From bfda8dc2b6ac03151f0f230ba938dbe5fb7ea5c9 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Petr=20Veli=C4=8Dka?= Date: Fri, 28 Mar 2025 09:46:23 +0100 Subject: [PATCH] prednaska 28.3.2025 --- linearni-rovnice.tex | 78 ++++++++++++++++++++++++++++++++++++++++++- skripta.pdf | Bin 123838 -> 136955 bytes skripta.tex | 3 ++ 3 files changed, 80 insertions(+), 1 deletion(-) diff --git a/linearni-rovnice.tex b/linearni-rovnice.tex index c83ee4e..a90bb65 100644 --- a/linearni-rovnice.tex +++ b/linearni-rovnice.tex @@ -48,7 +48,7 @@ V přednášce MA3 jste již studovali tento typ rovnic, teď se však budeme v \begin{theorem} \label{thm-unique-sol-lineq} - Nechť $t_0 \in (a, b), x_0 \in \mathbb{R}^n$ je dáno. Pak existuje jediné řešení rovnice \eqref{eq-linear-ode}, definované na celém $(a, b)$. + Nechť $t_0 \in (a, b), x_0 \in \mathbb{R}^n$ je dáno. Pak existuje jediné řešení rovnice \eqref{eq-linear-ode} definované na celém $(a, b)$ splňující počáteční podmínku $x(t_0) = x_0$. \end{theorem} \begin{proof} @@ -58,4 +58,80 @@ V přednášce MA3 jste již studovali tento typ rovnic, teď se však budeme v \hfill \textit{konec 5. přednášky (21.3.2025)} + Předpokládejme, že řešení není definované na celém $(a, b)$. Potom existují $\alpha, \beta \in (a, b)$ takové, že řešení je definováno na $(\alpha, \beta)$. Toto řešení musí opustit každý kompakt, tedy mimo jiné i $K = [t_0, \beta]\times \overline{B(0, R)}$, kde $R$ je dostatečně velké. + Řešení $x$ splňuje + $$ |x(t)| \leq |x(t_0)| \int_{t_0}^t \| A(s)\| |x(s)| + |g(s)| ds \overset{\begin{subarray}{c}\|A(s)\| \leq L\\|g(s)| \leq \tilde{C}\end{subarray}}{\leq} C + \int_{t_0}^t L|x(s)|C| ds \leq $$ + Z Gronwallova lemmatu dostaneme + $$ \leq \tilde{C} + C(\beta - t_0) + \int_{t_0}^t L|x(s)| ds \implies |x(t)| \leq \underbrace{[\tilde{C} + C(\beta - t_0) e^{L(\beta - t_0}]}_{R}. $$ + Došli jsme ke sporu s Větou \ref{thm-leaving-compact}, neboť řešení $x$ nemůže opustit kompakt $K$. \end{proof} + +Důležitá poznámka: řešení existuje globálně na oboru spojitosti $A(t), g(t)$. Ve skutečnosti předchozí věta i pro nelineární rovnice $x' = f(x, t)$ se sublineární pravou stranou, tj. pokud $|f(x, t)| \leq a(t)|x| + g(t)$, kde $a(\cdot), g(\cdot)$ jsou spojité. + +\begin{definition} + \textit{Homogenní rovnicí} rozumíme rovnici \eqref{eq-linear-ode} pro $g(t) \equiv 0$, tj. + \begin{equation} + \label{eq-homogenous-linear-ode} + x' = A(t)x, x(t_0) = x_0. + \end{equation} +\end{definition} + +Použijeme znalosti lineární algebry k tomu, abychom mohli formalizovat postup řešení lineárních ODR. + +\begin{theorem} + Množina $\mathcal{R}_H$ řešení homogenní rovnice \eqref{eq-homogenous-linear-ode} bez zadané počáteční podmínky tvoří $n$-dimenzionální podprostor $C^1((a, b), \mathbb{R}^n)$. +\end{theorem} + +\begin{proof} + Jádro lineárního zobrazení $Lx := x' - Ax$ je vektorový prostor. Dokážeme, že má dimenzi $n$. Nechť $i = 1,\dots,n$ a $x(t_0) = e_i$, pro tuto počáteční podmínku dostaneme řešení $x^i$. Potom $\{x^1, \dots, x^n\}$ tvoří bázi prostoru všech řešení. Skutečně, tyto vektory jsou lineárně nezávislé, mějme lineární kombinaci $c_1 x^1 + \dots + c_n x^n = 0$, speciálně v čase $t_0$ máme $c_1 e^1 + \dots + c_n x^n$, což implikuje, že $c_i = 0$ pro každé $i$. Navíc vezmeme libovolné řešení $z' = A(t) z$, opět zkoumejme stav v čase $t_0$. Máme $z(t_0) = d_1e^1 + \dots d_ne^n$ pro vhodná $d_1, \dots, d_n$. Definujme $y(t) := d_1 x^1(t) + \dots + d_n x^n(t)$, tedy $y$ řeší rovnici $y' = Ay$ a $y(t_0) = z(t_0)$, z čehož díky jednoznačnosti řešení dostáváme $y = z$. Tudíž jsme nalezli $n$-prvkovou bázi, tedy prostor $\mathcal{R}_H$ má dimenzi $n$. +\end{proof} + +\begin{definition} + \textit{Fundamentální systémem} pro \eqref{eq-homogenous-linear-ode} rozumíme libovolnou bázi $\mathcal{R}_H$. Matice, jejíž sloupce tvoří prvky libovolného fundamentálního systému, nazýváme \textit{fundamentální maticí} pro \eqref{eq-homogenous-linear-ode}. +\end{definition} + +Uvedeme si několik poznámek k definici fundamentální matice. Je-li $\Phi(t)$ nějaká fundamentální matice, pak +\begin{itemize} + \item $\Phi(t)$ splňuje ``maticový tvar \eqref{eq-homogenous-linear-ode}", tedy $\Phi'(t) = A\Phi(t)$. + \item $\Phi(t)$ je regulární pro každé $t \in (a, b)$. + \item Obecné řešení \eqref{eq-homogenous-linear-ode} má tvar $\Phi(t) c$, kde $c \in \mathbb{R}^n$. + \item $\tilde{\Phi}(t) := \Phi(t) \Phi^{-1}(t_0)$ je také fundamentální matice, která navíc splňuje $\tilde{\Phi}(t_0) = I$. +\end{itemize} + +\begin{theorem}[Variace konstant] + \label{thm-variation-of-constants} + Nechť $\Phi(t)$ je libovolná fundamentální matice pro \eqref{eq-homogenous-linear-ode}. Potom řešení nehomogenní rovnice \eqref{eq-linear-ode} lze napsat ve tvaru + $$ x(t) = \Phi(t)\Phi^{-1}(t_0)x_0 + \Phi(t) \int_{t_0}^t \Phi^{-1}(s) g(s) ds $$ + pro $t \in (a, b)$ +\end{theorem} + +\begin{proof} + Zderivováním dostaneme $x' = A(t) x + g(t)$, dále stačí ověřit počáteční podmínku dosazením. +\end{proof} + +\begin{definition} + \textit{Wronského determinant} (Wronskián) rovnice \eqref{eq-homogenous-linear-ode} je reálná funkce $w(t) := \det(\Phi(t))$, kde $\Phi$ je libovolná fundamentální matice příslušné rovnice. +\end{definition} + +\begin{theorem}[Liouvilleova formule] + \label{thm-liouville-formula} + Nechť $\Phi(t)$ je maticové řešení \eqref{eq-homogenous-linear-ode} a nechť $w(t) = \det \Phi(t)$. Potom + $$ w(t) = w(t_0) \exp \left( \int_{t_0}^t \tr A(s) ds \right), $$ + kde $\tr A$ je stopa matice $A$. +\end{theorem} + +\begin{proof} + Dokazovaná rovnost je ekvivalentní s + $$ w'(t) = w(t_0) \exp\left( \int_{t_0}^t \tr A(s) ds \right) \tr A(t) $$ + a tedy + $$ w'(t) = \tr A(t) w(t), w(t_0) = w(t_0) $$ + Dále + $$ \odv*{\det \Phi(t)}{t} = \odv*{\sum_\sigma (-1)^{\sgn \sigma} \Phi_{1, \sigma(1)}(t) \dots \Phi_{n, \sigma(n)}(t)}{t} = $$ + $$ \sum_{k = 1}^n \sum_\sigma (-1)^{\sgn \sigma} \overbrace{\Phi \dots \Phi}^{\Phi' \text{ je v $k$-tém řádku }} = \sum_{k=1}^n \det D_k, $$ + kde $D_k$ je matice $\Phi$ se zderivovaným $k$-tým řádkem. + + \hfill \textit{konec 6. přednášky (28.3.2025)} + +\end{proof} + +Pokud $\tr A(t) > 0$, potom wronskián roste, $=0$ zachovává objem a $<0$ objem klesá. diff --git a/skripta.pdf b/skripta.pdf index 21f56ffed63ed520e8b198ed77db75191d3d2bb8..e9c68beb180a091fc141d3aedb0410c8add82873 100644 GIT binary patch delta 63822 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