---
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&rsquo;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&rsquo;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&rsquo;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&hellip; \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}
