From f0b8e273bdf0cd37b18349b8539be77cbc689262 Mon Sep 17 00:00:00 2001 From: Petr Velycko Date: Tue, 20 May 2025 15:35:43 +0200 Subject: [PATCH] preklepy --- linearni-rovnice-konst-koef.tex | 2 +- maximalni-reseni.tex | 2 +- skripta.pdf | Bin 227262 -> 227250 bytes zavislost-na-podmince.tex | 2 +- 4 files changed, 3 insertions(+), 3 deletions(-) diff --git a/linearni-rovnice-konst-koef.tex b/linearni-rovnice-konst-koef.tex index 36afb64..2c593e2 100644 --- a/linearni-rovnice-konst-koef.tex +++ b/linearni-rovnice-konst-koef.tex @@ -13,7 +13,7 @@ Myšlenkou studia těchto rovnic je analogie s rovnicí $x' = ax$ pro $a \in \R$ s konvencí $A^0 = I$. \end{definition} -Řada s definice maticové exponenciály je dobře definovaná, neboť $\|\frac{1}{k!}A^k\| \leq \frac{1}{k!}\|A\|^k$, přičemž $\sum_{k=0}^\infty \frac{1}{k!}c^k = e^c$ konverguje pro každé $c \in \R$. Navíc z tohoto odhadu dostáváme $\|e^A\| \leq e^{\|A\|}$. +Řada z definice maticové exponenciály je dobře definovaná, neboť $\|\frac{1}{k!}A^k\| \leq \frac{1}{k!}\|A\|^k$, přičemž $\sum_{k=0}^\infty \frac{1}{k!}c^k = e^c$ konverguje pro každé $c \in \R$. Navíc z tohoto odhadu dostáváme $\|e^A\| \leq e^{\|A\|}$. \begin{example} Nechť $A = \begin{pmatrix}2 & 0\\0 & 1\end{pmatrix}$. Potom diff --git a/maximalni-reseni.tex b/maximalni-reseni.tex index ba3e1bf..5d08583 100644 --- a/maximalni-reseni.tex +++ b/maximalni-reseni.tex @@ -54,7 +54,7 @@ Na závěr si uvedeme jednu důležitou větu, která nám poskytne představu o \begin{proof} Pro spor budeme předpokládat, že takové $t_1$ neexistuje, chceme dojít ke sporu s maximalitou řešení. Mějme řešení $x$ na $(a, b)$ a $(x(t), t) \in K$ pro všechna $t \in [t_0, b)$. Ukážeme, že toto řešení můžeme prodloužit za $b$. Využijeme k tomu Lemma \ref{lemma-extension}. - Zřejmě platí $b < \infty$ (díky kompaktnosti $K$). Dále dokážeme, že existuje $\lim_{i \rightarrow b^-} x(t)$ pomocí Bolzanovy-Cauchyovy podmínky. Mějme $s, t \in (t_0, b)$. Dále díky Lagrangeově větě o střední hodnotě máme + Zřejmě platí $b < \infty$ (díky kompaktnosti $K$). Dále dokážeme, že existuje $\lim_{t \rightarrow b^-} x(t)$ pomocí Bolzanovy-Cauchyovy podmínky. Mějme $s, t \in (t_0, b)$. Dále díky Lagrangeově větě o střední hodnotě máme $$ \| x(s) - x(t) \| \leq \| x'(\xi) \| | s - t | = \| f(x(\xi), \xi) \| | s - t | \leq M | s - t |, $$ kde poslední nerovnost plyne z toho, že funkce $f$ je omezená na kompaktu $K$ konstantou $M$. 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b/zavislost-na-podmince.tex @@ -41,7 +41,7 @@ Jednoduchým důsledkem tohoto lemmatu je mj. jednoznačnost řešení (stačí uvažovat řešení s $x(t_0) = y(t_0)$). \begin{definition} - Nechť $f$ je spojitá a lokálně lipschitzovská vzhledem k $x$ v $\Omega$. Potom definujeme \textit{řešicí funcki} $\varphi : G \subset \mathbb{R}^{n + 2} \rightarrow \mathbb{R}^n$ předpisem $\varphi(t; t_0, x_0) := x(t)$, kde $x: I \rightarrow \mathbb{R}^n$ je řešení splňující počáteční podmínku $x(t_0) = x_0$ a $t \in I$. Zde $G$ je maximální možná, tj. obsahuje všechny trojice $(t; t_0, x_0) \in \mathbb{R}^{n + 2}$ pro něž výraz $\varphi(t; t_0, x_0)$ má smysl. + Nechť $f$ je spojitá a lokálně lipschitzovská vzhledem k $x$ v $\Omega$. Potom definujeme \textit{řešicí funkci} $\varphi : G \subset \mathbb{R}^{n + 2} \rightarrow \mathbb{R}^n$ předpisem $\varphi(t; t_0, x_0) := x(t)$, kde $x: I \rightarrow \mathbb{R}^n$ je řešení splňující počáteční podmínku $x(t_0) = x_0$ a $t \in I$. Zde $G$ je maximální možná, tj. obsahuje všechny trojice $(t; t_0, x_0) \in \mathbb{R}^{n + 2}$ pro něž výraz $\varphi(t; t_0, x_0)$ má smysl. \end{definition} Například, uvažujeme-li rovnici $x'(t) = x(t)$. Obecným řešením této rovnice je funkce $x(t) = ce^t$, vyřešením rovnice s počáteční podmínkou dostaneme řešicí funkci $\varphi(t; t_0, x_0) = x_0e^{t - t_0}$.