From 9d840e169e12f93baf369b7b5b15c6d5ff4bb4c4 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Petr=20Veli=C4=8Dka?= Date: Fri, 21 Feb 2025 09:41:42 +0100 Subject: [PATCH] prednaska 21.2.2025 --- lokalni-existence.tex | 80 ++++++++++++++++++++++++++++++++++++++++++ skripta.pdf | Bin 0 -> 60623 bytes skripta.tex | 34 ++++++++++++++++++ 3 files changed, 114 insertions(+) create mode 100644 lokalni-existence.tex create mode 100644 skripta.pdf create mode 100644 skripta.tex diff --git a/lokalni-existence.tex b/lokalni-existence.tex new file mode 100644 index 0000000..a0b910a --- /dev/null +++ b/lokalni-existence.tex @@ -0,0 +1,80 @@ +\section{Lokální existence řešení} + +Diferenciální rovnice nás doprovází v každé oblasti lidského života. Neexistuje obecná teorie, která by nám umožnila vyřešit všechny diferenciální rovnice najednou. Musíme se proto omezit jen na část rovnic. + +\begin{convention} +V této přednášce budeme studovat systém rovnic +\begin{equation} + \label{eq-ode} + x' = f(x, t) +\end{equation} + za trvalého předpokladu $\Omega \in \mathbb{R}^{n+1}$ otevřená, $f: \Omega \rightarrow \mathbb{R}^n$ spojitá. +\end{convention} + +\begin{definition} + Buď $I$ otevřený interval. Funkci $x(t): I \rightarrow \mathbb{R}^n$ nazveme \textit{řešením} diferenciální rovnice (\ref{eq-ode}) v $\Omega$, jestliže pro všechna $t \in I$ platí + \begin{enumerate}[(i)] + \item $(x(t),t)\in \Omega$, + \item existuje vlastní $x'(t)$, + \item $x'(t) = f(x(t),t)$. + \end{enumerate} +\end{definition} + +Takto definované řešení je nutně spojité a má spojitou derivaci (je třídy $C^1$), tzv. klasické řešení. Dále si poznamenejme, že platí tzv. princip nalepování: Pokud máme $x(t)$ řešení na $(a, t_0)$ a na $(t_0, b)$, pak už je řešením na celém $(a, b)$. To plyne z toho, že $x'_- (t_0) = \lim_{t\rightarrow t_0^-} x'(t) = \lim_{t\rightarrow t_0^-} f(x(t), t) = f(x(t_0), t_0)$, přičemž tatáž rovnost platí i pro derivaci zprava. + +\begin{lemma} + Nechť $I$ je otevřený interval, $x(t): I \rightarrow \mathbb{R}^n$ spojitá splňující $(x(t), t) \in \Omega$ pro každé $t \in I$ a nechť $t_0 \in I$. Potom je ekvivalentní + \begin{enumerate}[(i)] + \item $x$ je řešení (\ref{eq-ode}) splňující $x(t_0) = x_0$, + \item pro každé $t \in I$ platí $x(t) = x_0 + \int^t_{t_0} f(x(s), s)ds$. + \end{enumerate} + \begin{proof} + Víme, že platí $x'(s) = f(x(s), s)$ pro všechna $s \in I$, což je spojitá funkce, kterou můžeme zintegrovat na $[t_0, t]$. + Potom z Newtonova-Leibnizova vzorce máme $x(t) - x(t_0) = \int_{t_0}^t x'(s) ds = \int_{t_0}^t f(x(s), s) ds$. Tedy $x(t) = x_0 + \int_{t_0}^t f(x(s), s) ds$. + + Pro důkaz opačné strany si uvědomíme, ze pro každé $t\in I$ je pravá strana diferencovatelná, tedy $x'(t) = f(x(t), t)$ a po dosazení $t = t_0$ dostáváme $x(t_0) = x_0$. + \end{proof} +\end{lemma} + +Teď si zadefinujeme několik pojmů, které charakterizují množiny funkcí, které se chovají jistým způsobem podobně nebo stejně. + +\begin{definition} + Řekneme, že funkce množiny $M \subset C(K, \mathbb{R}^n)$ jsou + \begin{enumerate} + \item \textit{stejně spojité}, jestliže pro každé $x \in K$ a každé $\varepsilon > 0$ existuje $\delta > 0$ takové, že $\| f(x) - f(y) \| < \epsilon$ pro všechna $y \in (x - \delta, x + \delta)$ a všechny $f \in M$. + \item \textit{stejně omezené}, jestliže existuje $C > 0$ takové, že $\|f\| \leq C$ pro všechna $f \in M$. + \end{enumerate} +\end{definition} + +\begin{theorem}{\textbf{(Arzela-Ascoli)}} + \label{thm-arzela} + Nechť funkce $x_n(t)$ jsou stejně omezené a stejně spojité na $[0, T]$. Potom z nich lze vybrat stejnoměrně konvergující posloupnost. \textit{(bez důkazu)} +\end{theorem} + +Následující věta nám řiká, že na nějakém okolí libovolného bodu existuje řešení zkoumané diferenciální rovnice. + +\begin{theorem}{\textbf{(Peano)}} + \label{thm-peano} + Nechť $(x_0, t_0) \in \Omega$. Pak existuje $\delta > 0$ a funkce $x(t): (t_0 - \delta, t_0 + \delta) \rightarrow \mathbb{R}^n$, která je řešením (\ref{eq-ode}) a splňuje $x(t_0) = x_0$. + \begin{proof} + Nejdříve dokážeme pomocné tvrzení: + \begin{lemma} + Pokud $\Omega = \mathbb{R}^{n+1}$ a $f$ je omezená na $\Omega$, pak pro každé $T > 0$ existuje řešení (\ref{eq-ode}) na $(t_0 - T, t_0 + T)$ splňující $x(t_0) = x_0$. + \begin{proof} + Řešme ``porušenou" úlohu $(P_\lambda)$: $x(t) = x_0 + \int_{t_0}^t f(x(s - \lambda), s) ds$ pro $ t > t_0$ a $x(t) = x_0$ pro $t \in [t_0 - \lambda, t_0]$. + Na $I_1 := (t_0, t_0 + \lambda]$ definujeme $x(t) = x_0 + \int_{t_0}^t f(x(s - \lambda, s) ds$. + Na $I_2 := (t_0 + \lambda, t_0 + 2\lambda]$ definujeme $x(t)$ obdobně a indukcí pokračujeme až do nekonečna. + Tímto je ``porušená" úloha vyřešena na $[t_0-\lambda, t_0 + T]$. + + Položme $\lambda = \frac{1}{n}$ pro $n = 1,2,\dots$. Pišme dále jen $x_n$ namísto $x_{1/n}$, tedy máme posloupnost funkcí. + Ukážeme, že jsou stejně spojité a stejně omezené. + Stejná omezenost plyne z toho, že $\| x_n (t) \| = \| x_0 + \int_{t_0}^t f(x(s - \frac{1}{n}), s) ds \| \leq \| x_0 \| + \int_{t_0}^t f(x(s - \frac{1}{n}) \| ds$. Ale funkce $f$ je omezená, tedy máme $\| x_n(t) \| \leq \| x_0 \| + (T - t_0) \cdot K$, kde $K$ je příslušná konstanta omezenosti $f$. + Stejnou spojitost máme z odhadu $\| x_n(t) - x_n(r) \| = \| \int_r^t f(x(s - \frac{1}{n}), s) ds \| \leq |t - r| \cdot K$. V poslední nerovnosti jsme odhadli integrál součinem délky intervalu a konstantou omezenosti funkce $f$. Stačí položit $\delta = \frac{\varepsilon}{K}$, potom $\|x_n(t) - x_r(t)\| < \delta K = \epsilon$. + + Tedy dle Věty \ref{thm-arzela} můžeme z posloupnosti $x_n$ vybrat stejnoměrně konvergentní podposloupnost. Zbývá dokázat, že její limita řeší naši rovnici. + \hfill \textit{konec 1. přednášky (21.2.2025)} + \end{proof} + \end{lemma} + \end{proof} +\end{theorem} + diff --git a/skripta.pdf b/skripta.pdf new file mode 100644 index 0000000000000000000000000000000000000000..ba5f009b4e7529d4bdba312456bea661801849b6 GIT binary patch literal 60623 zcma&NQ?PB@wzWCew6&&f+qP}nwr$(CZQHhO+n%{^Mn>d4`Jc$xFC+TPcxhj4wf0g< z?Zk3IBGmLW%#g%Kch_%_^sIPvc(w)>kX&4}BIZ_3#tyV1R{BoHLdJ%+M#i*K#x|x- zW_ZlZj66J$j!q87`qq$c8^KDQHk$O%+n>l^e=5Bc0RiDaHUPH+JSaJaT(*kQ8J=Gy zb+#{u_#-ypizY+e+{7Sr`~eyKcg;KaRa|RJ)O@;J@mbKh_uwFU&C8%{35-`He<3|=5s^8e0xs5ka z(b1Dhccmx4GU)9*lAPTkY+p46<`ru~+sT@L>S~tKZ+&rkjBNmgtkdFOC!!e?FG4^$ z=UF`9wjCzBzHEd!Aw@?tor#MfwM z9o~BzaWdGPAri)Ro6|+ZWDsb}{OuFCa>tJiuYDr*hZ1(=MRwAilv|=vi{!t|oS;eM z_t9eSwdk7lx}@N?$!B44tJZo)l2vles%@1JsL1o)$!hHxg^~1eck9AG{d%KyXkwM0 zw76y)A7>c4C|DGlxDb3r!EJjUf4Wv1XRHI$Cyw7T_W>|N6 zU-8GCT1sA*h1rNXW?uCmGRWueFN{?UK79Gx{D)lXCg2yG3#g^P{O$0;NJD}9Hd0gc zXp(KA;3uQLux8>F#~ 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