Finite energy Lyapunov function candidate for fractional order general nonlinear systems

Abstract

The construction of Lyapunov function candidates and the norm of infinite energy solutions remain among the most elusive unsolved problems of fractional order systems. This paper develops a new framework to construct finite energy Lyapunov function candidates for fractional order general nonlinear systems with randomness, uncertainty, time-delay or memory. Fundamentals of fractional order system and equilibrium are revisited to start up the investigation, where the pseudo and true states of fractional order systems are considered. The process of constructing fractional order Lyapunov function candidate is mainly divided into three steps: Firstly, converting the original system into an equivalent Volterra integro-differential equation, where the weak singularity of fractional order system is included. Secondly, the fractional order Lyapunov function candidate is derived by canceling out the weak singularity that acts as the catalyst, and is absorbed into the fractional finite energy terms. Lastly, the first order derivative of the proposed Lyapunov function candidate is negative definite and bounded by power-law relevant terms. From finite energy aspect, the proposed fractional order Lyapunov function candidate is composed of potential, kinetic and/or Riesz potential energies in terms of physics. The fractional order Lyapunov’s theorem and asymptotic stability of equilibrium points are discussed, and some non-Lp stable cases have been shown as fractional order finite energy ones. The impacts of fractional order, region of attraction and initialization state on the stability of equilibrium points are presented as well. Some classical integer order and fractional order results can be deduced from this work. A number of examples are illustrated to substantiate the effectiveness of the proposed unified framework.