Gronwall-Bellman Inequalities Research Articles

Dynamic memory-based event-triggered security control for networked control systems subjected to false-injection-data attack

This study addresses the security control problem for networked control systems that are subjected to false-data-injection attacks. The designed controller rests on a dynamic memory-based event-triggered mechanism, which exploits historically transmitted data and an auxiliary term to improve transmission efficiency and ensure satisfactory control performance. Then, a controller that defends against the false-data-injection attacks is proposed by making full use of the historical information. With the aid of Cauchy–Schwarz and Gronwall–Bellman inequalities, the existence of such a controller is validated through a novel Lyapunov-based method, which allows the Lyapunov function to be nonpositive. Further, a test on a dynamic power system is provided to verify the effectiveness and superiority of the proposed method.

SOME NEW TYPES OF GRONWALL-BELLMAN INEQUALITY ON FRACTAL SET

Gronwall–Bellman-type inequalities provide a very effective way to investigate the qualitative and quantitative properties of solutions of nonlinear integral and differential equations. In recent years, local fractional calculus has attracted the attention of many researchers. In this paper, based on the basic knowledge of local fractional calculus and the method of proving inequality on the set of real numbers, some new Gronwall–Bellman inequalities are discussed and established on the fractal space. Our results are a powerful supplement to the existing literature.

Robust Stabilization for Uncertain Non-Minimum Phase Switched Nonlinear System under Arbitrary Switchings

This paper addresses the problem of stabilization of non-minimum phase switched nonlinear systems where the internal dynamics with symmetries or non-symmetries of each mode may be unstable. The authors initially build a stabilizing Lyapunov controller for each mode in order to stabilize its own unstable internal dynamics. The proposed approach is based on the exact input-output feedback linearization technique and the Lyapunov stability theory. The stability results for non-minimum phase switched nonlinear systems with arbitrary switching rules are then obtained using generalized Gronwall–Bellman inequalities. Finally, numerical examples are provided to demonstrate the efficacy of the achieved results.

Stochastic Sampled-Data Exponential Synchronization of Markovian Jump Neural Networks With Time-Varying Delays.

In this article, the exponential synchronization of Markovian jump neural networks (MJNNs) with time-varying delays is investigated via stochastic sampling and looped-functional (LF) approach. For simplicity, it is assumed that there exist two sampling periods, which satisfies the Bernoulli distribution. To model the synchronization error system, two random variables that, respectively, describe the location of the input delays and the sampling periods are introduced. In order to reduce the conservativeness, a time-dependent looped-functional (TDLF) is designed, which takes full advantage of the available information of the sampling pattern. The Gronwall-Bellman inequalities and the discrete-time Lyapunov stability theory are utilized jointly to analyze the mean-square exponential stability of the error system. A less conservative exponential synchronization criterion is derived, based on which a mode-independent stochastic sampled-data controller (SSDC) is designed. Finally, the effectiveness of the proposed control strategy is demonstrated by a numerical example.

Extensions of Gronwall-Bellman type integral inequalities with two independent variables

Abstract In this paper, we establish several kinds of integral inequalities in two independent variables, which improve well-known versions of Gronwall-Bellman inequalities and extend them to fractional integral form. By using these inequalities, we can provide explicit bounds on unknown functions. The integral inequalities play an important role in the qualitative theory of differential and integral equations and partial differential equations.

New results on the uniform exponential stability of nonautonomous perturbed dynamical systems

AbstractIn this article, we investigate the asymptotic behaviors of the solutions of nonlinear dynamic systems nearby an equilibrium point, when the nominal parts are subject to non‐necessarily small perturbations. We show that, under some estimates on the perturbation term, the equilibrium point remains (globally) uniformly exponentially stable. The obtained stability results can easily be applied in practice since they are based on the Gronwall–Bellman inequalities rather than the classical Lyapunov methods that require the knowledge of a Lyapunov function. Several numerical examples, as well as an application to control and mechanical systems, are given in illustration.

Some Gronwall–Bellman Inequalities on Time Scales and Their Continuous Forms: A Survey

Some generalizations of the Gronwall–Bellman (G–B) inequality are presented in this paper in continuous form and on time scales. After S. Hilger introduced the time scales theory in 1988, over the years many mathematicians have studied new versions of this inequality according to new results; the purpose of this paper is to present some of them. Therefore, in the Introduction, some generalizations of G–B inequality in continuous forms, linear and nonlinear are presented. In the second section, some important and interesting results on time scales theory are given. In the third and main part of our paper, G–B inequalities on time scales and their possible connection with G–B inequalities presented in the introduction are investigated. In particular, in the third section of this work, more attention is given to G–B type inequalities on time scales discussed in the last four years.

Some New Discrete Gronwall-Bellman Type Inequalities with Three Independent Variables and Applications

Some new discrete Gronwall-Bellman type inequalities with three independent variables which generalize some existing results and can be used as handy tools in the study of qualitative and quantitative properties of solutions of certain classes of difference equation are presented. As applications, some difference equations with the initial boundary conditions are also considered.

Some new stochastic forms of Gronwall–Bellman inequalities and their applications

Abstract In this work, we establish some new forms of stochastic Gronwall–Bellman inequalities. We illustrate our work with examples of applications.

A new type of the Gronwall-Bellman inequality and its application to fractional stochastic differential equations

This paper presents a new type of Gronwall-Bellman inequality, which arises from a class of integral equations with a mixture of nonsingular and singular integrals. The new idea is to use a binomial function to combine the known Gronwall-Bellman inequalities for integral equations having nonsingular integrals with those having singular integrals. Based on this new type of Gronwall-Bellman inequality, we investigate the existence and uniqueness of the solution to a fractional stochastic differential equation (SDE) with fractional order on (0, 1). This result generalizes the existence and uniqueness theorem related to fractional order (1/2 1) appearing in [1]. Finally, the fractional type Fokker-Planck-Kolmogorov equation associated to the solution of the fractional SDE is derived using It^o's formula.

Some Generalized Gronwall-Bellman Type Impulsive Integral Inequalities and Their Applications

This paper investigates some generalized Gronwall-Bellman type impulsive integral inequalities containing integration on infinite intervals. Some new results are obtained, which generalize some existing conclusions. Our result is also applied to study a boundary value problem of differential equations with impulsive terms.

On a Class of Gronwall-Bellman Type Inequalities

The aim of the present paper is to establish some new integral inequalities of Gronwall type involving functions of two independent variables which provide explicit bounds on unknown functions. The inequalities given here can be used as tools in the qualitative theory of certain partial differential and integral equations.

Some generalized Gronwall-Bellman type nonlinear delay integral inequalities for discontinuous functions

In this work, some new generalized Gronwall-Bellman type nonlinear delay integral inequalities for discontinuous functions are established, which can be used as a handy tool in the quantitative and qualitative analysis of solutions of certain integral equations. The established results generalize the main results of Gao and Meng in (Math. Pract. Theory 39(22):198-203, 2009).

Stability analysis of switched systems with general nonlinear disturbances

This paper studies the stability of switched systems with general nonlinear disturbances. By using generalized Gronwall–Bellman inequalities, we obtain stability results for the systems with arbitrary switching laws. Finally, an example is given to illustrate the effectiveness of the results obtained.

Some new integral inequalities on time scales containing integration on infinite intervals

Abstract In this paper, some new Gronwall-Bellman type integral inequalities on time scales containing integration on infinite intervals are established. They provide new bounds for unknown functions concerned and can be used as a handy tool in the research of qualitative and quantitative properties of solutions of certain dynamic equations on time scales. MSC:26E70, 26D15, 26D10.

Some new Gronwall-Bellman type nonlinear dynamic inequalities containing integration on infinite intervals on time scales

Abstract In this paper, some new Gronwall-Bellman-type nonlinear dynamic inequalities containing integration on infinite intervals on time scales are established, which provides new bounds on unknown functions and can be used as a handy tool in the qualitative analysis of solutions of certain dynamic equations on time scales. MSC:26E70, 26D15, 26D10.

Gronwall-Bellman type integral inequalities for abstract Lebesgue integral

In this paper, we consider Gronwall-Bellman type integral inequalities and the corresponding integral equations for scalar functions of several variables involving abstract Lebesque integrals. Some delay and advance effects are also included. Since the inequalities and the equations are in a general form, the results may have different applications.

A unified approach to continuous and discrete gronwall-bellman inequalities

A general method is given whereby a large class of Gronwall-Bellman type inequalities can be reduced to the well known classical case. The method is applicable to both the continuous case and the discrete case.

Solvability of the electromagneticelasticity problem with memory

Questions are studied of the solvability of boundary value problems of electromagneticelasticity for media with memory. Theorems are proved on the existence and uniqueness of the solution of the indicated problems in the spaces C(0, T; W21(Ω)) and C(0, T; {329-01}). In proving the existence theorem, we use the property of connected fields, the compactness method, monotonicity and generalized Gronwall-Bellman inequalities.

Abstract comparison principles and multivariable Gronwall-Bellman inequalities

On demontre deux resultats de comparaison contenant des auto-applications croissantes d'un espace uniforme metrisable