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Stabilization of nonlinear systems in compound critical cases. (English) Zbl 1039.93054

The aim of this paper is to study the stabilization of a critical nonlinear system \[ \begin{aligned}\dot\eta &= A_{11}\eta+ b_1u+ F(\eta,\xi),\tag{1}\\ \dot\xi &= A_{22}\xi+ b_2u+ G(\eta,\xi),\tag{2}\end{aligned} \] where the functions \(F\), \(G\) are sufficiently smooth with \(F(0,0)= 0\), \(DF(0,0)= 0\), \(G(0,0)= 0\), \(DG(0,0)= 0\).
The authors consider three degenerate cases in which \(A_{11}\) has exactly two zero eigenvalues with geometric multiplicity 1; one zero eigenvalue and a pair of purely imaginary eigenvalues; or two distinct pairs of purely imaginary eigenvalues.
The stability conditions for the low-dimensional critical system (1) with \(u= 0\), \(\xi= 0\) are derived by using the technique of normal form reduction and Lyapunov stability criteria. These stability conditions and the center manifold reduction technique are applied to study the stabilization of the system (1), (2).
A linear and/or nonlinear feedback stabilizing control law is proposed for linearly uncontrollable systems, while a purely nonlinear stabilizing control law is designed for linearly controllable systems.

MSC:

93D15 Stabilization of systems by feedback
93C15 Control/observation systems governed by ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
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