## Differential and Integral Equations

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

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

Tadayuki Hara, Yuko Matsumi, and Rinko Miyazaki

#### 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

**Permanent link to this document**

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