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Averaging method in the stability theory of functional-differential equations. (English. Russian original) Zbl 0907.34057

Differ. Equations 33, No. 4, 447-456 (1997); translation from Differ. Uravn. 33, No. 4, 448-457 (1997).
The author studies the stability of solutions to systems of functional-differential equations of retarded type \[ \dot x= F(t,x_t)= f(t,x(t))+ g(t, x_t), \] with \(x_t= x(t+ s)\), \(-h\leq s\leq 0\), the functional \(F: G_H\to \mathbb{R}^n\) is continuous and satisfies a Lipschitz condition with respect to \(x_t\) in the domain \(G_H= \mathbb{R}_+\times \Omega_H\), \(\Omega_H= \{\varphi\in C([-h,0],\mathbb{R}^n): \max_s \varphi(s)< H\}\). To this end, the author suggests a new approach based on the idea of estimating the behavior of the Lyapunov function by means of averaging its total derivative along a solution to a simplified system approximating the original one. The main theorem proved gives sufficient conditions for the uniform asymptotic stability of the trivial solution to the considered system. The obtained results are illustrated by typical examples. The references, sixteen in number, fully cover the topic.

MSC:

34K20 Stability theory of functional-differential equations
93D20 Asymptotic stability in control theory
34C29 Averaging method for ordinary differential equations
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