The method of several Lyapunov functions in the global stabilization problem for nonlinear systems (Q2704006)

The authors consider composite systems of the form NEWLINE\[NEWLINE\begin{cases} \dot x=f(x,y,v,u) \\ \dot y=g(x,y,v,u) \\ v^{(q)}=h(v,\dot v,\dots,v^{(q-1)}) \end{cases} \tag{1}NEWLINE\]NEWLINE with discontinuous feedback \(u=U(x,y)\). Assuming that \(V\) and \(W\) are known Lyapunov functions for the subsystems defined by the first two equations respectively, they prove that for all solutions of (1), \((x(t), y(t))\) converges to zero when \(t\to +\infty\). The result is applied to the problem of stabilizing an inverted pendulum on a controlled trolley subject to an unknown force.