## Differential and Integral Equations

### Exponential asymptotic stability in linear delay-differential equations with variable coefficients

#### Abstract

In this paper we give some new necessary and sufficient conditions under which the zero solution of the linear delay-differential equations with variable coefficients $$x'(t)=A(t)x(t-\tau) \tag L$$ is exponentially asymptotically stable. For example, in the case $$A(t) = -\rho(t)\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix},$$ where $\rho(t) > 0,$ $\lim_{t \to \infty} \int_{t-\tau}^t \rho(s)\,ds = q>0$ and $|\theta| < \frac{\pi}{2},$ the zero solution of (L) is exponentially asymptotically stable if and only if $q <\frac{\pi}{2}-|\theta|$.

#### Article information

Source
Differential Integral Equations, Volume 11, Number 2 (1998), 263-278.

Dates
First available in Project Euclid: 30 April 2013

https://projecteuclid.org/euclid.die/1367341070

Mathematical Reviews number (MathSciNet)
MR1741845

Zentralblatt MATH identifier
1017.34076

Subjects
Primary: 34K20: Stability theory

#### Citation

Hara, Tadayuki; Miyazaki, Rinko; Matsumi, Yuko. Exponential asymptotic stability in linear delay-differential equations with variable coefficients. Differential Integral Equations 11 (1998), no. 2, 263--278. https://projecteuclid.org/euclid.die/1367341070