From c70db175ba1884e69b4277b40edfe70d7698a483 Mon Sep 17 00:00:00 2001 From: flavis Date: Tue, 30 Jun 2020 18:57:28 +0200 Subject: [PATCH] finish peano proof in ana16, add ana17 --- ana16.tex | 65 +++++++++++++- ana17.tex | 236 +++++++++++++++++++++++++++++++++++++++++++++++++ analysisII.pdf | Bin 753554 -> 843404 bytes analysisII.tex | 1 + 4 files changed, 299 insertions(+), 3 deletions(-) create mode 100644 ana17.tex diff --git a/ana16.tex b/ana16.tex index 006e48f..0406533 100644 --- a/ana16.tex +++ b/ana16.tex @@ -91,7 +91,7 @@ Notationen: $x' = f(t,x), \dot x = f(t,x), \dv{x}{t} = f(t,x)$ (Dynamischer Proz [%minor tick num=4, %grid style={line width=.1pt, draw=gray!10}, %major grid style={line width=.2pt,draw=gray!50}, - %axis lines=middle, + axis lines=middle, %enlargelimits={abs=0.2}, %ymax=5, %ymin=0 @@ -120,12 +120,12 @@ Notationen: $x' = f(t,x), \dot x = f(t,x), \dv{x}{t} = f(t,x)$ (Dynamischer Proz } ] \addplot3 (x,y,0); - \addlegendentry{$f(t,x)$} + \addlegendentry{$f(t,y)$} \addplot{(x-0.25)^2+0.15}; \addlegendentry{$y(t)$} \end{axis} \end{tikzpicture} - \caption{Veranschaulichung: Richtungsfeld für DGL der Form $y' = f(x)$} + \caption{Veranschaulichung: Richtungsfeld für DGL der Form $y' = f(t,y)$} \end{figure} \begin{definition}[System erster Ordnung] @@ -398,6 +398,65 @@ Reminder: \[\max_{t\in I} \norm{y^{h_i} - y(t)} \xrightarrow{i\to \infty} 0\] Offenbar ist $\graph(y)\subset D$. \item \textbf{z.Z.} $y(t)$ erfüllt die Differentialgleichung $y'(t) = f(t,y(t))$ oder äquivalent dazu: $y(t)$ erfüllt die Integralgleichung \[y(t) = y_0 + \int_{t_0}^{t} f(s,y(s))\d s\] + + Sei dazu $t \in [t_{k-1}, t_k] \subseteq I$, $y^{i}(t) \coloneqq y^{h_i}(t)$. Für + ein $i$ gilt + \begin{salign*} + y^{i}(t) \stackrel{\text{\ \ \ \ }}{=}& y_{k-1}^{i} + (t - t_{k-1}) f(t_{k-1}, y_{k-1}^{i}) \\ + =& y_{k-2}^{i} + (t_{k-1} - t_{k-2})f(t_{k-2}, y_{k-2}^{i}) + + (t - t_{k-1}) f(t_{k-1}, y_{k-1}^{i}) \\ + \vdots \; & \\ + =& y_0 + \sum_{j=1}^{k-1} (t_j - t_{j-1}) f(t_{j-1}, y_{j-1}^{i}) + + (t-t_{k-1})f(t_{k-1}, y_{k-1}^{i}) \\ + =& y_0 + \sum_{j=1}^{k-1} \int_{t_{j-1}}^{t_j} f(t_{j-1}, y_{j-1}^{i}) \d s + + \int_{t_{k-1}}^{t} f(t_{k-1}, y_{k-1}^{i}) \d s \\ + &+ \int_{t_0}^{t} f(s, y^{i}(s)) \d s - \int_{t_0}^{t} f(s, y^{i}(s)) \d s \\ + =& y_0 + \sum_{j=1}^{k-1} \int_{t_{j-1}}^{t_j} (f(t_{j-1}, y_{j-1}^{i}) + - f(s, y^{i}(s)) \d s \\ + &+ \int_{t_{k-1}}^{t} (f(t_{k-1}, y_{k-1}^{i}) - f(s, y^{i}(s))) \d s + + \int_{t_0}^{t} f(s, y^{i}(s)) \d s + \tageq \label{eq:peano:1} + .\end{salign*} + Die Funktionen der Folge $(y^{i})_{i \in \N}$ sind gleichgradig stetig, d.h. + $\forall i$, $\forall \epsilon' > 0$, $\exists \delta _{\epsilon'}$ s.d. + \[ + |t-t'| < \delta_{\epsilon'} \implies \Vert y^{i}(t) - y^{i}(t') \Vert < \epsilon' + .\] + Da $D$ kompakt, ist die stetige Funktion $f(t,x)$ auch gleichmäßig stetig. Damit folgt + $\forall \epsilon > 0$, $\exists \epsilon' < \epsilon$, $\exists \delta_{\epsilon'}$, s.d. + \[ + |t-t'| < \delta_{\epsilon'}, \Vert y^{i}(t) - y^{i}(t') \Vert < \epsilon' + \implies \Vert f(t, y^{i}(t)) - f(t', y^{i}(t')) \Vert < \epsilon + .\] Falls $h_i$ hinreichend klein folgt damit $\forall k$ + \[ + \max_{s \in [t_{k-1}, t_k]} \Vert f(t, y^{i}(t)) - f(s, y^{i}(s) \Vert \le \epsilon + \tageq \label{eq:peano:2} + .\] Damit folgt + \begin{salign*} + \left\Vert y^{i}(t) - y_0 - \int_{t_0}^{t} f(s, y^{i}(s)) \d s \right\Vert + \kern -1mm \stackrel{\ref{eq:peano:1}}{=}& + \Bigg\Vert \sum_{j=1}^{k-1} \int_{t_{j-1}}^{t_j} + \left( f(t_{j-1}, y_{j-1}^{i}) - f(s, y^{i}(s)) \right) \d s \\ + &+ \int_{t_{k-1}}^{t} \left( f(t_{k-1} y_{k-1}^{i}) - f(s, y^{i}(s)) \right) \d s + \Bigg\Vert \\ + \le& \sum_{j=1}^{k-1} \int_{t_{j-1}}^{t_j} \Vert f(t_{j-1}, y_{j-1}^{i}) - f(s, y^{i}(s)) \Vert \d s \\ + &+ \int_{t_{k-1}}^{t} \Vert f(t_{k-1}, y_{k-1}^{i}) - f(s, y^{i}(s)) \Vert \d s \\ + \stackrel{\ref{eq:peano:2}}{\le}& \sum_{j=1}^{k-1} \epsilon \int_{t_{j-1}}^{t_j} \d s + + \epsilon \int_{t_{k-1}}^{t} \d s \\ + =& \epsilon |t - t_0| + \intertext{Damit folgt} + \Bigg\Vert \underbrace{y^{i}(t)}_{\xrightarrow{i \to \infty} y(t)} - y_0 + - \int_{t_0}^{t} \underbrace{f(s, y^{i}(s))}_{\xrightarrow{i \to \infty} f(s, y(s))} \d s + \Bigg\Vert + \kern -1mm \le& \epsilon |t-t_0| + \intertext{Also folgt} + \left\Vert y(t) - y_0 - \int_{t_0}^{t} f(s, y(s)) \d s \right\Vert \kern -1mm\le& \epsilon |t-t_0| + .\end{salign*} + Da $\epsilon$ beliebig ist, folgt damit + \[ + y(t) = y_0 + \int_{t_0}^{t} f(s, y(s)) \d s + .\] \end{enumerate} \end{proof} + \end{document} diff --git a/ana17.tex b/ana17.tex new file mode 100644 index 0000000..30f29a8 --- /dev/null +++ b/ana17.tex @@ -0,0 +1,236 @@ +\documentclass{lecture} + +\begin{document} + +\begin{bem}[Bedeutung des Existenzsatz von Peano] + Für die lokale Lösbarkeit des Systems $y' = f(t,y)$ reicht + die Stetigkeit der rechten Seite. Aber auch wenn $f(t,y)$ auf einem + Streifen $[a,b] \times \R^{n}$ stetig ist, kann nicht erwartet werden, dass die Lösung + der AWA im Intervall $[a,b]$ definiert ist. + + Zum Beispiel: $y' = 1 + y^2$. $f(t,y) = 1 + y^2$ ist stetig auf $\R \times \R$. Als Lösung + folgt $y = \tan(t + c)$, denn + \begin{align*} + y' = \frac{1}{\cos^2(t+c)} = \frac{\cos^2(t+c) + \sin^2(t+c)}{\cos^2(t+c)} + = 1 + \tan^2(t+c) = 1 + y^2 + .\end{align*} + Das heißt Lösungen sind nur in Intervallen der Länge $\pi$ definiert. Der Satz von Peano + macht eine Aussage über die Größe des Existenzintervalls (die nur von Stetigkeitseigenschaften + von $f(t,x)$ abhängig ist). +\end{bem} +\begin{figure}[h] + \label{fig:tan-dgl-solution} + \centering + \begin{tikzpicture} + \begin{axis}% + [default 2d plot, + ymax=4, + ymin=-4, + xmin=-4, + xmax=4, + xtick={-3.14, -1.57, 0, 1.57, 3.14}, + xticklabels={$\pi$, $-\frac{\pi}{2}$, $0$, $\frac{\pi}{2}$, $\pi$} + ] + \addplot[domain=1.58:4.70,samples=100,smooth,red] {tan(deg(x))}; + \addplot[domain=-1.56:1.56,samples=100,smooth,red] {tan(deg(x))}; + \addplot[domain=-4.72:-1.58,samples=100,smooth,red] {tan(deg(x))}; + \end{axis} + \end{tikzpicture} + \caption{Lösung $y = \tan(t+c)$ nur in Intervallen der Länge $\pi$ definiert.} +\end{figure} + +\begin{satz}[Fortsetzungssatz] + Sei $f(t,x)$ stetig auf $D \subseteq \R \times \R^{n}$, $D$ abgeschlossen und $(t_0, y_0) \in D$. + Sei weiter $y(t)$ Lösung der AWA für $t \in [t_0 - T, t_0 + T]$. + + Dann ist $y$ nach rechts und links auf ein maximales Existenzintervall $I_{\text{max}} = (t_0 - T^{*}, t_0 + T^{*})$ + bis zum Rand von $D$ (stetig diff'bar) fortsetzbar. +\end{satz} + +\begin{proof} + Wiederholte Anwendung des Satz von Peano (siehe R.R. S. 111). +\end{proof} + +\begin{bem} + Die maximal fortgesetzten Lösungen nach links und rechts laufen bis der Graph von $y$ an + den Rand von $D$ stößt. Dabei ist es möglich, dass + \[ + \text{Graph}(y) = \{ (t, y(t)) , t \in I_{\text{max}}\} + \] unbeschränkt ist, weil $t \to t_0 + T^{*} = \infty$ oder + $\Vert y(t) \Vert \xrightarrow{t \to t_0 + T^{*}} \infty$. +\end{bem} + +\begin{korollar}[Globale Existenz] + Sei $f(t,x)$ auf ganz $\R \times \R^{n}$ definiert und stetig. Seien alle lokalen Lösungen + $y(t)$ beschränkt durch eine stetige Funktion $\rho\colon \R \to \R$ mit + \[ + \Vert y(t) \Vert \le \rho(t), \quad t \in [t_0 - T, t_0 + T] + .\] Dann ist $y$ fortsetzbar auf ganz $\R$. +\end{korollar} +\begin{proof} + Wegen der Schranke, kann keine lokale Lösung auf einem beschränkten Zeitintervall + einen unbeschränkten Graphen haben. Also ist $y$ fortsetzbar auf ganz $\R$. +\end{proof} + +\begin{satz}[Regularitätssatz] + Sei $y$ eine Lösung der AWA $y' = f(t,y)$ auf dem Intervall $I$ und + sei $f \in C^{m}(D)$ mit $m \ge 1$. Dann gilt $y \in C^{m+1}(I)$. +\end{satz} + +\begin{proof} + Da $y(t)$ Lösung, folgt + \[ + y(t) = y_0 + \int_{t_0}^{t} f(s,y(s)) \d s, \quad t \in I + .\] Sei nun $f \in C^{1}(D)$. Dann ist $y$ zweimal stetig differenzierbar + mit + \begin{align*} + y''(t) = \frac{\mathrm{d}}{\mathrm{d}t} f(t, y(t)) + = \underbrace{\frac{\partial}{\partial t} f(t, y(t))}_{\text{stetig}} + + \underbrace{\frac{\partial}{\partial y} f(t, y(t))}_{\text{stetig}} + \underbrace{\frac{\mathrm{d}y}{\mathrm{d}t}}_{= f \text{ also stetig}} + .\end{align*} + Durch Wiederholung dieses Arguments folgt $y \in C^{m+1}$ falls $f \in C^{m}(D)$ ist. +\end{proof} + +\begin{bsp} + Was kann passieren, falls $f(t,x)$ nicht stetig ist? Beispiel: Coulomb Reibung. Sei + $c > 0$ und $v(0) = v_0$. + \begin{align*} + \dot{s} &= v \\ + \dot{v} &= -c \cdot \text{sign}(v) + .\end{align*} + Lösung: $v(t) = v_0 - ct$. Lösungen der DGL existieren ab $t = \frac{v_0}{c}$ nicht. + Abhilfe: Philipov Regel (siehe Literatur). + \begin{figure}[h] + \centering + \begin{tikzpicture}[declare function={f(\x) = -2;}] + \begin{axis}% + [%minor tick num=4, + %grid style={line width=.1pt, draw=gray!10}, + %major grid style={line width=.2pt,draw=gray!50}, + axis lines=middle, + legend pos=outer north east, + %enlargelimits={abs=0.2}, + %ymax=5, + %ymin=0 + width=0.6\textwidth, % Overall width of the plot + axis equal image, % Unit vectors for both axes have the same length + view={0}{90}, % We need to use "3D" plots, but we set the view so we look at them from straight up + xmin=0, xmax=2, % Axis limits + ymin=-1.1, ymax=1.1, + domain=0:2, y domain=-1:1, % Domain over which to evaluate the functions + xtick=\empty, ytick={1}, % Tick marks + xticklabels={$t_0$}, + yticklabels={$v_0$}, + xlabel=$t$, + ylabel=$v$, + samples=7, % How many arrows? + ] + \addplot3[ + y domain=1:0.1, + gray, + quiver={ + u={1}, v={-2}, % End points of the arrows + scale arrows=0.1, + every arrow/.append style={ + -latex % Arrow tip + }, + }] (x,y+0.1,0); + \addplot3[ + forget plot, + y domain=-1:-0.1, + gray, + quiver={ + u={1}, v={2}, % End points of the arrows + scale arrows=0.1, + every arrow/.append style={ + -latex % Arrow tip + }, + }] (x, y-0.1,0); + \addlegendentry{$f(t,v) = - c \cdot \text{sign}(v)$} + \addplot[blue, domain=0:1] {2 - 2*x}; + \addlegendentry{$v^{(1)}(t) = - c + v_0^{(1)}$} + \addplot[forget plot, blue, domain=0:0.5] {1 - 2*x}; + \addplot[red, domain=0:0.75] {-1.5 + 2*x}; + \addlegendentry{$v^{(2)}(t) = + c + v_0^{(2)}$} + \end{axis} + \end{tikzpicture} + \caption{Mögliche Lösungen der DGL $\dot{v} = - c \cdot \text{sign}(v)$ mit + $v_0^{(1)} > 0$ und $v_0^{(2)} < 0$. Ab $t = \frac{v_0}{c}$ existiert keine Lösung.} + \end{figure} +\end{bsp} + +\begin{bsp}[Uneindeutigkeit von AWA] + Sei $y' = f(t,y) \coloneqq \sqrt{|y(t)|}$ mit $y(t_0) = y_0$. Es + ist $f(t,y)$ stetig auf $\R \times \R$. + + Für $y_0 \ge 0$ gilt $\forall t_0 \in \R$: + \begin{align*} + y(t) &= \frac{(t-t_0 + 2 \sqrt{y_0})^2}{4} \qquad t_0 - 2 \sqrt{y_0} \le t < \infty + \intertext{Für $y_0 \le 0$ gilt $\forall t_0 \in \R$:} + y(t) &= - \frac{(t-t_0 - 2 \sqrt{-y_0})^2}{4} \qquad -\infty < t \le t_0 + 2 \sqrt{-y_0} + \intertext{Für $y_0 = 0$ ist jedoch $\forall t_0 \in \R$ auch} + y(t) &= 0 + .\end{align*} + eine Lösung der AWA. + + Falls $y_0 > 0$ oder $y_0 < 0$ ist $y(t; t_0, y_0)$ eindeutig bestimmt, aber für $y(t_0) = 0$ + existieren unendlich viele Lösungen. + + \begin{figure}[h] + \begin{subfigure}[b]{.5\linewidth} + \begin{tikzpicture} + \begin{axis}% + [default 2d plot, + grid=none, + ymax=10, + ymin=-10, + xmin=-10, + xmax=20, + xtick={6}, ytick={4,-4}, + yminorticks=false, + minor tick style={draw=none}, + xticklabels={$t_0$}, yticklabels={$y_0$, $y_0$} + ] + \addplot[domain=2:10,samples=100,smooth,red] {((x-6+4)^2)/4}; + \addplot[domain=-10:10,samples=100,smooth,blue] {-((x-6-4)^2)/4}; + \addplot[dashed,domain=10:20,samples=100,smooth,red] {((-(x-6)+4)^2)/4}; + \addplot[dashed,domain=-10:2,samples=100,smooth,blue] {-((x-6+4)^2)/4}; + \end{axis} + \end{tikzpicture} + \caption{Lösungen für $y_0 > 0$ (rot) bzw. $y_0 < 0$ (blau) und ihre Fortsetzungen.} + \end{subfigure} + \begin{subfigure}[b]{.5\linewidth} + \begin{tikzpicture} + \begin{axis}% + [default 2d plot, + grid=none, + ymax=10, + ymin=-10, + xmin=-10, + xmax=20, + xtick={6}, ytick=\empty, + xticklabels={$t_0$} + ] + \addplot[domain=-2:14,samples=100,smooth,orange] {0}; + \addplot[domain=14:20,samples=100,smooth,orange] {((x-18+4)^2)/4}; + \addplot[domain=-20:-2,samples=100,smooth,orange] {-((x+6-4)^2)/4}; + \addplot[domain=2:10,samples=100,smooth,blue] {0}; + \addplot[domain=10:20,samples=100,smooth,blue] {((x-14+4)^2)/4}; + \addplot[domain=-20:2,samples=100,smooth,blue] {-((x+2-4)^2)/4}; + \addplot[domain=4:8,samples=100,smooth,red] {0}; + \addplot[domain=8:20,samples=100,smooth,red] {((x-12+4)^2)/4}; + \addplot[domain=-20:4,samples=100,smooth,red] {-((x-4)^2)/4}; + %\addplot[dashed,domain=10:20,samples=100,smooth,red] {((-(x-6)+4)^2)/4}; + %\addplot[dashed,domain=-10:2,samples=100,smooth,blue] {-((x-6+4)^2)/4}; + \end{axis} + \end{tikzpicture} + \caption{Für $y_0 = 0$ existieren beliebig viele zusammengesetzte Lösungen.} + \end{subfigure} + \end{figure} + + Beobachtung: $f(t,x)$ ist stetig auf $\R \times \R$, aber $f(t,x)$ ist nicht Lipschitz stetig + in $(t,0)$ $\forall t \in \R$. +\end{bsp} + +\end{document} diff --git a/analysisII.pdf b/analysisII.pdf index 30ecc459f3a660296fa72299a3c8665fea6a7dda..814834cc43007d5f62ed0f24b982572aa80c79fc 100644 GIT binary patch delta 125839 zcmX_mb95hF)OBOqwrwYkoit`+HMWxpezDcqYMe$58r!xS+xC~|eb@W_x6irfoV#bw zp0(!An#2#L=yr|(5Euv?1Ob9@q!1F_#-)mAV`JDP>33!AnueHnDc`e8KgMivUdH%&pJ%rlUd 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