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Stability of perturbed delay homogeneous systems depending on a parameter. (English) Zbl 1499.34364

The authors consider a nonlinear system with delay \[\dot x=f(x(t),x(t-\alpha(t)),\theta),\,\,\, t\ge 0,\] where \(f(0,0,\theta)=0\), \(f\) is a continuous homogeneous function locally Lipschitz with respect to the first two arguments, \(x\in R^n\) is the state vector, \(\alpha:[t_0,\infty)\to[0,\infty)\) is a continuous bounded function, \(\theta\in\Omega\) is a parameter and \(\Omega\subset R^d\) is a compact set. Uniform stability of trivial solution is studied by Lyapunov-Krasovskii approach. More general systems \[\dot x=f(x(t),x(t-\alpha(t)),\theta)+g(x(t),x(t-\alpha(t)),\theta)\] where \(f\) and \(g\) are homogeneous functions of different degrees and \[\dot x=f(x(t),x(t-\alpha(t)),\theta)+R(x(t),x(t-\alpha(t)),\theta)+P(x(t),\theta)\] under different assumptions on \(R\) and \(P\) are considered as well.

MSC:

34K20 Stability theory of functional-differential equations
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