Gronwall Inequality Research Articles

Finite-time stability of fractional-order fuzzy cellular neural networks with time delays

The finite-time stability of fractional-order fuzzy cellular neural networks with time delays is investigated. One of the difficulties in stability analysis of fractional-order delayed systems lies in how to handle the delay terms in the systems. In view of this difficulty, firstly, a new fractional-order Gronwall inequality with time delay is developed; this can be widely used to study finite-time stability of a variety of fractional-order systems with time delays. Secondly, a new criterion for the finite-time stability of fractional-order fuzzy cellular neural networks with time delays is derived in terms of this inequality; it is less conservative than the one using the existing fractional-order Gronwall inequality with time delays. Finally, two examples are given to expound the effectiveness and less conservativeness of the proposed results.

Quantized feedback control for Markovian jumping singular systems driven by fractional Brownian motions

AbstractThe quantized feedback control for a type of delayed Markovian jumping singular systems driven by fractional Brownian motions is investigated. Firstly, the input is quantized by a uniform operator consisting of a linear part to deal with system stochastic stability and a nonlinear part to deal with the error caused by the quantizer. Secondly, to deal with the boundedness of fractional Brownian motions, a new type of Lyapunov function dependent of Hurst index is constructed. Then, utilizing generalized Finsler lemma and Gronwall inequality, some new criteria on the singular stochastic finite‐time boundedness are obtained for the considered systems with the designed quantized input. Finally, a numerical example and a DC motor example are presented to verify the effectiveness of the proposed methods and results.

Finite-time stability of linear stochastic fractional-order systems with time delay

This paper focuses on the finite-time stability of linear stochastic fractional-order systems with time delay for alpha in (frac{1}{2},1). Under the generalized Gronwall inequality and stochastic analysis techniques, the finite-time stability of the solution for linear stochastic fractional-order systems with time delay is investigated. We give two illustrative examples to show the interest of the main results.

Pairs of foliations and Mattei-Moussu's Theorem

We prove a reduction of singularities for pairs of foliations by blowing-up, and then investigate the analytic classification of the reduced models. Those reduced pairs of regular foliations are well understood. The case of a regular and a singular foliation is dealt with Mattei-Moussu's Theorem for which we provide a new proof, avoiding Gronwall's inequality. We end-up announcing results recently obtained by the first author in the case of a pair of reduced foliations sharing the same separatrices.

Finite-time fuzzy reliable controller design for fractional-order tumor system under chemotherapy

This article presents a stability analysis of fractional-order tumor system that correlates tumor-immune with chemotherapy under Lyapunov's theory. More particularly, a finite-time stabilization of prescribed system with the consequence of quantized input and actuator fault is investigated. Further, the nonlinear tumor system is converted to Takagi-Sugeno(T-S) fuzzy model. Then based on the utilization of Gronwall's inequality, the necessary and sufficient conditions are derived with the assist of Lyapunov's candidates. To overcome the effect of external disturbances such as unwanted smoke, intake habits and tower radiations, the optimal level of performance which is reduced from the generalized performance extended dissipative is included. Finally, a numerical simulation is presented with graphical results to expound the superiority and validity of the prescribed controller.

New results on finite‐time stability of fractional‐order Cohen–Grossberg neural networks with time delays

AbstractThe finite‐time stability (FTS) of fractional‐order delayed Cohen–Grossberg neural networks (FODCGNNs) with the order ℘ ∈ (1, 2) is investigated in this study. Based on the fractional‐order delayed Gronwall inequality (FODGI), a new sufficient condition to guarantee the FTS of FODCGNNs is established, which reduces the conservation of the existing criterion. Finally, one numerical example is exhibited to illustrate the effectiveness and less conservativeness of the obtained results.

Analysis of the Euler and trapezoidal discretization methods for the numerical solution of nonlinear functional Volterra integral equations of Urysohn type

In this paper, we investigate nonlinear functional Volterra–Urysohn integral equations, a class of nonlinear integral equations of Volterra type. The existence and uniqueness of the solution to the equation is proved by a technique based on the Picard iterative method. For the numerical approximation of the solution, the Euler and trapezoidal discretization methods are utilized which result in a system of nonlinear algebraic equations. Using a Gronwall inequality and its discrete version, first order of convergence to the exact solution for the Euler method and quadratic convergence for the trapezoidal method are proved. To prove the convergence of the trapezoidal method, a new Gronwall inequality is developed. Finally, numerical examples show the functionality of the methods.

Iterative learning consensus control with initial state learning for fractional order distributed parameter models multi‐agent systems

This paper considers the consensus control problem of multi‐agent systems (MAS) with distributed parameter models. Based on the framework of network topologies, a second‐order PI‐type iterative learning control (ILC) protocol with initial state learning is proposed by using the nearest neighbor knowledge. A discrete system for proposed ILC is established, and the consensus control problem is then converted to a stability problem for such a discrete system. Furthermore, by using generalized Gronwall inequality, a sufficient condition for the convergence of the consensus errors between any two agents is obtained. Finally, the validity of the proposed method is verified by two numerical examples.

Finite-time stability of a class of nonautonomous systems

In this paper, the finite-time stability of nonautonomous systems is studied by using Lyapunov Krasovskii method and Gronwall inequality. Firstly, a new lemma for the finite-time stability for nonautonomous systems is given by using Gronwall inequality. which make the derivative of Lyapunov function not to be negative all the time. Then, based on the new lemma and Lyapunov Krasovskii method, some finite-time stability criteria about nonautonomous linear systems and nonlinear systems are proposed. Based on this, by designing controllers, the finite-time stabilizable theorem is obtained. In addition, it is pointed out that the conclusion of this paper is more general than the existing literature in the remarks. Finally, several examples are simulated with MATLAB to verify the effectiveness of the results.

Existence and stability of some nonlinear ψ-Hilfer partial fractional differential equation

Abstract The main objective of this paper is to study properties of ψ -Hilfer partial fractional differential equations which are useful in modeling variety of phenomena in physical sciences. The existence, uniqueness, approximate solution and stability properties of solution are studied. The technique used for study is fixed point theorem and Gronwall inequality. Ulam–Hyers–Rassis stability of the solution is also investigated.

Stabilizing controller design for nonlinear fractional order systems with time varying delays

To deal with stabilizing of nonlinear affine fractional order systems subject to time varying delays, two methods for finding an appropriate pseudo state feedback controller are discussed. In the first method, using the Mittag-Lefler function, Laplace transform and Gronwall inequality, a linear stabilizing controller is derived, which uses the fractional order of the delayed system and the upper bound of system nonlinear functions. In the second method, at first a sufficient stability condition for the delayed system is given in the form of a simple linear matrix inequality (LMI) which can easily be solved. Then, on the basis of this result, a stabilizing pseudo-state feedback controller is designed in which the controller gain matrix is easily computed by solving an LMI in terms of delay bounds. Simulation results show the effectiveness of the proposed methods.

Some Existence and Dependence Criteria of Solutions to a Fractional Integro-Differential Boundary Value Problem via the Generalized Gronwall Inequality

The main intention of the present research study is focused on the analysis of a Caputo fractional integro-differential boundary problem (CFBVP) in which the right-hand side of supposed differential equation is represented as a sum of two nonlinear terms. Under the integro-derivative boundary conditions, we extract an equivalent integral equation and then define new operators based on it. With the help of three distinct fixed-point theorems attributed to Krasnosel’skiĭ, Leray–Schauder, and Banach, we investigate desired uniqueness and existence results. Additionally, the dependence criterion of solutions for this CFBVP is checked via the generalized version of the Gronwall inequality. Next, three simulative examples are designed to examine our findings based on the procedures applied in the theorems.

An Estimate of the Bound of the Lyapunov Exponents for Caputo–Hadamard Fractional Differential System

Abstract This paper is devoted to estimating the bound of the Lyapunov exponents for the Caputo–Hadamard fractional differential system. First, using the Gronwall inequality, we analyze the continuous dependence of the solution to the Caputo–Hadamard fractional initial value problem (IVP). Then, we define the Lyapunov exponents for the Caputo–Hadamard fractional differential system and estimate their bounds. Besides, numerical examples are displayed which support the theoretical results.

Solutions of linear uncertain fractional order neutral differential equations

Uncertain fractional order neutral differential equation is an important model to describe the evolution process of uncertain dynamical system. This paper devotes to studying linear uncertain fractional order neutral differential equations. After providing the analytic solutions for linear uncertain fractional order neutral differential equations by the Mittag-Leffler function, the author investigates the inverse uncertainty distribution of the solution to linear uncertain fractional order neutral differential equation by the α-path and has a discussion of the dependence of the solution on initial function based on the generalized Gronwall inequality.

A stochastic Gronwall inequality and applications to moments, strong completeness, strong local Lipschitz continuity, and perturbations

There are numerous applications of the classical (deterministic) Gronwall inequality. Recently, Michael Scheutzow discovered a stochastic Gronwall inequality which provides upper bounds for $p$-th moments, $p\in(0,1)$, of the supremum of nonnegative scalar continuous processes which satisfy a linear integral inequality. In this article we complement this with upper bounds for $p$-th moments, $p\in[2,\infty)$, of the supremum of general It\^o processes which satisfy a suitable one-sided affine-linear growth condition. As example applications, we improve known results on strong local Lipschitz continuity in the starting point of solutions of stochastic differential equations (SDEs), on (exponential) moment estimates for SDEs, on strong completeness of SDEs, and on perturbation estimates for SDEs.

A generalized fractional (q, h)–Gronwall inequality and its applications to nonlinear fractional delay (q, h)–difference systems

In this paper, a generalized fractional (q, h)–Gronwall inequality is investigated. Based on this inequality, the uniqueness theorem and the finite–time stability criterion of nonlinear fractional delay (q, h)–difference systems are derived. Finally, several examples are given to illustrate the effectiveness of our theoretical result.

A note on the mild solutions of Hilfer impulsive fractional differential equations

In this paper, we present a new type of Gronwall inequality and discuss some particular cases. We apply these results to investigate the uniqueness and δ-Ulam–Hyers–Rassias stability of mild solutions of a fractional differential equation with non-instantaneous impulses in a Pδ-normed Banach space. In this sense, we present an example, in order to elucidate one of the results discussed.

FINITE-TIME STABILITY IN MEAN FOR NABLA UNCERTAIN FRACTIONAL ORDER LINEAR DIFFERENCE SYSTEMS

In this paper, the finite-time stability in mean for the uncertain fractional order linear time-invariant discrete systems is investigated. First, the uncertain fractional order difference equations with the nabla operators are introduced. Then, some conditions of finite-time stability in mean for the systems driven by the nabla uncertain fractional order difference equations with the fractional order [Formula: see text] are obtained by the property of Riemann–Liouville-type nabla difference and the generalized Gronwall inequality. Furthermore, based on these conditions, the state feedback controllers are designed. Finally, some examples are presented to illustrate the effectiveness of the results.

Ulam‐Hyers‐Rassias stability for generalized fractional differential equations

In this paper, we present a generalized Gronwall inequality with singularity. Using this inequality, we investigate the existence, uniqueness, and Ulam‐Hyers‐Rassias stability for solutions of a class of generalized nonlinear fractional differential equations of order α (1 < α < 2). In this way, we improve and generalize several earlier outcomes.

Existence and global exponential stability of equilibrium for impulsive neural network models with generalized piecewise constant delay

In this paper, we investigate the models of the impulsive cellular neural network with generalized constant piecewise delay (IDEGPCD). To guarantee the existence, uniqueness and global exponential stability of the equilibrium state, several new adequate conditions are obtained, which extend the results of the previous literature. The method is based on utilizing Banach’s fixed point theorem and a new IDEGPCD’s Gronwall inequality. The criteria given are easy to check and when the impulsive effects do not affect, the results can be extracted from those of the non-impulsive systems. Typical numerical simulation examples are used to show the validity and effectiveness of the proposed results. We end the paper with a brief conclusion.