From cc7b3415176e574b01b87ed289342fe17d4edb29 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Petr=20Veli=C4=8Dka?= Date: Fri, 28 Feb 2025 13:20:50 +0100 Subject: [PATCH] prednaska 28.2.2025 --- jednoznacnost-reseni.tex | 60 +++++++++++++++++++++++++++++++++++++++ lokalni-existence.tex | 43 +++++++++++++++++++++------- skripta.pdf | Bin 60681 -> 81444 bytes skripta.tex | 2 ++ 4 files changed, 94 insertions(+), 11 deletions(-) create mode 100644 jednoznacnost-reseni.tex diff --git a/jednoznacnost-reseni.tex b/jednoznacnost-reseni.tex new file mode 100644 index 0000000..c9770b7 --- /dev/null +++ b/jednoznacnost-reseni.tex @@ -0,0 +1,60 @@ +\section{Jednoznačnost řešení} + +V této kapitole se budeme věnovat otázce jednoznačnosti řešení diferenciálních rovnic. V praxi to často požadujeme, například proto, aby nějaká simulace byla deterministická. + +\begin{definition} + Řekneme, že rovnice \eqref{eq-ode} má v $\Omega$ vlastnost \textit{globální jednoznačnosti}, jestliže pro libovolná řešení $(x, I), (y, J)$ splňující $x(t_0) = y(t_0)$ pro nějaké $t_0 \in I \cup J$, potom $x(t) = y(t)$ pro všechna $t \in I \cup J$. + Řekneme, že rovnice \eqref{eq-ode} má v $\Omega$ vlastnost \textit{lokální jednoznačnosti}, jestliže pro libovolná řešení $(x, I), (y, J)$ splňující $x(t_0) = y(t_0)$ pro nějaké $t_0 \in I \cup J$, potom existuje $\delta$ takové, že $x(t) = y(t)$ pro všechna $t \in (t_0 - \delta, t_0 + \delta)$. +\end{definition} + +\begin{theorem} + Rovnice \eqref{eq-ode} má v $\Omega$ vlastnost globální jednoznačnosti právě tehdy, když má vlastnost lokální jednoznačnosti. +\end{theorem} +\begin{proof} + Implikace směrem doprava je triviální (funkce, které se rovnají na celé množině se nutně musí rovnat i na nějakém okolí zkoumaného bodu). + + Pro důkaz opačné implikace nechť máme dvě řešení $(x, I), (y, J)$ splňující $x(t_0) = y(t_0) = x_0$ pro nějaké $t_0 \in I \cap J$. Bez újmy na obecnosti nechť $I \cup J = (a, b)$. Položme $M = \{t : x(t) = y(t)\}$. Tato množina je díky předpokladu neprázdná, nechť $c := \sup M$. + + Pro spor předpokládejme, že $c < b$. Potom platí $x(c) = \lim_{t\rightarrow c^-} x(t) = \lim_{t\rightarrow c^-} y(t)$, což se díky spojitosti $y$ rovná $y(c)$. Tedy $c$ je maximum $M$. Ale díky lokální jednoznačnosti existuje okolí $(c, x(c)$, na kterém platí $x = y$. Tedy $x(c + \delta) = y(c + \delta)$ pro nějaké $\delta > 0$, což je spor s tím, že $c = \sup M$. +\end{proof} + +\begin{definition} + Funkce $f$ se nazývá \textit{lokálně lipschitzovská vzhledem k $x$} v $\Omega$, jestliže pro každé $(x_0, t_0) \in \Omega$ existují $L$ a $\delta > 0$ takové, že + $$ | f(x, t) - f(y, t) | \leq L|x - y|$$ + pro všechna $(x, t), (y, t) \in U(x_0, \delta) \times (t_0 - \delta, t_0 + \delta)$. +\end{definition} + +\begin{theorem} + Nechť $f$ je lokálně lipschitzovská vzhledem k $x$ v $\Omega$. Potom rovnice \eqref{eq-ode} má v $\Omega$ vlastnost lokální jednoznačnosti. +\end{theorem} + +\begin{proof} + Volme $(x_0, t_0) \in \Omega$ a dvě řešení $(x, I), (y, J)$ taková, že $y(t_0) = x(y_0) = x_0$. Vezmeme $\delta_1 > 0$ tak, aby $f$ byla lipschitzovská na $\delta_1$-okolí $(x_0, t_0)$. + Nechť $\delta \leq \frac{1}{2L}$ je takové, že navíc $\delta < \delta_1$ a $t$ takové, aby $(x(t), t), (y(t), t) \in U(x_0, \delta) \times (t_0 - \delta, t_0 + \delta)$. Potom platí + $$ \| x(t) - y(t) \| = \| x_0 + \int_{t_0}^t f(x(s), s) ds - (x_0 + \int_{t_0}^t f(y(s),s) ds ) \| \leq $$ + $$ \left| \int_{t_0}^t \| f(x(s), s) - f(y(s), s) \| ds\right| \leq \left| \int L\|x(s) - y(s)\| ds \right| \leq L\cdot\gamma\cdot\delta \leq \frac{\gamma}{2}$$ + pro $\gamma := \sup \| x(s) - y(s) \|$. To platí pro všechna $t$, tedy $\gamma = \sup \|x(t) - y(t)\| \leq \frac{\gamma}{2}$, z čehož plyne $\gamma = 0$, což implikuje rovnost $x(t)$ a $y(t)$. +\end{proof} + +Zavedeme značení $f \in C_x^1(\Omega)$, jestliže $\pdv{f}{x_i}$ existují a jsou spojité v $\Omega$ pro každé $i$. + +\begin{lemma} + Nechť $f \in C_x^1(\Omega)$. Potom $f$ je lokálně lipschitzovská vzhledem k $x$ v $\Omega$. +\end{lemma} + +\begin{proof} + Mějme $(x_0, t_0) \in \Omega$. Nechť $\delta > 0$ je takové, že množina + $$M = \overline{U(x_0, \delta) \times (t_0 - \delta, t_0 + \delta)}$$ je podmnožinou $\Omega$. Z kompaktnosti $M$ máme, že její parciální derivace $\pdv{f}{x_i}$ jsou omezené konstantou $K$. + + Dále mějme dva body $(x, t),(y,t) \in M$. Potom $|f(x,t) - f(y, t)| = |f(x + 0(y - x), t) - f(x + 1(y - x)t)| = |\left[f(x+s(y - x), t)\right]^1_0| = |\int_0^1 \odv*{f(x+s(y-x),t)}{s} ds|$. Pro derivaci $f$ platí $\odv*{f(x + s(y - x), t)}{s} = \sum_{i=1}^n \pdv{f}{x_i}(x + s(y - x), t) (y_i - x_i)$. Z toho máme, že + $$\left|\int_0^1 \odv*{f(x+s(y-x),t)}{s} ds\right| \leq \int_0^1 \sum_{i=1}^n K|y_i - x_i| ds = \sum_{i=1}^n K \max_i |y_i - x_i| = $$ + $$n K \max |y_i - x_i| \leq nK | y - x |,$$ + kde poslední nerovnost plyne z faktu, že $|y -x| = \sqrt{\sum_{i = 1}^n |y_i - x_i|^2}$. + + Tedy $f$ je lokálně Lipschitzovská s konstantou $n \cdot K$. +\end{proof} + +\textit{Rule of thumb (just for fun)}: platí $f$ spojitá $\Rightarrow$ existuje řešení, $f \in C^1 \Rightarrow$ řešení je určeno jednoznačně. + +\hfill \textit{konec 2. přednášky (28.2.2025)} + diff --git a/lokalni-existence.tex b/lokalni-existence.tex index d0cd93c..ab991ab 100644 --- a/lokalni-existence.tex +++ b/lokalni-existence.tex @@ -12,7 +12,7 @@ V této přednášce budeme studovat systém rovnic \end{convention} \begin{definition} - Buď $I$ otevřený interval. Funkci $x(t): I \rightarrow \mathbb{R}^n$ nazveme \textit{řešením} diferenciální rovnice (\ref{eq-ode}) v $\Omega$, jestliže pro všechna $t \in I$ platí + Buď $I$ otevřený interval. Funkci $x(t): I \rightarrow \mathbb{R}^n$ nazveme \textit{řešením} diferenciální rovnice \eqref{eq-ode} v $\Omega$, jestliže pro všechna $t \in I$ platí \begin{enumerate}[(i)] \item $(x(t),t)\in \Omega$, \item existuje vlastní $x'(t)$, @@ -56,32 +56,53 @@ Následující věta nám říká, že na nějakém okolí libovolného bodu exi \begin{theorem}{\textbf{(Peano)}} \label{thm-peano} - Nechť $(x_0, t_0) \in \Omega$. Pak existuje $\delta > 0$ a funkce $x(t): (t_0 - \delta, t_0 + \delta) \rightarrow \mathbb{R}^n$, která je řešením (\ref{eq-ode}) a splňuje $x(t_0) = x_0$. + Nechť $(x_0, t_0) \in \Omega$. Pak existuje $\delta > 0$ a funkce $x(t): (t_0 - \delta, t_0 + \delta) \rightarrow \mathbb{R}^n$, která je řešením \eqref{eq-ode} a splňuje $x(t_0) = x_0$. \end{theorem} K důkazu této věty budeme potřebovat pomocné lemma: \begin{lemma} - Pokud $\Omega = \mathbb{R}^{n+1}$ a $f$ je omezená na $\Omega$, pak pro každé $T > 0$ existuje řešení (\ref{eq-ode}) na $(t_0 - T, t_0 + T)$ splňující $x(t_0) = x_0$. + \label{lemma-special-solution} + Pokud $\Omega = \mathbb{R}^{n+1}$ a $f$ je omezená na $\Omega$, pak pro každé $T > 0$ existuje řešení \eqref{eq-ode} na $(t_0 - T, t_0 + T)$ splňující $x(t_0) = x_0$. \end{lemma} \begin{proof} - Řešme ``porušenou" úlohu: $x(t) = x_0 + \int_{t_0}^t f(x(s - \lambda), s) ds$ pro $ t > t_0$ a $x(t) = x_0$ pro $t \in [t_0 - \lambda, t_0]$. + Řešme ``porušenou" úlohu $P_\lambda$: $x(t) = x_0 + \int_{t_0}^t f(x(s - \lambda), s) ds$ pro $ t > t_0$ a $x(t) = x_0$ pro $t \in [t_0 - \lambda, t_0]$. Na $I_1 := (t_0, t_0 + \lambda]$ definujeme $x(t) = x_0 + \int_{t_0}^t f(x(s - \lambda, s) ds$. Na $I_2 := (t_0 + \lambda, t_0 + 2\lambda]$ definujeme $x(t)$ obdobně a indukcí pokračujeme dokud $t_0 + k\lambda$ nebude větší než $T$. Tímto je ``porušená" úloha vyřešena na $[t_0-\lambda, t_0 + T]$. - Položme $\lambda = \frac{1}{n}$ pro $n = 1,2,\dots$. Pišme dále jen $x_n$ namísto $x_{1/n}$, tedy máme posloupnost funkcí. + Položme $\lambda = \frac{1}{n}$ pro $n = 1,2,\dots$. Pišme dále jen $x_n$ namísto $x_{1/n}$, tedy řešení úloh $P_\frac{1}{n}$ tvoří posloupnost funkcí. Ukážeme, že jsou stejně spojité a stejně omezené. - Stejná omezenost plyne z toho, že $\| x_n (t) \| = \| x_0 + \int_{t_0}^t f(x(s - \frac{1}{n}), s) ds \| \leq \| x_0 \| + \int_{t_0}^t f(x(s - \frac{1}{n}) \| ds$. Ale funkce $f$ je omezená, tedy máme $\| x_n(t) \| \leq \| x_0 \| + (T - t_0) \cdot K$, kde $K$ je příslušná konstanta omezenosti $f$. + Stejná omezenost plyne z toho, že $\| x_n (t) \| = \| x_0 + \int_{t_0}^t f(x(s - \frac{1}{n}), s) ds \| \leq \| x_0 \| + \int_{t_0}^t f(x(s - \frac{1}{n}), s) \| ds$. Ale funkce $f$ je omezená, tedy máme $\| x_n(t) \| \leq \| x_0 \| + (T - t_0) \cdot K$, kde $K$ je příslušná konstanta omezenosti $f$. Stejnou spojitost máme z odhadu $\| x_n(t) - x_n(r) \| = \| \int_r^t f(x(s - \frac{1}{n}), s) ds \| \leq |t - r| \cdot K$. V poslední nerovnosti jsme odhadli integrál součinem délky intervalu a konstantou omezenosti funkce $f$. Stačí položit $\delta = \frac{\varepsilon}{K}$, potom $\|x_n(t) - x_r(t)\| < \delta K = \varepsilon$. - - Tedy dle Věty \ref{thm-arzela} můžeme z posloupnosti $x_n$ vybrat stejnoměrně konvergentní podposloupnost. Zbývá dokázat, že její limita řeší naši rovnici. - + + Tedy dle Věty \ref{thm-arzela} můžeme z posloupnosti $x_n$ vybrat stejnoměrně konvergentní podposloupnost $x_{n_k}$. Zbývá dokázat, že její limita řeší naši rovnici. + \hfill \textit{konec 1. přednášky (21.2.2025)} + Zřejmě pro $k \rightarrow \infty$ platí $x_{n_k} \rightarrow x(t)$ a pokud $\int_{t_0}^t f(x_{n_k}(s - \frac{1}{n}), s) ds$ konverguje k $\int_{t_0}^t f(x(s - \frac{1}{n}),s)ds$, máme hotovo. + Tato vlastnost plyne z toho, že $\| \int_{t_0}^t f(x_{n_k}(s - \frac{1}{n_k}), s) - f(x(s), s) ds\| \leq \int_{t_0}^t \| f(x_{n_k}(s - \frac{1}{n_k}), s) - f(x(s - \frac{1}{n_k}), s) \| + \| f(x(s - \frac{1}{n_k}), s) - f(x(s), s \| ds$. + + Jelikož $f$ je spojitá, musí být stejnoměrně spojitá na kompaktní množině $[t_0, t_0 + T] \times \overline{B(0, r) \cap \Omega}$, jinými slovy platí, že pro $\varepsilon > 0$ existuje $\delta$ takové, že pro každé dva body $x, y$ takové, že $\|x - y\| < \delta$ máme, že $f(x, s) - f(y, \hat(s))$. + + Ze stejnoměrné konvergence $x_{n_k}$ máme, že pro $\delta > 0$ existuje $k_0$ takové, že pro všechna $k \geq k_0$ platí $\|x_{n_k}(s - \frac{1}{n_k}) - x(s - \frac{1}{n_k})\|<\delta$. + + Jelikož $x$ je spojitá, na kompaktním intervalu $[t_0, t_0 + T]$ je také stejnoměrně spojitá. Potom pro $\delta> 0$ existuje $k_1$ takové, že pro všechna $k \geq k_1$ platí $\| x(s - \frac{1}{n_k} - x(s) \| < \delta$. + + Potom pro všechna $k \geq \max\{k_0, k_1\}$ platí, že náš integrál je menší nebo roven $\int_{t_0}^t \varepsilon + \varepsilon ds \leq T\cdot 2\varepsilon$, tedy jsme opravdu nalezli požadované řešení. + + Existence řešení na $[t_0- T, t_0]$ se ukáže podobně. \end{proof} -% \begin{proof}[Důkaz Věty \ref{thm-peano}] -% \end{proof} +\begin{proof}[Důkaz Věty \ref{thm-peano}] + Uvažujme dvě koule kolem bodu $(x_0, t_0)$ takové, že $K_1 \subset K_2 \subset \Omega$. + Definujeme $\tilde{f}(x, t) = \begin{cases} + f(x, t) \text{ v } K_1,\\ + \text{spojitě v } K_2 \setminus K_1\\ + 0, (x, t) \in \mathbb{R}^{n+1} \setminus K_2 + \end{cases}$. + + Z Lemmatu \ref{lemma-special-solution} máme, že rovnice $x' = \tilde{f(x, t)}$ má řešení $x$ splňující počáteční podmínku $x(t_0) = x_0$. Nazveme toto řešení $\tilde{x}$. Potom ze spojitosti $\tilde{x}$. Tedy existuje $\delta > 0$ takové, že graf $\tilde{x}$ na $(t_0 - \delta, t_0 + \delta)$ leží v $K_1$. Restrikce $\tilde{x}$ na tento interval nám tedy dává řešení původní rovnice. +\end{proof} diff --git a/skripta.pdf b/skripta.pdf index db9a52f0b61dc0039a40587630899ee238066b1f..17da5e437b62481150ecf2c46e059a269c6f9821 100644 GIT binary patch delta 55260 zcmZUaQ*fYN&~7K3*q+$7ZQHh;OsqG_#I|kQww;MFv2Evkf7QA8Po2Ac(N%r5Yd^hK z@9=f7q7U#o7L<4-a)6VIm7gExw~MoxksXZ3`lX&utZA#0-)!y4i4}%0LE^AX{M`jF zr(+-|lp)1?NixWD6or2>x}lU{6_884jWAB$sW5A1;ns|Sv3SKF8d|EET1!uR_0^Ne zBlJCvH-J!V!29js{vOctPVn~Z(D|=>`oXa55cKrB{^D5e4tg;eEK}ikp5tt+2?N0trGNm0k;@Fu$AXT--hBUP`g>!Upfi zq)&~W>-osw06eYlEKO&+R9DwRH2?7WuhHK04mxf;+CABd6U<#oiJ=3EkbU2x0UAf2 zN{@hQZMp#kAMSYo8~r*>CKT}anK{34GR#ws0#bYRFA9>GRpffb>1krE@)`8OQ4|Tb zF^v8JyQ?t?)+@B9tBJ@<2U(xJ8g4;HI*bN9R1(G&An+90M$j-v96}iJPU__2>;ENk z%>>Z3=7~C5Gyu)_C#vVp`qwwa=m`!6W|>(W3YltjP?^gh=JRVA(!?zy|& zmSwrc6IE;I&r_p@QsC%+GA0R%OhFYbpV9m?fM3z36BX7@e>uqw9i-Z0Snxj0cy`#1 zl03lh(kNGlIh8aksp&SEI8m`y%viK$+#=uvNOgIz!%JI!N@@2|?(4)$x?|&Mknqw~ z3K8TMu+|#T*u&E`X(c65L*Fj78cV9_Dq=2oyotzG4% 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