Hierarchical matrix Lyapunov functions and stability of solutions of uncertain systems (Q2761394)

The author considers the uncertain system NEWLINE\[NEWLINE \frac{dx}{dt} = f(t,x,\alpha),\quad x(t_0) = x_0,\tag{1} NEWLINE\]NEWLINE where \(x\in \mathbb{R}^n\), \(t\in \mathbb{R}_+ = [0,+\infty)\), \(f\in C(\mathbb{R}_+\times \mathbb{R}^n\times {\mathcal S}, \mathbb{R}^n)\), \(\alpha\in\mathcal S\subseteq \mathbb{R}^d\), \(d\geq 1\) is the uncertainty parameter, \(\mathcal S\) is a compact set in \(\mathbb{R}^d\). In the space \((\mathbb{R}^n,\|\cdot\|)\) a moving set NEWLINE\[NEWLINE A(r) = \{x\in \mathbb{R}^n\: \|x\|=r(\alpha)\}\tag{2} NEWLINE\]NEWLINE is considered, where \(r(\alpha)>0\), \(r(\alpha)\to r_0 = \text{ const}>0\) as \(\|\alpha\|\to 0\) and \(r(\alpha)\to +\infty\) as \(\|\alpha\|\to+\infty\). For the class of system (1) admitting a mathematical decomposition into \(m\) interacting subsystems NEWLINE\[NEWLINE \frac{dx_i}{dt} = \widehat f_i(t,x_i)+g_i(t,x_1,\dots,x_m,\alpha),\tag{3} NEWLINE\]NEWLINE where \(x_i\in \mathbb{R}^{n_i}\), \(\widehat f_i\in C(\mathbb{R}_+\times \mathbb{R}^{n_i},\mathbb{R}^{n_i})\), \(g_i\in C(\mathbb{R}_+\times \mathbb{R}^{n_1}\times\dots\times \mathbb{R}^{n_m}\times\mathcal S, \mathbb{R}^{n_i})\), \(\sum_{i=1}^m n_i = n\), \(\mathcal S\subset \mathbb{R}^d\), sufficient stability conditions are established for solutions with respect to the set (2). Besides, the method of hierarchical Lyapunov functions used is proposed in [\textit{Y.-H.~Chen}, Dyn. Control 6, No. 2, 131-142 (1996; Zbl 0850.93731)].NEWLINENEWLINENEWLINERemark. When system (1) is nonlinearizable it is admissible to use the function \(r(\alpha)\to 0\) as \(\|\alpha\|\to 0\) [see \textit{V.~Lakshmikantham} and \textit{A.~S.~Vatsala}, in: S. Sivasundaram et al. (eds.), Advances in nonlinear dynamics. Langhorne, PA: Gordon and Breach. Stab. Control Theory Methods 5, 79-83 (1997; Zbl 0947.34039)].