I am trying to find Lipschitz constants for a system of equations that is placed in state-space form. For example, a double integrator has the following form
$\dot{x_1} = x_2\\\dot{x_2} = u$
According to the definition of Lipschitz constant, $\|f(y)-f(x)\|\leq L\|y-x\|$, how do I find the Lipschitz constants for the system above?
My intuition is $\|f(y)-f(x)\| = \|x_{2,y}-x_{2,x}\|\leq L \|x_{2,y}-x_{2,x}\|$, $L=1$?
Can I also think the problem as the following?
With $f_1=x_2$, $f_2=u$, we have $L=\|\frac{\partial f}{\partial x}\|=\begin{Vmatrix}\frac{\partial f_1}{\partial x_1}&\frac{\partial f_1}{\partial x_2}\\\frac{\partial f_2}{\partial x_1}&\frac{\partial f_2}{\partial x_2}\end{Vmatrix}=\begin{Vmatrix}0 &1\\0&0\end{Vmatrix}=1$
The norm that I am using is 2-norm.