Extension of a stochastic Gronwall lemma

We prove a stochastic Gronwall lemma of the convolution type. Our results extend that of Scheutzow [A stochastic Gronwall lemma, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 16 (2013) 1350019], and the related results established in the non-convolution case. The proofs of the present results are essentially based on the Métivier–Pellaumail inequality for semimartingales.

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On a discrete fractional stochastic Grönwall inequality and its application in the numerical analysis of stochastic FDEs involving a martingale

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2021-0100 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Ahmed S. Hendy ◽

Mahmoud A. Zaky ◽

Eid H. Doha

Keyword(s):

Numerical Analysis ◽

A Priori ◽

Gronwall Inequality ◽

Gronwall Lemma ◽

Discrete Form ◽

Left Hand ◽

Grönwall Inequality ◽

Difference Formula ◽

The Right ◽

The Given

Abstract The aim of this paper is to derive a novel discrete form of stochastic fractional Grönwall lemma involving a martingale. The proof of the derived inequality is accomplished by a corresponding no randomness form of the discrete fractional Grönwall inequality and an upper bound for discrete-time martingales representing the supremum in terms of the infimum. The release of a martingale term on the right-hand side of the given inequality and the graded L1 difference formula for the time Caputo fractional derivative of order 0 < α < 1 on the left-hand side are the main challenges of the stated and proved main theorem. As an example of application, the constructed theorem is used to derive an a priori estimate for a discrete stochastic fractional model at the end of the paper.

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A near-optimal decentralized control approach for polynomial nonlinear interconnected systems

Transactions of the Institute of Measurement and Control ◽

10.1177/0142331220977433 ◽

2020 ◽

pp. 014233122097743

Author(s):

Mohamed Sadok Attia ◽

Mohamed Karim Bouafoura ◽

Naceur Benhadj Braiek

Keyword(s):

Optimal Control ◽

Closed Loop ◽

Sufficient Conditions ◽

Open Loop ◽

System Stability ◽

Gronwall Lemma ◽

Interconnected System ◽

Control Approach ◽

State Dependent ◽

And Control

This article tackles the decentralized near-optimal control problem for the class of nonlinear polynomial interconnected system based on a shifted Legendre polynomials direct approach. The proposed method converts the interconnected optimal control problems into a nonlinear programming one with multiple constraints. In light of the formulated NLP optimization, state and control coefficients are used to design a nonlinear decentralized state feedback controller. Overall closed-loop system stability sufficient conditions are investigated with the help of Grönwall lemma. The triple inverted pendulum case is considered for simulation. Satisfactory results are obtained in both open-loop and closed-loop schemes with comparison to collocation and state-dependent Riccati equation techniques.

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A STOCHASTIC GRONWALL LEMMA

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025713500197 ◽

2013 ◽

Vol 16 (02) ◽

pp. 1350019 ◽

Cited By ~ 12

Author(s):

MICHAEL SCHEUTZOW

Keyword(s):

Simple Proof ◽

Integral Inequality ◽

Continuous Process ◽

Main Tool ◽

Local Martingale ◽

Gronwall Lemma ◽

Martingale Inequality ◽

Optimal Constant ◽

Linear Integral ◽

The Right

We prove a stochastic Gronwall lemma of the following type: if Z is an adapted non-negative continuous process which satisfies a linear integral inequality with an added continuous local martingale M and a process H on the right-hand side, then for any p ∈ (0, 1) the pth moment of the supremum of Z is bounded by a constant κp (which does not depend on M) times the pth moment of the supremum of H. Our main tool is a martingale inequality which is due to D. Burkholder. We provide an alternative simple proof of the martingale inequality which provides an explicit numerical value for the constant cp appearing in the inequality which is at most four times as large as the optimal constant.

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On the Hyers-Ulam Stability of First-Order Impulsive Delay Differential Equations

Journal of Function Spaces ◽

10.1155/2016/8164978 ◽

2016 ◽

Vol 2016 ◽

pp. 1-6 ◽

Cited By ~ 11

Author(s):

Akbar Zada ◽

Shah Faisal ◽

Yongjin Li

Keyword(s):

Continuous Functions ◽

Gronwall Lemma ◽

Ulam Stability ◽

Order Ordinary Differential Equation ◽

First Order ◽

Impulsive Delay Differential Equations ◽

Impulsive Delay ◽

Delay Differential ◽

Rassias Stability ◽

Single Constant

This paper proves the Hyers-Ulam stability and the Hyers-Ulam-Rassias stability of nonlinear first-order ordinary differential equation with single constant delay and finite impulses on a compact interval. Our approach uses abstract Gronwall lemma together with integral inequality of Gronwall type for piecewise continuous functions.

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