### Abstract

In this paper we study exponential stability of the trivial solution of the state-dependent delay system ẋ(t) = ∑_{i=1}^{m} A _{i}(t)x(t - τ_{i}(t, x_{t})). We show that under mild assumptions, the trivial solution of the state-dependent system is exponentially stable if and only if the trivial solution of the corresponding linear time-dependent delay system ẏ(t) = ∑_{i=1}^{m} A_{i}(t)y(t - τ_{i}(t, 0)) is exponentially stable. We also compare the order of the exponential stability of the nonlinear equation to that of its linearized equation. We show that in some cases, the two orders are equal. As an application of our main result, we formulate a necessary and sufficient condition for the exponential stability of the trivial solution of a threshold-type delay system.

Original language | English |
---|---|

Pages (from-to) | 773-791 |

Number of pages | 19 |

Journal | Discrete and Continuous Dynamical Systems |

Volume | 18 |

Issue number | 4 |

Publication status | Published - Aug 2007 |

### Fingerprint

### Keywords

- Exponential stability
- Order of exponential stability
- State-dependent delay
- Threshold delay

### ASJC Scopus subject areas

- Mathematics(all)
- Discrete Mathematics and Combinatorics
- Applied Mathematics
- Analysis

### Cite this

**Exponential stability of a state-dependent delay system.** / Győri, I.; Hartung, F.

Research output: Contribution to journal › Article

*Discrete and Continuous Dynamical Systems*, vol. 18, no. 4, pp. 773-791.

}

TY - JOUR

T1 - Exponential stability of a state-dependent delay system

AU - Győri, I.

AU - Hartung, F.

PY - 2007/8

Y1 - 2007/8

N2 - In this paper we study exponential stability of the trivial solution of the state-dependent delay system ẋ(t) = ∑i=1m A i(t)x(t - τi(t, xt)). We show that under mild assumptions, the trivial solution of the state-dependent system is exponentially stable if and only if the trivial solution of the corresponding linear time-dependent delay system ẏ(t) = ∑i=1m Ai(t)y(t - τi(t, 0)) is exponentially stable. We also compare the order of the exponential stability of the nonlinear equation to that of its linearized equation. We show that in some cases, the two orders are equal. As an application of our main result, we formulate a necessary and sufficient condition for the exponential stability of the trivial solution of a threshold-type delay system.

AB - In this paper we study exponential stability of the trivial solution of the state-dependent delay system ẋ(t) = ∑i=1m A i(t)x(t - τi(t, xt)). We show that under mild assumptions, the trivial solution of the state-dependent system is exponentially stable if and only if the trivial solution of the corresponding linear time-dependent delay system ẏ(t) = ∑i=1m Ai(t)y(t - τi(t, 0)) is exponentially stable. We also compare the order of the exponential stability of the nonlinear equation to that of its linearized equation. We show that in some cases, the two orders are equal. As an application of our main result, we formulate a necessary and sufficient condition for the exponential stability of the trivial solution of a threshold-type delay system.

KW - Exponential stability

KW - Order of exponential stability

KW - State-dependent delay

KW - Threshold delay

UR - http://www.scopus.com/inward/record.url?scp=35248831635&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=35248831635&partnerID=8YFLogxK

M3 - Article

VL - 18

SP - 773

EP - 791

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

SN - 1078-0947

IS - 4

ER -