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

Martynyuk, A. A.

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

Keywords:

exponential stability with respect to a part of the variables; Lyapunov function

Citations:

Zbl 0626.70021