Boundedness Properties For Time-Varying Nonlinear Systems

A Liapunov theorem guaranteeing uniform boundedness and uniform ultimate boundedness for a time-varying nonlinear system $\dot{x}(t) = f(x(t) , t)$ has been established. The study of uniform boundedness and uniform ultimate boundedness of particular classes of time-varying nonlinear systems $\dot{x} (t) =f(x(t) ,t )$ is reduced to the study of the corresponding time-invariant frozen systems $\dot{x} (t) = f(x(t) , \sigma )$ for all $\sigma \in \mathbb{R}$. This approach is illustrated for time-varying homogeneous systems with a positive order, for particular classes of time-varying nonhomogeneous systems and for time-varying Lotka--Volterra equations.