Impulsive stabilization of two kinds of second-order linear delay differential equations

Abstract

In this paper, we consider the impulsive stabilization problems for two kinds of 2nd-order linear delay differential equations. By the method of Lyapunov functionals, criteria on the exponential stabilization by impulses and exponential stabilization by periodical impulses are gained. The control effects of impulses are especially stressed in the definitions and results. Some examples are also given to show the applications of our results.

Impulsive stabilization of second-order delay differential equations

“…In the paper [3], the authors study the impulsive stabilization for the following second-order delay differential equations:…”

“…The problem of stabilizing the solutions by imposing proper impulsive control for second-order delay differential equations has attracted some authors' attentions and some results have been obtained. For example, see [1][2][3][4][5] and reference therein.…”

About sign-constancy of Green’s function of a two-point problem for impulsive second order delay equations

“…Oscillation/nonoscillation and stability of delay differential equations are considered in [1,5,6,8,9,25]. Delay impulsive differential equations of second order are considered concerning stabilization by impulses in [14,20]. For second order delay differential equations, we note the paper [24] where their nonoscillation is studied.…”

“…has only trivial solutions, then their solution can be represented in the form 20) where G(t, s) is called the Green's function of the corresponding problem. The form of Green's function G(t, s) of every problem can be obtained using the representation (1.11) of general solution of (1.1)-(1.3).…”

Advanced Impulsive Differential Equations With Piecewise Constant Arguments

“…Among these investigations stability and instability problems are very interesting. Impulses can make unstable systems stable and stable systems can become unstable after impulse effects [4,16,23]. To the best of our knowledge, there are only a few papers involving impulsive differential equations with piecewise constant arguments [15,26].…”