Several Inequalities of Gronwall and Their Proofs

It is well known that integral inequalities play a very important role in studying the properties of solutions to ordinary differential equations and integral equations. In 1919, Gronwall established a class of basic integral inequalities when he studied the dependence of differential equations on parameters, which is called Gronwall's inequalities. Gronwall's inequality play a very important role in ordinary differential equations, and it is also an important tool to study the properties of differential equations and integral equation solutions. There are several proofs of Gronwall's inequality, in particular, Agarwal, Deng and Zhang studied the Gronwall-Bellman inequality with multiple nonlinear terms, which made the adaptability of the Gronwall-Bellman inequality widely. Gronwall's inequality has various generalization forms and different proving methods, which is also a good tool for solving many mathematical problems. Different kinds of Gronwall's inequalities and their proofs are discussed in this paper. By researching the induction of Gronwall's inequality forms and their proofs, this paper aims to solve the problems of inequality as much as possible

Sampled-data synchronization control of dynamical networks with stochastic sampling

Copyright @ 2012 IEEEThis technical note is concerned with the sampled-data synchronization control problem for a class of dynamical networks. The sampling period considered here is assumed to be time-varying that switches between two different values in a random way with given probability. The addressed synchronization control problem is first formulated as an exponentially mean-square stabilization problem for a new class of dynamical networks that involve both the multiple probabilistic interval delays (MPIDs) and the sector-bounded nonlinearities (SBNs). Then, a novel Lyapunov functional is constructed to obtain sufficient conditions under which the dynamical network is exponentially mean-square stable. Both Gronwall's inequality and Jenson integral inequality are utilized to substantially simplify the derivation of the main results. Subsequently, a set of sampled-data synchronization controllers is designed in terms of the solution to certain matrix inequalities that can be solved effectively by using available software. Finally, a numerical simulation example is employed to show the effectiveness of the proposed sampled-data synchronization control scheme.This work was supported in part by the Engineering and Physical Sciences Research Council (EPSRC) of the UK under Grant GR/S27658/01, the Royal Society of the UK, the National Natural Science Foundation of China under Grants 61028008, 60974030, 61134009 and 61104125, the National 973 Program of China under Grant 2009CB320600, and the Alexander von Humboldt Foundation of Germany