Growth rate conditions for uniform asymptotic stability of cascaded time-varying systems (Q5930070)

The authors consider nonlinear time-varying cascade systems of the form NEWLINE\[NEWLINE\dot x_1= f_1(t, x_1)+ g(t, x_1, x_2)x_2,\quad \dot x_2= f_2(t, x_2).NEWLINE\]NEWLINE Besides some technical assumptions, the two subsystems \(\dot x_1= f_1(t, x_1)\), \(\dot x_2= f_2(t, x_2)\) are supposed to be uniformly globally asymptotically stable. The authors are interested in conditions which guarantee that the cascade connection is uniformly globally asymptotically stable, as well. They distinguish three cases, according to the way \(f_1\) and \(g\) grow when \(\|x\|\to +\infty\). The results are related to input-to-state stability and to certain growth rate conditions. In some cases, this approach allows to avoid the construction of a Lyapunov function.