Strict Lyapunov function and Chetaev function for stability/instability analysis of the pendulum

Stability analysis of nonlinear systems can be carried out by the Lyapunov's first method (linearizationtechnique). However, by utilizing this approach most insight about the physical system is lost. Alternativeapproaches are available to exploit systems characteristics such as those associated to energy conceptsfor stability and instability analysis. In this paper, with reference to a simple nonlinear system -thependulum-, we propose first, a simple strict Lyapunov function motivated by energy consideration tostudy asymptotic stability of some equilibria, and second a Chetaev function to analyze instability of theremaining ones. A remarkable point is that both functions have a common structure based on thesystems's energy, and differs only in a single scalar parameter.