This project is devoted to developing Liapunov direct method for fractional differential equations and systems. The method (constructing a system related scalar function) enables investigators to analyze the qualitative behavior of solutions of a differential equation without actually solving it. We are able to convert some fractional differential equations (semi-linear) to integral equations with singular kernels and construct Liapunov functionals for the integral equations to deduce conditions for boundedness and stability of solutions. Extending such a method to fully nonlinear equations presents a significant challenge to investigators and will be a major area of research for many years to come.
Zhang, Bo, "Stability and Liapunov Functionals for Fractional Differential Equations" (2012). Math and Computer Science Working Papers. 12.