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Lipschitz stability of nonlinear systems of differential equations. (English) Zbl 0595.34054
The authors define a new notion of stability - uniform Lipschitz stability (ULS) for nonlinear systems of differential equations (1) $\dot x=f(t,x)$, where $f\in C[J\times R\sp n,R\sp n]$ and $f\sb x=\partial f/\partial x$ exists and is continuous on $J\times R\sp n$, $J=[t\sb 0,\infty)$, $t\sb 0\ge 0$, $f(t,0)=0$, and $x(t,t\sb 0,x\sb 0)$ is the solution of (1) with $x(t\sb 0,t\sb 0,x\sb 0)=x\sb 0$. Definition. The zero solution of (1) is said to be (ULS) if there exists $M>0$ and $\delta >0$ such that $\vert x(t,t\sb 0,x\sb 0)\vert \le M\vert x\sb 0\vert,$ whenever $\vert x\sb 0\vert \le \delta$ and $t\ge t\sb 0\ge 0$. Several criteria for the (ULS) are obtained. It is shown by examples that the (ULS) coincides with the uniform stability in the linear case, which means that the (ULS) is a nonlinear phenomenon. The relationship between (ULS) and various types of stability notions is illustrated by a diagram.

34D10Stability perturbations of ODE
Full Text: DOI
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