The stability of a class of nonlinear systems (Q2761315)

In the domain \(\|X\|\leq h\), \(\|Y\|\leq h\), \(t\geq 0\), the author considers the system \(\dot X = F(X)+G(t,X,Y)\), \(\dot Y= D(t,X,Y)\). Here \(X\in \mathbb{R}^n\), \(Y\in \mathbb{R}^k\), \(F(X)\) is a homogeneous continuous function of order \(\mu\), \(\mu\geq 1\), \(\|G\|\leq c_1(X,Y)\|X\|^\mu\), \(\|D\|\leq c_2\|X\|^\lambda\), \(c_1(X,Y)\to 0\) for \(\|X\|+\|Y\|\to 0\), \(c_2>0\) and \(\lambda>0\). Using the direct Lyapunov method, the author derives sufficient conditions for stability of the system with respect to all variables, and conditions for asymptotic stability with respect to a part of variables.