From 472ca4ce30508280f308f713e1e2302fdf3c62af Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Petr=20Veli=C4=8Dka?= Date: Mon, 7 Apr 2025 07:47:48 +0200 Subject: [PATCH] prednaska 4.4.2025 --- linearni-rovnice-konst-koef.tex | 88 ++++++++++++++++++++++++++++++++ linearni-rovnice.tex | 15 +++++- skripta.pdf | Bin 136955 -> 152685 bytes skripta.tex | 4 ++ 4 files changed, 106 insertions(+), 1 deletion(-) create mode 100644 linearni-rovnice-konst-koef.tex diff --git a/linearni-rovnice-konst-koef.tex b/linearni-rovnice-konst-koef.tex new file mode 100644 index 0000000..e51c9d9 --- /dev/null +++ b/linearni-rovnice-konst-koef.tex @@ -0,0 +1,88 @@ +\section{Lineární rovnice s konstantními koeficienty} + +\begin{definition} + \textit{Lineární homogenní rovnice s konstantními koeficienty} a s maticí $A \in \R^{n\times n}$ je rovnice + \begin{equation}\label{eq-linhom-const}x' = Ax.\end{equation} +\end{definition} + +Myšlenkou studia těchto rovnic je analogie s rovnicí $x' = ax$ pro $a \in \R$, kde řešením je $x(t) = x_0e^{at}$. Ukážeme, že rovnice \eqref{eq-linhom-const} má řešení $x(t) = e^{At} x_0$. + +\begin{definition} + \textit{Maticovou exponenciálu} definujeme předpisem + $$ e^A = \sum_{k = 0}^\infty \frac{1}{k!} A^k $$ + 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 $\|e^A\| \leq e^{\|A\|}$. + +\begin{example} + Nechť $A = \begin{pmatrix}2 & 0\\0 & 1\end{pmatrix}$. Potom + $$e^A = \sum_{k=0}^\infty \frac{1}{k!} A^k = \sum_{k=0}^\infty \begin{pmatrix}\frac{2^k}{k!} & 0 \\ 0 & \frac{1^k}{k!}\end{pmatrix} = \begin{pmatrix}e^2 & 0\\0 & e\end{pmatrix}.$$ +\end{example} + +\begin{theorem} + Nechť $U(t) = e^{tA}$. Pak $U(t)$ je fundamentální matice rovnice \eqref{eq-linhom-const} a platí $U(0) = I$. +\end{theorem} + +\begin{proof} + Řada konverguje pro všechny matice, tedy i pro matici $tA$, což znamená, že $U$ je dobře definovaná. Platí + $$[U(t)]_{ij} = \left[\sum_{k=0}^\infty \frac{t^kA^k}{k!}\right]_{ij} = \sum_{k=0}^\infty \frac{1}{k!} [A^k]_{ij} t^k. $$ + Toto je mocninná řada s poloměrem konvergence $\infty$, tedy ji můžeme derivovat člen po členu (nultý člen se zderivuje na nulu). + $$U'(t) = \sum_{k=1}^\infty \frac{A^k}{k!} kt^{k-1} = \sum_{k=1}^\infty A \frac{1}{(k-1)!} A^{k-1}t^{k-1} = Ae^{tA} = A U(t).$$ + Vytknutí $A$ můžeme provést, neboť operátor násobení maticí $A$ je spojitý. + + Závěr ohledně $U(0)$ plyne z toho, že pro $t = 0$ je první člen sumy roven jednotkové matici a všechny ostatní jsou nulové. +\end{proof} + +Z obecného tvaru řešení dostáváme, že $x(t) = \Phi(t) \Phi^{-1}(t_0) x_0$, přičemž $t_0 = 0$ a tedy $U(0) = U^{-1}(0) = I$. Z toho již plyne $x(t) = e^{tA} x_0$. + +\begin{theorem}[Vlastnosti maticové exponenciály] + Platí následující vlastnosti maticové exponenciály + \begin{enumerate}[(i)] + \item $e^{aI} = e^a I$ pro $a \in \R$; + \item pokud $AB = BA$, pak $e^{A+B} = e^Ae^B$; + \item $e^{C^{-1}AC} = C^{-1}e^AC$; + \item $e^{-A} = (e^A)^{-1}$, speciálně $e^A$ je vždy regulární. + \end{enumerate} +\end{theorem} + +\begin{proof} + Budeme dokazovat postupně. + \begin{enumerate}[(i)] + \item Dosazením dostáváme + $$ e^{aI} = \sum_{k=0}^\infty \frac{a}{k!} I = I \sum_{k=1}^\infty \frac{a}{k!} Ie^a. $$ + \item Nejprve ukážeme, že $Be^{tA} = e^{tA}B$. To plyne z toho, že + $$Be^{tA} = B\sum_{k=0}^\infty \frac{t^kA^k}{k!} = \sum_{k=0}^\infty B\frac{t^kA^k}{k!} = \sum_{k=0}^\infty \frac{t^kA^k}{k!} B = e^{tA}B. $$ + Potom z definice $U(t) = e^{tA}e^{tB}$ a $U'(t) = Ae^{tA}e^{tB} + e^{tA}Be^{tB} = (A + B)U(t)$. Tedy $U(t)$ splňuje rovnici $x'(t) = (A+B)x(t)$, kterou také splňuje $\tilde U(t) = e^{(A+B)t}$. Z jednoznačnosti řešení této rovnice dostáváme $e^Ae^B = e^{A+B}$. + \item Z definice rozepíšeme + $$ e^{C^{-1}AC} = \sum_{k=1}^\infty \frac{1}{k!} (C^{-1}AC)^k = \sum_{k=1}^\infty \frac{1}{k!} C^{-1}A^kC = $$ + $$ C^{-1}\sum_{k=0}^\infty \frac{1}{k!}A^k C = C^{-1}e^A C. $$ + \item Okamžitě plyne z (ii), neboť $e^A e^{-A} = e^0 = I$ + \end{enumerate} +\end{proof} + +\begin{corollary}[Variace konstant pro \eqref{eq-linhom-const}] + Nechť $A \in \R^{n \times n}$, $g(t): (a, b) \to \R^n$ je spojitá, $t_0 \in (a, b)$ a $x_0 \in \R^n$ jsou dána. Potom řešení rovnice + $$ x' = Ax + g(t), x(t_0) = x_0$$ + má tvar + $$ x(t) = e^{(t - t_0)A}x_0 + \int_{t_0}^t e^{(t - s)A} g(s) ds. $$ +\end{corollary} + +Další otázka, kterou se budeme zabývat je hledání maticové exponenciály. K tomu použijeme takzvaný Jordanův kanonický tvar matice. + +\begin{theorem} + Nechť $A \in \R^{n \times n}$, $J$ její Jordanův kanonický tvar, $A = VJV^{-1}$ a $(\lambda_1, \dots, \lambda_n)$ je diagonála $J$. Potom $e^{tA} = Ve^{tJ}V^{-1}$, kde matici $e^{tJ}$ definujeme jako $diag(e^{t\lambda_1}, \dots, e^{t\lambda_n})$, přičemž $P(t)$ je blokově diagonální matice se stejně velkými a stejně uspořádanými bloky jako $J$ a blok velikosti $k$ matice $P(t)$ je roven + $$ \begin{pmatrix} + 1 & t & \frac{1}{2}t^2 & \cdots & \frac{1}{(k-1)!}t^{k-1} \\ + 0 & 1 & t & \cdots & \frac{1}{(k-2)!}t^{k-2} \\ + \vdots & \vdots & \vdots & \ddots & \vdots \\ + 0 & 0 & 0 & \cdots & 1 + \end{pmatrix}$$ +\end{theorem} + +\begin{corollary} + Buď $a = \max\{\Re \lambda: \lambda \in \sigma(A)\}$ a $m$ velikost největší Jordanovy buňky příslušné k vlastnímu číslu $\Re \lambda = a$. + Pak existuje $M > 0$, že $\|e^{tA}\| \leq Mt^{m-1}e^{at}$ pro všechna $t \geq 0$. Speciálně, pro všechna $\tilde a > a$ existuje $\tilde M > 0$ takové, že $\|e^{tA}\| \leq \tilde M e^{t\tilde a}$. +\end{corollary} + +\hfill \textit{konec 7. přednášky (4.4.2025)} diff --git a/linearni-rovnice.tex b/linearni-rovnice.tex index a90bb65..ef23626 100644 --- a/linearni-rovnice.tex +++ b/linearni-rovnice.tex @@ -132,6 +132,19 @@ Uvedeme si několik poznámek k definici fundamentální matice. Je-li $\Phi(t)$ \hfill \textit{konec 6. přednášky (28.3.2025)} + Dále si uvědomíme, že $\Phi'(t) = A(t)\Phi(t)$, přičemž násobení maticí zleva provádí řádkové úpravy na matici $\Phi(t)$. Konkrétně $\varphi_k^{j\prime}(t) = \sum_{i=1}^n a_{ki}(t)\varphi_i^j(t)$. + + Platí $\det D_k = A_{kk}(t) \det \Phi(t)$ (vlastnosti determinantu). Z toho dostáváme, že $w'(t) = \det \Phi(t) \sum_{k=1}^n A_{kk}(t) = w(t) = \tr A(t)$. \end{proof} -Pokud $\tr A(t) > 0$, potom wronskián roste, $=0$ zachovává objem a $<0$ objem klesá. +Pokud $\tr A(t) > 0$, potom wronskián roste, $=0$ množina možných hodnot řešení zachovává objem a pro $\tr A(t) <0$ v průběhu času objem klesá. + +\begin{example} + Řešme rovnici + $$ \begin{pmatrix}x \\ y\end{pmatrix}' = \begin{pmatrix}2 & 0\\0 & -2\end{pmatrix} \begin{pmatrix}x \\ y\end{pmatrix}. $$ + Dostáváme $x' = 2x, y' = -2y$, tedy $x = x(0)e^{2t}, y = y(0)e^{-2t}$. Nechť $x(0), y(0) \in [0, 1]$. Potom pro fixní $t_1 > 0$ dostáváme $x(t_1) \in [0, e^{2t_1}], y(t_1) \in [0, e^{-2t_1}]$. Obsah tohoto obdélníku je $e^{2t_1}e^{-2t_1} = 1$. Tedy, obsah je konstantní, což odpovídá pozorování z věty, neboť stopa matice ze zadání je nulová. +\end{example} + +\begin{example} + Mějme rovnici $x' = f(t, x)$. Ukážeme si, že roli stopy matice z předchozího příkladu tu hraje divergence $f$ v proměnné $x$. +\end{example} diff --git a/skripta.pdf b/skripta.pdf index e9c68beb180a091fc141d3aedb0410c8add82873..fada8a5e49c4e6284aaba3281e26576a91270400 100644 GIT binary patch delta 63493 zcmZsiQ*bU^&}L)v#I|kQ=80|FC&?Szwr$(CZQFL{tEsv8r)KWh4&uXaa zGU&Rd696+C7b8DEjI)cAsi7^5$HuXiOzc)0a?h#ySxR~!YQx)13yx$QWSj~*Pir&c z&G|Lqx{V(=oX6MgBO1Y<@)EBO+u4oCX2UkJZeN zf|87FqFEK)=EPeSuy}r=-6((}d9kl1seODw7q^X&@vG+2jlpJLXu3`1{mBhjMwSp%nC}V^x7ZCir zgf}O*p@enwcs|^+i79V&#bMyK$^CE55_R)fH-=J*QQr64Ba08RQ$H>d81H^alBk)W za8qKQWUv%&1zO>~d@7$*h&31vMMOLVhvpBcELdS!9Zs=(A`s;l*3smr-c{paaX34P zFk0ooZ>dO(L(u>;*HZWb6yvy~C!ov1zwFEmZUx7;0M1P@uI9_cy7o8-Io_N9Q*Sgb znU!N0-BJ!W$78uvL~lCYScJn2hmR=w9AtR+wJ}?a+OY_NkLy-;tFrU=2^6%87YygY zqd;IsqewjTOpw0`y0AX_xapj8!N5O4kKH@&JR}yAI4Gv}tFgBz%?^1l1Q1VdOI5`A z+Dl36kxgV!e1FnAX;Od%?1|^Ak9K`I;nHVvbogn4?4$bn=E` zjK5{1?$|WKrvu5XL$HQZ1sW80LpG7ifTj%BjElj 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RNDr. Tomáš Bárta, Ph.D. \footnote{\href{mailto:barta@karlin.mff.cuni.cz}{barta@karlin.mff.cuni.cz}}} \date{LS 2024/25} @@ -39,5 +42,6 @@ \include{maximalni-reseni} \include{zavislost-na-podmince} \include{linearni-rovnice} +\include{linearni-rovnice-konst-koef} \end{document}