From 6d5e80692a76767c1ea7e770491cd5fdca5b367a Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Petr=20Vely=C4=8Dko?= Date: Thu, 15 May 2025 08:56:03 +0200 Subject: [PATCH] prednaska 9.5.2025 --- ljapunovske-funkce.tex | 95 +++++++++++++++++++++++++++++++++++++++++ skripta.pdf | Bin 194399 -> 207782 bytes skripta.tex | 1 + 3 files changed, 96 insertions(+) create mode 100644 ljapunovske-funkce.tex diff --git a/ljapunovske-funkce.tex b/ljapunovske-funkce.tex new file mode 100644 index 0000000..d146383 --- /dev/null +++ b/ljapunovske-funkce.tex @@ -0,0 +1,95 @@ +\section{Ljapunovské funkce a stabilita} + +V této kapitule se pokusíme vybudovat pokročilejší teorii, která nám umožní určit stabilitu u většího počtu možných obyčejných diferenciálních rovnic. Budeme uvažovat evoluční rovnici \eqref{eq-ode} +$$ x' = f(x, t), $$ +kde $f: \Omega \times I \to \R^n$ a $\Omega\subset\R^n$ je otevřená. Bez újmy na obecnosti dále nechť $f(0, t) = 0$ pro všechna $t \in I$ a $I = [0, +\infty)$, neboli jinými slovy $0$ je stacionární řešení této rovnice. Budeme zkoumat stabilitu tohoto nulového řešení. + +\begin{definition} + Funkci $\omega: \Omega \to [0, +\infty)$ nazveme \textit{pozitivně definitní}, pokud je spojitá, $\omega(0) = 0$ a $\omega(x) > 0$ pro všechna $x \in \Omega \setminus \{0\}$. +\end{definition} + +Pozitivně definitní funkce jsou například $f(x, y) = x^2 + y^2$ nebo $g(x) = \sum_{i=1}^n a_i x_i^{2k_i}$ pro $k_i \in \N$ a $a_i > 0$. + +\begin{definition} + Funkci $V(x, t): \Omega \times I \to [0, +\infty)$ nazveme \textit{ljapunovskou} pro \eqref{eq-ode} v $\Omega$, jestliže + \begin{enumerate}[(i)] + \item $V$ je spojitá a $V(0, t) = 0$ pro všechna $t \in I$; + \item $t \mapsto V(x(t), t)$ je nerostoucí pro každé řešení $x$ rovnice \eqref{eq-ode}; + \item existuje pozitivně definitní funkce $\omega$ v $\Omega$ taková, že $V(x, t) \geq \omega(x)$ pro všechna $x \in \Omega$ a $t \in I$. + \end{enumerate} +\end{definition} + +V některých případech první integrál může být kandidátem na ljapunovskou funkci, ale ne vždy to tak vyjde. Dále, je-li $V$ třídy $C^1$, je podmínka (ii) ekvivalentní nekladnosti orbitální derivace +$$ \pdv{V}{t}(x, t) + \nabla_x V(x, t)\cdot f(x, t) \leq 0 $$ +pro všechna $(x, t) \in \Omega \times I$. Důkaz je analogický důkazu Věty \ref{thm-first-integral}. + +\begin{theorem}[Ljapunovská funkce a stabilita] + \label{thm-ljapunov} + Nechť \eqref{eq-ode} má ljapunovskou funkci pro bod $0$, pak nulové řešení je stabilní. +\end{theorem} + +\begin{proof} + Volme $\varepsilon > 0$ a chceme najít $\delta > 0$ tak, aby $|x_0| < \delta$ implikovalo $|x(t)| < \varepsilon$ v každém čase $t \geq 0$. Bez újmy na obecnosti nechť $\overline{U(0, \varepsilon)} \subset \Omega$. Dále budeme uvažovat ljapunovskou funkci $V$ a příslušnou pozitivně definitní funkci $\omega$ (takovou, že $V(x, t) \geq \omega(x)$ pro každé $x \in \Omega$). + + Na sféře $S_\varepsilon := \{x \in \R^n: |x| = \varepsilon\}$ je funkce $\omega$ kladná a spojitá a $S_\varepsilon$ je kompaktní, tedy $\omega$ nabývá na $S_\varepsilon$ svého minima $\alpha > 0$ v nějakém bodě $\xi \in S_\varepsilon$. + + Máme, že $\omega(x) \geq \alpha$ pro všechna $x \in S_\varepsilon$ a chceme, aby $V(x_0, t_0) < \alpha$. Pro dané $t_0$ je $V(\cdot, t_0)$ spojitá a $V(0, t_0) = 0$. Díky tomu existuje $\delta > 0, \delta < \varepsilon$ takové, že $V(x, t_0) < \alpha$ na $U(0, \delta)$. + + Nechť nyní $x_0 \in U(0, \delta)$, tedy $V(x_0, t_0) < \alpha$, a tedy $V(x(t), t) \leq V(x_0, t_0) < \alpha$ pro všechna $t \geq t_0$. Pro spor předpokládejme, že existuje čas $t_1$ takový, že $|x(t_1)| \geq \varepsilon$, ale pak by díky spojitosti musel existovat čas $t_2$ takový, že $|x(t_2)| = \varepsilon$, tedy $x(t_2) \in S_\varepsilon$, ale + $V(x(t_2, t_2)) \geq \omega(x(t_2)) \geq \alpha$, což je spor s volbou vhodného $\delta$. +\end{proof} + +\begin{theorem}[Ljapunovská funkce a asymptotická stabilita] + \label{thm-asymptotic-ljapunov} + Nechť rovnice \eqref{eq-ode} má v $\Omega \times I$ ljapunovskou funkci $V$. Nechť navíc existují v $\Omega$ pozitivně definitní funkce $\lambda$ a $\eta$ takové, že + \begin{enumerate}[(i)] + \item $V(x, t) \leq \lambda(x)$ pro každé $(x, t) \in \Omega \times I$, + \item $\odv*{V(x(t), t)}{t} \leq -\eta(x(t))$, kdykoli $x$ řeší \eqref{eq-ode} v $\Omega$. + \end{enumerate} + Potom $0$ je asymptoticky stabilní v $I$. +\end{theorem} + +K důkazu této věty budeme potřebovat následující lemma o zachovávání konvergence při aplikaci pozitivně definitní funkce (poznamenejme si, že opačná implikace platí také díky větě o limitě složené funkce). + +\begin{lemma} + Nechť $\omega$ je pozitivně definitní v $\Omega$ a $\varepsilon > 0$ je takové, že $\overline{U(0, \varepsilon)} \subset \Omega$. Buď $(x_n)_{n=1}^\infty$ posloupnost v $\overline {U(0, \varepsilon)}$ splňující $\omega(x_n) \to 0$. Potom $x_n \to 0$. +\end{lemma} + +\begin{proof} + Volme $\tilde \varepsilon > 0$, $\tilde \varepsilon < \varepsilon$ libovolně. Potom $\overline{U(0, \varepsilon)} \setminus U(0, \tilde\varepsilon)$ je kompaktní, $\omega$ je spojitá, tedy zde nabývá svého minima $\omega(x_0) > 0$. Potom existuje $n_0$ takové, že pro všechna $n \geq n_0$ máme $\omega(x_n) < \omega(x_0)$, a tedy $x_n \in U(0, \tilde\varepsilon)$. +\end{proof} + +\begin{proof}[Důkaz Věty \ref{thm-asymptotic-ljapunov}] + Stabilitu jsme již dokázali ve Větě \ref{thm-ljapunov}, stačí tedy dokázat, že počátek je lokální atraktor. Mějme $\varepsilon > 0$, k tomu ze stability nalezneme $\delta > 0$ takové, že pro $|x_0| < \delta$ máme $|x(t)| < \varepsilon$ v každém čase $t \geq 0$. + + Dále uvažujme $x_0$ takové, že $|x_0| < \delta$. Funkce $t \mapsto V(x(t), t)$ je neklesající a nezáporná, tedy existuje $\lim_{t \to +\infty} V(x(t), t) = a \geq 0$. Dále platí + $$ \int_{t_0}^t \eta(x(s)) ds \leq - \int_{t_0}^t \odv*{V(x(s), s)}{s} ds = V(x(t_0), t_0) - V(x(t), t), $$ + pro $t$ jdoucí do nekonečna dostáváme $V(x(t_0), t_0) - V(x(t), t) \to V(x(t_0), t_0) - a$, + a tedy + $$\int_{t_0}^\infty \eta(x(s)) ds \leq V(x(t_0), t_0). $$ + Jmenovitě tento integrál je konečný, a tedy existuje posloupnost $t_n \nearrow +\infty$ taková, že $\eta(x(t_n)) \to 0$ (pozor, $\eta(x(t))$ samotná nemusí konvergovat k nule). + Z právě dokázaného lemmatu dostáváme, že $x(t_n) \to 0$, z čehož díky spojitosti a pozitivní definitnosti funkce $\lambda$ v $0$ dostáváme $\lambda(x(t_n)) \to 0$. + + Dále můžeme psát $0 \leq V(x(t_n), t_n) \leq \lambda(x(t_n)) \to 0$, + a tedy $V(x(t_n), t_n) \to 0$. Díky větě o limitě podposloupnosti dostáváme, že $V(x(t), t) \to 0 = a$ a jelikož $0 \leq \omega(x(t)) \leq V(t, x(t)) \to 0$, musí nutně platit $\omega(x(t)) \to 0$. + + Jelikož posloupnost $t_n$ byla volena libovolně, máme díky předchozímu lemmatu $x(t_n) \to 0$ pro libovolnou $t_n \nearrow +\infty$, a tedy dle Heineovy věty $x(t) \to 0$, což jsme chtěli dokázat. + +\end{proof} + +Na závěr si uvedeme větu, která nám poskytuje několik možných charakterizaci pro asymptotickou stabilitu nulového řešení. + +\begin{theorem}[Ekvivalentní podmínky pro rovnici \eqref{eq-linhom-const}] + Je dána rovnice $x' = Ax$, kde $A \in \R^{n\times n}$. Následující výroky jsou ekvivalentní: + \begin{enumerate}[(i)] + \item $0$ je asymptoticky stabilní v $[0, +\infty)]$; + \item $\Re \lambda < 0$ pro všechna $\lambda \in \sigma(A)$, neboli $A$ je hurwitzovská; + \item existují $\alpha, c > 0$ taková, že $\|e^{tA}\| \leq ce^{-\alpha t}$ pro všechna $t \geq 0$; + \item existuje symetrická pozitivně definitní matice $B \in \R^{n \times n}$ taková, že + $$ A^T B + BA = -I. $$ + \end{enumerate} +\end{theorem} + +Poslední rovnosti se běžně říká \textit{Ljapunovova rovnice}. 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+\include{ljapunovske-funkce} \end{document}