Liu, Xinzhi; Shen, Xuemin; Zhang, Yi Exponential stability of singularly perturbed systems with time delay. (English) Zbl 1044.34031 Appl. Anal. 82, No. 2, 117-130 (2003). 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. Reviewer: Tamaz Tadumadze (Tbilisi) Cited in 16 Documents MSC: 34K20 Stability theory of functional-differential equations 34K26 Singular perturbations of functional-differential equations Keywords:singular perturbation; time delay; exponential stability PDFBibTeX XMLCite \textit{X. Liu} et al., Appl. Anal. 82, No. 2, 117--130 (2003; Zbl 1044.34031) 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 · doi:10.1137/0310031 [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 · doi:10.1137/1026104 [6] Konyaev Y.A., Mat. Zametki 62 pp 494– (1997) [7] DOI: 10.1093/imamat/60.1.91 · Zbl 0902.34050 · doi:10.1093/imamat/60.1.91 [8] Shen X., Contr. Theory and Advan. Tech. 9 pp 759– (1993) [9] DOI: 10.1109/9.59817 · Zbl 0721.93059 · doi:10.1109/9.59817 [10] DOI: 10.1049/el:19960644 · 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 · doi:10.1016/S0005-1098(01)00265-5 [13] DOI: 10.1006/jmaa.2000.6955 · Zbl 0965.93049 · doi:10.1006/jmaa.2000.6955 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.