What is the definition of 'Lyapunov function'?

Question

I interested in the Lyapunov stability of fractional order system, so I would like to know what is the definition of Lyapunov functions in the sense of fractional order system?. Please advise me any books to read. Thanks

Answer 1

I am not familiar with fractional order systems, can't help there. In differential equations, here's Lyapunov function, hopefully it's not so different from what you're after:

Given an ODE $\dot x = f(x)$ with fixed point $x = 0$, a $C^1$ function $V$ with $ImV \subseteq \Bbb R$ is said to be positive (or negative in the opposite case) if there exists a Neighbourhood of $0$ where $V(x) \ge 0$. If in addition $V$ gets the value $0$ only in the point $x = 0$, then we write $V \gt 0$. Now consider $\dot V(z) = \sum_{i=1}^n\frac{\partial V}{\partial x_i}(z)f_i(z)$.

If a $C^1$ function, $V \gt 0$ and $\dot V \le 0$ then $V$ is a Lyapunov function. If $\dot V \lt 0$ then $V$ is a strong Lyapunov function.

A helpful theorem: If an ODE as above, has a Lyapunov function then $x = 0$ is stable. If the Lyapunov function is a strong Lyapunov function then $x=0$ is asymptotically stable.