Introducing some classes of stable systems without any smooth Lyapunov functions

This paper proves that certain classes of stable nonlinear systems do not admit any smooth Lyapunov functions. In fact, the stability analysis of two different classes of nonlinear dynamical systems is provided, and it is demonstrated that their stability properties cannot be established by any convex Lyapunov functions or smooth Lyapunov functions. In this regard, we begin by studying our first class of autonomous dynamical systems and prove that, despite stability, there are no convex Lyapunov functions in the form of V(x) or V(t,x) to establish the stability properties. For the second class, we consider a much more general form of nonlinear autonomous dynamical systems with a stable origin, and prove that this class, too, does not admit any smooth Lyapunov functions in the form of V(x) or V(t,x). Furthermore, another general class of stable non-autonomous dynamical systems is presented, and it is demonstrated that there is no smooth Lyapunov function in the form of V(x) to establish the stability properties. Finally, for the second class of more general stable non-autonomous systems, it is proved that the class does not admit even a continuous Lyapunov function in the form of V(x) or V(t,x).