Global Asymptotic Stability of Impulsive CNNs with Proportional Delays and Partially Lipschitz Activation Functions

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Introduction
Cellular neural networks (CNNs) introduced by Chua and Yang [1,2] have found many important applications in biology, the solving of optimization problem, image processing, and pattern recognition [3].In fact, CNNs can be characterized by an array of identical nonlinear dynamical systems (called cells) locally interconnected in the paper [4] which presented a set of sufficient conditions ensuring the existence of at least one stable equilibrium point in terms of the template elements.As we know, time delays are inevitable in electronic implementation of CNNs [5].However, time delays may destroy stability of the networks and even lead to the oscillation behaviors.Hence, it is necessary to study the stability of CNNs with different types of delays.Time delays may be proportional delays; that is to say, the delay function () = (1−) is a monotonically increasing function with respect to  > 0, where  is a constant and satisfies 0 <  < 1.The type of proportional delays is usually required in Web quality of service routing decision and one may be convenient to control the network's running time according to the network allowed delays.Moreover, one can refer to the paper [6] about more information on the proportional delay engineering.Proportional delays [7][8][9][10] are unbounded time-varying ones different from constant delays [11], bounded time-varying delays [12][13][14][15][16][17][18], and unbounded distributed delay [19][20][21][22][23].It is relatively difficult to deal with this class of the unbounded time-varying delays because none of any other assumptions are imposed on it compared with other unbounded time-varying delays, such as, unbounded distributed delays often require that the delay kernel functions   : R + → R + satisfy ∫ ∞ 0   () = 1, ∫ ∞ 0   () < ∞, or there exists a positive number  such that ∫ ∞ 0   ()   < ∞ [20][21][22][23].Several stability criteria of CNNs with proportional delays have been obtained [7].Moreover, the abrupt changes in the voltages produced by faulty circuit elements are exemplary of impulse phenomena which can affect the transient behavior of the network [24].Hence, it is significant to discuss the stability of the CNNs with impulses and proportional delays.However, to the best of the authors' knowledge, few authors have handled the stability of CNNs with impulses and proportional delays.
Among the existing research results about neural networks, some activation functions are assumed to be globally Lipschitz continuous [25][26][27][28][29][30], bounded and monotonic [31], and bounded [24,32].However, these assumptions make these existing results unapplicable to some important engineering problems.For example, when the neural networks are used to solve optimization problems with the presence of constraints (linear, quadratic, or more general programming problems), unbounded (or nonmonotonic, non-globally Lipspchitz continuous) activations modeled by diode-like exponential-type functions are needed such that constraints are satisfied [33].Motivated by this, we attempt to abandon these assumptions and only require activation functions to be partially Lipschitz continuous.Moreover, the relative nonlinear measure is more efficient than the nonlinear measure for exponential stability analysis of different classes of neural networks without delays where the equilibrium points are given [20,34].
According to the foregoing analysis, this paper is devoted to analyzing stability of impulsive CNNs with proportional delays and Lipschitz continuous activation functions by relative nonlinear measure.The remainder of this paper is arranged as follows.Section 2 describes the model of proportion-delayed impulsive CNNs with partial Lipschitz continuous activation functions and provides its equivalent form by some transformation.Being preliminaries, Section 3 is devoted to uniqueness and exponential stability of equilibrium point of a nonlinear impulsive functional differential equation with variable coefficients and constant delays by means of relative nonlinear measure.In Section 4, a sufficient condition is obtained for global asymptotic stability of equilibrium point of impulsive proportion-delayed CNNs with partially Lipschitz continuous activation functions by results derived in Section 3. Furthermore, an example and its simulations are presented to illustrate that our method is valid and that our derived results are new and correct.Conclusions are given in Section 5.

Model Description and Its Equivalent Form
We consider the following CNNs with impulses and multiproportional delays: for  = is the impulse at moments   and 1 =  1 <  2 < ⋅ ⋅ ⋅ is a strictly increasing sequences such that lim  → +∞   = +∞;   ,   , and ℎ  are the nonlinear activation functions;   > 0 denotes the th component of an external input source introduced from outside the network to the th cell at time .

Preliminaries
Let -dimensional real vector space R  be endowed with 1norm ‖ ⋅ ‖ 1 defined by where the superscript  denotes the transpose.Let ⟨⋅, ⋅⟩ denote the inner product in R  and sign() = (sign( 1 ), sign( 2 ), . . ., sign(  ))  the sign vector of  ∈ R  , where sign() represents the sign function of  ∈ R. Obviously, the relations hold for all ,  ∈ R  .In order to discuss the stability of the neural networks (1), we firstly consider exponential stability of the following differential equation with variable coefficients, delays, and impulses where  > 0, C([−, 0], Ω) denotes the space of all continuous functions from [−, 0] into the open subset Ω of R  ;   ∈ C([−, 0], Ω) is defined by   () = ( + ) for all  ∈ [−, 0] and ‖  ‖ C = sup −≤≤0 ‖( + )‖ 1 ;  and  : Ω → R  are nonlinear operators; 0 =  0 <  1 <  2 < ⋅ ⋅ ⋅ is a strictly increasing sequence such that lim  → +∞   = +∞;   () is defined as follows: The nonlinear operators  and  are defined, respectively, by Definition 1 (see [20]).(1) A nonlinear operator  : Ω → R  is called be Lipschitz continuous on Ω if there exists a nonnegative constant  such that where  is called the Lipschitz constant of  on Ω.The constant is called the minimal Lipschitz constant (MLC) of  on Ω.Furthermore, the operator  is called globally Lipschitz continuous if Ω = R  .
(2) A nonlinear operator  : Ω → R  is said to be partially Lipschitz continuous on Ω if, for any  ∈ Ω, there exists a constant   > 0 such that The constant is called minimal partial Lipschitz constant (MPLC) of  on Ω with respect to .Furthermore, the operator  is called partially Lipschitz continuous if Ω = R  .
From the paper [20] we conclude that every Lipschitz continuous operator on Ω is partially Lipschitz continuous on Ω and   Ω (, ) ≤  Ω () for any Lipschitz continuous operator  and  ∈ Ω.
Definition 2 (see [34]).Assume that Ω is an open subset of R  ,  is a nonlinear operator from Ω into R  , and  0 ∈ Ω is any vector.The constant is called relative nonlinear measure of  at  0 .Definition 3.  * is said to be an equilibrium point of ( 15) if ( + ) * = 0 and I  ( * ) = 0 for all  ∈ N.
Definition 4. Let  * be an equilibrium point of (15) and Ω an open neighborhood of  * . * is exponentially stable on Ω if there exist two positive constants  and  such that holds for  ≥ 0, where () is the unique solution of (15) initiated from the function  ∈ C([−, 0], Ω).
Particularly, if Ω = R  holds, then  * is the unique equilibrium point and ( 15) is said to be globally exponentially stable.
Remark 9. Our proof idea mainly comes from Theorem 2 of [20] investigating the exponential stability of the special case of (15) (i.e., (15) with constant coefficients).However, they are essentially different because Theorem 8 in this paper has to deal with time-varying coefficients.Consequently, Theorem 8 in this paper is a generalization of Theorem 2 in [20].Moreover, it needs to point out that the exponential stability criterion (30) and exponential decay index  in (31) are independent of time  although the abstract equation ( 15) enjoys time-varying coefficients, which means that our method is essential to qualitatively and quantitatively characterize exponential stability of (15).Moreover, Theorem 8 is not only generalization and improvement of Theorem 1 in [35] because there indeed exists a nonlinear Lipschitz continuous map  on Ω such that   Ω (, ) is strictly less than  Ω () for any  ∈ Ω and (15) enjoys time-varying coefficients.
It is obvious that the CNNs model ( 12) can be changed into the form of (15).By Theorem 8, we can obtain the exponential stable criterion of equilibrium point of the CNNs model (12).Since the model ( 1) is equivalent to the model (12) in the sense of solution, models (12) and (1) enjoy the same equilibrium point V * =  * , where V * = (V * 1 , V * 2 , . . ., V *  )  and  * = ( * 1 ,  * 2 , . . .,  *  )  are the equilibrium point of models ( 12) and (1), respectively.What qualitative property of the model ( 1) can be derived from the global exponential stability of the model ( 12)?The next theorem can answer this problem.
This implies that the equilibrium point  * of the model ( 1) is globally asymptotic stable.
Remark 11.It need point out that the paper [7] has obtained not exponential stability, but asymptotic stable criteria of CNNs with multi-proportional delays because it mistakes asymptotic stability as exponential stability, which can be easily seen from the Remark 3.2 in [7] and Theorem 10 in this paper.

Uniqueness and Global Asymptotic Stability of Equilibrium Point of Model (1)
In this subsection, we firstly prove that model (1) has a unique equilibrium point in R  .It is enough to prove that model (12) has a unique equilibrium point in R  because models ( 12) and (1) enjoy the same equilibrium point.For this, we define that  = ( 1 ,  2 , . . .,   )  and  = ( 1 ,  2 , . . .,   )  : R  → R  are defined, respectively, by Proof.Obviously, it is enough to prove that model (12) has no other equilibrium point in R  different from  * if the inequality (54) holds.Define  = diag( 1 ,  2 , . . .,   ) and we need only prove   −1 (R  ) ( −1 ( + ),  * ) < 0 according to Theorem 7. In detail, for V ∈  −1 (R  ), we enjoy The combination of (55) and (54) implies that   −1 (R  ) ( −1 ( + ),  * ) < 0, which implies that model (12) enjoys no other equilibrium point in R  different from  * .That is to say,  * is the unique equilibrium point in R  of model (1).

Illustrative Example
In this section, we present an illustrative example to verify effectiveness of our method.
Remark 2. It needs to be pointed out that the impulsive instants are only selected as 2, 4, 6, . . . in the simulation of this example to simplify the simulation, which is obviously not enough to illustrate the impulsive effect.In order to accurately characterize wider of impulses, the papers [39,40] proposed the concepts of average dwell time and average impulsive interval.Moreover, the papers [41,42] presented single impulsive controller and pinning impulsive stabilization criterion, respectively.Consequently, their methods are recommended to simulations with more general impulses.

Conclusions and Further Work
By means of relative nonlinear measure and transformation, this paper has discussed global asymptotic stability of impulsive cellular neural networks with proportional delays and partially Lipschitz activation functions.We have obtained the novel criterion of uniqueness and global asymptotic stability of the equilibrium point of this CNNs model.Our method does not require conventional assumptions on global Lipschitz continuity, boundedness, and monotonicity of activation functions and proportional delays to meet other requirements, which demonstrates that our criteria derived are less restrictive than some existing ones and that they are generalizations and improvements of some existing ones.Finally, the example with three cells has illustrated that our method is effective and that our results are correction.Our method only requires the activation functions of the cellular networks to be partially Lipschitz continuous.The relative weak assumption makes our results applicable to more general engineering problems.In the future, we attempt to design a cellular neural networks model to solve optimization problems with some constraints, where unbounded (or nonmonotonic, nonglobally Lipspchitz continuous) activations are required such that these constraints are satisfied.
the resting state when isolated from other cells and inputs at time ;   ,   , and   denote the strengths of connectivity between the th and the th cells at time ,   , and   , respectively;   and   are proportional delay factors and satisfy 0 <   ,   < 1,  = min 1≤≤ {  ,   } and    =  − (1 −   ),    =  − (1 −   ), in which (1 −   ), (1 −   ) correspond to the time delays required in processing and transmitting a signal from the th cell to the th cell, and (1 −   ) → +∞, (1 −   ) → +∞ as  → +∞; Δ 1, 2, . . ., , where  ≥ 2 is the number of cells in the networks;   () denotes the potential of the th cell at time ;   > 0 represents the rate with which the th cell resets its potential to