Lipschitz stability of nonlinear systems of differential equations (Q1077612)

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 \(J\times R^ n\), \(J=[t_ 0,\infty)\), \(t_ 0\geq 0\), \(f(t,0)=0\), 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 \(| x(t,t_ 0,x_ 0)| \leq M| x_ 0|,\) whenever \(| x_ 0| \leq \delta\) and \(t\geq t_ 0\geq 0\). 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.