The definition of passivity for control systems is: $\langle y,u \rangle _T \geq 0 \: \: \: \forall T >0 \: \: \: \forall u$. Considering signals $u,y :[0,T] \rightarrow R^m$.
As I understand it, passivity is when the system is still stable and does not change the direction and lose stability. If we look at the vectors I understand that in the scalar multiplication $\cos\theta \geq 0$. Does that mean that the direction or phase doesn't change from positive to negative and why?
Another definition using the storage function is:
$$V(x(T)) \leq \int_0^Ty(t)u(t)dt + V(x(0)), \: \: \forall T > 0 $$
Passivity idea: “Increase in stored energy ≤ Added energy”
How do I describe the passivity for a system in words without the storage function? I mean, what is passivity and why is it useful? In what way is it different from stability?