From 09b9d6cd30819fdd0d4f03bfed172222fbd74016 Mon Sep 17 00:00:00 2001 From: jakobus2205 <61044581+jakobus2205@users.noreply.github.com> Date: Wed, 8 Jul 2020 12:43:26 +0200 Subject: [PATCH] add rest of ana19 week 12 --- ana19.tex | 238 +++++++++++++++++++++++++++++++++++++++++++++++++ analysisII.pdf | Bin 1041586 -> 1062199 bytes 2 files changed, 238 insertions(+) diff --git a/ana19.tex b/ana19.tex index 79a610b..9f0cd17 100644 --- a/ana19.tex +++ b/ana19.tex @@ -149,6 +149,7 @@ \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} + \label{satz:lineare-awa} \end{satz} \begin{proof} @@ -177,6 +178,243 @@ \end{enumerate} \end{proof} +\begin{satz}[Homogene lineare Systeme] + Ein homogenes lineares System von DGLs ist der Form + \begin{equation} + y'(t) = A(t)y(t). \label{eqq:homog-lin-syst} + \end{equation} + \begin{enumerate}[1)] + \item Die Menge der Lösungen bildet einen Vektorraum $H$. + \item Sei $\{y_{0}^{1},...,y_{0}^{n}\}$ eine Basis des $\R^{n}$. Seien $\{y^{1},...,y^{n}\}$ die Lösungen von AWA: + \[ + (y^{i})' = A(t)y^{i}, \ \ \ y^{i}(t_{0}) = y_{0}^{i}, \ \ \ i\in \{1,...,n\} + \] + Dann ist $\{y^{1},...,y^{n}\}$ eine Basis von $H$ und es gilt $\dim(H) = n$. + \item Sei $\{y^{1},...,y^{n}\}$ eine Basis des Lösungsraum $H$, dann ist $\{y^{1}(t),...,y^{n}(t)\}$ für $\forall t \geq t_{0}$ eine Basis des $\R^{n}$. + \end{enumerate} +\end{satz} + +\begin{proof} + \begin{enumerate}[1)] + \item Sei $H$ die Menge der Lösungen von (\ref{eqq:homog-lin-syst}). $H$ ist ein $\R$-VR, denn: + \begin{itemize} + \item Nullfunktion erfüllt $0'(t) = A(t)0(t)$, also $0 \in H$. + \item Seien $\alpha, \beta \in \R$, $u,v \in H$, dann gilt: + \[ + (\alpha u + \beta v)' = \alpha u' + \beta v' = \alpha A(t)u(t) + \beta A(t)v(t) = A(t)\left(\alpha u + \beta v\right). + \] + Also $\left(\alpha u + \beta v\right) \in H$. + \end{itemize} + + \item Sei $\{y_{0}^{1},...,y_{0}^{n}\}$ eine Basis des $\R^{n}$. Seien $\{y^{1},...,y^{n}\}$ die eindeutigen Lösungen der AWAn (nach Satz \ref{satz:lineare-awa}). Seien $\alpha_{i} \in \R$, $i \in \{1,...,n\}$ sodass für $t \geq t_{0}$ gilt: + \[ + \sum_{i=1}^{n} \alpha_{i} y^{i}(t) = 0. + \] + Für $t = t_{0}$ gilt dann $\sum_{i=1}^{n} \alpha_{i} y_{0}^{i} = 0$, da die $y_{0}^{i}$ linear unabhängig sind, folgt $\alpha_{i} = 0$ für alle $i \in \{1,...,n\}$. Daraus folgt, dass die $y^{i}(t)$ ($i \in \{1,...,n\}$) linear unabhängig sind, also bereits, dass die $y^{i}$ ($i \in \{1,...,n\}$) linear unabhängig sind. \\ + Da es höchstens $n$ linear unabhängige Anfangswerte gibt, sind nicht mehr als $n$ Funktionen aus $H$ linear abhängig, also $\dim(H) = n$. + \item Analog zu 2). + \end{enumerate} +\end{proof} + +\begin{definition} + Eine Basis $\{\varphi^{1},...,\varphi^{n}\}$ des Lösungsraums $H$ von $y'(t) = A(t)y(t)$ zu den Anfangswerten $\varphi^{i}(t_{0}) = e^{i}$ heißt \underline{Fundamentalsystem} des linearen Systems von DGLs. \\ + Die Matrix $\phi = \left[\varphi^{1},...,\varphi^{n} \right] $ der Spaltenvektoren $\varphi^{i}$ heißt \underline{Fundamentalmatrix} des linearen Systems von DGLs. \\ + Diese Matrix ist regulär und löst die AWA (komponentenweise): + \[ + \phi'(t) = A(t)\phi(t), \ \ \ \ t \geq t_{0}, \ \ \ \ \phi(t_{0}) = I. + \] +\end{definition} + +\begin{satz}[Inhomogene lineare Systeme] + Ein inhomogenes lineare System von DGLs ist der Form + \[ + y'(t) = A(t))y(t) + b(t). + \] + 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 partikuläre Lösung ist für $\text{const} \ c \in R^{n}$: + \[ + y_{b}(t) = \phi(t) \left( \int_{t_{0}}^{t} \phi^{-1}(s)b(s) \d{s} + c \right). + \] + \item Alle Lösungen der inhomogenen Gleichung haben die Form: + \[ + y(t) = y_{b}(t) + v(t) + \] + wobei $v \in H$ (Lösungsraum des assoziierten homogenen Systems). + \item Gilt $c = y_{0}$, dann gilt $y_{b}(t_{0}) = y_{0}$. + \end{enumerate} +\end{satz} + +\begin{proof} + \begin{enumerate}[1)] + \item Sei $\psi \coloneqq \int_{t_{0}}^{t} \phi^{-1}(s)b(s) \d{s} + c$. Dann gilt für $t\geq t_{0}$: $\psi' = \phi^{-1}(t)b(t)$. Für $y_{b} = \phi\psi$ gilt dann: + \[ + y_{b}' = \phi'\psi + \phi\psi' = A\phi\psi = \phi\phi^{-1}b = Ay_{b} + b. + \] + Also ist $y_{b}$ eine Lösung des inhomogenen Systems und gilt $c=y_{0}$, dann löst $y_{b}$ die AWA $y' = Ay + b$, $y(t_{0}) = y_{0}$. Daraus folgt 1) und 3). + \item Sei $y$ eine zweite Lösung des inhomogenen Systems. Für $w=y-y_{b}$ gilt dann: + \[ + w' = y' - y_{b}' = Ay + b - (Ay_{b} + b) = A(y-y_{0}) = Aw. + \] + Also bereits $w \in H$. + \end{enumerate} +\end{proof} + +\begin{bem} + \begin{enumerate}[(1)] + \item Die Lösung $y_{b}(t) = \phi(t) \left( \int_{t_{0}}^{t} \phi^{-1}(s)b(s) \d{s} + y_{0} \right)$ entspricht genau der Lösung + \[ + y(t) = \exp\left( \int_{t_{0}}^{t} a(s) \d{s} \right) \left(y_{0} + \int_{t_{0}}^{t} \exp\left( -\int_{t_{0}}^{t} a(s) \d{s} \right)b(t) \d{s} \right) + \] + der skalaren linearen AWA $y'(t) = a(t)y(t) + b(t)$ ($t \geq t_{0}$) (Variation der Konstanten). + \item Für lineare DGL mit konstanten Koeffizienten + \[ + y'(t) = Ay(t), \ \ \ \ \ A \in \R^{n\times n} + \] + gibt es eine Lösungstheorie, die auf algebraischen Argumente zurückgreift. + \end{enumerate} +\end{bem} + +\section{Randwertaufgaben} +\begin{bem} + Wir betrachten nun sogenannte Randwertaufgaben/Randwertprobleme der Form: $f\colon I \times \R^{n} \to \R^{n}$, $r \colon \R^{n} \times \R^{n} \to \R^{n}$ + \begin{salign*} + & y'(t) = f(t,y(t)), && t \in I = [a,b] \\ + & r(y(a),y(b)) = 0 + \end{salign*} + Gesucht ist eine stetig differenzierbare Lösung $y \colon I \to \R^{n}$ die beide Bedingungne erfüllt. Die zweite Bedingung lässt sich verallgemeinern zu einer Mehrpunkt RWA: + \[ + r\left( y(t_{1}),...,y(t_{k}) \right) = 0. + \] +\end{bem} + +\begin{bsp} + Wann existieren solche Lösungen? Wir betrachten folgendes Beispiel: + \[ + y'' + y = 0 + \] + für $t \in [0,\pi]$. Dies ist äquivalent zu folgenden System: + \begin{salign*} + y_{1} = y, y_{2} = y', && \begin{cases} + y_{1}' = y_{2} \\ + y_{2}' = -y_{1} + \end{cases}. + \end{salign*} + Dieses Problem hat die allgemeine Lösung + \[ + y(t) = c_{1}\sin(t) + c_{2}\cos(t). + \] + \begin{enumerate}[1)] + \item Für $y(0) = y(\pi)$, $y'(0)=y'(\pi)$. $r = \begin{pmatrix} + y_{1}(0)-y_{1}(\pi) \\ y_{2}(0)-y_{2}(\pi) + \end{pmatrix} = 0$ ist die Lösung des RWP $y(t) \equiv 0$, $t \in [0,\pi]$ eindeutig. + \item Für $y(0) = 0$, $y(\pi) = 0$, $r = \begin{pmatrix} + y_{1}(0) \\ y_{1}(\pi) + \end{pmatrix} = 0$ hat das RWP unendlich viele Lösungen $y(t) = c_{1}\sin(t)$ ($t \in [0,t]$). + \item Für $y(0) = 0$, $y(\pi) = 1$, $r = \begin{pmatrix} + y_{1}(0) \\ y_{1}(\pi) -1 + \end{pmatrix} = 0$ hat das RWP keine Lösung. + \end{enumerate} +\end{bsp} + +\begin{definition}[Allgemeine inhomogene lineare RWA] + Seien $B_{a}, B_{b} \in \R^{n\times n}$, $g \in \R^{n}$ sowie $A \colon I \to \R^{n\times n}$, $f \colon I \to \R^{n}$ stetig. Dann ist eine allgemeine inhomogene lineare RWA der Form: + \begin{salign*} + & y'(t) = A(t)y(t) + f(t), && t\in I \\ + & B_{a}y(a) + B_{b}y(b) = g. + \end{salign*} +\end{definition} + +\begin{bem} + Lösung des inhomogenen DLG-System ist der Form: + \[ + y(t,s) = \varphi^{0}(t) + \sum_{i=1}^{n} s_{i}\varphi^{i}(t) = \varphi^{0}(t) + \phi(t)s. + \] + Hier löst $\varphi^{0}$ die AWA: + \[ + (\varphi^{0})'(t) = A(t)\varphi^{0}(t) + f(t), \ \ \ \ t \geq a, \ \varphi^{0}(a) = 0. + \] + $\varphi^{i}$ ($i \in \{1,...,n\}$) lösen die AWA: + \[ + (\varphi^{i})'(t) = A(t)\varphi^{i}(t), \ \ \ \ t \geq a, \ \varphi^{i}(a) = e^{i}. + \] + $\varphi^{0}, \varphi^{i}$ ($i \in \{1,...,n\}$) sind eindeutige Lösungen, außerdem: + \[ + \phi(t) = \left[\varphi^{1}(t),...,\varphi^{n}(t) \right]. + \] + Offenbar löst $y(t,s)$ die DGL: + \begin{salign*} + y'(t,s) = (\varphi^{0})'(t) + \sum_{i=1}^{n} s_{i}(\varphi^{i})'(t) = A(t)\underbrace{\left(\varphi^{0}(t) + \sum_{i=1}^{n} s_{i}\varphi^{i}(t) \right)}_{y(t,s)} + f(t), \ \ \ t \geq t_{0}. + \end{salign*} + Wie muss $s \in \R^{n}$ gewählt werden? Ziel: $s \in \R^{n}$ so zu bestimmen, sodass gilt: + \[ + B_{a}y(a,s) + B_{b}y(b,s) = g. + \] + Dies lässt sich umformen zu: + \[ + B_{a}(\varphi^{0}(a) + \phi(a)s) + B_{b}(\varphi^{0}(b) + \phi(b)s) = g. + \] + Also: + \[ + \left( B_{a} + B_{b}\phi(b) \right)s = g - B_{b}\varphi^{0}(b). + \] +\end{bem} + +\begin{satz}[Existenzsatz für lineare RWA] + Die lineare RWA besitzt eine eindeutige Lösung $y(t)$ für beliebige $f(t)$ und $g$ genau dann wenn: $B_{a} + B_{b}\phi(b) \in \R^{n\times n}$ regulär ist, bzw. die assoziierte RWA nur die triviale Lösung $y \equiv 0$ hat. +\end{satz} + +\begin{proof} + \glqq $\Leftarrow$\grqq: Ist $B_{a} + B_{b}\phi(b) \in \R^{n\times n}$ regulär, so ist das System + \[ + \left( B_{a} + B_{b}\phi(b) \right)s = g - B_{b}\varphi^{0}(b) + \] + eindeutig lösbar für $s \in \R^{n}$ und somit löst $y(t,s)$ die RWA. \\ + \glqq $\Rightarrow$ \grqq: Die Lösung der RWA kann man darstellen als + \[ + y(t) = \varphi^{0}(t) + \phi(t)s, \ \ \ s \in \R^{n}. + \] + weil die Funktionen $\{ \varphi^{1},...,\varphi^{n}\}$ eine Basis des Lösungsraum des assoziierten homogenen DGL bilden. \\ + Für homogene RWA gilt $g - B_{b}\varphi^{0}(b) = 0$ und die Gleichung für $s$ lautet + \[ + \left( B_{a} +B_{b}\phi(b)\right)s = 0- + \] + Daraus folgen alle Behauptungen. +\end{proof} + +\begin{bem} + Die Lösung einer linearen RWA ist im Kern die Lösung eines LGS. +\end{bem} + +\begin{bem} + Nun betrachten wir die nichtlineare RWA + \[ + y' = f(t,y), \ \ \ t \in [a,b], \ \ \ \ \ r(y(a),y(b)) = 0. + \] + Frage: falls die nichtlineare RWA eine Lösung $y(t)$ besitzt, ist diese lokal eindeutig? \\ + Also existiert eine Umgebung $U_{R}(y) = \{ v \in C[a,b] \ | \ \norm{y-v}_{\infty} < R\}$, sodass es keine andere Lösung $\tilde{y} \neq y$ existiert? \\ + Wir führen folgende Notationen ein: + \begin{salign*} + f_{x}'(t,x) &= \left( \frac{\partial f_{i}(t,x)}{\partial x_{j}} \right)_{i,j=1}^{n} \\ + r_{x}'(t,x) &= \left( \frac{\partial r_{i}(x,y)}{\partial x_{j}} \right)_{i,j=1}^{n} \\ + r_{y}'(t,x) &= \left( \frac{\partial r_{i}(x,y)}{\partial y_{j}} \right)_{i,j=1}^{n} + \end{salign*} + für die Jacobi-Matrizen von $f(t,\cdot)$, $r(\cdot,\cdot)$ ein. +\end{bem} + +\begin{satz}[Lokale Eindeutigkeit] + Eine Lösung $y$ von nichtlinearen RWA ist lokal eindeutig genau dann wenn: \\ + die lineare RWA + \begin{salign*} + & v'(t) = f_{x}'(t,y(t))v(t), \ \ \ \ \ \ t \in I, \\ + & r_{x}'(y(a),y(b))v(a) + r_{y}'(y(a),y(b))v(b) = 0. + \end{salign*} + nur die triviale Lösung $v \equiv 0$ besitzt. +\end{satz} + +\begin{proof} + Siehe Skript von Rolf Rannacher Seite 128. +\end{proof} diff --git a/analysisII.pdf b/analysisII.pdf index 5242f1c4c04fda1233a06aea8bea8b9ea0a41eb8..5c41f33e9014991744010b7204d273f05ca76a5d 100644 GIT binary patch delta 123870 zcma%k1z1$g_rKdq$1V*Lf=UQ%!0w8IprR5g76PK!-GvISU}1xc42p$~g&1JBVs~L< zD}sRC|1)!Um%EGa_xFFE_j#R}8++%>IiHg=>^>h-@~+jcuMQ|?JWId|kV%jz&(KS` zEPBa(fAcvXei=GfwhsH{^tE_oUu3w$tNpY$H(KAzO&aOno&Tuv&yVJ(^Ji@7S+FI2 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