The asymptotic stability of the equilibrium of nonautonomous systems (Q2714634)

The author considers the system NEWLINE\[NEWLINE\dot x_s = f_s(X)+ \sum_{j=1}^k b_{sj}(t)h_{sj}(X),\quad s=1,\dots,n,\qquad X=[x_1,\dots,x_n]^T, \tag{1}NEWLINE\]NEWLINE with \(f_s\in C(\mathbb{R}^n,\mathbb{R})\), \(h_{sj}\in C(\mathbb{R}^n,\mathbb{R})\), \(b_{sj}\) are continuous and bounded for \(t\geq 0\). NEWLINENEWLINENEWLINEIt is assumed that the solution \(X=0\) to system NEWLINE\[NEWLINE\dot x_s = f_s(X)\tag{2} NEWLINE\]NEWLINE is asymptotically stable. The author establishes conditions for the stability of \(X=0\) to system (1) in case when \(f_s(X)\) and \(h_{sj}(X)\) are generalized homogeneous functions and the integrals NEWLINE\[NEWLINE I_{sj}(t) = \int_0^t b_{sj}(\tau) d\tau NEWLINE\]NEWLINE are not generally bounded for \( t\geq 0\).\flushpar The general result is illustrated by an example from solid dynamics.