.It is worth pointing out that in order to avoid the chattering phenomenon, the complex sign function has been implemented using a boundary layer approach [28], [38], which replaces it by a saturation function, namely The performance of the proposed controllers is now compared among them and also with the one proposed in [24] and presented in Theorem 2 (with θ R , θ 10), through the key performance indicators (KPI) detailed in Table I.In the rest of the section they have been named, respectively, as real/imaginary SMC, simplified real/imaginary SMC, and complex-valued SMC.
In Fig. 7 (top), the evolution of the ∞-norm of the control input vector is reported, respectively, for the controllers of Theorem 2, Theorem 3, and Proposition 1.In turn, Fig. 7 (bottom) pictures the area of the maximum value of the control inputs.Although the maximum control effort in the third case is almost halved with respect to the first and second one, it is clear that, from an energy point of view, the three controllers generate a similar control input for the current settings.This ensures a fair comparison of errors and settling times carried out next.
The evolution of the average of the state errors is displayed in Fig. 8 (left).The real/imaginary SMC and its simplified version exhibit almost identical behavior, with a 0.4-s settling time.Recall, however, that the advantage of the latter in terms of  implementation effort and the computational burden has already been pointed out in Section V.In turn, as expected after Fig. 6, the complex-valued SMC has a much faster convergence speed, with the settling time being 0.1 s, and the finite-time feature is again observed.
The last KPI we report is the area of the ∞-norm of the error vector, shown in Fig. 8 (right).As happens with its left counterpart, the similarity between original and simplified real/imaginary strategies, plus the higher performance of the complex-valued SMC algorithm, is further confirmed.
Example II: The simplified real/imaginary and the fully complex sliding mode controllers have also been validated for the case of complex-valued Hopfield neural networks, with the purpose of illustrating the onset of inner synchronization.The network model is with the Laplacian matrix given by ( 13), and identical neurons.Notice that this matches the CVNN model ( 5) except for the activation function, which is now nonlinear, namely, h(x) = tanh(|x|)e j arg(x) .However, Corollaries 1 and 2 also hold in this case after an appropriate redefinition of P i in (22).The values of the CVNN parameters are set as τ = 1, ω = 1, σ = −1 + j, and the external input is I i (t) = sin(t) + j cos(t).Inner synchronization is not achieved in open-loop behavior, as shown in Fig. 9 (left).However, this is effectively induced using the fully complex-valued SMC of Corollary 2 with control gain vector components set to K i = 5, i = 1, . . ., 10, as  The counterparts of Figs. 9 (right) and 10 for the simplified real/imaginary SMC of Corollary 1 have been omitted for the sake of brevity.However, its performance is illustrated in the sequel, where a comparison between the three strategies presented in Sections V and VI is carried out.
In Example I, control signals of similar energy resulted in the fully complex-valued strategy requiring less maximum control input to render better performances (recall Figs.7 and 8).In a complementary way, in this case, we illustrate in Figs.11 and 12 that, with equal maximum control effort, the fully complex-valued SMC strategy is able to significantly shorten the transient time with less control energy.In turn, the simplified real/imaginary algorithm is shown to enforce inner synchronization, as predicted in Corollary 1. Notice, finally, that the real/imaginary SMC controller proposed in [24] for outer synchronization can also induce inner synchronization in CVNNs, the performance results being similar to those of its simplified version.

VIII. CONCLUSION
This article presents two sliding mode controllers capable of achieving synchronization in CVNNs.The first one was a much  simpler version of an already existing algorithm that splits the action into real and imaginary parts, while the second stems from the multiinput version of a fully complex-valued SMC.It was theoretically proved that both algorithms might induce inner and outer synchronization, the second one in finite time.Numerical results comparing the three strategies in two examples validated the theoretical predictions and showed the superior performance of the fully complex-valued architecture in terms of control effort and transient response times.
Further research is aimed at applying the fully complexvalued algorithm to broader classes of CVNNs and multiagent systems.

Fig. 5 .
Fig. 5. Example I. Evolution of the errors e i (t) (left), and control inputs u i (t) (right) when applying the simplified real/imaginary SMC of Theorem 3.

Fig. 6 .
Fig. 6.Example I. Evolution of the errors e i (t) (left), and control inputs u i (t) (right) when applying the complex-valued SMC of Proposition 1.

Fig. 8 .
Fig. 8. Example I. Evolution of the average of the errors 1 N N i=1 |e i (t)| (top), and the integral of the ∞-norm of the maximum error T 0e ∞ dt (bottom), for the real/imaginary SMC (blue), simplified real/imaginary SMC (red), and complex-valued SMC (yellow).

Fig. 9 .
Fig. 9. Example II.Complex-valued Hopfield neural network: state variables x i (t) in open-loop (left), and closed-loop (right) applying the complex-valued SMC of Corollary 2.

Fig. 10 .
Fig. 10.Example II.Complex-valued Hopfield neural network: evolution of the errors e i (t) (left), and control inputs u i (t) (right) when applying the complex-valued SMC of Corollary 2.