Asymptotical Stability of Riemann‐Liouville Nonlinear Fractional Neutral Neural Networks with Time‐Varying Delays

Korkmaz, Erdal; Özdemir, Abdulhamit; Yildirim, Kenan

doi:10.1155/2022/6832472

Cited by 6 publications

(5 citation statements)

References 35 publications

(15 reference statements)

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“…Remark 6. Note that the Caputo version of the inequality (47) cannot be applied to study Riemann-Liouville fractional differential equations (as is done, for example, in Lemma 2.2 [33], in Lemma 3 [34], in Lemma 2.2 [35], and in Lemma 2 [36,37]).…”

Section: Tempered Riemann-liouville Fractional Derivativementioning

confidence: 99%

Inequalities for Riemann–Liouville-Type Fractional Derivatives of Convex Lyapunov Functions and Applications to Stability Theory

Agarwal,

Hristova,

O’Regan

2023

Mathematics

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In recent years, various qualitative investigations of the properties of differential equations with different types of generalizations of Riemann–Liouville fractional derivatives were studied and stability properties were investigated, usually using Lyapunov functions. In the application of Lyapunov functions, we need appropriate inequalities for the fractional derivatives of these functions. In this paper, we consider several Riemann–Liouville types of fractional derivatives and prove inequalities for derivatives of convex Lyapunov functions. In particular, we consider the classical Riemann–Liouville fractional derivative, the Riemann–Liouville fractional derivative with respect to a function, the tempered Riemann–Liouville fractional derivative, and the tempered Riemann–Liouville fractional derivative with respect to a function. We discuss their relations and their basic properties, as well as the connection between them. We prove inequalities for Lyapunov functions from a special class, and this special class of functions is similar to the class of convex functions of many variables. Note that, in the literature, the most common Lyapunov functions are the quadratic ones and the absolute value ones, which are included in the studied class. As a result, special cases of our inequalities include Lyapunov functions given by absolute values, quadratic ones, and exponential ones with the above given four types of fractional derivatives. These results are useful in studying types of stability of the solutions of differential equations with the above-mentioned types of fractional derivatives. To illustrate the application of our inequalities, we define Mittag–Leffler stability in time on an interval excluding the initial time point. Several stability criteria are obtained.

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Section: Tempered Riemann-liouville Fractional Derivativementioning

confidence: 99%

Inequalities for Riemann–Liouville-Type Fractional Derivatives of Convex Lyapunov Functions and Applications to Stability Theory

Agarwal,

Hristova,

O’Regan

2023

Mathematics

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“…Remark 3. The inequality (17) was applied by several authors to the RLFD, but the authors used (inappropriately) the version for Caputo fractional derivatives, proven in [16] (see, for example, Lemma 2.2 [17], Lemma 3 [18], Lemma 2.2 [19], and Lemma 2 [20]).…”

Section: Lemma 11 Suppose the Functionmentioning

confidence: 99%

Lyapunov Functions and Stability Properties of Fractional Cohen–Grossberg Neural Networks Models with Delays

Agarwal,

Hristova,

O’Regan

2023

Fractal Fract

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Some inequalities for generalized proportional Riemann–Liouville fractional derivatives (RLGFDs) of convex functions are proven. As a special case, inequalities for the RLGFDs of the most-applicable Lyapunov functions such as the ones defined as a quadratic function or the ones defined by absolute values were obtained. These Lyapunov functions were combined with a modification of the Razumikhin method to study the stability properties of the Cohen–Grossberg model of neural networks with both time-variable and continuously distributed delays, time-varying coefficients, and RLGFDs. The initial-value problem was set and studied. Upper bounds by exponential functions of the solutions were obtained on intervals excluding the initial time. The asymptotic behavior of the solutions of the model was studied. Some of the obtained theoretical results were applied to a particular example.

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“…Over the years, many papers studying different dynamic properties of FONNs have appeared. Recently, the asymptotic stability and synchronization properties of FONNs were discussed, for example, in [20][21][22][23][24][25][26][27]. Then, the equivalent of the exponential stability and synchronization from integer-order neural networks, namely Mittag-Leffler stability and synchronization, were discussed in [28][29][30][31][32][33].…”

Section: Introductionmentioning

confidence: 99%

“…It is natural that this type of delay was also added to neural network models, because they have been seen to appear when VLSI circuits are used for implementing neural networks. FONNs with neutral-type delays were the focus of the following recent papers: [20,21,26,[57][58][59].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic and Mittag–Leffler Synchronization of Fractional-Order Octonion-Valued Neural Networks with Neutral-Type and Mixed Delays

Popa

2023

Fractal Fract

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Very recently, a different generalization of real-valued neural networks (RVNNs) to multidimensional domains beside the complex-valued neural networks (CVNNs), quaternion-valued neural networks (QVNNs), and Clifford-valued neural networks (ClVNNs) has appeared, namely octonion-valued neural networks (OVNNs), which are not a subset of ClVNNs. They are defined on the octonion algebra, which is an 8D algebra over the reals, and is also the only other normed division algebra that can be defined over the reals beside the complex and quaternion algebras. On the other hand, fractional-order neural networks (FONNs) have also been very intensively researched in the recent past. Thus, the present work combines FONNs and OVNNs and puts forward a fractional-order octonion-valued neural network (FOOVNN) with neutral-type, time-varying, and distributed delays, a very general model not yet discussed in the literature, to our awareness. Sufficient criteria expressed as linear matrix inequalities (LMIs) and algebraic inequalities are deduced, which ensure the asymptotic and Mittag–Leffler synchronization properties of the proposed model by decomposing the OVNN system of equations into a real-valued one, in order to avoid the non-associativity problem of the octonion algebra. To accomplish synchronization, we use two different state feedback controllers, two different types of Lyapunov-like functionals in conjunction with two Halanay-type lemmas for FONNs, the free-weighting matrix method, a classical lemma, and Young’s inequality. The four theorems presented in the paper are each illustrated by a numerical example.

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Asymptotical Stability of Riemann‐Liouville Nonlinear Fractional Neutral Neural Networks with Time‐Varying Delays

Cited by 6 publications

References 35 publications

Inequalities for Riemann–Liouville-Type Fractional Derivatives of Convex Lyapunov Functions and Applications to Stability Theory

Inequalities for Riemann–Liouville-Type Fractional Derivatives of Convex Lyapunov Functions and Applications to Stability Theory

Lyapunov Functions and Stability Properties of Fractional Cohen–Grossberg Neural Networks Models with Delays

Asymptotic and Mittag–Leffler Synchronization of Fractional-Order Octonion-Valued Neural Networks with Neutral-Type and Mixed Delays

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