## 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

### Keywords:

singular perturbation; time delay; exponential stability
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### References:

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