From daad49908f353b77162301b1e299dcbca30ef454 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Petr=20Veli=C4=8Dka?= Date: Fri, 11 Apr 2025 09:45:23 +0200 Subject: [PATCH] prednaska 11.4.2025 --- linearni-rovnice-konst-koef.tex | 28 +++++++++++++ skripta.pdf | Bin 152685 -> 165390 bytes skripta.tex | 3 ++ stabilita.tex | 67 ++++++++++++++++++++++++++++++++ 4 files changed, 98 insertions(+) create mode 100644 stabilita.tex diff --git a/linearni-rovnice-konst-koef.tex b/linearni-rovnice-konst-koef.tex index e51c9d9..c4ca47d 100644 --- a/linearni-rovnice-konst-koef.tex +++ b/linearni-rovnice-konst-koef.tex @@ -62,6 +62,7 @@ Z obecného tvaru řešení dostáváme, že $x(t) = \Phi(t) \Phi^{-1}(t_0) x_0$ \end{proof} \begin{corollary}[Variace konstant pro \eqref{eq-linhom-const}] + \label{thm-variation-hom-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 @@ -86,3 +87,30 @@ Další otázka, kterou se budeme zabývat je hledání maticové exponenciály. \end{corollary} \hfill \textit{konec 7. přednášky (4.4.2025)} + +\begin{definition} + Pro matici $A \in \R^{n \times n}$ a její spektrum $\sigma(A)$ definujeme $\sigma_-(A) = \sigma(A) \cap \{\Re < 0\}$, $\sigma_0(A) = \sigma(A) \cap \{\Re = 0\}$, $\sigma_+(A) = \sigma(A) \cap \{\Re > 0\}$. Příslušné podprostory generované příslušnými (zobecněnými) vlastními vektory značíme $X_-(A), X_0(A), X_+(A)$ (nazýváme je \textit{stabilní, centrální} a \textit{nestabilní} podprostor). +\end{definition} + +Zřejmě $\R^n = X_+(A) \oplus X_-(A) \oplus X_0(A)$. Tyto prostory jsou invariantní vzhledem k $A$ a též vzhledem k $e^{tA}$. + +\begin{theorem}[Asymptotické chování podprostorů] + Nechť $A$ je daná matice. Potom existují kladná $\alpha, \beta, M$ a $c$ taková, že platí: + \begin{enumerate} + \item Pokud $x_0 \in X_-(A)$, pak $|e^{tA}x_0| \leq ce^{-\alpha t}|x_0|$ pro každé $t \geq 0$. + \item Pokud $x_0 \in X_+(A)$, pak $|e^{tA}x_0| \leq ce^{\beta t}|x_0|$ pro každé $t \leq 0$. + \item Pokud $x_0 \in X_0(A)$, pak $|e^{tA}x_0| \leq c(1 + |t|)^M|x_0|$ pro každé $t \in \R$. + \end{enumerate} +\end{theorem} + +\begin{proof} + Nejdříve nechť $x_0 \in X_-(A)$. Potom $x_0 = \sum_{i=1}^k a_i v_i$, kde $v_i$ jsou zobecněné vlastní vektory příslušné $\lambda_i \in \sigma_-(A)$. + Dále máme, že $e^{tA} x_0 = Ve^{tJ}V^{-1} x_0$. Spočteme $V^{-1}x_0$. Jestliže $v$ je sloupec matice $V$, potom $V^{-1}v$ je jeden ze sloupců jednotkové matice, tedy má tvar $(0,\cdots,0,1,0,\cdots,0)^T$. + Tedy $V^{-1}x_0$ má nenulové hodnoty jen v řádcích příslušných $\Re \lambda < 0$. Můžeme odhadovat normu + $$ \|e^{tA} x_0 \| \leq \|V\|\|e^{tJ} \text{: řádky s } e^{-at}\| \| V^{-1} x_0\| \leq C e^{-\alpha t} \|x_0\|. $$ + Zde jsme využili faktu, že ``polynom" $e^{-\lambda t} t^k$ lze odhadnout $e^{-\lambda t} t^k \leq e^{(-\lambda + \varepsilon)t}c$ pro vhodná $c$ a $\varepsilon$. + + Důkaz ostatních implikací je podobný. +\end{proof} + +V předchozí větě platí i opačná implikace, a to ve smyslu, že uvedené vlastnosti charakterizují dané podprostory. diff --git a/skripta.pdf b/skripta.pdf index fada8a5e49c4e6284aaba3281e26576a91270400..4fe74023d532a5a9d53bac84a3669bac3dc506eb 100644 GIT binary patch delta 62563 zcmV)OK(@c_s0ogY3V^f$rVW2JF)%PbJ_>Vma%Ev{3V7P>J!@}dw{_q9E2jM4@9^0V_`iQW{O;A^H-8Y~fIk^9>hS7xu!etG0a6~02ADO< z$iu6L|GMEE-f}MPIaiK@04}t+Ju0s77cDq9o^!`JxA=d`#F6ljN6-@*Apt*mdZYYm zfPTsU0!hE&=P8a?ckQnI^5Ak)HCM<;iXU(kx8s~2R z`_*@cqfpE%=?#7p%xf)dg0s@MM=9w9w39IHUEBphOv*4xyD9<;oQtc)Lsy{Hi2~|O z1Rw!SCxEGka5MyDo?OW^FOQ(gqX^tI6TY2Cn*3`$e*|6JF#dnu%$wdVE1Npc1oU)@ 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diferenciální rovnice (NMMA336)} \author{Petr Velička \footnote{\href{mailto:petrvel@matfyz.cz}{petrvel@matfyz.cz}}\\přednášející: doc. RNDr. Tomáš Bárta, Ph.D. \footnote{\href{mailto:barta@karlin.mff.cuni.cz}{barta@karlin.mff.cuni.cz}}} \date{LS 2024/25} @@ -43,5 +45,6 @@ \include{zavislost-na-podmince} \include{linearni-rovnice} \include{linearni-rovnice-konst-koef} +\include{stabilita} \end{document} diff --git a/stabilita.tex b/stabilita.tex new file mode 100644 index 0000000..5aa4659 --- /dev/null +++ b/stabilita.tex @@ -0,0 +1,67 @@ +\section{Stabilita} + +Lemma \ref{lemma-sol-dist} nám teoreticky poskytuje spojitost řešící funkce v proměnné $x_0$, pro větší $t$ však kvůli exponenciálnímu růstu nemá význam. Budeme proto zkoumat okolnosti, za nichž existují odhady, které se nezhoršují pro $t \in \infty$. + +\begin{definition} + Nechť $f = f(x, t)$ je spojitá v otevřené $\Omega \in \R^{n+1}$ a navíc lokálně lipschitzovská vůči $x$. Nechť $\Omega \supset \{0\} \times I$ kde $I = (\tau, \infty)$ a nechť $f(0, t) = 0$ pro všechna $t \in I$. Řekneme, že nulové řešení rovnice $x' = f(t, x)$ \eqref{eq-ode} je + \begin{enumerate}[(i)] + \item \textit{stabilní}, jestliže pro všechna $t_0 \in I$ a $\varepsilon > 0$ existuje $\delta > 0$ takové, že $|x_0| < \delta$ implikuje, že $\varphi(t, t_0, x_0)$ je definováno a splňuje $|\varphi(t, t_0, x_0)| < \varepsilon$ pro $t \geq t_0$; + \item \textit{nestabilní}, jestliže není stabilní; + \item \textit{lokální atraktor}, jestliže $\forall t_0 \in I$ existuje $\eta > 0$ tak, že $|x_0| < \eta$ implikuj, že $\varphi(t, t_0, x_0)$ je definováno pro všechna $t \geq t_0$ a navíc $\varphi(t, t_0, x_0) \to 0$ pro $t \to +\infty$; + \item \textit{asymptoticky stabilní}, jestliže je stabilní a navíc lokální atraktor; + \item \textit{uniformně stabilní}, jestliže pro všechna $\varepsilon > 0$ existuje $\delta > 0$ takové, že pro všechna $t_0 \in I$ z $|x_0| < \delta$ plyne $\varphi(t, t_0, x_0)$ je definováno a splňuje $|\varphi(t, t_0, x_0)| < \varepsilon$ pro $t \geq t_0$; + \item \textit{uniformě asymptoticky stabilní}, jestliže je uniformně stabilní a navíc existuje $\eta < 0$ takové, že $\forall \varepsilon > 0$ existuje $T > 0$ takové, že pro všechna $t_0 \in I$ z $|x_0| < \eta$ plyne, že $\varphi(t, t_0, x_0)$ je definováno pro všechna $t \geq t_0$ a $|\varphi(t, t_0, x_0)| \leq \varepsilon|$ pro $t \geq t_0 + T$. + \end{enumerate} +\end{definition} + +Pojem asymptotické stability zavádíme proto, že lokální atraktor nutně nemusí implikovat stabilitu. Konstrukci takového řešení můžeme nahlédnout pomocí tzv. Vinogradovova systému. +V případě autonomní rovnice splývají pojmy (asymptotické) stability a uniformní (asymptotické) stability, neboť můžeme psát $\varphi(t, t_0, x_0) = \varphi(t - t_0, 0, x_0)$. + +Obecněji řešeno, řešení $\tilde x(t)$ rovnice $x' = f(x, t)$ se nazve stabilní (resp. uniformně stabilní atd.), jestliže má analogickou vlastnost nulové řešení rovnice $u' = g(u, t)$ kde $g(u, t) = f(\tilde x(t) + u, t) - f(\tilde x(t), t)$. + +V případě řešení lineární rovnice \eqref{eq-linear-ode}, tj. $x' = A(t)x + g(t)$ je stabilita libovolného řešení příslušné homogenní rovnice \eqref{eq-homogenous-linear-ode}. + +\begin{theorem} + Je dána rovnice $x' = A(t) x$, kde $A(t)$ je spojitá v $I = (\tau, \infty)$. Nechť $\Phi(t)$ je (libovolná) fundamentální matice. Potom nulové řešení je + \begin{enumerate} + \item stabilní, právě když pro $\forall t_0 \in I$ je $\|\Phi(t)\|$ omezená v $[t_0, \infty)$; + \item asymptoticky stabilní, právě když $\|\Phi(t)\| \to 0$ pro $t \to \infty$; + \item uniformně stabilní, právě když existuje $c > 0$ takové, že pro všechna $s < t \in I$ je $\|\Phi(t)\Phi^{-1}(s)\| \leq c$. + \item uniformně asymptoticky stabilní, právě když existují kladná $\alpha$ a $c$ taková, že pro všechna $s < t \in I$ je $\|\Phi(t)\Phi^{-1}(s)\| \leq ce^{-\alpha(t-s)}$. + \end{enumerate} +\end{theorem} + +\begin{theorem} + Nechť $A$ je konstantní matice. Potom nulové řešení rovnice $x' = Ax$ je + \begin{enumerate} + \item (uniformně) stabilní, právě když $\Re \lambda \leq 0$ pro všechna vlastní čísla $\lambda \in \sigma(A)$, přičemž $\Re \lambda = 0$ pouze pro polojednoduchá vlastní čísla (tedy příslušné Jordanovy buňky mají velikost 1). + \item (uniformě) asymptoticky stabilní, právě když $\Re \lambda \leq 0$ pro všechna vlastní čísla $\lambda \in \sigma(A)$. + \end{enumerate} +\end{theorem} + +\begin{proof} + Plyne ihned z tvaru maticové exponenciály. +\end{proof} + +Matice $A$ splňující $\Re \lambda < 0$ pro všechna $\lambda \in \sigma(A)$ se nazývá \textit{Hurwitzowská}. + +\begin{lemma} + Je dána rovnice $x' = Ax + r(x, t)$. Nechť existují kladná $\alpha, c$ tak, že $\|e^{tA}\| \leq ce^{-t\alpha}$ pro $t \geq 0$. Nechť dále $r(x, t): \R^{n+1} \to \R^n$ je spojitá a $|r(x, t)| \leq \gamma |x|$ pro všechna $x, y$ kde $\gamma < \frac{\alpha}{c}$. Pak každé řešení splňuje + $$ |x(t)| \leq c|x(t_0)| \exp(-\beta(t - t_0)) $$ + pro $t \geq t_0$, kde $\beta = \alpha - c\gamma > 0$. +\end{lemma} + +\begin{proof} + Nechť $x$ řeší $x' = Ax + r(x, t)$ na $(0, +\infty)$. Pak $x$ řeší $x' = Ax + g(t)$, kde $g(t) := r(x(t), t)$. Z variace konstant (Důsledek \ref{thm-variation-hom-const}) dostáváme, že + $$ x(t) = e^{(t - t_0)A} x_0 + \int_{t_0}^t e^{(t-s)A} g(s) ds. $$ + Pro $t > t_0$ dostaneme + $$ \|x(t)\| \leq ce^{-(t-t_0)\alpha} \|x_0\| + \int_{t_0}^t ce^{-(t-s)\alpha} \gamma \|x(s)\| ds. $$ + Jinými slovy, + $$ \|x(t)\| e^{t\alpha} \leq ce^{t_0\alpha} \|x_0\| + \int_{t_0}^t ce^{-(t-s)\alpha} \gamma \|x(s)\| ds. $$ + Z Gronwallova lemmatu (Lemma \ref{lemma-gronwall}) dostáváme + $$ e^{t\alpha} \| x(t) \| \leq ce^{t_0\alpha} \|x_0\| e^{c\gamma(t - t_0}. $$ + Po opětovném přenásobení exponenciálou nakonec máme + $$ \|x(t)\| \leq c \|x_0\| e^{(t - t_0)(c\gamma - \alpha)} = ce^{-\beta(t - t_0)} \| x_0 \|. $$ +\end{proof} + +\hfill \textit{konec 8. přednášky (11.4.2025)}