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 $f\in C[J\times {R}^{n},{R}^{n}]$ and ${f}_{x}=\partial f/\partial x$ exists and is continuous on $J\times {R}^{n}$, $J=[{t}_{0},\infty )$, ${t}_{0}\ge 0$, $f(t,0)=0$, and $x(t,{t}_{0},{x}_{0})$ is the solution of (1) with $x({t}_{0},{t}_{0},{x}_{0})={x}_{0}$. Definition. The zero solution of (1) is said to be (ULS) if there exists $M>0$ and $\delta >0$ such that $|x(t,{t}_{0},{x}_{0})|\le M|{x}_{0}|,$ whenever $|{x}_{0}|\le \delta $ and $t\ge {t}_{0}\ge 0$. 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.

##### MSC:

34D10 | Stability perturbations of ODE |