I am trying to figure out how to find the "lower bound" on the Lipschitz constant for a LTV dynamic system, i.e $\dot{x} = A(t)x + B(t)u$. Where x is 6x1, and u is 2x1. All states and inputs are upper and lower bounded. I am fairly new to this, but my reasoning so far is:
Standard Lipschitz condition is:
||f($x_1$,t) - f($x_2$,t)|| $\le$ L||$x_1$-$x_2$||. The left hand side for LTV becomes:
||f($x_1$,t) - f($x_2$,t)|| = ||A(t)$x_1$ + B(t)u - (A(t)$x_2$ + B(t)u)|| = ||A(t)$x_1$ - A(t)$x_2$|| = ||A(t)|| ||$x_1$ - $x_2$||. Now this gives the Lipschitz condition as:
||A(t)|| ||$x_1$ - $x_2$|| $\le$ L ||$x_1$ - $x_2$|| => ||A(t)|| $\le$ L.
So in order to find the "lower bound" on L, i need to find the upper bound on the norm of the time varying A matrix, and then set L accordingly? Since I have the analytical jacobian for A (and the states and inputs are bounded) I can examine the entries and find the upper bound on A. Is this mathematically correct? I am quite rusty and I'm not sure I'm doing things correctly here.
On a side note, I know that usually we are only concerned by finding just any lipschitz constant, but in this case it is to be used in further assumptions on the control system and it would be beneficial to find the constant which fulfills the lipschitz condition for any A (well, A is bounded so the "largest" A is of concern)
Best Regards MC
This looks like the way to go. Here are a few comments:
There is a slight error in $$\|A(t)(x_1(t)-x_2(t))\|=\|A(t)\|\|x_1(t)-x_2(t)\|$$ It should be $$\|A(t)(x_1(t)-x_2(t))\|\leq\|A(t)\|\|x_1(t)-x_2(t)\|$$ Also you cannot use every matrix norm in this inequality. Usually one uses the Operator norm (see e.g. https://en.wikipedia.org/wiki/Operator_norm) $$\|A\|_{op}=\sup_{x\neq 0}\frac{\|Ax\|}{\|x\|}.$$
Another thing: Are you sure about your Lipschitz condition? I mean it is usually stated without functions on the right side. So it should look like this (see e.g. https://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem) $$\|f(x_1,t)-f(x_2,t)\|\leq L\|x_1-x_2\|$$