From 47c95c2640843f4e9931444dca2059b3a0d52f3f Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Petr=20Veli=C4=8Dka?= Date: Tue, 11 Mar 2025 23:30:55 +0100 Subject: [PATCH 01/10] preklep --- lokalni-existence.tex | 2 +- skripta.pdf | Bin 95860 -> 95845 bytes 2 files changed, 1 insertion(+), 1 deletion(-) diff --git a/lokalni-existence.tex b/lokalni-existence.tex index f4ff2bb..fc0dd50 100644 --- a/lokalni-existence.tex +++ b/lokalni-existence.tex @@ -103,5 +103,5 @@ K důkazu této věty budeme potřebovat pomocné lemma: 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. + 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 40be9d848fd5d53bf2c9406593a57132d727d281..35f8248f35a8763dcdbfa36c143070da9044d498 100644 GIT binary patch delta 7581 zcmai(RaBG>pmtG^Mnbx!8wQXLX^`&jA*8$ErID6K8iwxf5Qpw&=OKS;`RE1quX6vjSnievG8;FPL70sL0~pdGMFXg_+r-lfj9@W+f_vXOq_to(BL;dGIB z(?o5EMS}BpQb|E2duILh!24c=e8LE9IJ7K@U(d7t!asus7%|LwLZhnzXe#Gcgk}ua zf+Z&CQh{^_RW`T|r#7GPBC#giZ_zURY;5<+BKY9By75VNHp_GJ-?6VQH)&!?|7amI z#`+;6jCG*Tj}l&p)Vf`3K)h}ZXQ9QG>F@52XcIdIZ;#m zG24sV(IT1;R$`rvZGjXK-&pkoXBf1FRoNG+t7g0b|KSia%YHlYy8$f}75N#nfi-PI z^Y0%cXHMVLMf3P~xYWkQ8e{amcH%Az?iBKd*8^_`pFwT-dYDS$a7r;cGos(TSX%?u zq^LW2<&0)jHouI#*I>yHpTgHM-n8^7&70=nd$-3av85ajjSq3=f85%D7uj^0zUH9o z`;1TkY(0-tlJsHIlwO4QU>cv|BceKqWrytka-&V8QrP9Jk*=vyp!<}tQDp~tP7$ED zlyUZl-EH32hBb0H;wl8{2D3`nFXymym;jpP_l#w1HtI#I$~qI3vGNwNO34k@}>o zp*Z7Sbp|!D->eZF^wJh=7kZjyi&q^I4}E0OC)a*|6%QBd&?Efkw0NgvrG~pm_j>R{ 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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)} From f3e663a2616e62e33c398c44403fd35779ed91e5 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Petr=20Veli=C4=8Dka?= Date: Fri, 14 Mar 2025 09:42:55 +0100 Subject: [PATCH 03/10] prednaska 14.3.2025 - pdf --- skripta.pdf | Bin 110246 -> 110247 bytes 1 file changed, 0 insertions(+), 0 deletions(-) diff --git a/skripta.pdf b/skripta.pdf index 9015c8e6ed5e16e31c2a37ddf67298784affb89c..a2181bd042dee6b64bbe693719e6b80419ce85c9 100644 GIT binary patch delta 3666 zcmV-Y4z2N~-3F)K27t5y_KpEFxA%?#I}LwPCOj14zw!SdkMZiVnjEH?poSHk$InN`$wn!&Hw9<){*IR+9l3^znK!B1xVhwHZ2PjYc!g^D2LxdD9O(I8x!x-$@Xd`IrL4lVJPW@^J%m5*>>tQpi4Q zq;UBt71n|l$B>UIr0Yu9G_lro8Gc;svp65|JR&NI~}*;nS6ndPjc?5hW&OnXbZF 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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 $x$. + Bez újmy na obecnosti nechť $t_0 > t$. Vezměme $(t; t_0, x_0) \in G$, buď $x$ maximální řešení s počáteční podmínkou $x(t_0) = x_0$. 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 $s \in [s_0, t + \delta]$. 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zukT5)J_sAle*6F)nV9r?J~k)Y;`H6t1-Evdo+vk?6|AhW=w`_qs_R2Cdi#qeJuSnwb-;XfGBVp2p{P8ihbb<|ZC#D?G`_QwAj`>DpqP zqMz4O>fT)^Mt6Ii=cxNM+XNFre;(Ejvw!mHNk)%ZI)48OniQM7)`dZz>HPfOVjGb@ z`@$9dg)eUsWA&H2dAN_k%_S&|Nj?}c2Wf*32<`%??VU2(z1%GA$sU-3$w`I@R)5- zN{w`l^tGIFVxypKBljp>V(364cdTfyJ?>a_-l8uEFXL#vzh8u@Tz#I$+wcE;8E0O_ z#UrCW#RE;Su7)MjLo(8X*3-i}JYepB{eC_SpQ|Jlv=-6{RImS0cc9uG0&v}M?u@*@ zAQK&^_{2)Qp#FS~P;;?=p!YPk>}-zIZ}d%KZO5N9;<=(ZyQ599+RXWCM+K`7lzq|Z z!VRxtfexAv!%!uC-pCIk+ntVJ6k z<@iZ$4>J#74_ z8b|!tKQL>MRQu%Fu`t*XztY3mm|-^e<>W>+*AUGQ@c#u}vS7X&2?77&24 zJh_Rm3}CPNOWKjK4sdls=^QZZhF#kU+YZdODxElo_@Kz0>)GfgGCl_~RF*S-o|JJj zg>2MWx5*A@x1mF>HAKZ8oYO$yaOdn65V}9tq5TuT7_Z(&nXh9NklDBbNI- zx=(Vo9OX|ud$fJr@BqNbrl0iDGfCTrOaq1P(Y3(E5bX%cl@DE(`6SA+lh8IQ`X&6= zcSTWUFZ_jbB}rFJTt%l4vG2~s`pO?wZ4)hhC0iUDUr`R;(YM0Y?`xil4UW0D##NOX z#fqo6rrzU6f{VpU8MYdGug~?jXsdIp3(>khjXc%u^=*Y;07ISqM($qDrE&H3UjZNg zxEmP3AuN6oRcF+HmJ}z>IP^TOV5;{+;$I- 0$ 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 $x(t_0) = x_0$. 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 $x$. + Bez újmy na obecnosti nechť $t_0 > t$. Vezměme $(t; t_0, x_0) \in G$, buď $x$ maximální řešení s počáteční podmínkou $x(t_0) = x_0$. Pak $[t_0, t] \subset D_x$ (řešení je definováno na celém tomto intervalu). Vezměme $\delta > 0$ tak malé, aby $[t_0, t + 2\delta] \subset D_x$ (to můžeme, neboť $D_x$ je otevřená) a zároveň $K_\delta := \{ (y, s) \in \mathbb{R}^n \times \mathbb{R} : s \in [t_0 - \delta, t + \delta] \land |y(s) - x(s)| \leq \delta \} \subset \Omega$. 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 $$ |y(s_0) - x(s_0)| \leq |y(s_0) - x(t_0)| + |x(t_0) - x(s_0)| \leq $$ - $$ |y_0 - x_0| + |x'(\xi)| |t_0 - s_0| \leq (1 + C_0) \varepsilon, $$ + $$ |y_0 - x_0| + |x'(\xi)| |t_0 - s_0| \leq (1 + c_0) \varepsilon, $$ kde $\xi$ je konstanta z Lagrangeovy věty, která ve vícerozměrném prostoru platí pouze jako neostrá nerovnost. Dále odhadujme (použijeme Lemma \ref{lemma-sol-dist}) - $$ |y(s) - x(s)| \leq |y(s_0) - x(s_0)| e^{L|s - s_0|} \leq (1 + C_0) \varepsilon e^{L|s - s_0|} \leq (1+C_0)\varepsilon e^{L(t - t_0 + 2\delta)}, $$ + $$ |y(s) - x(s)| \leq |y(s_0) - x(s_0)| e^{L|s - s_0|} \leq (1 + c_0) \varepsilon e^{L|s - s_0|} \leq (1+c_0)\varepsilon e^{L(t - t_0 + 2\delta)}, $$ 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}. + |y(s) - x(s)| \leq (1 + c_0) \varepsilon e^{L(t - t_0 + 2\delta)} < \frac{\delta}{2}. \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 $G$. @@ -72,7 +72,7 @@ Například, uvažujeme-li rovnici $x'(t) = x(t)$. Obecným řešením této rov Dokážeme spojitost $\varphi$ na $G$. Vezměme dva body $(t; t_0, x_0)$ a $(s; s_0, y_0)$ jako minule a uvažujme rozdíl $$ | \varphi(t;t_0, x_0) - \varphi(s, s_0, y_0) | \leq |\varphi(t; t_0, x_0) - \varphi(s; t_0, x_0)| + $$ $$|\varphi(s; t_0, x_0) - \varphi(s; s_0, y_0)| \leq |x(t) - x(s)| + |x(s) - y(s)| \leq $$ - $$ C_0(t - s) + |x(s_0) - y(s_0)e^{L|s - s_0|}| \leq C_0|t - s| + (1 + C_0)e^{L|s - s_0|} |x_0 - y_0|,$$ + $$ c_0(t - s) + |x(s_0) - y(s_0)e^{L|s - s_0|}| \leq c_0|t - s| + (1 + c_0)e^{L|s - s_0|} |x_0 - y_0|,$$ čímž jsme ukázali lipschitzovskost, a tedy spojitost $\varphi$ na $G$. \end{proof} From bcd3b1aabd15da2e466efb81203a179ce935f4ab Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Petr=20Veli=C4=8Dka?= Date: Fri, 21 Mar 2025 09:42:11 +0100 Subject: [PATCH 06/10] prednaska 21.3.2025 --- linearni-rovnice.tex | 61 ++++++++++++++++++++++++++++++++++++++ skripta.pdf | Bin 110339 -> 123826 bytes skripta.tex | 1 + zavislost-na-podmince.tex | 41 +++++++++++++++++++++++++ 4 files changed, 103 insertions(+) create mode 100644 linearni-rovnice.tex diff --git a/linearni-rovnice.tex b/linearni-rovnice.tex new file mode 100644 index 0000000..c83ee4e --- /dev/null +++ b/linearni-rovnice.tex @@ -0,0 +1,61 @@ +\section{Lineární rovnice} + +\begin{definition} + \textit{Normu matice} $A \in \mathbb{R}^{n \times n}$ definujeme + $$ \| A \| = \sup \{ |Ax|; x \in \mathbb{R}^n, |x| \leq 1 \}, $$ + kde $|x| = \sqrt{x_1^2 + \cdots + x_n^2}$ je norma vektoru $x \in \mathbb{R}^n$. +\end{definition} + +\begin{theorem}[Vlastnosti normy matice] + \label{thm-matrix-norm-properties} + Nechť $A, B \in \mathbb{R}^{n\times n}$. Potom: + \begin{enumerate}[(i)] + \item $\| A \| \geq 0$ a $\|A \| = 0$ právě když $A = 0$. + \item $\| aA \| = |a| \| A \|$ pro všechna $a \in \mathbb{R}$. + \item $\| A + B \| \leq \|A\| + \|B\|$. + \item $\| AB \| \leq \|A\|\|B\|$. + \item $|Ax| \leq \|A\| |x|$ pro $x \in \mathbb{R}^n$. + \item Je-li $A$ regulární, pak $Ay \geq \frac{|y|}{\|A^{-1}\|}$ pro $y \in \mathbb{R}^n$. + \end{enumerate} +\end{theorem} + +\begin{proof} + První tři vlastnosti říkají, že operátor $\|\cdot\|$ je norma (cvičení). + + Dokážeme vlastnost (v). Případ $x = 0$ je triviální, nechť tedy $x \neq 0$. Položme $y = \frac{x}{|x|}$. Potom můžeme psát + $$ |Ax| = |A(|x|y)| = \left||x|Ay\right| = |x| |Ay| \leq |x| \|A\|. $$ + + K důkazu vlastnosti (iv) můžeme psát + $ |ABx| \leq \|A\|\|B\||x| $, kde jsme dvakrát použili již dokázanou vlastnost (v). + Potom + $$ \|AB\| = \sup_{|x| \leq 1} |ABx| \leq \sup_{|x| \leq 1} \|A\|\|B\||x| \leq \|A\|\|B\|\cdot1. $$ + + Nakonec, pro vlastnost (vi) položme $v := Ay$, tedy $y = A^{-1} v$. Potom + $$ |y| = |A^{-1} v| \leq \|A^{-1}\| |v| = \|A^{-1}\| |Ay|, \textrm{ tedy } |Ay| \geq \frac{|y|}{\|A^{-1}\|}, $$ + čímž je důkaz ukončen. +\end{proof} + +\begin{definition} + \textit{Lineární rovnicí} rozumíme rovnici + \begin{equation} + \label{eq-linear-ode} + x' = A(t)x + g(t), x(t_0) = x_0, + \end{equation} + kde $A(t): (a, b) \rightarrow \mathbb{R}^{n\times n}, g(t): (a, b) \rightarrow \mathbb{R}^n$ jsou spojité. +\end{definition} + +V přednášce MA3 jste již studovali tento typ rovnic, teď se však budeme věnovat obecnějšímu případu, kdy $A$ a $g$ závisí na $t$. + +\begin{theorem} + \label{thm-unique-sol-lineq} + Nechť $t_0 \in (a, b), x_0 \in \mathbb{R}^n$ je dáno. Pak existuje jediné řešení rovnice \eqref{eq-linear-ode}, definované na celém $(a, b)$. +\end{theorem} + +\begin{proof} + Rovnice \eqref{eq-linear-ode} je ekvivalentní rovnici \eqref{eq-ode}, kde $f(x, t) = A(t)\cdot x + g(t)$. Můžeme psát + $$ |f(x, t) - f(y, t)| = |A(t)x - A(t)y| \leq \|A(t)\| |x - y|. $$ + Funkce $A(t)$ je omezená na kompaktních intervalech, tedy $f$ je lipschitzovská. Tedy pro každou počáteční podmínku existuje právě jedno maximální řešení. Dokážeme, že toto řešení je definované na celém $(a, b)$. + + \hfill \textit{konec 5. přednášky (21.3.2025)} + +\end{proof} diff --git a/skripta.pdf b/skripta.pdf index 583b2fb89a154d511fe465b88379a545b4c30b1a..16bf8a1210a5de2f0c5a631786851c21960f70e9 100644 GIT binary patch delta 61722 zcmY(qQ+O^+u&o>0wrv|Tw(ZQ=w!YZ5ZQHhO+jcU8{qJ?o#oBlM^j%kV*LdC<)j_)u zjbo7YgDJo)NeRs405&c*0Rb3i7bnxdwlE%>H+ouecx}jjb9L!_p5Vq^Da3d3h!Yf1u0F2uo4#$VyfeQU z_=5;?s!BiC`aLJ+zo&f$w?FSbZ(V&@O-FSgU~S%h>)=I9fcTLe0ifU0z#V?T_ndW2 zg5B$?*w4+1bq)RfUeM~UMg9fuo3i!dM_EV~fd98D`VQlW7ab46j~~(dMHu(Dnhn;L zRry>uykrA&&PxQlCjO6z<+F%8QC*?X;Nt1h!XF;RVo&9QR-O4#>c!Z}VvmmdP6j?v zVzKXFSs_d;i{Qx9(Z5R4dJvj+zOA7SwM;4~nL7HQkLkqZ|uO9Dfj5=^$%lIF6I+ z{fdwI&h9ndz3WSvlb!YW^qHCl6>iz~y_nY<1O6x|$5X&rCXXO?OT0T7$CW<}H}I+v zIa$LSz|~ykW~!pJ=W&sM#rOi=RVreIY3GGQ>^0Kkv(tj9p_gNbUeMHDp@YHa_2TJ{ zSB6KFvBH(_E+p1`T#j#HpBy!-Y*i(36f*u~rQ`QwV=vtPSs~bNLyUzNLm(gXbGTth z+0(Eq&&$m^BAvhr%hMgazK2+f5tv^06!Gl~$RlvgevhOsd#;C>}YZ!_Y^>~@5FS|V8;a1ZmgDY{Q+xN|%>zhCmR<*FpW?U#<~ z)L!GUzhS8Zs%-yxGC4X>bV6>nYT0=K>UKnkP7e)hM7>m-{RLco*pj1M){Y4iy%`rz2$#Qhxcm0jlsWakRLs|1vB=V zB)@7}pQBQ9_pXPK!|1|5O+}dOX43q?vzRRWX9?&aVPS~Cw3lX0U2O{Rt?LQ_bUsoY zFV+!lyLXwXdv2a!Xy9X`u6sx)$$``$s+g{{ErDempLh-3kTrqEHBPx5vXN&aDO#5K z8ZTM9?HKRvx4^TAmC*TZjZ!ETc!`h;=du0}WyYLiup4c$P z_jb^7P9Sx~tJ3Q3q&=y^gIN`TmamFp;t$H>V$(_7HN2GP=TnLk8^yG@VBB)*>C8@9 z%O%(R&f**C+A;y(y>2WZ+|6^jb=hb!F}Fnm%15K=-Bhc^t46kiyu7>KEoxkB>Yl9! z#1q+Y6l5{w?c9))o5uzvAxN{T-|L@-e^x7;=|CKuBsU3NJU3V++N&&pr=^+PV$IGL zE%%zZHQy3s@!T_~(WtR0YCip3v!nLFU=? ztHFp)x$3jj6xI)H+?}tC#d24l3*sJHg)_&J5=mM7J~{ujME`97ZN=iLn~Zf%2}qPG zl=F$F*~er}cOBg#F-{U7^9B;P-MG?o{Th?ZIILlbahdZ;x{+*-H8$#{1U-1|`&p1z zd<>!Ej>c>%`^;6&7NzvBr`jV*<*OB>CrvFueJo`!HGoxOo zuN|DOFL$1hJKHRS8$6SxtYkINWe6>0*2$qHFh%uiW)iJh-1G$=tbxvhTs{@5r>-~DIMon4C zp(xO`riB^v`KK08I*!q+dDkIs1Z@~@3oLsLID6=gI@~#h&$D?^49494;#%E3aa84#DLy*EBR=TP{h^zV&`Q>BpZ2gu%E+q~B16iZw}h8H zTJ2r1?zqcM0C`K2eR$MKxW2XUR!kYEm!eT}_`R!RMg;vr$ z@%m`bE%VkaE=Tr*=^%8pOS{pxRIY^U{h3CZ3CyWvMl6zM7I%~z(GTUIv`|;CXhUu` zx1JUcl;6N=eW2^JX+3BMIbEF%r;-8|M5W-YmdcJ9Q{&KQv=&^FS1*gu2FkP&JAq#3 z!d#xjOCkk`tVAZ|a#uN-1;khi(tb5pxYRz@U(-Dq=#LferxA2RAErPd88d+#R>PPA zer*n1xSZe)$(t^b$zTRU^*t%ccab%#!;%Nr^@tXMLG8?mlHl zI3P@bsqX+4qM3wOvG+(VhuyY^<}TZ~GOhhq@stL0%sG3-e|Xf3!&0IJ>e_^{Fe&((TG zmTSN-1ipFjWuj&K!=fPddJsgtFFJNgs~N~Z7;+oXC^TY!IR<}Yn-26>nLvPbbN&g& zl6iFx*Iac~pwW&A=Rr2{&=lk@a(F9H7UpI@QDnw55p5y$W^#9)dy!8ZP=bTBvdK*w zX7WlX4J1)@Z*uan$9p<{+ypWW`pLH>-96xVq?x(tk#MQJMxhf1Rfst5HdsW->@V5+ zpigCW3iONu=d&rRpeuY6xcMAY0Ni$USc@<9w5IIMQt)#Z2b{SNx^q)MJSC?L_ zkH_tB-`7?5S=>07qKjWvHvUQED z-Z3N8C`BpPUjz0xv6SlJaS>o}dh+*`ZVB&%QiglG3_A)C?wAHXjlmf3hR1=>x`p*$!@4do!It)`yG zrYT3BQq&~(&emg#pmMs#lUWo~^qlZ6icpC`=rKDCgd4zte^#Cys{+{n0qMgQ@ZYl; 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RNDr. Tomáš Bárta, Ph.D. \footnote{\href{mailto:barta@karlin.mff.cuni.cz}{barta@karlin.mff.cuni.cz}}} \date{LS 2024/25} From 472ca4ce30508280f308f713e1e2302fdf3c62af Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Petr=20Veli=C4=8Dka?= Date: Mon, 7 Apr 2025 07:47:48 +0200 Subject: [PATCH 09/10] prednaska 4.4.2025 --- linearni-rovnice-konst-koef.tex | 88 ++++++++++++++++++++++++++++++++ linearni-rovnice.tex | 15 +++++- skripta.pdf | Bin 136955 -> 152685 bytes skripta.tex | 4 ++ 4 files changed, 106 insertions(+), 1 deletion(-) create mode 100644 linearni-rovnice-konst-koef.tex diff --git a/linearni-rovnice-konst-koef.tex b/linearni-rovnice-konst-koef.tex new file mode 100644 index 0000000..e51c9d9 --- /dev/null +++ b/linearni-rovnice-konst-koef.tex @@ -0,0 +1,88 @@ +\section{Lineární rovnice s konstantními koeficienty} + +\begin{definition} + \textit{Lineární homogenní rovnice s konstantními koeficienty} a s maticí $A \in \R^{n\times n}$ je rovnice + \begin{equation}\label{eq-linhom-const}x' = Ax.\end{equation} +\end{definition} + +Myšlenkou studia těchto rovnic je analogie s rovnicí $x' = ax$ pro $a \in \R$, kde řešením je $x(t) = x_0e^{at}$. Ukážeme, že rovnice \eqref{eq-linhom-const} má řešení $x(t) = e^{At} x_0$. + +\begin{definition} + \textit{Maticovou exponenciálu} definujeme předpisem + $$ e^A = \sum_{k = 0}^\infty \frac{1}{k!} A^k $$ + s konvencí $A^0 = I$. +\end{definition} + +Řada s definice maticové exponenciály je dobře definovaná, neboť $\|\frac{1}{k!}A^k\| \leq \frac{1}{k!}\|A\|^k$, přičemž $\sum_{k=0}^\infty \frac{1}{k!}c^k = e^c$ konverguje pro každé $c \in \R$. Navíc z tohoto odhadu dostáváme $\|e^A\| \leq e^{\|A\|}$. + +\begin{example} + Nechť $A = \begin{pmatrix}2 & 0\\0 & 1\end{pmatrix}$. Potom + $$e^A = \sum_{k=0}^\infty \frac{1}{k!} A^k = \sum_{k=0}^\infty \begin{pmatrix}\frac{2^k}{k!} & 0 \\ 0 & \frac{1^k}{k!}\end{pmatrix} = \begin{pmatrix}e^2 & 0\\0 & e\end{pmatrix}.$$ +\end{example} + +\begin{theorem} + Nechť $U(t) = e^{tA}$. Pak $U(t)$ je fundamentální matice rovnice \eqref{eq-linhom-const} a platí $U(0) = I$. +\end{theorem} + +\begin{proof} + Řada konverguje pro všechny matice, tedy i pro matici $tA$, což znamená, že $U$ je dobře definovaná. Platí + $$[U(t)]_{ij} = \left[\sum_{k=0}^\infty \frac{t^kA^k}{k!}\right]_{ij} = \sum_{k=0}^\infty \frac{1}{k!} [A^k]_{ij} t^k. $$ + Toto je mocninná řada s poloměrem konvergence $\infty$, tedy ji můžeme derivovat člen po členu (nultý člen se zderivuje na nulu). + $$U'(t) = \sum_{k=1}^\infty \frac{A^k}{k!} kt^{k-1} = \sum_{k=1}^\infty A \frac{1}{(k-1)!} A^{k-1}t^{k-1} = Ae^{tA} = A U(t).$$ + Vytknutí $A$ můžeme provést, neboť operátor násobení maticí $A$ je spojitý. + + Závěr ohledně $U(0)$ plyne z toho, že pro $t = 0$ je první člen sumy roven jednotkové matici a všechny ostatní jsou nulové. +\end{proof} + +Z obecného tvaru řešení dostáváme, že $x(t) = \Phi(t) \Phi^{-1}(t_0) x_0$, přičemž $t_0 = 0$ a tedy $U(0) = U^{-1}(0) = I$. Z toho již plyne $x(t) = e^{tA} x_0$. + +\begin{theorem}[Vlastnosti maticové exponenciály] + Platí následující vlastnosti maticové exponenciály + \begin{enumerate}[(i)] + \item $e^{aI} = e^a I$ pro $a \in \R$; + \item pokud $AB = BA$, pak $e^{A+B} = e^Ae^B$; + \item $e^{C^{-1}AC} = C^{-1}e^AC$; + \item $e^{-A} = (e^A)^{-1}$, speciálně $e^A$ je vždy regulární. + \end{enumerate} +\end{theorem} + +\begin{proof} + Budeme dokazovat postupně. + \begin{enumerate}[(i)] + \item Dosazením dostáváme + $$ e^{aI} = \sum_{k=0}^\infty \frac{a}{k!} I = I \sum_{k=1}^\infty \frac{a}{k!} Ie^a. $$ + \item Nejprve ukážeme, že $Be^{tA} = e^{tA}B$. To plyne z toho, že + $$Be^{tA} = B\sum_{k=0}^\infty \frac{t^kA^k}{k!} = \sum_{k=0}^\infty B\frac{t^kA^k}{k!} = \sum_{k=0}^\infty \frac{t^kA^k}{k!} B = e^{tA}B. $$ + Potom z definice $U(t) = e^{tA}e^{tB}$ a $U'(t) = Ae^{tA}e^{tB} + e^{tA}Be^{tB} = (A + B)U(t)$. Tedy $U(t)$ splňuje rovnici $x'(t) = (A+B)x(t)$, kterou také splňuje $\tilde U(t) = e^{(A+B)t}$. Z jednoznačnosti řešení této rovnice dostáváme $e^Ae^B = e^{A+B}$. + \item Z definice rozepíšeme + $$ e^{C^{-1}AC} = \sum_{k=1}^\infty \frac{1}{k!} (C^{-1}AC)^k = \sum_{k=1}^\infty \frac{1}{k!} C^{-1}A^kC = $$ + $$ C^{-1}\sum_{k=0}^\infty \frac{1}{k!}A^k C = C^{-1}e^A C. $$ + \item Okamžitě plyne z (ii), neboť $e^A e^{-A} = e^0 = I$ + \end{enumerate} +\end{proof} + +\begin{corollary}[Variace konstant pro \eqref{eq-linhom-const}] + Nechť $A \in \R^{n \times n}$, $g(t): (a, b) \to \R^n$ je spojitá, $t_0 \in (a, b)$ a $x_0 \in \R^n$ jsou dána. Potom řešení rovnice + $$ x' = Ax + g(t), x(t_0) = x_0$$ + má tvar + $$ x(t) = e^{(t - t_0)A}x_0 + \int_{t_0}^t e^{(t - s)A} g(s) ds. $$ +\end{corollary} + +Další otázka, kterou se budeme zabývat je hledání maticové exponenciály. K tomu použijeme takzvaný Jordanův kanonický tvar matice. + +\begin{theorem} + Nechť $A \in \R^{n \times n}$, $J$ její Jordanův kanonický tvar, $A = VJV^{-1}$ a $(\lambda_1, \dots, \lambda_n)$ je diagonála $J$. Potom $e^{tA} = Ve^{tJ}V^{-1}$, kde matici $e^{tJ}$ definujeme jako $diag(e^{t\lambda_1}, \dots, e^{t\lambda_n})$, přičemž $P(t)$ je blokově diagonální matice se stejně velkými a stejně uspořádanými bloky jako $J$ a blok velikosti $k$ matice $P(t)$ je roven + $$ \begin{pmatrix} + 1 & t & \frac{1}{2}t^2 & \cdots & \frac{1}{(k-1)!}t^{k-1} \\ + 0 & 1 & t & \cdots & \frac{1}{(k-2)!}t^{k-2} \\ + \vdots & \vdots & \vdots & \ddots & \vdots \\ + 0 & 0 & 0 & \cdots & 1 + \end{pmatrix}$$ +\end{theorem} + +\begin{corollary} + Buď $a = \max\{\Re \lambda: \lambda \in \sigma(A)\}$ a $m$ velikost největší Jordanovy buňky příslušné k vlastnímu číslu $\Re \lambda = a$. + Pak existuje $M > 0$, že $\|e^{tA}\| \leq Mt^{m-1}e^{at}$ pro všechna $t \geq 0$. Speciálně, pro všechna $\tilde a > a$ existuje $\tilde M > 0$ takové, že $\|e^{tA}\| \leq \tilde M e^{t\tilde a}$. +\end{corollary} + +\hfill \textit{konec 7. přednášky (4.4.2025)} diff --git a/linearni-rovnice.tex b/linearni-rovnice.tex index a90bb65..ef23626 100644 --- a/linearni-rovnice.tex +++ b/linearni-rovnice.tex @@ -132,6 +132,19 @@ Uvedeme si několik poznámek k definici fundamentální matice. Je-li $\Phi(t)$ \hfill \textit{konec 6. přednášky (28.3.2025)} + Dále si uvědomíme, že $\Phi'(t) = A(t)\Phi(t)$, přičemž násobení maticí zleva provádí řádkové úpravy na matici $\Phi(t)$. Konkrétně $\varphi_k^{j\prime}(t) = \sum_{i=1}^n a_{ki}(t)\varphi_i^j(t)$. + + Platí $\det D_k = A_{kk}(t) \det \Phi(t)$ (vlastnosti determinantu). Z toho dostáváme, že $w'(t) = \det \Phi(t) \sum_{k=1}^n A_{kk}(t) = w(t) = \tr A(t)$. \end{proof} -Pokud $\tr A(t) > 0$, potom wronskián roste, $=0$ zachovává objem a $<0$ objem klesá. +Pokud $\tr A(t) > 0$, potom wronskián roste, $=0$ množina možných hodnot řešení zachovává objem a pro $\tr A(t) <0$ v průběhu času objem klesá. + +\begin{example} + Řešme rovnici + $$ \begin{pmatrix}x \\ y\end{pmatrix}' = \begin{pmatrix}2 & 0\\0 & -2\end{pmatrix} \begin{pmatrix}x \\ y\end{pmatrix}. $$ + Dostáváme $x' = 2x, y' = -2y$, tedy $x = x(0)e^{2t}, y = y(0)e^{-2t}$. Nechť $x(0), y(0) \in [0, 1]$. Potom pro fixní $t_1 > 0$ dostáváme $x(t_1) \in [0, e^{2t_1}], y(t_1) \in [0, e^{-2t_1}]$. Obsah tohoto obdélníku je $e^{2t_1}e^{-2t_1} = 1$. Tedy, obsah je konstantní, což odpovídá pozorování z věty, neboť stopa matice ze zadání je nulová. +\end{example} + +\begin{example} + Mějme rovnici $x' = f(t, x)$. 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RNDr. Tomáš Bárta, Ph.D. \footnote{\href{mailto:barta@karlin.mff.cuni.cz}{barta@karlin.mff.cuni.cz}}} \date{LS 2024/25} @@ -39,5 +42,6 @@ \include{maximalni-reseni} \include{zavislost-na-podmince} \include{linearni-rovnice} +\include{linearni-rovnice-konst-koef} \end{document} From daad49908f353b77162301b1e299dcbca30ef454 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Petr=20Veli=C4=8Dka?= Date: Fri, 11 Apr 2025 09:45:23 +0200 Subject: [PATCH 10/10] prednaska 11.4.2025 --- linearni-rovnice-konst-koef.tex | 28 +++++++++++++ skripta.pdf | Bin 152685 -> 165390 bytes skripta.tex | 3 ++ stabilita.tex | 67 ++++++++++++++++++++++++++++++++ 4 files changed, 98 insertions(+) create mode 100644 stabilita.tex diff --git a/linearni-rovnice-konst-koef.tex b/linearni-rovnice-konst-koef.tex index e51c9d9..c4ca47d 100644 --- a/linearni-rovnice-konst-koef.tex +++ b/linearni-rovnice-konst-koef.tex @@ -62,6 +62,7 @@ Z obecného tvaru řešení dostáváme, že $x(t) = \Phi(t) \Phi^{-1}(t_0) x_0$ \end{proof} \begin{corollary}[Variace konstant pro \eqref{eq-linhom-const}] + \label{thm-variation-hom-const} Nechť $A \in \R^{n \times n}$, $g(t): (a, b) \to \R^n$ je spojitá, $t_0 \in (a, b)$ a $x_0 \in \R^n$ jsou dána. Potom řešení rovnice $$ x' = Ax + g(t), x(t_0) = x_0$$ má tvar @@ -86,3 +87,30 @@ Další otázka, kterou se budeme zabývat je hledání maticové exponenciály. \end{corollary} \hfill \textit{konec 7. přednášky (4.4.2025)} + +\begin{definition} + Pro matici $A \in \R^{n \times n}$ a její spektrum $\sigma(A)$ definujeme $\sigma_-(A) = \sigma(A) \cap \{\Re < 0\}$, $\sigma_0(A) = \sigma(A) \cap \{\Re = 0\}$, $\sigma_+(A) = \sigma(A) \cap \{\Re > 0\}$. Příslušné podprostory generované příslušnými (zobecněnými) vlastními vektory značíme $X_-(A), X_0(A), X_+(A)$ (nazýváme je \textit{stabilní, centrální} a \textit{nestabilní} podprostor). +\end{definition} + +Zřejmě $\R^n = X_+(A) \oplus X_-(A) \oplus X_0(A)$. Tyto prostory jsou invariantní vzhledem k $A$ a též vzhledem k $e^{tA}$. + +\begin{theorem}[Asymptotické chování podprostorů] + Nechť $A$ je daná matice. Potom existují kladná $\alpha, \beta, M$ a $c$ taková, že platí: + \begin{enumerate} + \item Pokud $x_0 \in X_-(A)$, pak $|e^{tA}x_0| \leq ce^{-\alpha t}|x_0|$ pro každé $t \geq 0$. + \item Pokud $x_0 \in X_+(A)$, pak $|e^{tA}x_0| \leq ce^{\beta t}|x_0|$ pro každé $t \leq 0$. + \item Pokud $x_0 \in X_0(A)$, pak $|e^{tA}x_0| \leq c(1 + |t|)^M|x_0|$ pro každé $t \in \R$. + \end{enumerate} +\end{theorem} + +\begin{proof} + Nejdříve nechť $x_0 \in X_-(A)$. Potom $x_0 = \sum_{i=1}^k a_i v_i$, kde $v_i$ jsou zobecněné vlastní vektory příslušné $\lambda_i \in \sigma_-(A)$. + Dále máme, že $e^{tA} x_0 = Ve^{tJ}V^{-1} x_0$. Spočteme $V^{-1}x_0$. Jestliže $v$ je sloupec matice $V$, potom $V^{-1}v$ je jeden ze sloupců jednotkové matice, tedy má tvar $(0,\cdots,0,1,0,\cdots,0)^T$. + Tedy $V^{-1}x_0$ má nenulové hodnoty jen v řádcích příslušných $\Re \lambda < 0$. Můžeme odhadovat normu + $$ \|e^{tA} x_0 \| \leq \|V\|\|e^{tJ} \text{: řádky s } e^{-at}\| \| V^{-1} x_0\| \leq C e^{-\alpha t} \|x_0\|. $$ + Zde jsme využili faktu, že ``polynom" $e^{-\lambda t} t^k$ lze odhadnout $e^{-\lambda t} t^k \leq e^{(-\lambda + \varepsilon)t}c$ pro vhodná $c$ a $\varepsilon$. + + Důkaz ostatních implikací je podobný. +\end{proof} + +V předchozí větě platí i opačná implikace, a to ve smyslu, že uvedené vlastnosti charakterizují dané podprostory. diff --git a/skripta.pdf b/skripta.pdf index fada8a5e49c4e6284aaba3281e26576a91270400..4fe74023d532a5a9d53bac84a3669bac3dc506eb 100644 GIT binary patch delta 62563 zcmV)OK(@c_s0ogY3V^f$rVW2JF)%PbJ_>Vma%Ev{3V7P>J!@}dw{_q9E2jM4@9^0V_`iQW{O;A^H-8Y~fIk^9>hS7xu!etG0a6~02ADO< z$iu6L|GMEE-f}MPIaiK@04}t+Ju0s77cDq9o^!`JxA=d`#F6ljN6-@*Apt*mdZYYm zfPTsU0!hE&=P8a?ckQnI^5Ak)HCM<;iXU(kx8s~2R z`_*@cqfpE%=?#7p%xf)dg0s@MM=9w9w39IHUEBphOv*4xyD9<;oQtc)Lsy{Hi2~|O z1Rw!SCxEGka5MyDo?OW^FOQ(gqX^tI6TY2Cn*3`$e*|6JF#dnu%$wdVE1Npc1oU)@ 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diferenciální rovnice (NMMA336)} \author{Petr Velička \footnote{\href{mailto:petrvel@matfyz.cz}{petrvel@matfyz.cz}}\\přednášející: doc. RNDr. Tomáš Bárta, Ph.D. \footnote{\href{mailto:barta@karlin.mff.cuni.cz}{barta@karlin.mff.cuni.cz}}} \date{LS 2024/25} @@ -43,5 +45,6 @@ \include{zavislost-na-podmince} \include{linearni-rovnice} \include{linearni-rovnice-konst-koef} +\include{stabilita} \end{document} diff --git a/stabilita.tex b/stabilita.tex new file mode 100644 index 0000000..5aa4659 --- /dev/null +++ b/stabilita.tex @@ -0,0 +1,67 @@ +\section{Stabilita} + +Lemma \ref{lemma-sol-dist} nám teoreticky poskytuje spojitost řešící funkce v proměnné $x_0$, pro větší $t$ však kvůli exponenciálnímu růstu nemá význam. Budeme proto zkoumat okolnosti, za nichž existují odhady, které se nezhoršují pro $t \in \infty$. + +\begin{definition} + Nechť $f = f(x, t)$ je spojitá v otevřené $\Omega \in \R^{n+1}$ a navíc lokálně lipschitzovská vůči $x$. Nechť $\Omega \supset \{0\} \times I$ kde $I = (\tau, \infty)$ a nechť $f(0, t) = 0$ pro všechna $t \in I$. Řekneme, že nulové řešení rovnice $x' = f(t, x)$ \eqref{eq-ode} je + \begin{enumerate}[(i)] + \item \textit{stabilní}, jestliže pro všechna $t_0 \in I$ a $\varepsilon > 0$ existuje $\delta > 0$ takové, že $|x_0| < \delta$ implikuje, že $\varphi(t, t_0, x_0)$ je definováno a splňuje $|\varphi(t, t_0, x_0)| < \varepsilon$ pro $t \geq t_0$; + \item \textit{nestabilní}, jestliže není stabilní; + \item \textit{lokální atraktor}, jestliže $\forall t_0 \in I$ existuje $\eta > 0$ tak, že $|x_0| < \eta$ implikuj, že $\varphi(t, t_0, x_0)$ je definováno pro všechna $t \geq t_0$ a navíc $\varphi(t, t_0, x_0) \to 0$ pro $t \to +\infty$; + \item \textit{asymptoticky stabilní}, jestliže je stabilní a navíc lokální atraktor; + \item \textit{uniformně stabilní}, jestliže pro všechna $\varepsilon > 0$ existuje $\delta > 0$ takové, že pro všechna $t_0 \in I$ z $|x_0| < \delta$ plyne $\varphi(t, t_0, x_0)$ je definováno a splňuje $|\varphi(t, t_0, x_0)| < \varepsilon$ pro $t \geq t_0$; + \item \textit{uniformě asymptoticky stabilní}, jestliže je uniformně stabilní a navíc existuje $\eta < 0$ takové, že $\forall \varepsilon > 0$ existuje $T > 0$ takové, že pro všechna $t_0 \in I$ z $|x_0| < \eta$ plyne, že $\varphi(t, t_0, x_0)$ je definováno pro všechna $t \geq t_0$ a $|\varphi(t, t_0, x_0)| \leq \varepsilon|$ pro $t \geq t_0 + T$. + \end{enumerate} +\end{definition} + +Pojem asymptotické stability zavádíme proto, že lokální atraktor nutně nemusí implikovat stabilitu. Konstrukci takového řešení můžeme nahlédnout pomocí tzv. Vinogradovova systému. +V případě autonomní rovnice splývají pojmy (asymptotické) stability a uniformní (asymptotické) stability, neboť můžeme psát $\varphi(t, t_0, x_0) = \varphi(t - t_0, 0, x_0)$. + +Obecněji řešeno, řešení $\tilde x(t)$ rovnice $x' = f(x, t)$ se nazve stabilní (resp. uniformně stabilní atd.), jestliže má analogickou vlastnost nulové řešení rovnice $u' = g(u, t)$ kde $g(u, t) = f(\tilde x(t) + u, t) - f(\tilde x(t), t)$. + +V případě řešení lineární rovnice \eqref{eq-linear-ode}, tj. $x' = A(t)x + g(t)$ je stabilita libovolného řešení příslušné homogenní rovnice \eqref{eq-homogenous-linear-ode}. + +\begin{theorem} + Je dána rovnice $x' = A(t) x$, kde $A(t)$ je spojitá v $I = (\tau, \infty)$. Nechť $\Phi(t)$ je (libovolná) fundamentální matice. Potom nulové řešení je + \begin{enumerate} + \item stabilní, právě když pro $\forall t_0 \in I$ je $\|\Phi(t)\|$ omezená v $[t_0, \infty)$; + \item asymptoticky stabilní, právě když $\|\Phi(t)\| \to 0$ pro $t \to \infty$; + \item uniformně stabilní, právě když existuje $c > 0$ takové, že pro všechna $s < t \in I$ je $\|\Phi(t)\Phi^{-1}(s)\| \leq c$. + \item uniformně asymptoticky stabilní, právě když existují kladná $\alpha$ a $c$ taková, že pro všechna $s < t \in I$ je $\|\Phi(t)\Phi^{-1}(s)\| \leq ce^{-\alpha(t-s)}$. + \end{enumerate} +\end{theorem} + +\begin{theorem} + Nechť $A$ je konstantní matice. Potom nulové řešení rovnice $x' = Ax$ je + \begin{enumerate} + \item (uniformně) stabilní, právě když $\Re \lambda \leq 0$ pro všechna vlastní čísla $\lambda \in \sigma(A)$, přičemž $\Re \lambda = 0$ pouze pro polojednoduchá vlastní čísla (tedy příslušné Jordanovy buňky mají velikost 1). + \item (uniformě) asymptoticky stabilní, právě když $\Re \lambda \leq 0$ pro všechna vlastní čísla $\lambda \in \sigma(A)$. + \end{enumerate} +\end{theorem} + +\begin{proof} + Plyne ihned z tvaru maticové exponenciály. +\end{proof} + +Matice $A$ splňující $\Re \lambda < 0$ pro všechna $\lambda \in \sigma(A)$ se nazývá \textit{Hurwitzowská}. + +\begin{lemma} + Je dána rovnice $x' = Ax + r(x, t)$. Nechť existují kladná $\alpha, c$ tak, že $\|e^{tA}\| \leq ce^{-t\alpha}$ pro $t \geq 0$. Nechť dále $r(x, t): \R^{n+1} \to \R^n$ je spojitá a $|r(x, t)| \leq \gamma |x|$ pro všechna $x, y$ kde $\gamma < \frac{\alpha}{c}$. Pak každé řešení splňuje + $$ |x(t)| \leq c|x(t_0)| \exp(-\beta(t - t_0)) $$ + pro $t \geq t_0$, kde $\beta = \alpha - c\gamma > 0$. +\end{lemma} + +\begin{proof} + Nechť $x$ řeší $x' = Ax + r(x, t)$ na $(0, +\infty)$. Pak $x$ řeší $x' = Ax + g(t)$, kde $g(t) := r(x(t), t)$. Z variace konstant (Důsledek \ref{thm-variation-hom-const}) dostáváme, že + $$ x(t) = e^{(t - t_0)A} x_0 + \int_{t_0}^t e^{(t-s)A} g(s) ds. $$ + Pro $t > t_0$ dostaneme + $$ \|x(t)\| \leq ce^{-(t-t_0)\alpha} \|x_0\| + \int_{t_0}^t ce^{-(t-s)\alpha} \gamma \|x(s)\| ds. $$ + Jinými slovy, + $$ \|x(t)\| e^{t\alpha} \leq ce^{t_0\alpha} \|x_0\| + \int_{t_0}^t ce^{-(t-s)\alpha} \gamma \|x(s)\| ds. $$ + Z Gronwallova lemmatu (Lemma \ref{lemma-gronwall}) dostáváme + $$ e^{t\alpha} \| x(t) \| \leq ce^{t_0\alpha} \|x_0\| e^{c\gamma(t - t_0}. $$ + Po opětovném přenásobení exponenciálou nakonec máme + $$ \|x(t)\| \leq c \|x_0\| e^{(t - t_0)(c\gamma - \alpha)} = ce^{-\beta(t - t_0)} \| x_0 \|. $$ +\end{proof} + +\hfill \textit{konec 8. přednášky (11.4.2025)}

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ccf4ad8..8e02217 100644 --- a/zavislost-na-podmince.tex +++ b/zavislost-na-podmince.tex @@ -1,6 +1,7 @@ \section{Závislost na počáteční podmínce} \begin{lemma}[Gronwall] + \label{lemma-gronwall} Nechť $w(t), g(t)$ jsou nezáporné a spojité na nějakém intervalu $I$ a nechť $t_0 \in I, K \geq 0$. Nechť pro každé $t \in I$ platí $$ w(t) \leq K + \left| \int_{t_0}^t w(s) g(s) ds \right|. $$ Potom pro každé $t \in I$ platí @@ -44,6 +45,7 @@ Jednoduchým důsledkem tohoto lemmatu je mj. jednoznačnost řešení (stačí 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} + \label{thm-cont-dep} Množina $G$ z předchozí definice je otevřená a $\varphi$ je spojitá na $G$. \end{theorem} @@ -89,3 +91,42 @@ Označme pro účely následující věty $\pdv{}{w}$ derivaci ve směru $w \in \end{theorem} \hfill \textit{konec 4. přednášky (14.3.2025)} + +\begin{proof} + Větu dokážeme za silnějšího předpokladu $f \in C_x^2(\Omega)$. + + Vezmeme pevně bod $(x_0, t_0)$ a víme, že tímto bodem prochází právě jedno maximální řešení, označíme ho $x(t)$. Dále označme $A(t) = \nabla_x f(x(t), t)$. Potom $A(t)$ je matice $n \times n$. Vezmeme pevné $w \in \mathbb{R}^n$ a označme $u(t)$ maximální řešení počáteční úlohy \eqref{eq-thm-diff-dep}. + + Dle Věty \ref{thm-unique-sol-lineq} existuje právě jedno řešení a je definované na celém intervalu, kde je definovaná $A(t)$. Chceme dokázat, že $u(t) = \pdv*{\varphi}{w}(t, t_0, x_0)$. + Z definice máme, že + $$ \pdv*{\varphi}{w}(t, t_0, x_0) = \lim_{h \rightarrow 0} \frac{1}{h} (\varphi(t, t_0, x_0 + hw) - \varphi(t, t_0, x_0). $$ + Vezmeme $t$ pevné tak, aby $(t, t_0, x_0) \in G$, tedy $x(t)$ je dobře definované. Vezmeme dost malé tak, aby $\varphi(t, t_0, x_0 + hw)$ bylo definované. Položme $y_h(t) := \varphi(t, t_0, x_0 + hw)$. + + Definujeme funkci $\eta_h(t) = \frac{1}{h}(y_h(t) - x(t)) - u(t)$. Ukážeme, že $\lim_{h \rightarrow 0} \eta_h(t) = 0$. Pišme + $$ \eta'_h(t) = \frac{1}{h}(y_h'(t) - x'(t)) - u'(t) = \frac{1}{h} (f(y_h(t), t) - f(x(t), t)) - \nabla_x f(x(t), t) u(t). $$ + Použijeme Taylorův rozvoj prvního řádu pro funkci $f$, dostaneme + $$ \eta'_h(t) = \frac{1}{h} (\nabla_x f(x(t), t) (y_h(t) - x(t)) + $$ + $$ \frac{1}{2}(y_h(t) - x(t))^T \pdv*[order={2}]{f}{x}(x(t), t) (y_n(t) - x(t)) - \nabla_x f(t, x(t)) u(t). $$ + Tedy máme, že + $$ \eta_h'(t) = \nabla_x f(x(t), t) \left[\frac{1}{h}(y_h(t) - x(t)) - u(t)\right] + \frac{1}{h} z_n(t).$$ + Potom $\eta_h'(t) = A(t)\eta_h(t) + z_n(t)$. + + Pro $h$ malé je vše v $K_\delta$ z Věty \ref{thm-cont-dep}. Na $K_\delta$ jsou $\nabla_x f$ a $\nabla^2_x f$ omezené $\leq M$. Zde předpokládáme, že $f \in C_x^2(\Omega)$. + Potom z Lemmatu \ref{lemma-sol-dist} můžeme psát + $$ \| z_h(t) \| \leq \frac{1}{2} M \| y_h(t) - x(t) \|^2 \leq \frac{1}{2}M \|y_h(t_0) - x(t_0) \|^2 e^{2M|t - t_0|} \leq Ch^2 \|w\|^2. $$ + + Uvědomíme si, že $\eta_h(t_0) = 0$ a napíšeme integrální rovnici odpovídající diferenciální rovnici pro $\eta'_h$ + $$ \eta_h(t) = \eta_h(t_0) + \int_{t_0}^t A(s) \eta_h(s) + z_n(s) ds, $$ + Tedy $\| \eta_h(t) \| \leq \left| \int_{t_0}^t M \| \eta_h(s) \| + Chds \right| = C|t - t_0| h + \left|\int_{t_0}^t M \| \eta_h(s) ds \right|$. + Použijeme Gronwallovo lemma (Lemma \ref{lemma-gronwall}), dostaneme. + $$ \| \eta_h(t) \| \leq \tilde{C} h e^{M|t - t_0|}, $$ + tedy $\eta_h(t) \rightarrow 0$ pro $h \rightarrow 0$, čímž je důkaz ukončen. +\end{proof} + +Ukážeme si jednu aplikaci následující věty pro výpočet derivace řešící funkce. + +\begin{example} + Mějme rovnici $x' = x$, její řešící funkce má tvar $\varphi(t, t_0, x_0) = x_0 e^{t - t_0}$. Potom $\odv*{\varphi(t, t_0, x_0)}{x} = e^{t - t_0}$. Totéž můžeme spočítat z předchozí věty. Hledaná funkce řeší diferenciální rovnici $u' = u$ s počáteční podmínkou $u(t_0) = t$. Jejím řešením je $e^{t - t_0}$, což jsme chtěli dokázat. +\end{example} + +Za uvedených předpokladů dokonce $\odv{\varphi}{w}$ závisí spojitě na $x_0$ tj. řešicí funkce je diferencovatelná (má totální diferenciál) vzhledem k $x_0$. Lze též ukázat, že $\varphi$ je diferencovatelná vůči $t$ a $t_0$. 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z_N%_A@*&-o@7t_gQu*=Km7klfTvquN=*sVjR<5Z0`R#7hlAWwvRrSwxC2)(CYpOvp zu0+1Fa$U7*vMX^Po$(E=-%w4M>`LPQ|JqH}r2pr+rJCmFR_i~va$7Y$%$25nof#Xg z-%)M$(B>zqEza6}SAR7#+Rfgb39&F$^nUI9A38jKVmqjtQ8EX{w)pi2MWYET;|%Wo~41beEgs0U8N5H3}sqMNdWw Drq4-B diff --git a/zavislost-na-podmince.tex b/zavislost-na-podmince.tex index 8e02217..8be8f4b 100644 --- a/zavislost-na-podmince.tex +++ b/zavislost-na-podmince.tex @@ -100,7 +100,7 @@ Označme pro účely následující věty $\pdv{}{w}$ derivaci ve směru $w \in Dle Věty \ref{thm-unique-sol-lineq} existuje právě jedno řešení a je definované na celém intervalu, kde je definovaná $A(t)$. Chceme dokázat, že $u(t) = \pdv*{\varphi}{w}(t, t_0, x_0)$. Z definice máme, že $$ \pdv*{\varphi}{w}(t, t_0, x_0) = \lim_{h \rightarrow 0} \frac{1}{h} (\varphi(t, t_0, x_0 + hw) - \varphi(t, t_0, x_0). $$ - Vezmeme $t$ pevné tak, aby $(t, t_0, x_0) \in G$, tedy $x(t)$ je dobře definované. Vezmeme dost malé tak, aby $\varphi(t, t_0, x_0 + hw)$ bylo definované. Položme $y_h(t) := \varphi(t, t_0, x_0 + hw)$. + Vezmeme $t$ pevné tak, aby $(t, t_0, x_0) \in G$, tedy $x(t)$ je dobře definované. Vezmeme dost malé $h$ tak, aby $\varphi(t, t_0, x_0 + hw)$ bylo definované. Položme $y_h(t) := \varphi(t, t_0, x_0 + hw)$. Definujeme funkci $\eta_h(t) = \frac{1}{h}(y_h(t) - x(t)) - u(t)$. Ukážeme, že $\lim_{h \rightarrow 0} \eta_h(t) = 0$. Pišme $$ \eta'_h(t) = \frac{1}{h}(y_h'(t) - x'(t)) - u'(t) = \frac{1}{h} (f(y_h(t), t) - f(x(t), t)) - \nabla_x f(x(t), t) u(t). $$ From bfda8dc2b6ac03151f0f230ba938dbe5fb7ea5c9 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Petr=20Veli=C4=8Dka?= Date: Fri, 28 Mar 2025 09:46:23 +0100 Subject: [PATCH 08/10] prednaska 28.3.2025 --- linearni-rovnice.tex | 78 ++++++++++++++++++++++++++++++++++++++++++- skripta.pdf | Bin 123838 -> 136955 bytes skripta.tex | 3 ++ 3 files changed, 80 insertions(+), 1 deletion(-) diff --git a/linearni-rovnice.tex b/linearni-rovnice.tex index c83ee4e..a90bb65 100644 --- a/linearni-rovnice.tex +++ b/linearni-rovnice.tex @@ -48,7 +48,7 @@ V přednášce MA3 jste již studovali tento typ rovnic, teď se však budeme v \begin{theorem} \label{thm-unique-sol-lineq} - Nechť $t_0 \in (a, b), x_0 \in \mathbb{R}^n$ je dáno. Pak existuje jediné řešení rovnice \eqref{eq-linear-ode}, definované na celém $(a, b)$. + Nechť $t_0 \in (a, b), x_0 \in \mathbb{R}^n$ je dáno. Pak existuje jediné řešení rovnice \eqref{eq-linear-ode} definované na celém $(a, b)$ splňující počáteční podmínku $x(t_0) = x_0$. \end{theorem} \begin{proof} @@ -58,4 +58,80 @@ V přednášce MA3 jste již studovali tento typ rovnic, teď se však budeme v \hfill \textit{konec 5. přednášky (21.3.2025)} + Předpokládejme, že řešení není definované na celém $(a, b)$. Potom existují $\alpha, \beta \in (a, b)$ takové, že řešení je definováno na $(\alpha, \beta)$. Toto řešení musí opustit každý kompakt, tedy mimo jiné i $K = [t_0, \beta]\times \overline{B(0, R)}$, kde $R$ je dostatečně velké. + Řešení $x$ splňuje + $$ |x(t)| \leq |x(t_0)| \int_{t_0}^t \| A(s)\| |x(s)| + |g(s)| ds \overset{\begin{subarray}{c}\|A(s)\| \leq L\\|g(s)| \leq \tilde{C}\end{subarray}}{\leq} C + \int_{t_0}^t L|x(s)|C| ds \leq $$ + Z Gronwallova lemmatu dostaneme + $$ \leq \tilde{C} + C(\beta - t_0) + \int_{t_0}^t L|x(s)| ds \implies |x(t)| \leq \underbrace{[\tilde{C} + C(\beta - t_0) e^{L(\beta - t_0}]}_{R}. $$ + Došli jsme ke sporu s Větou \ref{thm-leaving-compact}, neboť řešení $x$ nemůže opustit kompakt $K$. \end{proof} + +Důležitá poznámka: řešení existuje globálně na oboru spojitosti $A(t), g(t)$. Ve skutečnosti předchozí věta i pro nelineární rovnice $x' = f(x, t)$ se sublineární pravou stranou, tj. pokud $|f(x, t)| \leq a(t)|x| + g(t)$, kde $a(\cdot), g(\cdot)$ jsou spojité. + +\begin{definition} + \textit{Homogenní rovnicí} rozumíme rovnici \eqref{eq-linear-ode} pro $g(t) \equiv 0$, tj. + \begin{equation} + \label{eq-homogenous-linear-ode} + x' = A(t)x, x(t_0) = x_0. + \end{equation} +\end{definition} + +Použijeme znalosti lineární algebry k tomu, abychom mohli formalizovat postup řešení lineárních ODR. + +\begin{theorem} + Množina $\mathcal{R}_H$ řešení homogenní rovnice \eqref{eq-homogenous-linear-ode} bez zadané počáteční podmínky tvoří $n$-dimenzionální podprostor $C^1((a, b), \mathbb{R}^n)$. +\end{theorem} + +\begin{proof} + Jádro lineárního zobrazení $Lx := x' - Ax$ je vektorový prostor. Dokážeme, že má dimenzi $n$. Nechť $i = 1,\dots,n$ a $x(t_0) = e_i$, pro tuto počáteční podmínku dostaneme řešení $x^i$. Potom $\{x^1, \dots, x^n\}$ tvoří bázi prostoru všech řešení. Skutečně, tyto vektory jsou lineárně nezávislé, mějme lineární kombinaci $c_1 x^1 + \dots + c_n x^n = 0$, speciálně v čase $t_0$ máme $c_1 e^1 + \dots + c_n x^n$, což implikuje, že $c_i = 0$ pro každé $i$. Navíc vezmeme libovolné řešení $z' = A(t) z$, opět zkoumejme stav v čase $t_0$. Máme $z(t_0) = d_1e^1 + \dots d_ne^n$ pro vhodná $d_1, \dots, d_n$. Definujme $y(t) := d_1 x^1(t) + \dots + d_n x^n(t)$, tedy $y$ řeší rovnici $y' = Ay$ a $y(t_0) = z(t_0)$, z čehož díky jednoznačnosti řešení dostáváme $y = z$. Tudíž jsme nalezli $n$-prvkovou bázi, tedy prostor $\mathcal{R}_H$ má dimenzi $n$. +\end{proof} + +\begin{definition} + \textit{Fundamentální systémem} pro \eqref{eq-homogenous-linear-ode} rozumíme libovolnou bázi $\mathcal{R}_H$. Matice, jejíž sloupce tvoří prvky libovolného fundamentálního systému, nazýváme \textit{fundamentální maticí} pro \eqref{eq-homogenous-linear-ode}. +\end{definition} + +Uvedeme si několik poznámek k definici fundamentální matice. Je-li $\Phi(t)$ nějaká fundamentální matice, pak +\begin{itemize} + \item $\Phi(t)$ splňuje ``maticový tvar \eqref{eq-homogenous-linear-ode}", tedy $\Phi'(t) = A\Phi(t)$. + \item $\Phi(t)$ je regulární pro každé $t \in (a, b)$. + \item Obecné řešení \eqref{eq-homogenous-linear-ode} má tvar $\Phi(t) c$, kde $c \in \mathbb{R}^n$. + \item $\tilde{\Phi}(t) := \Phi(t) \Phi^{-1}(t_0)$ je také fundamentální matice, která navíc splňuje $\tilde{\Phi}(t_0) = I$. +\end{itemize} + +\begin{theorem}[Variace konstant] + \label{thm-variation-of-constants} + Nechť $\Phi(t)$ je libovolná fundamentální matice pro \eqref{eq-homogenous-linear-ode}. Potom řešení nehomogenní rovnice \eqref{eq-linear-ode} lze napsat ve tvaru + $$ x(t) = \Phi(t)\Phi^{-1}(t_0)x_0 + \Phi(t) \int_{t_0}^t \Phi^{-1}(s) g(s) ds $$ + pro $t \in (a, b)$ +\end{theorem} + +\begin{proof} + Zderivováním dostaneme $x' = A(t) x + g(t)$, dále stačí ověřit počáteční podmínku dosazením. +\end{proof} + +\begin{definition} + \textit{Wronského determinant} (Wronskián) rovnice \eqref{eq-homogenous-linear-ode} je reálná funkce $w(t) := \det(\Phi(t))$, kde $\Phi$ je libovolná fundamentální matice příslušné rovnice. +\end{definition} + +\begin{theorem}[Liouvilleova formule] + \label{thm-liouville-formula} + Nechť $\Phi(t)$ je maticové řešení \eqref{eq-homogenous-linear-ode} a nechť $w(t) = \det \Phi(t)$. Potom + $$ w(t) = w(t_0) \exp \left( \int_{t_0}^t \tr A(s) ds \right), $$ + kde $\tr A$ je stopa matice $A$. +\end{theorem} + +\begin{proof} + Dokazovaná rovnost je ekvivalentní s + $$ w'(t) = w(t_0) \exp\left( \int_{t_0}^t \tr A(s) ds \right) \tr A(t) $$ + a tedy + $$ w'(t) = \tr A(t) w(t), w(t_0) = w(t_0) $$ + Dále + $$ \odv*{\det \Phi(t)}{t} = \odv*{\sum_\sigma (-1)^{\sgn \sigma} \Phi_{1, \sigma(1)}(t) \dots \Phi_{n, \sigma(n)}(t)}{t} = $$ + $$ \sum_{k = 1}^n \sum_\sigma (-1)^{\sgn \sigma} \overbrace{\Phi \dots \Phi}^{\Phi' \text{ je v $k$-tém řádku }} = \sum_{k=1}^n \det D_k, $$ + kde $D_k$ je matice $\Phi$ se zderivovaným $k$-tým řádkem. + + \hfill \textit{konec 6. přednášky (28.3.2025)} + +\end{proof} + +Pokud $\tr A(t) > 0$, potom wronskián roste, $=0$ zachovává objem a $<0$ objem klesá. diff --git a/skripta.pdf b/skripta.pdf index 21f56ffed63ed520e8b198ed77db75191d3d2bb8..e9c68beb180a091fc141d3aedb0410c8add82873 100644 GIT binary patch delta 63822 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