From 4a8568092d61277a96481f294464fd68434f90a5 Mon Sep 17 00:00:00 2001 From: flavis Date: Wed, 1 Jul 2020 11:02:59 +0200 Subject: [PATCH] add ana18 --- ana13.tex | 1 + ana17.tex | 1 + ana18.tex | 193 +++++++++++++++++++++++++++++++++++++++++++++++++ analysisII.pdf | Bin 847625 -> 862805 bytes analysisII.tex | 1 + 5 files changed, 196 insertions(+) create mode 100644 ana18.tex diff --git a/ana13.tex b/ana13.tex index cffcdc6..523d912 100644 --- a/ana13.tex +++ b/ana13.tex @@ -30,6 +30,7 @@ \begin{salign*} f(x+h) - f(x) = \left( \int_{0}^{1} J_{f}(x + sh) \d{s} \right) h. \end{salign*} + \label{satz:mittelwertsatz} \end{satz} \begin{proof} Sei $f: D \to \R^{m}$. Sei $g_{j}\colon [0,1] \to \R, \ g_{j}(s) \coloneqq f_{j}(x+sh)$. Dann gilt: diff --git a/ana17.tex b/ana17.tex index 43bc572..8814b5a 100644 --- a/ana17.tex +++ b/ana17.tex @@ -197,6 +197,7 @@ \end{bsp} \begin{bsp}[Uneindeutigkeit von AWA] + \label{bsp:dgl-uneindeutig} Sei $y' = f(t,y) \coloneqq \sqrt{|y(t)|}$ mit $y(t_0) = y_0$. Es ist $f(t,y)$ stetig auf $\R \times \R$. diff --git a/ana18.tex b/ana18.tex new file mode 100644 index 0000000..99191d8 --- /dev/null +++ b/ana18.tex @@ -0,0 +1,193 @@ +\documentclass{lecture} + +\begin{document} + +\section{Eindeutigkeit und lokale Stabilität} + +\begin{definition}[Lipschitz-Stetigkeit] + Sei $D \subseteq \R \times \R^{n}$ und $f\colon D \to \R^{n}$. $f$ ist in $D$ + Lipschitz-stetig bzgl. $x$ mit Lipschitz-Konstante $L \ge 0$, falls $\forall (t,x), (t, \tilde{x}) \in D$ gilt + \[ + \Vert f(t,x) - f(t, \tilde{x}) \Vert \le L \Vert x - \tilde{x} \Vert + .\] $f$ ist lokal Lipschitz-stetig bzgl. $x$ in $D$, falls für + alle Punkte $(t, x) \in D$ eine Umgebung $U$ existiert, s.d. + $f$ in $D \cap U$ Lipschitz-stetig bezüglich $x$ ist. +\end{definition} + +\begin{bsp} + \begin{enumerate}[(1)] + \item Sei $D = \R \times G$, $G \subseteq \R^{n}$ konvex + und $f\colon D \to \R^{n}$ stetig partiell differenzierbar nach $x$ + mit $\displaystyle \sup_{x \in G} \left\Vert \frac{\partial f(t,x)}{\partial x} \right\Vert \le L(t)$, + dann ist $f$ Lipschitz-stetig bezüglich $x$. + \begin{proof} + \begin{align*} + \Vert f(t,x) - f(t, \tilde{x}) \Vert + \quad \stackrel{\text{\ref{satz:mittelwertsatz}}}{=} \quad + \left\Vert \int_{0}^{1} \frac{\partial f}{\partial x} (t,s) \d s (x - \tilde{x}) \right\Vert + \le L(t) \Vert x - \tilde{x}\Vert + .\end{align*} + \end{proof} + \item $f(y) = \sqrt{y}$, $f\colon [0, \infty[\; \to \R$ ist nicht Lipschitz-stetig, denn + $|f(y) - f(0)| = \sqrt{y} $ und $\forall L \ge 0$ $\exists y \in [0, \infty[$ mit + $|\sqrt{y}| \ge L \cdot |y|$. + \item $f(y) = \sqrt{y} $, $f\colon \;]0, \infty[\; \to \R$ ist lokal Lipschitz-stetig. + \begin{proof} + Sei $y_0 \in ]0, \infty[$ fest. Betrachte + $U = \left[ \frac{y_0}{2}, \infty \right] \subseteq \R$. Es gilt + $\left| f'(y) \right| = \left| \frac{1}{2 \sqrt{y} } \right| $. Dann folgt + \[ + \max_{x \in U} \left| \frac{1}{2 \sqrt{y} } \right| = \frac{1}{2} \sqrt{\frac{2}{y_0}} + .\] Damit ist mit (1) $f$ Lipschitz-stetig auf $U$. + \end{proof} + \end{enumerate} +\end{bsp} + +\begin{lemma}[von Gronwall] + Sei $w(t) \ge 0$ stückweise stetig und genüge für $a, b \in \R$ der Integralgleichung + \[ + w(t) \le a \int_{t_0}^{t} w(s) \d s + b, \quad t \ge t_0 + .\] Dann gilt + \[ + w(t) \le e^{a(t-t_0)}b, \quad t \ge t_0 + .\] + \label{lemma:gronwall} +\end{lemma} + +\begin{proof} + Sei $t \ge t_0$. Betrachte die Funktion $\psi(t) \coloneqq a \int_{t_0}^{t} w(s) \d s + b$. Es gilt + $\psi'(t) = a w(t)$ und nach Voraussetzung + \[ + \psi'(t) = a w(t) \le a \left( a \int_{t_0}^{t} w(s) \d s + b \right) = a \psi(t) + .\] Betrachte nun $e^{-at}\psi(t)$ und berechne + \begin{salign*} + (e^{-at} \psi(t))' = - a e^{-at}\psi(t) + e^{-at}\psi'(t) + = e^{-at} \underbrace{\left( \psi'(t) - a \psi(t) \right)}_{\le 0} \le 0 \tageq\label{eq:gronwall-1} + .\end{salign*} + Die Funktion $e^{-at}\psi(t)$ ist also monoton fallend. Damit folgt + \begin{salign*} + e^{-at}\psi(t) &\stackrel{\text{mon. fallend}}{\le } e^{-a t_0} \psi(t_0) = e^{-a t_0} b \\ + e^{-at} w(t) &\stackrel{\text{(\ref{eq:gronwall-1})}}{\le} e^{-at} \psi(t) \le e^{- a t_0} b + \intertext{Insgesamt folgt also} + w(t) &\le e^{a(t - t_0)} b + .\end{salign*} +\end{proof} + +\begin{satz}[Stabilitäts und Eindeutigkeitssatz] + Sei $f(t,x)$ stetig auf $D \subseteq \R \times \R^{n}$ und lokal Lipschitz-stetig + bezüglich $x$. Seien $y, v$ zwei beliebige Lösungen der Differentialgleichung + \[ + y'(t) = f(t, y(t)), \quad t \in I + \] auf einem gemeinsamen Existenzintervall $I$. Dann gilt + \[ + \Vert y(t) - v(t) \Vert \le e^{L(t-t_0)} \Vert y(t_0) - v(t_0)\Vert, \quad t \in I, t \ge t_0 + ,\] wobei $t_0 \in I$ und $L$ die Lipschitz-konstante $L = L_K$ von $f(t,x)$ auf + $K \subseteq D$ mit $K$ beschränkt und $\text{Graph}(y) \subseteq K$, $\text{Graph}(v) \subseteq K$. + + Weiterhin: Falls $y(t_0) = v(t_0)$ (d.h. $y$ und $v$ lösen AWA $y' = f(t,y), y(t_0) = y_0)$, dann + gilt $y(t) = v(t)$, $\forall t \in I$, d.h. die lokale Lösung der AWA (nach Peano) ist eindeutig bestimmt. +\end{satz} + +\begin{proof} + Sei $K \subseteq D$ eine beschränkte Teilmenge und $\text{Graph}(y) \subseteq K$, $\text{Graph}(v) \subseteq K$. + Betrachte + \begin{salign*} + y(t) - v(t) &\stackrel{\text{Integralform}}{=} + y(t_0) + \int_{t_0}^{t} f(s, y(s)) \d s - v(t_0) - \int_{t_0}^{t} f(s, v(s)) \d s \\ + &= \int_{t_0}^{t} \left( f(s, y(s)) - f(s, v(s)) \right) \d s + y(t_0) - v(t_0) + \intertext{Dann folgt} + \Vert h(t) \Vert &\le \int_{t_0}^{t} \underbrace{\Vert f(s, y(s)) - f(s, v(s)) \Vert}_{\le L_k \Vert y(s) - v(s) \Vert} \d s + \Vert y(t_0) - v(t_0)\Vert \\ + &\stackrel{\text{Lipschitz}}{\le } L_K \int_{t_0}^{t} \Vert h(s) \Vert \d s + \Vert y(t_0) - v(t_0) \Vert + \intertext{Damit folgt mit Lemma \ref{lemma:gronwall}} + \Vert h(t) \Vert &\le e^{L(t-t_0)} \Vert y(t_0) - v(t_0)\Vert, \quad t \ge t_0 + .\end{salign*} + Seien nun $y(t)$ und $v(t)$ zwei Lösungen der AWA + \[ + \begin{cases} + y' = f(t,y) & t \in I = [t_0, t_0 + T], T \text{ aus Peano} \\ + y(t_0) = y_0 + \end{cases} + .\] Aus + \[ + \Vert y(t) - v(t) \Vert \le e^{L(t-t_0)} \Vert y_0 - y_0\Vert = 0, \quad t \in I + ,\] folgt $y(t) = v(t)$ auf dem gemeinsamen Existenzintervall $I$. +\end{proof} + +\begin{satz}[Existenzsatz von Picard-Linderlöf] + Sei $f\colon D \to \R^{n}$ stetig, lokal Lipschitz-stetig bezüglich $x$. Dann + gilt $\forall (t_0, y_0) \in D$, $\exists \epsilon > 0$ und eine Lösung der AWA + \begin{align*} + &y\colon I = [t_0 - \epsilon, t_0 + \epsilon] \to \R^{n} \\ + &y'(t) = f(t, y(t)), \quad t \in I \\ + &y(t_0) = y_0 + .\end{align*} +\end{satz} + +\begin{proof} + Unabhängig vom Satz von Peano, basiert auf dem Fixpunktsatz von Banach. + \begin{enumerate}[1)] + \item Sei $\delta > 0$ s.d. + \[ + K\coloneqq \{ (t,x) \in \R \times \R^{n} \mid |t-t_0| \le \delta, \Vert x - y_0 \Vert \le \delta \} \subseteq D + \] und $f(t,x)$ auf $K$ Lipschitz-stetig ist, d.h. es ex. ein $L_K \ge 0$, s.d. + \[ + \Vert f(t,x) - f(t, \tilde{x}) \Vert \le L_K \Vert x - \tilde{x} \Vert, + \quad (t,x), (t, \tilde{x}) \in K + .\] $K$ ist kompakt und $f$ stetig, d.h. $f$ ist beschränkt auf $K$, d.h. $\exists M \ge 0$ + s.d. $\Vert f(t,x) \Vert \le M$, $(t,x) \in K$. Wir setzen + \[ + \epsilon \coloneqq \text{min}\left( \delta , \frac{\delta }{M}, \frac{1}{2L_K} \right), + \quad I_{\epsilon} \coloneqq [t_0 - \epsilon, t_0 + \epsilon] + .\] Definiere Vektorraum $V \coloneqq C(I_{\epsilon})$ mit Norm + $\Vert y \Vert_{\infty} = \max_{t \in I_\epsilon} \Vert y(t) \Vert$. Dann ist + $V$ ein Banach-Raum. + \item Definiere auf $V$ die Abbildung $g\colon V \to V$ + \[ + g(y)(t) \coloneqq y_0 + \int_{t_0}^{t} f(s, y(s)) \d s, \quad t \in I_{\epsilon} + .\] Betrachte Teilmenge + \[ + V_0 \coloneqq \left\{ v \in V \;\Big|\; \max_{t \in I_{\epsilon}} \Vert v(t) - y_0 \Vert \le \delta + \right\} + \subseteq V + .\] Für $y \in V_0$ gilt für $t \in I_{\epsilon}$ + \begin{salign*} + \Vert g(y)(t) - y_0 \Vert &= \left\Vert \int_{t_0}^{t} f(s, y(s)) \d s \right\Vert \\ + &\le \int_{t_0}^{t} \left\Vert f(s, y(s))\right\Vert \d s \\ + &\stackrel{f \text{ beschr.}}{\le } M \int_{t_0}^{t} \d s \\ + &= M|t-t_0 + \le M\epsilon + \le \delta + .\end{salign*} + Damit folgt also $g(V_0) \subseteq V_0$. Seien nun $y, v \in V_0$: + \begin{salign*} + \Vert g(y)(t) - g(v)(t) \Vert + &= \left\Vert \int_{t_0}^{t} \left( f(s, y(s)) - f(s, v(s)) \right) \d s \right\Vert \\ + &\le \int_{t_0}^{t} \Vert f(s, y(s)) - f(s, v(s))\Vert \d s \\ + &\stackrel{\text{Lipschitz}}{\le } \int_{t_0}^{t} L_K \Vert y(s) - v(s) \Vert \d s \\ + &\le L_k \max_{s \in [t_0, t]} \Vert y(s) - v(s) \Vert \int_{t_0}^{t} \d s \\ + &\le L_k \Vert y - v \Vert_{\infty} \underbrace{|t - t_0|}_{\le \epsilon} \\ + &\stackrel[\epsilon \le \frac{1}{2L_K}]{}{\le} \frac{1}{2} \Vert y - v\Vert_{\infty} + .\end{salign*} + Damit ist $g$ auf $V_0$ eine Kontraktion und hat damit mit Satz \ref{satz:banach-fix} + genau einen Fixpunkt $y^{*}$, d.h. + \[ + y^{*}(t) = g(y^{*})(t) = y_0 + \int_{t_0}^{t} f(s, y^{*}(s)) \d s, \quad t \in I_{\epsilon} + .\] Damit ist $y^{*}(t)$, $t \in I_{\epsilon}$ eindeutige lokale Lösung der + AWA $y'= f(t,y)$, $y(t_0 ) = y_0$. + \end{enumerate} +\end{proof} + +\begin{bem} + \begin{enumerate}[(1)] + \item Der Beweis liefert ein Verfahren für beliebiges $y_0$ und $t \in I_{\epsilon}$: + \begin{align*} + y^{k}(t) \coloneqq y_0 + \int_{t_0}^{t} f(s, y^{k-1}(s)) \d s + \xrightarrow{k \to \infty} \text{Lösung der AWA} + .\end{align*} + \item Ohne Die Forderung der Lipschitz-stetigkeit geht die Eindeutigkeit der Lösung verloren + (siehe Beispiel \ref{bsp:dgl-uneindeutig}), aber die Existenz gilt nach dem Satz von Peano + immer noch. + \end{enumerate} +\end{bem} + +\end{document} diff --git a/analysisII.pdf b/analysisII.pdf index 08eba110f64e6c3ec8afab7a97aa51bc9ce56e95..9156729593d57ffc88256bdba8429a8c76dacd85 100644 GIT binary patch delta 159667 zcmZsiWl&vP*QMbc+}+&??oMzE?(XjHaBz1B2iF7%?(P=cU4lb!*8rXS-nYN*s_s9# zR;`+Q)vnq<<``o>D|(A#ae=dz1{N&31i%Dv@_|LoVLyQu(-fT6S&TNL zn)F)u>>?=e2%I9h1bCZF5j{4@U6!hQy4@??Wzh$)uWxL@-FohHe0+M&}zK`}T$n@+eS(;il=|x^>c?b0mnlMX1Aizx;ZLjP`2_;>rOb{z?n3U_TKo%-v5VYK^3{c}#<4 zTaZh_>otW;r$|0PPy*HHwYUdSp|`rDN;nH*>S`&RI|&rKR=nR7=MRasV%I`?o!+)r 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diff --git a/analysisII.tex b/analysisII.tex index 726434d..12e03f0 100644 --- a/analysisII.tex +++ b/analysisII.tex @@ -40,5 +40,6 @@ Rui Yang (\href{mailto:rui.yang@stud.uni-heidelberg.de}{rui.yang@stud.uni-heidel \input{ana15.tex} \input{ana16.tex} \input{ana17.tex} +\input{ana18.tex} \end{document}