From 6ce0bb9a8f9fd7d169cbb414a9537d68c5290aae Mon Sep 17 00:00:00 2001 From: Prefetch Date: Fri, 14 Oct 2022 23:25:28 +0200 Subject: Initial commit after migration from Hugo --- .../concept/gronwall-bellman-inequality/index.md | 204 +++++++++++++++++++++ 1 file changed, 204 insertions(+) create mode 100644 source/know/concept/gronwall-bellman-inequality/index.md (limited to 'source/know/concept/gronwall-bellman-inequality') diff --git a/source/know/concept/gronwall-bellman-inequality/index.md b/source/know/concept/gronwall-bellman-inequality/index.md new file mode 100644 index 0000000..5fd9f38 --- /dev/null +++ b/source/know/concept/gronwall-bellman-inequality/index.md @@ -0,0 +1,204 @@ +--- +title: "Grönwall-Bellman inequality" +date: 2021-11-07 +categories: +- Mathematics +layout: "concept" +--- + +Suppose we have a first-order ordinary differential equation +for some function $u(t)$, and that it can be shown from this equation +that the derivative $u'(t)$ is bounded as follows: + +$$\begin{aligned} + u'(t) + \le \beta(t) \: u(t) +\end{aligned}$$ + +Where $\beta(t)$ is known. +Then **Grönwall's inequality** states that the solution $u(t)$ is bounded: + +$$\begin{aligned} + \boxed{ + u(t) + \le u(0) \exp\!\bigg( \int_0^t \beta(s) \dd{s} \bigg) + } +\end{aligned}$$ + +
+ + + +
+ +Grönwall's inequality can be generalized to non-differentiable functions. +Suppose we know: + +$$\begin{aligned} + u(t) + \le \alpha(t) + \int_0^t \beta(s) \: u(s) \dd{s} +\end{aligned}$$ + +Where $\alpha(t)$ and $\beta(t)$ are known. +Then the **Grönwall-Bellman inequality** states that: + +$$\begin{aligned} + \boxed{ + u(t) + \le \alpha(t) + \int_0^t \alpha(s) \: \beta(s) \exp\!\bigg( \int_s^t \beta(r) \dd{r} \bigg) \dd{s} + } +\end{aligned}$$ + +
+ + + +
+ +In the special case where $\alpha(t)$ is non-decreasing with $t$, +the inequality reduces to: + +$$\begin{aligned} + \boxed{ + u(t) + \le \alpha(t) \exp\!\bigg( \int_0^t \beta(s) \dd{s} \bigg) + } +\end{aligned}$$ + +
+ + + +
+ + + +## References +1. U.H. Thygesen, + *Lecture notes on diffusions and stochastic differential equations*, + 2021, Polyteknisk Kompendie. -- cgit v1.2.3