×

Exponential stability of singularly perturbed systems with time delay. (English) Zbl 1044.34031

For the linear system \[ \begin{alignedat}{2} & \dot{x}(t)=A_{11}(t)x(t)+A_{12}(t)x(t-\tau)+B_{12}(t)z(t) + B_{12}(t)z(t-\tau),&\quad&x(t)\in \mathbb R^n,\\ & \varepsilon\dot{z}(t)=A_{21}(t)+A_{22}(t)+x(t-\tau)+B_{21}(t)z(t),&\quad&z(t)\in \mathbb R^m, \end{alignedat} \] and for the nonlinear system \[ \begin{aligned} & \dot{x}(t)=A_{11}x(t)+g(x(t),x(t-\tau),z(t),z(t-\tau)),\\ &\varepsilon \dot{z}(t)=B_{21}z(t)+B(x(t),x(t-\tau)), \end{aligned} \] criteria for the exponential stability are derived.

MSC:

34K20 Stability theory of functional-differential equations
34K26 Singular perturbations of functional-differential equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Nevanlinna O., Numerical Solution of a Singularly Perturbed Nonlinear Volterra Equation (1978)
[2] DOI: 10.1137/0310031 · Zbl 0241.49006
[3] Goering H., Singularly Perturbed Differential Equation (1983) · Zbl 0522.35003
[4] Kokotovic P.V., Singular Perturbation Methods in Control: Analysis and Design (1986)
[5] DOI: 10.1137/1026104 · Zbl 0548.93001
[6] Konyaev Y.A., Mat. Zametki 62 pp 494– (1997)
[7] DOI: 10.1093/imamat/60.1.91 · Zbl 0902.34050
[8] Shen X., Contr. Theory and Advan. Tech. 9 pp 759– (1993)
[9] DOI: 10.1109/9.59817 · Zbl 0721.93059
[10] DOI: 10.1049/el:19960644
[11] Fridman E., DCDIS (Series B) 9 pp 201– (2002)
[12] DOI: 10.1016/S0005-1098(01)00265-5 · Zbl 1014.93025
[13] DOI: 10.1006/jmaa.2000.6955 · Zbl 0965.93049
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.