Gronwall Inequality Research Articles

Closed‐loop consensus control of partial difference multi‐agent systems via variable gain iterative learning strategy

AbstractIn the research on consensus control of multi‐agent systems (MASs), it is the key to ensure that the state quantities (speed, displacement, etc.) of all agents tend to be consensus. Under this premise, how to accelerate the convergence rate of the consensus error is also an important issue. Aiming at the accelerated consensus problem of MASs composed of partial difference equations, based on the network topology and the output form of each follower, a distributed closed‐loop accelerated iterative learning control (ILC) protocol with variable gain was proposed, which was designed to improve the speed of consensus error. With the help of basic mathematical tools such as discrete Gronwall inequality and contraction mapping method, the sufficient conditions for the convergence of the consensus error are derived and analyzed. Finally, numerical simulations show that the proposed acceleration control protocol is more effective than the traditional open‐loop and closed‐loop ILC protocols.

Hyers–Ulam stability for a class of Hadamard fractional Itô–Doob stochastic integral equations

Our goal in this work is to demonstrate the existence and uniqueness of the solution to a class of Hadamard Fractional Itô–Doob Stochastic integral equations (HFIDSIE) of order φ∈(0,1) via the fixed point technique (FPT). Hyers–Ulam stability (HUS) is investigated for HFIDSIE according to the Gronwall inequality. Two theoretical examples are provided to illustrate our results.

A variety of Gronwall Inequalities of Fractional Variable Order

A variety of Gronwall Inequalities of Fractional Variable Order

Finite-time annular domain stability and stabilisation of linear positive systems

This paper is concerned with the finite-time annular domain stability and stabilisation for the positive systems. A new analysis method is proposed to obtain less conservative finite-time annular domain stability criteria and its superiority to modified Gronwall inequality is analysed. Moreover, conditions for the existence of state feedback and observer-based controllers that guarantee the closed-loop system to be positive and finite-time annular domain stable are given under the linear programming framework. Finally, two numerical examples are provided to show the effectiveness and superiority of the theoretical results.

Quantized control for finite-time synchronization of delayed fractional-order memristive neural networks: The Gronwall inequality approach

This paper is concerned with the finite-time synchronization (FTS) of delayed fractional-order memristive neural networks (FMNNs) by using quantized control. Firstly, two kinds of quantized controllers are designed to eliminate the residual synchronization errors that appear owing to the characteristics of memristive connection weights. Then, to derive α-dependent FTS criteria for drive–response FMNNs, the Gronwall inequality technique is used by combination with the direct estimation method. Compared with the existing results, the advantages of the derived synchronization criteria are discussed from the aspects of both theory and simulation. In addition, the relationships between the finite-time parameter and the quantization parameters are displayed. It reveals that there is a compromise between the admissible finite-time and the saved network bandwidth. Finally, two examples are given to illustrate the effectiveness of the obtained results.

Analytic resolvent semilinear integro‐differential systems: Existence and optimal control

The primary contributions of this article are the existence of mild solutions and optimal control for analytic resolvent semilinear integro‐differential systems. Using Lipschitz nonlinearity and the Banach fixed point theorem, we demonstrate results for mild solutions. Using the ‐semigroup, minimizing sequence, Gronwall's inequality, and fundamentals of functional operator theory, we obtain the results for optimal control.

Iterative learning control for conformable stochastic impulsive differential systems with randomly varying trial lengths

Abstract In this paper, we introduce a new kind of conformable stochastic impulsive differential systems (CSIDS) involving discrete distribution of Bernoulli. For random discontinuous trajectories, we modify the tracking error of piecewise continuous variables by a zero-order holder. First, the improved P-type and PD α -type learning laws of the random iterative learning control (ILC) scheme are designed through global and local averaging operators. Next, we establish sufficient conditions for convergence of the tracking error in the expectation sense and prove the main results by using the impulsive Gronwall inequality and mathematical analysis tools. Finally, the theoretical results are verified by two numerical examples, and the tracking performance is compared for different conformable order of α.

Properties of Certain Volterra type ABC Fractional Integral Equations

In this paper we study existence, uniqueness and other properties of solutions of Volterra type ABC fractional integral equations. We have used Banach fixed point theorem with Bielecki type norm and Gronwall inequality in the frame of ABC fractional integral for proving our results.

Finite-time guaranteed cost control for uncertain delayed switched nonlinear stochastic systems

This paper studies the finite-time guaranteed cost control problem for switched nonlinear stochastic systems with parameter uncertainties and time-varying delays. By choosing a model-dependent and delay-dependent Lyapunov-Krasovskii functional, applying the average dwell time approach and the Gronwall inequality, some novel sufficient conditions are derived to ensure that the switched nonlinear stochastic closed-loop system is finite-time stochastically stable and an upper bound is given on the performance index. The obtained nonlinear matrix is transformed into a linear matrix form, and then the feedback controller gains of the switched nonlinear stochastic systems with time-varying delay are obtained. Finally, two simulation examples are designed to verify the effectiveness of the suggested approach.

Ulam–Hyers stability of pantograph fractional stochastic differential equations

In this paper, we investigate the existence and uniqueness theorem (EUT) of Pantograph fractional stochastic differential equations (PFSDE) using the Banach fixed point theorem (BFPT). We show the Ulam–Hyers stability (UHS) of PFSDE by the generalized Gronwall inequalities (GGI). We illustrate our results by two examples.

On Some Dynamic (ΔΔ)∇- Gronwall–Bellman–Pachpatte-Type Inequalities on Time Scales and Its Applications

In the present paper, some new generalizations of dynamic inequalities of Gronwall–Bellman–Pachpatte-type on time scales are established. Some integral and discrete Gronwall–Bellman–Pachpatte-type inequalities that are given as special cases of main results are original. The main results are proved by using the dynamic Leibniz integral rule on time scales. To highlight our research advantages, several implementations of these findings are presented. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities.

Convergence analysis of a fast second‐order time‐stepping numerical method for two‐dimensional nonlinear time–space fractional Schrödinger equation

AbstractIn this article, we consider the two‐dimensional nonlinear time–space fractional Schrödinger equation with space described by the fractional Laplacian. A second‐order fractional backward difference formula in the temporal direction while Fourier spectral method in the spatial direction is proposed to solve the model numerically. In the numerical implementation, a fast method is applied based on a globally uniform approximation of the trapezoidal rule for the integral on the real line to decrease the memory requirement and computational cost. By using the generalized discrete Gronwall inequality developed by Dixon and McKee and the temporal–spatial error splitting argument, the convergence of the fast time‐stepping numerical method is also proved in a simple manner without imposing the Courant‐Friedrichs‐Lewy (CFL) condition. Finally, some numerical results are provided to support the theoretical analysis.

Stability Analysis of Stochastic Differential Equation Driven by G-Brownian Motion under Non-Lipschitz Condition

This paper is devoted to studying the p-th moment exponential stability for a class of stochastic differential equation (SDE) driven by G-Brownian motion under non-Lipschitz condition. The delays considered in this paper are time-varying delays τ i t 1 ≤ i ≤ 3 . Since the coefficients are non-Lipschitz, the normal enlargement on the coefficients is not available and the Gronwall inequality is not suitable in this case. By Bihari inequality and Itô integral formula, it is pointed out that there exists a constant τ ∗ such that the p-th moment exponential stability holds if the time-varying delays are smaller than τ ∗ .

Caputo-Hadamard fractional differential equations on time scales: Numerical scheme, asymptotic stability, and chaos.

This study investigates Caputo-Hadamard fractional differential equations on time scales. The Hadamard fractional sum and difference are defined for the first time. A general logarithm function on time scales is used as a kernel function. New fractional difference equations and their equivalent fractional sum equations are presented by the use of fundamental theorems. Gronwall inequality, asymptotical stability conditions, and two discrete-time Mittag-Leffler functions of Hadamard type are obtained. Numerical schemes are provided and chaos in fractional discrete-time logistic equation and neural network equations are reported.

Linkage-constraint criteria for robust exponential stability of nonlinear BAM system with derivative contraction coefficients and piecewise constant arguments

This paper presents linkage-constraint criteria to guarantee the robustness of global exponential stability (RoGES) for a class of generalized nonlinear bidirectional associative memory (BAM) systems in the presence of derivative contraction coefficients and piecewise constant arguments (NBAM-dp system). However, because of the appearance of the tricky derivative compression coefficients and coupling relation with piecewise constant arguments, the previous Gronwall inequality is not applicable here. Thus, by means of generalized Gronwall inequality and norm inequality techniques, an independent parameters and interdependent variables (IPIV) principle is designed to solve the implicit bivariate transcendental equations to provide the respective supremum of the two interference factors, and the major criteria for guaranteeing the RoGES for NBAM-dp systems are obtained accordingly. Meanwhile, according to the IPIV principle, the relationship between the above two interference factors reflected in the linkage-constraint criteria is ensured to be mutually constrained and dynamically interconnected. Finally, comparative simulations and auxiliary data are provided to demonstrate the effectiveness of the theoretical findings.

Iterative learning control for time-delay nonlinear hyperbolic distributed parameter systems

In this paper, we study the application of iterative learning control to a class of nonlinear varying time-delay equations. The handled procedure is the distributed parameter system, hyperbolic type. The selected plan is based on PD-type iterative learning control with initial state learning. Initially, we present the equation and the control law utilised. Subsequently, we presented some dilemmas. Then, sufficient conditions for monotone convergence of the error under the proper assumption are found. Also, a detailed convergence analysis is given based on the once-offered lemmas and Gronwall inequality. Finally, we show the usefulness of the method utilising a numerical model. Abbreviations DPS: distributed parameter system; ILC: iterative learning control; PDE:partial differential equation; DPS: partial difference system; PD: proportional derivative

Impulsive effects on stabilization of stochastic nonlinear reaction-diffusion systems with time delays and boundary feedback control

In this paper, we investigate the stabilization of stochastic nonlinear impulsive reaction-diffusion systems (SNIRDSs) with time delays and boundary feedback control via average impulsive interval approach. Boundary feedback control strategy are designed to stabilization of SNIRDSs. By constructing a Lyapunov-Krasovskii functional (LKF), and using Wirtinger's inequality, Gronwall inequality, average impulsive interval approach, sufficient conditions are derived to guarantee the finite-time stability (FTS) of proposed systems. We investigate the stabilization results by designing the control gain matrices for boundary feedback controller. The criterions are expressed in terms of linear matrix inequalities (LMIs) that can be verified by Matlab LMI toolbox. Finally, numerical example are given to verify the efficiency and superiority of proposed stabilization criterions.

The Existence of Weak Solutions for the Vorticity Equation Related to the Stratosphere in a Rotating Spherical Coordinate System

In this paper, based on the Euler equation and mass conservation equation in spherical coordinates, the ratio of the stratospheric average width to the planetary radius and the ratio of the vertical velocity to the horizontal velocity are selected as parameters under appropriate boundary conditions. We establish the approximate system using these two small parameters. In addition, we consider the time dependence of the system and establish the governing equations describing the atmospheric flow. By introducing a flow function to code the system, a nonlinear vorticity equation describing the planetary flow in the stratosphere is obtained. The governing equations describing the atmospheric flow are transformed into a second-order homogeneous linear ordinary differential equation and a Legendre’s differential equation by applying the method of separating variables based on the concepts of spherical harmonic functions and weak solutions. The Gronwall inequality and the Cauchy–Schwartz inequality are applied to priori estimates for the vorticity equation describing the stratospheric planetary flow under the appropriate initial and boundary conditions. The existence and non-uniqueness of weak solutions to the vorticity equation are obtained by using the functional analysis technique.

Error estimate of the fast L1 method for time-fractional subdiffusion equations

This paper analyzes the convergence of the fast L1 method for the semi-linear time-factional subdiffusion equations. The tool we use is the generalized discrete Gronwall inequality. We show that the difference between the solution of the L1 method and the solution of the fast L1 method can be arbitrarily small and is independent of the sizes of the time and/or space grids. Our proof is very simple and is helpful to be used to simplify the convergence analysis of the fast methods for time-fractional evolution equations.

A Mass-Preserving Two-Step Lagrange–Galerkin Scheme for Convection-Diffusion Problems

A mass-preserving two-step Lagrange–Galerkin scheme of second order in time for convection-diffusion problems is presented, and convergence with optimal error estimates is proved in the framework of L^2-theory. The introduced scheme maintains the advantages of the Lagrange–Galerkin method, i.e., CFL-free robustness for convection-dominated problems and a symmetric and positive coefficient matrix resulting from the discretization. In addition, the scheme conserves the mass on the discrete level if the involved integrals are computed exactly. Unconditional stability and error estimates of second order in time are proved by employing two new key lemmas on the truncation error of the material derivative in conservative form and on a discrete Gronwall inequality for multistep methods. The mass-preserving property is achieved by the Jacobian multiplication technique introduced by Rui and Tabata in 2010, and the accuracy of second order in time is obtained based on the idea of the multistep Galerkin method along characteristics originally introduced by Ewing and Russel in 1981. For the first time step, the mass-preserving scheme of first order in time by Rui and Tabata in 2010 is employed, which is efficient and does not cause any loss of convergence order in the ell ^infty (L^2)- and ell ^2(H^1_0)-norms. For the time increment Delta t, the mesh size h and a conforming finite element space of polynomial degree k in {mathbb {N}}, the convergence order is of O(Delta t^2 + h^k) in the ell ^infty (L^2)cap ell ^2(H^1_0)-norm and of O(Delta t^2 + h^{k+1}) in the ell ^infty (L^2)-norm if the duality argument can be employed. Error estimates of O(Delta t^{3/2}+h^k) in discrete versions of the L^infty (H^1_0)- and H^1(L^2)-norm are additionally proved. Numerical results confirm the theoretical convergence orders in one, two and three dimensions.