Gronwall-Bellman type integral inequalities and applications to global uniform asymptotic stability

Abstract

In this paper, some new nonlinear generalized Gronwall-Bellman-Type integral inequalities are established. These inequalities can be used as handy tools to research stability problems of perturbed dynamic systems. As applications, based on these new established inequalities, some new results of practical uniform stability are also given. A numerical example is presented to illustrate the validity of the main results. RESUMEN En este artículo, establecemos algunas desigualdades integrales nolineales nuevas de tipo Gronwall-Bellman. Estas desigualdades pueden ser usadas como herramientas utiles para estudiar problemas de estabilidad de sistemas dinámicos perturbados. Como aplicaciones, basados en las nuevas desigualdades establecidas, también damos algunos resultados nuevos de estabilidad uniforme prácticos. Un ejemplo numérico es presentado para ilustrar la validez de los resultados principales.

h-stability and boundedness results for solutions to certain nonlinear perturbed systems

“…Lemma 2.16. (Gronwall-Bellman type integral inequality, see [9]) Let ϕ, λ and be non-negative continuous functions on R + for which the following inequality holds.…”

“…In mathematics, this kind of inequalities allows one to bound a function that is known to satisfy a certain differential or integral inequality by the solution to the corresponding differential or integral equation. This type of inequalities can be used in the study of existence, uniqueness and stability properties of solutions 10 H. DAMAK, M. A. HAMMAMI and A. KICHA to differential perturbed equations (see [8,9]). This work focuses on the study of the global uniform h-stability and the global uniform boundedness of time-varying perturbed differential equations under conditions on the perturbed terms and we will see how we can use the Gronwall's inequalities in this research.…”

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

“…30 Another approach to study the stability of perturbed systems consists of using the behavior of the solutions of the associated unperturbed system in the vicinity of the equilibrium point. [31][32][33][34][35][36][37] Being formulated in terms of integral inequalities of Gronwall type, it is a type of stability which is easy to verify in practice, and it extends the class of systems for which the effect of perturbations can be measured. In this article, we make use of such an approach in order to derive some conditions that ensures the (global) uniform exponential stability of perturbed systems.…”

A New Gronwall-Bellman Type Integral Inequality and Its Application to Fractional Stochastic Differential Equation

“…Among others Gronwall-Bellman integral inequality plays a significant role to discuss the boundedness, global existence, uniqueness, stability, and continuous dependence of solutions to some certain differential equations, fractional differential equations, stochastic differential equations. Such inequalities have gained much attention of many researchers [12,6,3,1,8,7,10,2,9,11,5]. Recent paper is a motivation of an idea given by Q-X Kong et al [4].…”