Lyapunov stability theorem about fractional system without and with delay

The difficulty of fractional direct Lyapunov stable theorem lies in that how to design a positive definite function V and easily ascertain whether fractional derivative of the function V is less than zero. In view of this difficulty, we propose a Lyapunov stability theorem for fractional system without delay and extend the newly proposed theorem to fractional system with delay. The obvious difference of the proposed theory with the fractional Lyapunov direct theory is taking the integer derivative instead of the fractional derivative of the positive definite function V. Four examples are provided to illustrate the proposed theorem. The studying results in this paper show that the proposed theorem is not only applicable to the fractional autonomous system with and without delay, but also applicable to the fractional non-autonomous system with and without delay.