Uniform ultimate boundedness of nonlinear nonstationary systems

A certain class of nonlinear, nonstationary systems of differential equations is studied. It is assumed that the right-hand sides of the equations under consideration are homogeneous functions of order smaller than one with respect to the phase variables. The purpose of this paper is to obtain sufficient conditions for the uniform ultimate boundedness of systems of this form. A method for constructing nonstationary Lyapunov functions is suggested and applied to prove that the asymptotic stability of the zero solution of the corresponding averaged system implies the uniform ultimate boundedness of the initial nonstationary system. Classes of perturbations that do not violate uniform ultimate boundedness, even in the case where the order of the perturbations exceeds the homogeneity order of the unperturbed equations, are described. Unlike in previous works, where the results are based on the averaging method, the presence of a small parameter on the right-hand sides of the equations under examination is not assumed. Dissipativity is ensured at the expense of homogeneity orders.