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Construction of Lyapunov functions for homogeneous second-order systems

Finding an explicit Lyapunov function for stability analysis of a given dynamical system entails the nontrivial task of solving a partial differential inequality. Although many methods for finding Lyapunov functions are available, much remains to be done in this regard since there isn't a universal constructive method for finding simple, explicit Lyapunov functions for dynamical systems with stable equilibria. Homogeneity properties of systems may be used to address this problem since they are capable of reducing the complexity of the equations involved. The present work outlines a method to obtain homogeneous Lyapunov functions for homogeneous second-order systems. In comparison with previous results, the method described here provides explicit Lyapunov functions for a larger set of dynamical systems and greatly reduces the sign-definiteness analysis of the underlying equations.