On the exponential multistability of separating motions. (English. Russian original) Zbl 0839.34059

Martynyuk, A. A.

Russ. Acad. Sci., Dokl., Math. 49, No. 3, 528-531 (1994); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 336, No. 4, 446-447 (1994).

The author considers nonlinear systems (1) \(dx/dt = f(t,x)\), \(x \in \mathbb{R}^n\), which admit a decomposition into two subsystems \(dx_i/dt = g_i (t,x_i) + h_i (t,x)\), \(i = 1,2\), in such a way that the unperturbed subsystems \(dx_i/dt = g_i (t,x_i)\) are independent, i.e., \(x = (x_1, x_2)\), \(x_i \in \mathbb{R}^{n_i}\), \(\mathbb{R}^n = \mathbb{R}^{n_1} \oplus \mathbb{R}^{n_2}\). He defines the notions of local and global exponential multistability (which allow different decay rates for the components \(x_i)\) and gives sufficient conditions for the solutions of (1) to be locally or globally exponentially multistable.

Reviewer: W.Müller (Berlin)

MSC:

34D20	Stability of solutions to ordinary differential equations
93D05	Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory

Keywords:

nonlinear systems; decomposition; exponential multistability; decay rates