On Stability and Stabilization of Perturbed Time Scale Systems with Gronwall Inequalities

Abstract

In the paper, new nonlinear time scale integral inequalities are given. By means of the explicit integral bounds, we derive sufficient conditions for the uniform asymptotic stability of perturbed systems on time scales. In the sequel, basing on Lyapunov's direct method, we develop the required type of stability.

Some refinements of certain Gamidov integral inequalities on time scales and applications

“…During the past few years, many such new inequalities have been discovered, which are motivated by certain applications. For example, see [2][3][4][7][8][9][10][11][12][13][14][15][16][17][18][19][20] and the references therein. In [14], Sh.…”

Stability of abstract dynamic equations on time scales by Lyapunov's second method

“…See [5,19]. In the Hilbert space setting, the Lyapunov function is chosen to be V (t, x) =< P (t)x, x >, where P (t) is a bounded linear operator on a Hilbert space and < ., .…”

“…Cui [7] gave generalizations for the boundedness theorems on R n . Nasser et al [19] established some sufficient conditions for the existence of the quadratic Lyapunov function that ensure the desired asymptotic convergence of trajectories. The difficulty of the Lyapunov technique is to construct a Lyapunov function.…”

“…The Lyapunov function is usually chosen to be V (t, x) = x T P (t)x, where P (t) is an n × n matrix and x T is the transpose of x ∈ R n . See [5,19]. In the Hilbert space setting, the Lyapunov function is chosen to be V (t, x) =< P (t)x, x >, where P (t) is a bounded linear operator on a Hilbert space and < ., .…”

Further nonlinear integral inequalities in two independent variables on time scales and their applications

“…During the few years, a lot of research related to studies and the extension of some fundamental integral inequalities used in the theory of differential and integral equations on time scales. For example, we refer the reader to the papers [1][2][3][4][5][8][9][10][11][12][13][14][15][16][17][18][19]. The purpose of this note is to illustrate some time scale Pachpattetype inequalities by extending some continuous inequalities given in [15].…”