From 1c757da35ca7b07be03c1dabc3fef9bcfe03b7e2 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Petr=20Veli=C4=8Dka?= Date: Fri, 14 Mar 2025 09:42:37 +0100 Subject: [PATCH] prednaska 14.3.2025 --- maximalni-reseni.tex | 3 +- skripta.pdf | Bin 95845 -> 110246 bytes zavislost-na-podmince.tex | 72 ++++++++++++++++++++++++++++++++++++++ 3 files changed, 74 insertions(+), 1 deletion(-) diff --git a/maximalni-reseni.tex b/maximalni-reseni.tex index 51850a6..a61cbcb 100644 --- a/maximalni-reseni.tex +++ b/maximalni-reseni.tex @@ -46,7 +46,8 @@ V případě $f$ lipschitzovské se důkaz dá výrazně zjednodušit. Budeme uv Na závěr si uvedeme jednu důležitou větu, která nám poskytne představu o tom, jak vypadají maximální řešení diferenciálních rovnic. -\begin{theorem}[o opuštění kompaktu] +\begin{theorem}[Opuštění kompaktu] + \label{thm-leaving-compact} Nechť $K \subset \Omega$ je kompaktní, nechť $(x, I)$ je maximální řešení rovnice \eqref{eq-ode} splňující $(x(t_0), t_0) \in K$ pro nějaké $t_0 \in I$. Potom existují $t_1 > t_0 > t_2$ taková, že $(x(t_1), t_1) \notin K$ a $(x(t_2), t_2) \notin K$. \end{theorem} diff --git a/skripta.pdf b/skripta.pdf index 35f8248f35a8763dcdbfa36c143070da9044d498..9015c8e6ed5e16e31c2a37ddf67298784affb89c 100644 GIT binary patch delta 55810 zcmZs>Q;;q^)TX`L=5E`zZQHhO>uKz^ZQHhO+qUiQd1vZBn6KtPOI4DCRMx7>b>}+1 z0xQ`DuZuMSVoi)Q5eN97Tz(%sEf$eU{#K6nAE_V){ATWaOWgiQ{{FCh|2#h};{%a> znDk$qKYKPvaa_*JLclzwws3Uw`2v!Ve+c`*DVvj(x)T>XDWd^zo0t5HSW?Z_d1xtO zgxB#WR#^7(PqC+qwTe&dcaMWWL7XM@r_9$6Z+~KP?!W%<^#a%{qL@DR5%nJ{-yr62 zz9~>^$Eb-J{<-q-B4y2_)P4=0xSkg;vFse{(=S@!dGe*k&)nYT%)~ngorceK!UOM= zLadpiWRZB}2JqbzPLzx=P*GWEiHYEwP2nXW_hTG0h3sZ;m|6f~mpl7Ig3Z&}jWrd1 zUzEh2qs#FmumNWD9}<_ifU?a0%x!O_Ftug z{F#K*qmJQd)qc1qQaogYsoY3WC6q8sW+%J9*v{MVCQ;#HUnKaWeU z`n%_2m>}MZ00lrqQ_KJ(el~#(JVKNuJq(xU+EPX`#=`xg-Qs3WyYviIiJ>;$K_m*5 z4j#L8O>20aAYEa_uBwD}UWYxrtQo0cEAAMpCFxmIH+ezR zyHS&7gSXzThVBbd$Pl;}DBTD&IK5U_C@nfh5y#X+zaxaBZS9=o6a|a;yQHDyB@afH 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ztGzmhMu|<`Dh*(jL&=JJvDA31>C<#MgNG`=pDt<6vD@d6)GOE$@Wk9_te2b9WejJ*l%9*47BhN?`nTOvV33 YhNef?!ee6!@^W%P(7|A76&dvZ0if3sBme*a diff --git a/zavislost-na-podmince.tex b/zavislost-na-podmince.tex index bce1697..af12efd 100644 --- a/zavislost-na-podmince.tex +++ b/zavislost-na-podmince.tex @@ -17,3 +17,75 @@ \end{proof} \hfill \textit{konec 3. přednášky (7.3.2025)} + +\begin{lemma} + \label{lemma-sol-dist} + Nechť $f$ je globálně $L$-lipschitzovská v $\Omega$ vzhledem k $x$. Potom pro libovolná dvě řešení $(x, I), (y, J)$ v $\Omega$ a body $t, t_0 \in I \cap J$ platí + $$ \| x(t) - y(t) \| \leq \| x(t_0) - y(t_0) \| \exp(L|t - t_0|). $$ +\end{lemma} + +\begin{proof} + Můžeme psát + $$ \| x(t) - y(t) \| = \| x(t_0) + \int_{t_0}^t f(x(s), s) ds - (y(t_0) + \int_{t_0}^t f(y(s), s) ds) \| \leq $$ + $$ \| x(t_0) - y(t_0) \| + \| \int_{t_0}^t \left| f(x(s), s) - f(y(s), s) \right| ds \| \leq $$ + $$ \| x(t_0) - y(t_0) \| + \| \int_{t_0}^t L | x(s) - y(s) | ds.$$ + + Poté z Gronwallova lemmatu dostáváme, že + $$ \| x(t) - y(t) \| \leq K e^{|\int_{t_0}^t L ds|} = K e^{|t - t_0| L}, $$ + kde funkci $w(s)$ ze znění lemmatu odpovídá výraz $\|x(s) - y(s)\|$. +\end{proof} + +Jednoduchým důsledkem tohoto lemmatu je mj. jednoznačnost řešení (stačí uvažovat řešení s $x(t_0) = y(t_0)$). + +\begin{definition} + Nechť $f$ je spojitá a lokálně lipschitzovská vzhledem k $x$ v $\Omega$. Potom definujeme \textit{řešicí funcki} $\varphi : G \subset \mathbb{R}^{n + 2} \rightarrow \mathbb{R}^n$ předpisem $\varphi(t; t_0, x_0) := x(t)$, kde $x: I \rightarrow \mathbb{R}^n$ je řešení splňující počáteční podmínku $x(x_0) = t_0$ a $t \in I$. Zde $G$ je maximální možná, tj. obsahuje všechny trojice $(t; t_0, x_0) \in \mathbb{R}^{n + 2}$ pro něž výraz $\varphi(t; t_0, x_0)$ má smysl. +\end{definition} + +Například, uvažujeme-li rovnici $x'(t) = x(t)$. Obecným řešením této rovnice je funkce $x(t) = ce^t$, vyřešením rovnice s počáteční podmínkou dostaneme řešicí funkci $\varphi(t; t_0, x_0) = x_0e^{t - t_0}$. + +\begin{theorem} + Množina $G$ z předchozí definice je otevřená a $\varphi$ je spojitá na $G$. +\end{theorem} + +\begin{proof} + Intermezzo: otevřenost $G$ znamená, že pro každé $(x_0, t_0) \in \Omega$ a $t$ existuje $r > 0$ takové, že pokud $\| (y_0, s_0) - (t_0, x_0) \|$, potom řešení $y$ procházející bodem $(y_0, s_0)$ je definované v bodech $(t - r, t + r)$, spojitost pak odpovídá tomu, že toto řešení bude po celou dobu ``blízko" toho původního. + + Bez újmy na obecnosti nechť $t_0 > t$. Vezměme $(t; t_0, x_0) \in G$, buď $x$ maximální řešení s počáteční podmínkou $x(t_0), x$. Pak $[t_0, t] \subset D_x$ (řešení je definováno na celém tomto intervalu). Vezměme $\delta > 0$ tak malé, aby $[t_0, t + 2\delta] \subset D_x$ a zároveň $K_\delta := \{ (y, s) \in \mathbb{R}^n \times \mathbb{R} : s \in [t_0 - \delta, t + \delta] \land |y - x(s)| \leq \delta \} \subset \Omega$. Takto definovaná množina $K_\delta$ je kompaktní a tedy $f$ je na $K_\delta$ omezená konstantou $c_0$ (spojitá funkce na kompaktu) díky čemuž z lokální lipschitzovskosti plyne globální $L$-lipschitzovskost vzhledem k $x$. + + Dokážeme, že řešení ``blízko" toho původního neopustí ``rouru" $K_\delta$. Zvolme $\varepsilon > 0$ takové, aby $\varepsilon < \frac{\delta}{2(1 + c_0)e^{L(t - t_0 + 2\delta)}}$. Vezměme $y_0, s_0$ tak, aby $|s_0 - t_0| < \varepsilon$, $|x_0 - y_0| < \varepsilon$. Dále vezmeme $y$ maximální řešení s podmínkou $y(s_0) = y_0$. Chceme dokázat, že $y$ je definované aspoň na intervalu $[s_0, t + \delta]$ a platí $|y(s) - x(s)| \leq \delta$ pro všechna $s \in [s_0, t + \delta]$. + + Můžeme psát + $$ |y(s_0) - x(s_0)| \leq |y(s_0) - x(t_0)| + |x(t_0) - x(s_0)| \leq $$ + $$ |y_0 - x_0| + |x'(\xi)| |t_0 - s_0| \leq (1 + C_0) \varepsilon, $$ + kde $\xi$ je konstanta z Lagrangeovy věty, která ve vícerozměrném prostoru platí pouze jako neostrá nerovnost. + + Dále odhadujme (použijeme Lemma \ref{lemma-sol-dist}) + $$ |y(s) - x(s)| \leq |y(s_0) - x(s_0)| e^{L|s - s_0|} \leq (1 + C_0) \varepsilon e^{L|s - s_0|} \leq (1+C_0)\varepsilon e^{L(t - t_0 + 2\delta)}, $$ + kde uvažujeme pouze body $s$, pro které existuje $y(s)$ a $y$ leží v $K_\delta$ na $[s_0, s]$. Z volby $\varepsilon$ dostáváme navíc + \begin{equation} + \label{eq-estimate-thm-sol-fun} + |y(s) - x(s)| \leq (1 + C_0) \varepsilon e^{L(t - t_0 + 2\delta} < \frac{\delta}{0}. + \end{equation} + + Maximální řešení $y$ opustí kompakt (Věta \ref{thm-leaving-compact}) $K_\delta$ někde za časem $s_0$. Označme $\gamma$ čas prvního opuštění (přesněji řečeno infimum všech časů, kdy to už není v tom kompaktu). Na intervalu $[s_0, \gamma]$ platí odhad \eqref{eq-estimate-thm-sol-fun}, tedy $|y(\gamma) - x(\gamma)| < \frac{\delta}{2}$, z čehož máme $\gamma = t + \delta$, to znamená, že kompakt nemůžeme opustit jinak než za časem $t$. Tím jsme dokázali otevřenost $G$. + + Dokážeme spojitost $\varphi$ na $G$. Vezměme dva body $(t; t_0, x_0)$ a $(s; s_0, y_0)$ jako minule a uvažujme rozdíl + $$ | \varphi(t;t_0, x_0) - \varphi(s, s_0, y_0) | \leq |\varphi(t; t_0, x_0) - \varphi(s; t_0, x_0)| + $$ + $$|\varphi(s; t_0, x_0) - \varphi(s; s_0, y_0)| \leq |x(t) - x(s)| + |x(s) - y(s)| \leq $$ + $$ C_0(t - s) + |x(s_0) - y(s_0)e^{L|s - s_0|}| \leq C_0|t - s| + (1 + C_0)e^{L|s - s_0|} |x_0 - y_0|,$$ + čímž jsme ukázali lipschitzovskost, a tedy spojitost $\varphi$ na $G$. +\end{proof} + +Z hlediska praktických aplikací často uvažujeme rovnici \eqref{eq-ode} ve tvaru $x' = f(x, t, \lambda)$ závislém na hodnotě parametru $\lambda$. Přidejme druhou rovnici $\lambda' = 0$ a počáteční podmínky $x(t_0) = 0$ a $\lambda(t_0) = \lambda_0$, čímž jsme závislost na parametru převedli na závislost na počáteční podmínce (v případě, že $f$ je závisí na $\lambda$ lipschitzovsky). + +Označme pro účely následující věty $\pdv{}{w}$ derivaci ve směru $w \in \mathbb{R}^n$ dle proměnné $x_0$. + +\begin{theorem} + Nechť $f \in C_x^1(\Omega), w \in \mathbb{R}^n$. Potom $\pdv{\varphi}{w}(t, t_0, x_0)$ existuje v každém bodě $G$. Označíme-li $x(t) = \varphi(t, t_0, x_0)$ a $u(t) = \pdv{\varphi}{w}(t, t_0, x_0)$, pak funkce $u$ je řešením rovnice ve variacích + \begin{equation} + \label{eq-thm-diff-dep} + u' = \nabla_x f(x(t), t)u, u(t_0) = w. + \end{equation} +\end{theorem} + +\hfill \textit{konec 4. přednášky (14.3.2025)}