Stability of solutions of nonlinear systems with unbounded perturbations (Q1272024)

The author considers the nonautonomous system \[ \dot x_s= f_s(X)+ r_s(t,X),\quad s= 1,\dots, n,\tag{1} \] where \(f_s\), \(r_s\) are defined in the domain (2) \(t\geq 0\), \(\| X\|< H\), \(H\geq 0\) and satisfy conditions that guarantee existence and uniqueness of solutions to (1) and the continuous dependence of solutions on the initial data. Moreover, \(f_s(X)\) are continuously differentiable \(\mu\)th-order homogeneous functions, where \(\mu>1\) is a rational number with odd denominator, \(r_s(t,X)\) are unbounded functions. The problem, to find out by how much the order of the perturbation must exceed the order of the functions \(f_s(X)\) so as to ensure that the asymptotic stability of the zero solution to \(\dot x_s= f_s(X)\) implies the asymptotic stability of the zero solution to (1), is solved. A few theorems are proved. The conditions of these theorems have a character of verifiable inequalities, and to determine them a method for constructing Lyapunov functions for nonautonomous systems is suggested.