P-Moment Exponential Stability of Differential Equations With Random Noninstantaneous Impulses and the Erlang Distribution

Agarwal, Ravi P.; Hristova, Snezhana; O’Regan, Donal; Kopanov, Peter

doi:10.12732/ijpam.v109i1.3

Cited by 5 publications

(6 citation statements)

References 12 publications

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“…We will give some results about Erlnag distributed moments of impulses studied in Reference [18] and applied to study ordinary differential equation with random non-instantaneous impulses.…”

Section: Preliminary Results For Erlang Distributed Moments Of Impulsesmentioning

confidence: 99%

“…Impulsive differential equations with random impulsive moments differ from the study of stochastic differential equations with impulses (see, for example, Reference [12][13][14][15][16]). Some stability properties of differential equations with non-instantaneous impulses, starting at randomly distributed points, are studied in Reference [17,18]. Fractional differential equations with non-instantaneous impulses have recently been studied in Reference [19] but the meaning of the impulses are not random (as they are called in the paper) but arbitrary and the solutions are deterministic functions.…”

Section: Introductionmentioning

confidence: 99%

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Caputo Fractional Differential Equations with Non-Instantaneous Random Erlang Distributed Impulses

Hristova

Ivanova

2019

Fractal Fract

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The p-moment exponential stability of non-instantaneous impulsive Caputo fractional differential equations is studied. The impulses occur at random moments and their action continues on finite time intervals with initially given lengths. The time between two consecutive moments of impulses is the Erlang distributed random variable. The study is based on Lyapunov functions. The fractional Dini derivatives are applied.

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Section: Preliminary Results For Erlang Distributed Moments Of Impulsesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Caputo Fractional Differential Equations with Non-Instantaneous Random Erlang Distributed Impulses

Hristova

Ivanova

2019

Fractal Fract

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“…One of them is considering stochastic models (see, for example, the review paper [21] and the references cited therein). Another is considering impulsive perturbation in the neural networks occurring at random times (see, for example, [3,4]).…”

Section: Introductionmentioning

confidence: 99%

Mean-square stability of Riemann–Liouville fractional Hopfield’s graded response neural networks with random impulses

et al. 2021

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In this paper a model of Hopfield’s graded response neural network is investigated. A network whose neurons are subject to a certain impulsive state displacement at random times is considered. The model is set up and studied. The presence of random moments of impulses in the model leads to a change of the solutions to stochastic processes. Also, we use the Riemann–Liouville fractional derivative to model adequately the long-term memory and the nonlocality in the neural networks. We set up in an appropriate way both the initial conditions and the impulsive conditions at random moments. The application of the Riemann–Liouville fractional derivative leads to a new definition of the equilibrium point. We define mean-square Mittag-Leffler stability in time of the equilibrium point of the model and study this type of stability. Some sufficient conditions for this type of stability are obtained. The general case with time varying self-regulating parameters of all units and time varying functions of the connection between two neurons is studied.

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“…So, we define for the first time the generalization of Hopfield neural network with impulses at random times, briefly give an explanation of the solutions being stochastic processes and study stability properties. Note the stability problem for differential equation with impulses at random time are studied in [1]. [2].…”

Section: Introductionmentioning

confidence: 99%

Stability of Neural Networks With Random Impulses

Hristova¹,

Kopanov²

2018

DSA

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One of the main properties of solutions of neural networks is stability and often the direct Lyapunov method is used to study stability properties. We consider the Hopfield's graded response neural network in the case when the neurons are subject to a certain impulsive state displacement at random exponentially distributed moments. It changes significantly the behavior of the solutions because they are not deterministic ones but they are stochastic processes. We examine the stability of the equilibrium of the model. Some sufficient conditions for p-moment stability of equilibrium of neural networks with time varying self-regulating parameters of all units and time varying functions of the connection between two neurons in the network are obtained. These sufficient conditions are explicitly expressed in terms of the parameters of the system and hence they are easily verifiable. We illustrate our theory on a particular nonlinear neural network.

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P-Moment Exponential Stability of Differential Equations With Random Noninstantaneous Impulses and the Erlang Distribution

Cited by 5 publications

References 12 publications

Caputo Fractional Differential Equations with Non-Instantaneous Random Erlang Distributed Impulses

Caputo Fractional Differential Equations with Non-Instantaneous Random Erlang Distributed Impulses

Mean-square stability of Riemann–Liouville fractional Hopfield’s graded response neural networks with random impulses

Stability of Neural Networks With Random Impulses

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