Neural networks with distributed delays and Hölder continuous activation functions

Math. Model. Nat. Phenom.

Khemmoudj

2021

In this paper, we examine a Bidirectional Associative Memory neural network model with distributed delays. Using a result due to Cid [J. Math. Anal. Appl. 281 (2003) 264–275], we were able to prove an exponential stability result in the case when the standard Lipschitz continuity condition is violated. Indeed, we deal with activation functions which may not be Lipschitz continuous. Therefore, the standard Halanay inequality is not applicable. We will use a nonlinear version of this inequality. At the end, the obtained differential inequality which should imply the exponential stability appears ‘state dependent’. That is the usual constant depends in this case on the state itself. This adds some difficulties which we overcome by a suitable argument.

Section: Numerical Illustrationmentioning

confidence: 99%

“…In view of the importance of non-Lipschitz activation functions in implementations [15], a relaxation of the Lipschitz condition is necessary. This has motivated some researchers to consider discontinuous functions and Hölder-type functions, one can refer to [2,3,7,11,[27][28][29][30][31][32][33]35].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic behavior of a BAM neural network with delays of distributed type

Othmani

Math. Model. Nat. Phenom.

Khemmoudj

2021

“…Such cases arise naturally in applications. Some efforts have been made in this respect such as in Tatar and Wu et al [14][15][16][17] and Forti et al and Tatar 8,14,[18][19][20][21][22] through the treatment of Hölder continuous activation functions. One of the difficulties encountered is lost of the global stability due to the use of some integral inequalities.…”

Section: Introductionmentioning

confidence: 99%

Time‐dependent neural networks with activation functions violating the standard Lipschitz condition

Math Methods in App Sciences

Sakthivel

2022

and Minerals, Grant/Award Number: SB201006A Hopfield neural network system with discrete (or variable delays) is considered in this paper. The standard assumption of Lipschitz continuity of the activation functions is dropped partially. Having in mind that this condition is needed not only for the uniqueness of solutions but also for the stability of the system, the present work improves the existing ones in the literature. The other feature here, which is in fact the main one, is the treatment of time-dependent activation functions. The time-independent case has been discussed by one of the authors in Tatar (2020). Unfortunately, it is not applicable to the present situation. Indeed, when applied, it will require a uniform boundedness condition.To overcome this difficulty, we provide here a new argument in addition to the introduction of some functionals.

“…Namely, in, 28 the activation functions, of a neural network system without delays, are supposed to be merely bounded by non-decreasing functions. The Hölder continuity has been assumed in , 29 for problems without delays, in 30 for discrete delays and in 31 for distributed delays. The Hölder continuity of f j and g j is assumed in, 32 for the problem…”

Section: Introductionmentioning

confidence: 99%

A nonlinear version of the distributed Halanay inequality and its application

Math Methods in App Sciences

Sakthivel

2021

In this paper, first we prove a nonlinear version of Halanay inequality with distributed delay. Then, we use it to establish a local exponential stability result for a problem appearing in neural networks theory under the name Hopfield neural network system. This is done for activation functions violating the usual standard condition of Lipschitz continuity. One of the activation functions may be non-Lipschitz continuous. Finally, an example is provided to illustrate the developed theory.