3
$\begingroup$

Suppose that we have two differential inclusions

$$\frac{dY^1}{dt}(t)\in b_1(Y^1,t)$$

with $Y^1(0)\in Y_0^1$ and

$$\frac{dY^2}{dt}(t)\in b_2(Y^2,t)$$

with $Y^2(0)\in Y_0^2$.

Can we then control $d(Y^1(t),Y^2(t))$ by the distance of $b_1$ and $b_2$ and the initial conditions? The distance here would be some appropriate set distance such as Kuratowski or Hausdorff distance or some other distance?

Improve this question
$\endgroup$
3
  • $\begingroup$ I feel like the appropriate distance between $b_1 (x, t)$ and $b_2 (x, t)$ should just be the simple $\sup_{a_1 \in b_1 (x, t), a_2 \in b_2(x, t)} |a_1 - a_2|$. And then you probably want $\text{dist}(b_1, b_2)$ to be the sup over $x, t$ of the above distance. The reason being that someone who wanted to ruin our day could just choose for $dY_1/dt$ and $dY_2/dt$ the worst possible choice (i.e. the one that maximises the difference). This is the locally optimal way to maximise the distance. (1/2) $\endgroup$
    – Nate River
    Commented Nov 24, 2023 at 12:47
  • $\begingroup$ If we set up $b_1$ and $b_2$ so that the greedy choice is also globally optimal, we see that the bound given by the above is sharp. (2/2) $\endgroup$
    – Nate River
    Commented Nov 24, 2023 at 12:48
  • $\begingroup$ You probably also want a uniform in $t$ Lipschitz condition in $x$ for both $b_1$ and $b_2$, with respect to the above pointwise distance. $\endgroup$
    – Nate River
    Commented Nov 24, 2023 at 12:52

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .