[[
wikihub
]]
Search
⌘K
Explore
People
For Agents
Sign in
Explore
People
For Agents
Sign in
@jemoka / Jemoka Knowledge Base / raw/course/cs238v/kbhsu_cs238v_feb132025.md
Suggest edit
Cancel
Submit suggestion
Title
Name
Note
--- title: "SU-CS238V FEB132025" source: https://www.jemoka.com/posts/kbhsu_cs238v_feb132025/ date: 2025-02-13 --- Key Sequence Notation New Concepts Important Results / Claims overapproximate inclusion functions Questions Interesting Factoids reachability for non-linear systems Standard reachability analysis for Linear Dynamical System is not great, because polytopes don’t stay polytopes when we apply non-linear operations. The general vibe, then, is to take a non-linear thing and bound them using a polytope. interval arithmetic We can’t propagate polytopes though non linear systems; but we can propagate intervals. Suppose we have an interval: \begin{equation} [x] = \qty {x \mid x_1 \leq x \leq x_2} \end{equation} Let’s define some operations interval counterpart of addition \begin{equation} [x] + [y] = \qty {x+y \mid x \in [x], y \in [y]} \end{equation} We could actually compute the interval explicitly: \begin{equation} [x] + [y] = [x_1 + y_1, x_2 + y_2] \end{equation} we can just add the intervals together interval counter part of binary operators \begin{equation} [x] \cdot [y] = \qty {x \cdot y \mid x \in [x], y \in [y]} \end{equation} specifically… \begin{equation} [x] + [y] = [x_1 - y_2, x_2 - y_1] \end{equation} \begin{equation} [x] \times [y] = [\min \qty(x_1y_1, x_1y_2, x_2y_1, x_2y_2), \max \qty(x_1y_1, x_1y_2, x_2y_1, x_2y_2)] \end{equation} notably! this last thing is not defined if any of the intervals contains \(0\). for monotone function f: \begin{equation} f\qty([x]) = [f\qty(x_1), f\qty(x_2)] \end{equation}