## On exponential stability with respect to some of the variables.(English. Russian original)Zbl 0814.34035

Russ. Acad. Sci., Dokl., Math 48, No. 1, 17-20 (1994); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 331, No. 1, 17-19 (1993).
Using results and notation from [V. V. Rumyantsev and A. S. Oziraner, Stability and stabilization of motion with respect to a part of the variables. Moskva (1987; Zbl 0626.70021)] the author gives definitions for (local and global) exponential $$x_ 1$$-stability of the equilibrium state $$x= 0$$ of a system $$dx/dt= f(t, x)$$ with $$f(t,0)\equiv 0$$, where $$x\in \mathbb{R}^ n$$ splits into two subvectors $$x_ i\in \mathbb{R}^{n_ i}$$, $$i= 1,2$$, $$n_ 1+ n_ 2= n$$. In two theorems he presents sufficient conditions for these types of exponential stability with respect to a part of the variables in terms of a Lyapunov function $$V(t, x)$$ which is locally Lipschitz continuous in $$x$$ and satisfies the condition $$V(t, x_ 1, x_ 2)= 0$$ if $$x_ 1= 0$$ and $$x_ 2\in D$$ where $$D= \{x_ 2\in \mathbb{R}^{n_ 2}: 0< \| x_ 2\|< \infty\}$$.
Reviewer: W.Müller (Berlin)

### MSC:

 34D20 Stability of solutions to ordinary differential equations 70K20 Stability for nonlinear problems in mechanics 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $$L^p, l^p$$, etc.) in control theory

Zbl 0626.70021