From 1b9576423327e9e3c1c1a419abae5bad07871901 Mon Sep 17 00:00:00 2001 From: jakobus2205 <61044581+jakobus2205@users.noreply.github.com> Date: Tue, 7 Jul 2020 18:38:37 +0200 Subject: [PATCH] added ana19 part1 --- ana19.tex | 191 +++++++++++++++++++++++++++++++++++++++++++++++++ analysisII.pdf | Bin 863395 -> 1036024 bytes analysisII.tex | 1 + 3 files changed, 192 insertions(+) create mode 100644 ana19.tex diff --git a/ana19.tex b/ana19.tex new file mode 100644 index 0000000..79a610b --- /dev/null +++ b/ana19.tex @@ -0,0 +1,191 @@ +\documentclass{lecture} + +\begin{document} +\section{Globale Stabilität} + +\begin{definition}[Exponentielle Stabilität]s + Sei $y(t)$ die globale Lösung einer AWA. $y(t)$ heißt \underline{exponentiell stabil}, falls Konstanten $\delta, \alpha, A > 0$ sodass $\forall t_{*} \geq t_{0}$ und $\forall w_{*} \in \R^{n}$ mit $\norm{w_{*}} < \delta$ die gestörte AWA + \[ + v'(t) = f\left(t,v(t)\right), \ \ \ t \geq t_{*}, \ \ \ v(t_{*}) = y(t_{*}) + w_{*} + \] + eine globale Lösung $v(t)$ hat, für welche gilt: + \[ + \norm{v(t) -y(t)} \leq Ae^{-\alpha (t-t_{*})} \norm{w_{*}}, \ \ \ t \geq t_{*}. + \] +\end{definition} + +\begin{definition}[Monotone AWA] + Die Funktion $f(t,x)$ heißt \underline{stark monoton} falls ein $\lambda > 0$ existiert, sodass für alle $(t,x), (t,y) \in D$ gilt: + \[ + \left( f(t,x) - f(t,y), x-y \right) \geq \lambda \norm{x-y}^{2} + \] +\end{definition} + +\begin{satz}[Globaler Stabilitätssatz] + Sei $f(t,x)$ einer AWA stetig, lokal Lipschitz-stetig bezüglich $x$ und stark monoton. Dann sind alle Lösungen des AWA global und exponentiell stabil, mit $\delta$ beliebig und $\alpha = A = 1$. \\ + Zusatz: gilt $\sup_{t \geq t_{0}} \norm{f(t,0)} < \infty$, dann sind alle Lösungen gleichmäßig beschränkt. + \label{satz:global-stabil} +\end{satz} +\begin{proof} + \begin{enumerate}[1)] + \item Die Lösungen des AWA + \[ + y'(t) = f(t,y(t)), \ \ \ t \geq t_{0}, \ \ \ y(t_{0}) = y_{0}, + \] + und des gestörten AWA + \[ + v'(t) = f(t,v(t)), \ \ \ t \geq t_{*} \geq t_{0}, \ \ \ v(t_{*}) = y(t_{*}) + w_{*} + \] + existieren global und sind eindeutig. \\ + Da $f$ Lipschitz-stetig ist gilt: + \begin{salign*} + \norm{f(t,x)} \leq \norm{f(t,x) - f(t,0)} + \norm{f(t,0)} \leq L\norm{x} + \norm{f(t,0)}. + \end{salign*} + Für die Lösung $y(t)$ gilt: + \begin{salign*} + y(t) &= y_{0} + \int_{t_{0}}^{t} f(s,y(s)) \d{s} + \intertext{und damit} + \norm{y(t)} &\leq \norm{y_{0}} + \int_{t_{0}}^{t} \norm{f(s,y(s))} \d{s} \\ + &\leq \norm{y_{0}} + \int_{t_{0}}^{t} L \norm{y(s)} \d{s} + \int_{t_{0}}^{t} \norm{f(s,0)} \d{s} + \intertext{$\norm{f(s,0)}$ stetig auf $[t,t_{0}]$, also gleichmäßig beschränkt, d.h. es existiert ein $M_{t} > 0$ sodass für alle $s \in [t_{0},t]$ gilt $\norm{f(s,0)} \leq M_{t}$ und damit:} + &\leq L \int_{t_{0}}^{t} \norm{y(s)} \d{s} + \norm{y_{0}} + M_{t}\left| t-t_{0}\right| + \end{salign*} + Mit dem Lemma von Gronwall für $w(t) = \norm{y(t)}$ folgt: + \[ + \norm{y(t)} \leq e^{L(t-t_{0})} \left( \norm{y_{0}} + M_{t}\left| t-t_{0}\right| \right). + \] + Somit liefern der Satz von Peano und der Fortsetzungssatz, dass $y(t)$ auf dem maximalen Existenzintervall $I_{\max} = [t_{0},t_{\max})$ existiert, wobei entweder $t_{\max} = \infty$ gilt, oder $t_{\max} < \infty$ mit $\max_{[t_{0},t]} \norm{y(t)} \oldstackrel{t \to \infty}{\longrightarrow} \infty $. \\ + Wir führen $t_{\max} < \infty$ mit $\max_{[t_{0},t]} \norm{y(t)} \oldstackrel{t \to \infty}{\longrightarrow} \infty $ zum Widerspruch. Bereits gezeigt wurde, dass gilt: + \begin{salign*} + \norm{y(t)} &\leq e^{L(t-t_{0})}\left( \norm{y_{0}} + M_{t}\left| t-t_{0} \right| \right) \leq e^{L(t_{\max}-t_{0})} \left( \norm{y_{0}} + M_{t_{\max}}\left| t_{\max} -t_{0} \right| \right) + \end{salign*} + Also ist $\norm{y(t)}$ beschränkt für $t \to t_{\max}$. Dies ist ein Widerspruch, also gilt bereits $t_{\max} = \infty$. Damit existiert $y(t)$ für $t \in [t_{0},\infty)$. + + \item $y(t)$ ist exponentiell stabil. Dafür gilt es zu zeigen, dass für $t\geq t_{*}$: + \[ + \norm{v(t) - y(t)} \leq e^{-\lambda(t-t_{*})}\norm{w_{*}}. + \] + Sei $w(t) \coloneqq v(t) - y(t)$. Dann gilt: + \begin{align*} + &\frac{\d{}}{\d{t}} w(t) = 2\left(w'(t),w(t)\right)= 2 \left(f(t,v(t))-f(t,y(t)), v(t) - y(t) \right) + \intertext{und da $f$ stark monoton ist, folgt:} + &\frac{\d{}}{\d{t}} \norm{w(t)}^{2} \leq -2\lambda \norm{w(t)}^{2} \ \ \ \ \ \implies \ \ \ \ \ \frac{\d{}}{\d{t}} \norm{w(t)}^{2} +2\lambda \norm{w(t)}^{2} \leq 0. + \end{align*} + Ferner gilt: + \begin{align*} + \frac{\d{}}{\d{t}} e^{2\lambda (t-t_{*})} \norm{w(t)}^{2} = \underbrace{e^{2\lambda (t-t_{*})}}_{> 0} \underbrace{\left( \frac{\d{}}{\d{t}}\norm{w(t)}^{2} + 2\lambda\norm{w(t)}^{2} \right)}_{\leq 0} \leq 0 + \end{align*} + Woraus wir folgern: + \begin{salign*} + & \int_{t_{*}}^{t} \frac{\d{}}{\d{s}} \left( e^{2\lambda(s-t_{*})}\norm{w(s)}^{2} \right) \d{s} = e^{2\lambda(t-t_{*})}\norm{w(t)}^{2} - \norm{w(t_{*})}^{2} \leq 0 \\ + \implies \ & \norm{w(t)}^{2} \leq e^{-2\lambda(t-t_{*})}\norm{w(t)}^{2} \\ + \implies \ & \norm{w(t)} \ \leq e^{-\lambda(t-t_{*})}\norm{w(t^{*})} \\ + \implies \ & \norm{v(t) - y(t)} \leq e^{-\lambda(t-t_{*})}\norm{w_{*}}. + \end{salign*} + + \item $y(t)$ ist gleichmäßig beschränkt falls $\sup_{t \geq t_{0}} \norm{f(t,0)} < \infty$. Denn: \\ + Es gilt: + \[ + y'(t) = f(t,y(t)) \ \ \ \ \ \implies \ \ \ \ \ y'(t) - f(t,y(t)) + f(t,0) = f(t,0). + \] + Womit wir erhalten: + \[ + \left(y(t), y'(t)\right) - \left(f(t,y(t)) - f(t,0), y(t) - 0\right) = \left(f(t,0),y(t)\right). + \] + Wegen Monotonie gilt: + \[ + \left(y(t), y'(t)\right) = \frac{1}{2}\frac{\d{}}{\d{t}} \norm{y(t)}^{2} + \left(f(t,y(t)) - f(t,0), y(t)\right) \leq -\lambda\norm{y(t)}^{2}. + \] + Somit können wir folgern: + \begin{salign*} + \frac{1}{2} \cdot \frac{\d{}}{\d{t}} \norm{y(t)}^{2} + 2\lambda \norm{y(t)}^{2} &\leq \frac{1}{2} \cdot \frac{\d{}}{\d{t}} \norm{y(t)}^{2} - \left( f(t,y(t)) - f(t,0), y(t)\right)\\ + &= \left(f(t,0), y(t)\right) \oldstackrel{\text{CSU}}{\leq} \norm{f(t,0)}\cdot \norm{y(t)} \\ + &\leq \frac{1}{2\lambda} \norm{f(t,0)}^{2} + \frac{\lambda}{2} \norm{y(t)}^{2} + \end{salign*} + woraus folgt + \begin{salign*} + & \frac{1}{2} \cdot \frac{\d{}}{\d{t}} \norm{y(t)}^{2} + \lambda \norm{y(t)}^{2} - \frac{\lambda}{2} \norm{y(t)}^{2} \leq \frac{1}{2\lambda} \norm{f(t,0)}^{2} \\ + \implies \ \ & \frac{\d{}}{\d{t}} \norm{y(t)}^{2} + \lambda \norm{y(t)}^{2} \leq \frac{1}{\lambda} \norm{f(t,0)}^{2}. + \end{salign*} + Multiplikation mit $e^{\lambda(t-t_{0})}$ liefert: + \begin{salign*} + \frac{\d{}}{\d{t}} \left( e^{\lambda(t-t_{0})}\norm{y(t)}^{2}\right) = e^{\lambda(t-t_{0})} \left( \frac{\d{}}{\d{t}} \norm{y(t)}^{2} + \lambda \norm{y(t)}^{2} \right) \leq \frac{1}{\lambda} e^{\lambda(t-t_{0})}\norm{f(t,0)}^{2} + \end{salign*} + woraus folgt: + \begin{salign*} + \int_{t_{0}}^{t} \frac{\d{}}{\d{s}} \left( e^{\lambda(s-t_{0})}\norm{y(s)}^{2}\right) \d{s} = e^{\lambda(t-t_{0})}\norm{y(t)}^{2} - \norm{y(t_{0})}^{2} \leq \frac{1}{\lambda} \int_{t_{0}}^{t} e^{\lambda(s-t_{0})}\norm{f(s,0)}^{2}\d{s} + \end{salign*} + und ferner sogar + \begin{salign*} + \norm{y(t)}^{2} &\leq e^{-\lambda(t-t_{0})}\norm{y_{0}}^{2} + \frac{1}{\lambda} e^{-\lambda(t-t_{0})}\int_{t_{0}}^{t} e^{\lambda(s-t_{0})}\norm{f(s,0)}^{2}\d{s} \\ + &\leq e^{-\lambda(t-t_{0})}\norm{y_{0}}^{2} +\frac{1}{\lambda} \max_{s \in [t_{0},t]} \norm{f(s,0)}^{2}e^{-\lambda(t-t_{0})}\int_{t_{0}}^{t} e^{\lambda(s-t_{0})} \d{s}. + \end{salign*} + Wir halten fest: + \begin{salign*} + e^{-\lambda(t-t_{0})}\int_{t_{0}}^{t} e^{\lambda(s-t_{0})} \d{s} = e^{-\lambda(t-t_{0})}\left( \frac{1}{\lambda}e^{\lambda(t-t_{0})} -\frac{1}{\lambda}\right) \leq \frac{1}{\lambda}. + \end{salign*} + Somit können wir schließen: + \begin{salign*} + \norm{y(t)}^{2} \leq e^{-\lambda(t-t_{0})}\norm{y_{0}}^{2} + \frac{1}{\lambda^{2}} \max_{s \in [t_{0},t]} \norm{f(s,0)}^{2} + \end{salign*} + \end{enumerate} +\end{proof} + + + +\section{Lineare Systeme von Differentialgleichungen} + +\begin{definition}[Lineare AWA] + Sei $A(\cdot) \colon I \to \R^{n \times n}$ eine Matrixfunktion, sowie $b(\cdot) \colon I \to \R^{n}$ eine Vektorfunktion. Dann ist eine Lineares AWA der Form: + \begin{salign*} + y'(t) &= A(t)y + b(t), \ \ \ t\geq t_{0} \\ + y(t_{0}) &= y_{0}. + \end{salign*} +\end{definition} + +\begin{satz}[Lösung einer linearen AWA] + Seien $A \colon [t_{0}, \infty) \to \R^{n \times n}$, $b \colon [t_{0}, \infty) \to \R^{n}$ stetig. Dann gilt: + \begin{enumerate}[1)] + \item Die lineare AWA besitzt eine eindeutige globale Lösung $y \colon [t_{0},\infty) \to \R^{n}$. + \item Falls $A(\cdot)$ gleichmäßig negativ definit auf $[t_{0},\infty)$ ist und $b(\cdot)$ beschränkt ist, dann ist $y(t)$ beschränkt und exponentiell stabil. + \end{enumerate} +\end{satz} + +\begin{proof} + \begin{enumerate}[1)] + \item Der Satz von Peano liefert die Existenz eines $T >0$ sodass eine lokale Lösung $y \colon [t_{0},t_{0}+T] \to \R^{n}$ der linearen AWA existieren, für welche gilt ($t \in [t_{0},t_{0}+\infty]$): + \begin{salign*} + & y(t) = y_{0} + \int_{t_{0}}^{t} \left( A(s)y(s) + b(s)\right) \d{s} \\ + \implies \ \ & \norm{y(t)} \leq \norm{y_{0}} + \int_{t_{0}}^{t} \left( \norm{A(s)}\cdot\norm{y(s)} + \norm{b(s)}\right) \d{s}. + \end{salign*} + Das Lemma von Gronwall für $w(t) = \norm{y(t)}$ liefert für $t \in [t_{0},t_{0}+T]$: + \begin{salign*} + \norm{y(t)} \leq \exp\left( \int_{t_{0}}^{t} \norm{A(s)} \d{s}\right)\left(\norm{y_{0}} + \int_{t_{0}}^{t}\norm{b(s)} \d{s} \right) + \end{salign*} + Also ist $\norm{y(t)}$ beschränkt durch ein $C(T, A(\cdot), b(\cdot)) > 0$. Nach Fortsetzungssatz ist der Graph von $y(t)$ fortsetzbar bis an den Rand von $D$. Damit existiert $y(t)$ für alle $t \geq t_{0}$. \\ + $f(t,x)$ ist Lipschitz-stetig bezüglich $x$. Denn: + \begin{salign*} + \norm{f(t,x) - f(t,y)} & = \norm{A(t)(x-y)} \leq \norm{A(t)}\cdot\norm{x-y}. + \end{salign*} + Damit folgt die Eindeutigkeitsaussage aus dem Satz von Picard-Lindelöf. + + \item Sei $A(t)$ negativ definit, dann existiert ein $\lambda > 0$ sodass: + \begin{salign*} + -\left(f(t,x) - f(t,y),(x-y)\right) = - \left(A(t)(x-y), (x-y)\right) \geq \lambda \norm{x-y}^{2}. + \end{salign*} + Sei $b(t)$ beschränkt, dann gilt $\sup_{t \in [t_{0},\infty)} \norm{f(t,0)} = \sup_{t \in [t_{0},\infty)} \norm{b(t)} < \infty$. Damit folgt nach Satz \ref{satz:global-stabil}, dass $y(t)$ beschränkt und exponentiell stabil ist. + \end{enumerate} +\end{proof} + + + + + + + + + + + + +\end{document} \ No newline at end of file diff --git a/analysisII.pdf b/analysisII.pdf index e46854acf28878722a42934e94b3b791cfece9f9..e41301f791cd717066f30aba7dd9e04aeedbd4a7 100644 GIT binary patch delta 283973 zcmcFMcOcc>`ynHe5n08xS6%LQDJ`i)Br~JznXQm~N-0GeDoGMqQG`lGQK+o8h!R;5 zp=gia`P`Au9m@N@-`^kaTV0-eo^#H#*E#zf|2cK>B4slJ!vz>LfnWLI)0!53JW3no z@9fRLa3M--JxXh(zlA?a%K&9Z0H2$qv<$sM98miD{BC}(@NXp*_3?=nDZb!H3gO}q zh0L5h-1xPY`}>8s`Go|dFjyFfU(4Jr*grJL#SMIdp?+fG=IZ6N%s&ierwu+JkO^ok ziHOp{W6(Ne2YxLxry%eJC=7}E+e`iY{K41Rp@{ImQItR;@BlZI7UgUBwU+q@xw-{W7S?vqTBWsGYdQD_2896&o4dJ$pzN>!51bAj zg(rY>(Z-_iI1-wmLqOq)Bs3n61HZRE)Hwuvr*5*Y6&&L4j`nZ^fAd%p91`k^4)ya|;OXY&>gMb07UY3;byEk( zPv8&l@*m<4O>96Ljin<_0ElC>(c0QXupt~8PbN&dfywaq^K$V=dpbF}{~2?LMiubY z{6o}{m=m?hC_EO2CTf$wp9D0C0RF_0(F8o{&zM7Oj^5ipIMfO46YAmX7W8NAX~@GA 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