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The exponential asymptotic stability of singularly perturbed delay differential equations with a bounded lag. (English) Zbl 1014.34062
The author considers the singularly perturbed delay differential system: \[ \varepsilon y'(t,\varepsilon) = f(t,y(t,\varepsilon),y(t-\tau(t),\varepsilon)), \quad t\geq t_0,\qquad y(t,\varepsilon)=\varphi(t), \quad t\leq t_0, \] with \(f:[0,+\infty)\times \mathbb{C}^s\times \mathbb{C}^s \to \mathbb{C}^s\) and \(y(t,\varepsilon):\mathbb{R}^+ \times \mathbb{R}^+ \to \mathbb{C}^s\).
Sufficient conditions are provided to ensure that any solution to this system with a bounded lag is exponentially stable uniformly for sufficiently small \(\varepsilon>0\). As a preliminary result, a generalized Halanay inequality is derived.

34K20 Stability theory of functional-differential equations
34K26 Singular perturbations of functional-differential equations
34K12 Growth, boundedness, comparison of solutions to functional-differential equations
Full Text: DOI
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