an:01001223
Zbl 0863.34013
Nikitin, S.
Decoupling normalizing transformations and local stabilization of nonlinear systems
EN
Math. Bohem. 121, No. 2, 225-248 (1996).
00040419
1996
j
34A34 34D05 34D35 93C10
nonlinear system; stabilization; center manifold; normalizing transformation; smooth feedback
Systems of the form
\[
\dfrac{\text{d}}{\text{d}t}x = Ax +\Phi(x,y),
\]
\[
\dfrac{\text{d}}{\text{d}t}y = By +\Psi(x,y)
\]
are investigated where \(x\in\mathbb{R}^m,y\in\mathbb{R}^n\), \(A\in L(\mathbb{R}^m, \mathbb{R}^m)\) is a linear operator on \(\mathbb{R}^n\) with \(A=-A^T\), the eigenvalues of \(B\in L(\mathbb{R}^n, \mathbb{R}^n)\) have negative parts, \(\Phi, \Psi\) are at least \(C^3\) vanishing together with their derivatives at the origin. It is shown that there is a normalizing transform completely decoupling the stable and center manifold dynamics of the system into two independent systems of the form
\[
\dfrac{\text{d}}{\text{d}t}\widetilde x = A\widetilde x +\widetilde\Phi(\widetilde x,h(\widetilde x)),
\]
\[
\dfrac{\text{d}}{\text{d}t}\widetilde y = B\widetilde y +\widetilde\Psi(\widetilde x,\widetilde y).
\]
Some conditions for the local stabilization of the system are presented.
??.Schwabik (Praha)