Lipschitz Stability in Time for Riemann–Liouville Fractional Differential Equations

Hristova, Snezhana; Tersian, Stepan; Terzieva, Radoslava

doi:10.3390/fractalfract5020037

Cited by 14 publications

(12 citation statements)

References 18 publications

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“…This requires, for the stability to be defined, the initial time point 0 to be excluded, i.e., the special type of stability is deeply connected with the applied type of Riemann-Liouville fractional derivative (see, for example, refs. [2,4,38]). In some papers, for example, in [39], the authors do not exclude the initial time point and this leads to misunderstandings (see, for example the main condition of Theorem 1, t γ−1 E γ,γ (At γ ) ≤ Me −γt , which is not satisfied on the whole interval [0, ∞)).…”

Section: Consider the Quadratic Lyapunov Function Given Bymentioning

confidence: 99%

“…These types of derivatives have a singularity at the initial time point and give us a new tool for modeling anomalies in the dynamics of processes. The study of linear systems of fractional differential equations with Riemann-Liouville-type fractional derivatives was considered in [1], nonlinear systems in [2], existence and Ulam stability was studied in [3], and for basic concepts on stability for Riemann-Liouville fractional differential equations, we refer the reader to [4]. A general fractional derivative of arbitrary order in the Riemann-Liouville sense was defined and applied to Cauchy problems for single-and multi-term linear fractional differential equations by Luchko in [5].…”

Section: Introductionmentioning

confidence: 99%

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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: Consider the Quadratic Lyapunov Function Given Bymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Agarwal,

Hristova,

O’Regan

2023

Mathematics

Self Cite

View full text Add to dashboard Cite

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“…Fractional calculus is a generalization of the classical integer-order derivatives and integrals. The common definitions of fractional calculus are included in Riemann-Liouville (R-L) [22], Grunwald-Letnikov (G-L), and Caputo definitions [23,24], which have been applied in different fields, such as mathematics, engineering, computer science, etc. [25,26].…”

Section: Fractional Derivativesmentioning

confidence: 99%

Multi-AUV Dynamic Maneuver Countermeasure Algorithm Based on Interval Information Game and Fractional-Order DE

et al. 2022

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The instability of the underwater environment and underwater communication brings great challenges to the coordination and cooperation of the multi-Autonomous Underwater Vehicle (AUV). In this paper, a multi-AUV dynamic maneuver countermeasure algorithm is proposed based on the interval information game theory and fractional-order Differential Evolution (DE), in order to highlight the features of the underwater countermeasure. Firstly, an advantage function comprising the situation and energy efficiency advantages is proposed on account of the multi-AUV maneuver strategies. Then, the payoff matrix with interval information is established and the payment interval ranking is achieved based on relative entropy. Subsequently, the maneuver countermeasure model is presented along with the Nash equilibrium condition satisfying the interval information game. The fractional-order DE algorithm is applied for solving the established problem to determine the optimal strategy. Finally, the superiority of the proposed multi-AUV maneuver countermeasure algorithm is verified through an example.

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“…In recent years, fractional-order systems have become the focus of research, and articles [18][19][20][21] provide stability analysis of Riemann-Liouville neural networks. In the article, 22 the finite-time stability analysis of fractional-order amnestic fuzzy cell neural networks (MFFCNNs) with time delays and leakage terms was studied by using generalized Bernoulli's inequality and Holder's inequality.…”

Section: Introductionmentioning

confidence: 99%

Convergence analysis of Riemann‐Liouville fractional neural network

Dong

Liao

et al. 2022

Math Methods in App Sciences

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Many fractional order calculus researchers believe that fractional order calculus is a good way to solve information processing as well as certain physical system modeling problems. In the training of neural networks, there is the problem of long convergence time. In order to shorten the convergence time of the network, an R-L gradient descent method is proposed in this study. The article begins with a theoretical proof of the convergence of fractional order derivatives using function approximation and interpolation inequality theorems. Finally, through multiple simulations, it can be obtained that the fractional-order neural network can maintain a higher accuracy rate compared with the integer-order neural network, and also can well solve the problem of longer convergence time of the neural network. The convergence time can be reduced by nearly 10% compared to the integer order.

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Lipschitz Stability in Time for Riemann–Liouville Fractional Differential Equations

Cited by 14 publications

References 18 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

Multi-AUV Dynamic Maneuver Countermeasure Algorithm Based on Interval Information Game and Fractional-Order DE

Convergence analysis of Riemann‐Liouville fractional neural network

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