Stabilization of a class of nonlinear systems with uncontrollable linearization (Q2739033)

The stabilization problem of a class of nonlinear systems with uncontrollable linearization is discussed. It is assumed that the nonlinear systems considered can be reduced to the form NEWLINE\[NEWLINE\left[\begin{matrix} \dot\xi_1\\ \dot\xi_2\\ \dot\xi_3\end{matrix}\right]= \left[\begin{matrix} \widehat A_{11} & 0 & 0\\ 0 &\widehat A_{22} & 0\\ 0 & 0 & \widehat A_{33}\end{matrix}\right] \left[\begin{matrix} \xi_1\\ \xi_2\\ \xi_3\end{matrix}\right]+ \left[\begin{matrix}\widehat\varphi_1(\xi_1, \xi_2, \xi_3)\\ \widehat\varphi_2(\xi_1, \xi_2, \xi_3)\\ \widehat\varphi_3(\xi_1, \xi_2,\xi_3)\end{matrix}\right]+ \left[\begin{matrix} b_1\\ b_2\\ b_3\end{matrix}\right] u,\tag{\(*\)}NEWLINE\]NEWLINE where \(\xi_1\in R^p\), \(\xi_2\in R^q\), \(\xi_3\in R^r\), \(\sigma(\widehat A_{11})\subset C_-\), \(\sigma(\widehat A_{22})\subset C_+\), \(\sigma(\widehat A_{33})\subset \{j\omega\mid \omega\in \mathbb{R}\}\), and the subsystem \((\widehat A_{22},\widehat b_2)\) is controllable and \((\widehat A_{33}, \widehat b_3)\) is completely uncontrollable, i.e., \(\widehat b_3= 0\). So, by the transformation \(u= v+\widehat k\widehat\xi_2\) with \(\sigma(\widehat A_{22}+\widehat b_2\widehat k)\subset C_-\), system \((*)\) can be rewritten as NEWLINE\[NEWLINE\left[\begin{matrix} \dot\xi_1\\ \dot\xi_2\end{matrix}\right]= \left[\begin{matrix} A_{11} & 0\\ 0 & A_{22}\end{matrix}\right] \left[\begin{matrix} \xi_1\\ \xi_2\end{matrix}\right]+ \left[\begin{matrix} \varphi_1(\xi_1, \xi_2)\\ \varphi_2(\xi_1, \xi_2)\end{matrix}\right]+ \left[\begin{matrix} b\\ 0\end{matrix}\right],\tag{\(**\)}NEWLINE\]NEWLINE where \(\xi_1= [\widehat\xi_1, \widehat\xi_2]^T\), \(\xi_2= \widehat\xi_3\), \(\sigma(A_{11})\subset C_-\), \(A_{22}= \widehat A_{33}\).NEWLINENEWLINENEWLINEApplying four theorems due to \textit{S. Behtash} and \textit{S. Sastry} [IEEE Trans. Autom. Control 33, 585-590 (1988; Zbl 0647.93054)] directly to system \((**)\), the authors derive a necessary condition and a sufficient condition for the local asymptotic stability of the zero solution of the original system \((*)\).NEWLINENEWLINENEWLINESince many notations and symbols from the above-mentioned paper by S. Behtash et al. are involved, it is too long to cite these conditions in detail here.