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] 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)$$``$s+g{{ErDempLh-3kTrqEHBPx5vXN&aDO#5K z8ZTM9?HKRvx4^TAmC*TZjZ!ETc!`h;=du0}WyYLiup4c$P z_jb^7P9Sx~tJ3Q3q&=y^gIN`TmamFp;t$H>V$(_7HN2GP=TnLk8^yG@VBB)*>C8@9 z%O%(R&f**C+A;y(y>2WZ+|6^jb=hb!F}Fnm%15K=-Bhc^t46kiyu7>KEoxkB>Yl9! z#1q+Y6l5{w?c9))o5uzvAxN{T-|L@-e^x7;=|CKuBsU3NJU3V++N&&pr=^+PV$IGL zE%%zZHQy3s@!T_~(WtR0YCip3v!nLFU=? ztHFp)x$3jj6xI)H+?}tC#d24l3*sJHg)_&J5=mM7J~{ujME`97ZN=iLn~Zf%2}qPG zl=F$F*~er}cOBg#F-{U7^9B;P-MG?o{Th?ZIILlbahdZ;x{+*-H8$#{1U-1|`&p1z zd<>!Ej>c>%`^;6&7NzvBr`jV*<*OB>CrvFueJo`!HGoxOo zuN|DOFL$1hJKHRS8$6SxtYkINWe6>0*2$qHFh%uiW)iJh-1G$=tbxvhTs{@5r>-~DIMon4C zp(xO`riB^v`KK08I*!q+dDkIs1Z@~@3oLsLID6=gI@~#h&$D?^49494;#%E3aa84#DLy*EBR=TP{h^zV&`Q>BpZ2gu%E+q~B16iZw}h8H zTJ2r1?zqcM0C`K2eR$MKxW2XUR!kYEm!eT}_`R!RMg;vr$ z@%m`bE%VkaE=Tr*=^%8pOS{pxRIY^U{h3CZ3CyWvMl6zM7I%~z(GTUIv`|;CXhUu` zx1JUcl;6N=eW2^JX+3BMIbEF%r;-8|M5W-YmdcJ9Q{&KQv=&^FS1*gu2FkP&JAq#3 z!d#xjOCkk`tVAZ|a#uN-1;khi(tb5pxYRz@U(-Dq=#LferxA2RAErPd88d+#R>PPA zer*n1xSZe)$(t^b$zTRU^*t%ccab%#!;%Nr^@tXMLG8?mlHl zI3P@bsqX+4qM3wOvG+(VhuyY^<}TZ~GOhhq@stL0%sG3-e|Xf3!&0IJ>e_^{Fe&((TG zmTSN-1ipFjWuj&K!=fPddJsgtFFJNgs~N~Z7;+oXC^TY!IR<}Yn-26>nLvPbbN&g& zl6iFx*Iac~pwW&A=Rr2{&=lk@a(F9H7UpI@QDnw55p5y$W^#9)dy!8ZP=bTBvdK*w zX7WlX4J1)@Z*uan$9p<{+ypWW`pLH>-96xVq?x(tk#MQJMxhf1Rfst5HdsW->@V5+ zpigCW3iONu=d&rRpeuY6xcMAY0Ni$USc@<9w5IIMQt)#Z2b{SNx^q)MJSC?L_ zkH_tB-`7?5S=>07qKjWvHvUQED z-Z3N8C`BpPUjz0xv6SlJaS>o}dh+*`ZVB&%QiglG3_A)C?wAHXjlmf3hR1=>x`p*$!@4do!It)`yG zrYT3BQq&~(&emg#pmMs#lUWo~^qlZ6icpC`=rKDCgd4zte^#Cys{+{n0qMgQ@ZYl; 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ccf4ad8..8e02217 100644 --- a/zavislost-na-podmince.tex +++ b/zavislost-na-podmince.tex @@ -1,6 +1,7 @@ \section{Závislost na počáteční podmínce} \begin{lemma}[Gronwall] + \label{lemma-gronwall} Nechť $w(t), g(t)$ jsou nezáporné a spojité na nějakém intervalu $I$ a nechť $t_0 \in I, K \geq 0$. Nechť pro každé $t \in I$ platí $$ 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 $t_0$.