Good Day everyone. I was assigned to show that given an autonomous system of Differential Equations and a function $V$, I need to show that $V$ is Lyapunov function.
To show that $V$ is Lyapunov. I will show the following: (there is no problem here)
$V$ and its first partial derivatives are continuous.
$V$ is positive definite.
$\dot{V}$ (given by certain equation) is negative definite.
On a domain $D$ on the xy-plane containing $(0,0)$, then $V$ is a Lyapunov Function. However, the question that I need to answer is followed by: is it a strong Lyapunov Function (I do not know how to show that a function is strong Lyapunov)?
My question is, what is the difference between Lyapunov and Strong Lyapunov Function?
It is better to refer the exact definition given in your course, because it is not a standard notion.
One of possibilities is that if $\vec x(t)$ is the solution of your differential equation, then you will have $$\frac{d}{dt}V(\vec x(t))\le 0$$ for Lyapunov function and $$\frac{d}{dt}V(\vec x(t))< 0 \text{ whenever }x\ne 0$$ for strong Lyapunov function.
Another possibility would be to say that we impose $\nabla V( x)\cdot x \le -c\|x\|^2$ for strong Lyapunov functions (as in strong convexity).