From d81703f774b82ec32480476999915c53e7fd67c2 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Petr=20Veli=C4=8Dka?= Date: Fri, 7 Mar 2025 09:49:50 +0100 Subject: [PATCH] prednaska 7.3.2025 --- maximalni-reseni.tex | 62 ++++++++++++++++++++++++++++++++++++++ skripta.pdf | Bin 78615 -> 95860 bytes skripta.tex | 2 ++ zavislost-na-podmince.tex | 20 ++++++++++++ 4 files changed, 84 insertions(+) create mode 100644 maximalni-reseni.tex create mode 100644 zavislost-na-podmince.tex diff --git a/maximalni-reseni.tex b/maximalni-reseni.tex new file mode 100644 index 0000000..10d4fc0 --- /dev/null +++ b/maximalni-reseni.tex @@ -0,0 +1,62 @@ +\section{Maximální řešení} + +V této kapitole se budeme věnovat otázce rozšíření řešení na co největší podmnožinu prostoru, v němž toto řešení hledáme. Bez újmy na obecnosti budeme dále předpokládat, že $f$ je spojitá (ne nutně lipschitzovská) na $\Omega$ (což znamená, že nutně nemusíme mít jednoznačnost řešení). + +\begin{definition} + Řešení $(\hat{x}, \hat{I})$ diferenciální rovnice \eqref{eq-ode} nazýváme \textit{prodloužením} řešení $(x, I)$, jestliže $\hat{I} \supset I$ a $\hat{x}(t) = x(t)$ pro každé $t \in I$. Řešení $(x, I)$ se nazve \textit{maximální}, jestliže nemá žádné netriviální ($\hat{I} \supsetneq I$) prodloužení. +\end{definition} + +\begin{theorem} + \label{thm-max-extension} + Každé řešení rovnice \eqref{eq-ode} má alespoň jedno maximální prodloužení. +\end{theorem} + +\begin{proof} + Mějme řešení $(x, I)$ takové, že $I = (a, b)$. Budeme induktivně prodlužovat za bod $b$ (na druhou stranu se to pak udělá analogicky). Položme $x_0 = x$, $b_0 = b$, $I_0 = I$. V $n$-tém kroku dostaneme řešení $(x_n, I_n)$, kde $I_n = (a, b_n)$. Dále definujeme $\omega_n = \sup \{z > b_n; (x_n, I_n) \text{ lze prodloužit na } (a, z) \}$. Pokud příslušná množina je prázdná, jsme hotovi, neboť řešení již nejde prodloužit, tedy je maximální. + + V opačném případě můžeme definovat $b_{n + 1} = \frac{b_n + \omega_n}{2}$ (pokud $\omega_n < \infty$), případně $b_{n + 1} = b_n + 1$. Tímto postupem získám rostoucí posloupnost $b_n$, která musí mít limitu. Označme tuto limitu $\beta$. Dále položme $\tilde{I} = (a, \beta)$, $\tilde{x} = x_n(t)$, pro všechna $t \in \tilde{I}$ zvolím $n$ tak, aby $t \in I_n$. Na volbě $n$ nezávisí, neboť na příslušných intervalech jsou funkce $x_n$ stejné. + + Dokážeme, že takto definované řešení $(\tilde{x}, \tilde{I})$ je maximální. Pro spor budeme předpokládat, že existuje rozšíření na $(a, \hat{\beta})$ takové, že $\hat{\beta}$. Okamžitě vidíme, že $\beta < \infty$. Vezmeme $n$ takové, aby $\beta - b_n < \hat{\beta} - \beta$ a $\beta - b_n < 1$ (existuje díky tomu, že $b_n$ konvergují k $\beta$). V tom případě $(x_n, I_n)$ má prodloužení až do $\hat{\beta}$, tedy $\omega_n \geq \hat{\beta}$. Pak ale (pokud $\omega_n = \infty$) $b_{n + 1} = b_n + 1 > \beta$, máme spor, případně pro $\omega_n$ konečné máme $b_{n + 1} = \frac{b_n + \omega_n}{2} > \frac{2\beta - \hat{\beta} + \hat{\beta}}{2} = \beta$, opět jsme došli ke sporu. +\end{proof} + +V případě $f$ lipschitzovské se důkaz dá výrazně zjednodušit. Budeme uvažovat všechna prodloužení řešení $x$ (platí jednoznačnost), dostaneme lineárně uspořádanou množinu, potom díky Zornovu lemmatu existuje maximální prvek. + +\begin{theorem}[Picard] + \label{thm-picard} + Nechť $f \in C_x^1(\Omega)$. Pak pro každé $(x_0, t_0) \in \Omega$ existuje právě jedno maximální řešení $x$ diferenciální rovnice \eqref{eq-ode} v $\Omega$ splňující $x(t_0) = x_0$. +\end{theorem} + +\begin{proof} + Plyne z Peanovy věty (Věta \ref{thm-peano}) a Věty \ref{thm-max-extension}. +\end{proof} + +\begin{lemma} + \label{lemma-extension} + Řešení $(x, I)$ diferenciální rovnice \eqref{eq-ode} lze prodloužit za bod $b$ právě tehdy, když platí všechny + \begin{enumerate}[(i)] + \item $b < \infty$; + \item existuje $\lim_{t \rightarrow b^{-}} x(t) =: x_0 \in \mathbb{R}^n$; + \item $(x_0, b) \in \Omega$. + \end{enumerate} +\end{lemma} + +\begin{proof} + Nutnost těchto podmínek plyne triviálně z podstaty prodloužení (cvičení). Dokážeme, že jde o podmínky postačující. Nechť tedy máme $(x_0, b)$ jakou novou počáteční podmínku, dle Peanovy vety existuje řešení $\hat{x}$ na $(b - \delta, b + \delta)$ splňující tuto počáteční podmínku. Definujeme $\hat{x}(t) = \begin{cases} x(t), t < b \\ \tilde{x}(t), t \geq b \end{cases}$. Potom $\hat{x}$ je řešení (díky principu nalepování) a navíc prodlužuje $x$ za bod $b$, což jsme chtěli dokázat. +\end{proof} + +Na závěr si uvedeme jednu důležitou větu, která nám poskytne představu o tom, jak vypadají maximální řešení diferenciálních rovnic. + +\begin{theorem}[o opuštění kompaktu] + Nechť $K \subset \Omega$ je kompaktní, nechť $(x, I)$ je maximální řešení rovnice \eqref{eq-ode} splňující $(x(t_0), t_0) \in K$ pro nějaké $t_0 \in I$. Potom existují $t_1 > t_0 > t_2$ taková, že $(x(t_1), t_1) \notin K$ a $(x(t_2), t_2) \notin K$. +\end{theorem} + +\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 + $$ \| x(s) - x(t) \| \leq \| x'(\xi) \| | s - t | = \| f(x(\xi), \xi) \| | s - t | \leq M | s - t |, $$ + kde poslední nerovnost plyne z toho, že funkce $f$ je omezená na kompaktu $K$ konstantou $M$. Nakonec $(x_0, b) = \lim_{t \rightarrow b^-} (x(t), t)$, tedy z uzavřenosti $K$ máme, že $(x_0, b) \in K \subset \Omega$. + + Zjistili jsme, že řešení lze prodloužit za bod $b$, což je spor s jeho maximalitou. Důkaz pro $t_2$ se udělá obdobně. +\end{proof} + diff --git a/skripta.pdf b/skripta.pdf index 2791a9514f797130639f046a16725fd1fdc4ff22..08fdccb10063adbff94836bd2154c0f6d932c12a 100644 GIT binary patch delta 54389 zcmY(pQ*bT}uq+yzE4FRhSg~yzD|Wuvwr$(CZQHhU{$2N;s(s%2d3tK5yJu2Qz@Dza z>w*+OIGEy*$pDNjto-~iPR@>|hPE*78^=15u_o;fJ)i1D_S=}G!%#P_<%rBCB0b|b znWG@wohls`5J-fAAqbC~c>UQj52j|SZq|g;c^wUU4-U{H55vMaKh6P(MZfz;cSnR_ zeZSfKzk;(rf&#w|AIqbQ3_pBXg0RJOU|t_@jtzua&wz>S88v7sJ$It^C-eD5CzjAj0L_#pHzR4*>`xpnQXJ<(JH59f*A*@FG^nn7z?b{00~6G0H!_W^K{y9pg-NSonKiH zb+b!U1IonSnkColez{`AqbX@nX#x(l@m&}9=+v8F8qED7r<@XD6!+H$Uow9^cSz<|4=GWr#6vWF zCqVp-$?RkYlCpavujN$YIr_R^Z?sIjCM~l{k+T(EE9dLXpOqt~X5$P-z9`$nA-bu_ z#_xwd?tIC4G-7WC;8bTIDxfNuWTdVvQulE4MTMEveeB))U5CJ&L)0*FN+6NGbkGU2 z>|}1@Sf#bd_hRd06)9zd@?$c-XesLFDgcfE+R3%~&%1}&#n*}WN7|ZVkGoIrPnr!> z`BjYXbdX<)Fv=ksPC)PMKpN!oT2qPtm}zdC(?>t|SQlT)J1ZFFCMZKH2?@)cpYT3i zO101Hcjv3g$-OHf7jUqPpXYrJ9SMe+srH3V!zdU7cXF;9jzeAn@&t`>CgI<}=l-#x0XWeL z!lCDU+S%d2kpT4;&32D5? 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Nechť pro každé $t \in I$ platí + $$ w(t) \leq K + \left| \int_{t_0}^t w(s) g(s) ds \right|. $$ + Potom pro každé $t \in I$ platí + $$ w(t) \leq K \exp\left(\left| \int_{t_0}^t g(s) ds \right|\right) .$$ +\end{lemma} + +\begin{proof} + Definujeme $\Phi(t) := K + \int_{t_0}^t w(s) g(s) ds + \varepsilon$ pro $t > t_0$. Okamžitě z předpoklady vidíme, že $w(t) \leq \Phi(t)$. Zderivujeme funkci $\Phi(t)$, dostáváme + $\Phi'(t) = w(t) g(t) \leq \Phi(t) g(t)$ což po vydělení $\Phi(t)$ (je nenulové díky přičtení $\varepsilon$) nám dává $\frac{\Phi'(t)}{\Phi(t)} \leq g(t)$, což můžeme přeintegrovat od $t_0$ do $t$, čímž dostaneme + $\int_{t_0}^t \frac{\Phi'(s)}{\Phi(s)} ds \leq \int_{t_0}^t g(s) ds$. Po vyčíslení integrálů dostaneme + $ \log (\Phi(t) - \Phi(t_0)) \leq \int_{t_0}^t g(s) ds$. Jelikož $\exp$ je rostoucí funkce, můžeme psát $ \frac{\Phi(t)}{\Phi(t_0)} \leq \exp\left(\int_{t_0}^t g(s) ds\right)$. + + Nakonec dostáváme $w(t) \leq \Phi(t) \leq (K+\varepsilon) \exp\left(\int_{t_0}^t g(s) ds\right)$. Požadované tvrzení získáme posláním $\varepsilon$ do $0$. +\end{proof} + +\hfill \textit{konec 3. přednášky (7.3.2025)} +