Abstract
In this paper, we study the finite time stability of delay differential equations via a delayed matrix cosine and sine of polynomial degrees. Firstly, we give two alternative formulas of the solutions for a delay linear differential equation. Secondly, we obtain a norm estimation of the delayed matrix sine and cosine of polynomial degrees, which are used to establish sufficient conditions to guarantee our finite time stability results. Meanwhile, a numerical example is presented demonstrating the validity of our theoretical results. Finally, we extend our study to the same issue of a delay differential equation with nonlinearity by virtue of the Gronwall inequality approach.
Highlights
1 Introduction Generally speaking, it is not an easy task to seek the fundamental matrix for linear differential delay systems due to the memory accumulated by the long-tail effects that the time-delay term introduces
There has been a rapid development on the representation of solutions, which lead to results on asymptotic stability, finite time stability and control problems for linear/nonlinear continuous delay systems and discrete delay systems or fractional order delay systems
The concept of finite time stability of delay differential equations arises from the fields of multibody mechanics, automatic engines and physiological systems as introduced by Dorato [ ], which characterizes the system state by not exceeding a certain bounded for a given finite time interval and this seems more appropriate from practical considerations
Summary
It is not an easy task to seek the fundamental matrix for linear differential delay systems due to the memory accumulated by the long-tail effects that the time-delay term introduces. We study the finite time stability of the following second order linear differential equations with a pure delay term: Delayed matrix cosine and sine of polynomial degrees play an important role in studying second order delay differential equations since they can act as the fundamental matrix to seeking some possible representation of solutions to the problem by using a variation of constants formula.
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