From 9825dc0cdc39fd11921e70eda61fdbf30c929782 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Petr=20Vely=C4=8Dko?= Date: Wed, 14 May 2025 22:31:41 +0200 Subject: [PATCH] prednaska 25.4.2025 --- linearni-rovnice-konst-koef.tex | 1 + skripta.pdf | Bin 165390 -> 180910 bytes skripta.tex | 1 + stabilita.tex | 81 +++++++++++++++++++++++++++++++- 4 files changed, 82 insertions(+), 1 deletion(-) diff --git a/linearni-rovnice-konst-koef.tex b/linearni-rovnice-konst-koef.tex index c4ca47d..1685296 100644 --- a/linearni-rovnice-konst-koef.tex +++ b/linearni-rovnice-konst-koef.tex @@ -82,6 +82,7 @@ Další otázka, kterou se budeme zabývat je hledání maticové exponenciály. \end{theorem} \begin{corollary} + \label{corollary-existence-of-constant} 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} diff --git a/skripta.pdf b/skripta.pdf index 4fe74023d532a5a9d53bac84a3669bac3dc506eb..ba2e5da188a074d1ef3855aefc916a29242b2b7d 100644 GIT binary patch delta 63578 zcmZshLvWxCu(gAUF|lpiwryu7wyigu*mg3polI=owr%VG{>5F~s=GbguIkfIpA!=W zTO9)D-w3)!49?2Jk-A9?iUSNXRqho?X(7jfNbhW*Tuf#LH6=15k%EG)`zyX)QVV2m zin={VHU6ZmC%&%yGwtQ4ZZaDR5bhAY8gd~KKzH2i=Jvk>ch_C}xq|q9-%mH0f^Xm! z`Hc|vKkcWX-~uaEUAT165Tn#XTT1K+3LD>Up;g9i;=daWWIPzjX#i^WyI5JEcw6x6 z@-o-c#vfeKdiIQ|kq-|!SQW%nPF|&a=9T*MAO`TrOyRh`s=7hJCx}sitDs-yi`b(^ zP_)n0@B+_c(xGvm)z2+F*y`{fCiU`=@`fH~`JKz+i`WhK-d`eFI1|}Tr0a!6f!G$h z0WrQ@mU#%%h}gdX12`;^1JJgxFymZHp#m>NyB`7!KS0TY?Qf|+(z}x*?)Y|xi^-iL z`OJUejLxNuBDJ7Pz^Do4ItzYr3!*Qbrg)9?{gTEVmoq8f{PfAy$>(Fzl;XY=VI{59 zG{BD`YFw5(pV8QQ4i<)+v^||^gqIbNN28C7g5#l4>yx?G0+>L9IU%W@ccTX{|JX8} zvIM-X3kgrvzZB_ztmvns>+qbbW^(!AvC6a~;<;6RVSsA{l#V(lX%$jTX^6*pUq0=PIV?`Mg#)PI02PU 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100644 --- a/stabilita.tex +++ b/stabilita.tex @@ -19,7 +19,7 @@ V případě autonomní rovnice splývají pojmy (asymptotické) stability a uni 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}. +V případě řešení lineární rovnice \eqref{eq-linear-ode}, tj. $x' = A(t)x + g(t)$ je stabilita ekvivalentní stabilitě 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 @@ -46,6 +46,7 @@ V případě řešení lineární rovnice \eqref{eq-linear-ode}, tj. $x' = A(t)x Matice $A$ splňující $\Re \lambda < 0$ pro všechna $\lambda \in \sigma(A)$ se nazývá \textit{Hurwitzowská}. \begin{lemma} + \label{lemma-sol-eq-est} 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 $\beta = \alpha - c\gamma > 0$. @@ -65,3 +66,81 @@ Matice $A$ splňující $\Re \lambda < 0$ pro všechna $\lambda \in \sigma(A)$ s \end{proof} \hfill \textit{konec 8. přednášky (11.4.2025)} + +\begin{theorem}[o linearizované stabilitě] + \label{thm-linearized-stability} + Je dána rovnice $x' = f(x)$. Nechť $f(x_0) = 0$ a $f(x)$ je $C^1$ na okolí $x_0$ a nechť $\Re\lambda < 0$ pro každé $\lambda \in \sigma(A)$, kde $A = \nabla f(x_0)$. Potom $x_0$ je (uniformně) asymptoticky stabilní. +\end{theorem} + +\begin{proof} + Bez újmy na obecnosti uvažujme $x_0 = 0$. Dále definujeme $g(x) = f(x) - Ax$. Potom naši rovnici přepíšeme do tvaru $x' = Ax + g(x)$. + + O matici $A$ můžeme říct, že $\|e^{tA}\| \leq ce^{-\alpha t}$. Přesněji řečeno, nechť $\lambda_0 := \max \{\Re\lambda: \lambda\in\sigma(A)\}$. Nechť $-\alpha \in (\lambda_0, 0)$ a k tomuto $\alpha$ nalezneme $c > 0$ takové, že platí výše uvedená rovnost. Existenci takového $c$ nám zaručuje Důsledek \ref{corollary-existence-of-constant}. + + Dále o funkci $g$ víme, že $g(x) = o(\|x\|)$ a $\lim_{x \to 0} \frac{g(x)}{\|x\|} = 0$. Nechť $\gamma < \frac{\alpha}{c}$, vezmeme $\delta > 0$ dost malé, aby $\frac{\|g(x)\|}{\|x\|} < \gamma$ na $B(0, \delta)$. Pak na tomto okolí platí $\|g(x)\| \leq \delta\|x\|$. + + Použijeme seřezávací funkci + $$\eta(t) = \begin{cases}1, t < \frac{\delta}{2};\\ + 0, t > \delta;\\ + \textit{spojité prodloužení na }(\frac{\delta}{2}, \delta). + \end{cases}.$$ + Dále definujeme $h(x) := \eta(\|x\|)g(x)$. Podíváme se na rovnici $x' = Ax + h(x)$. Pro $\|x\| < \frac{\delta}{2}$ platí $h(x) = g(x)$, dále pro $\|x\| > \delta$ je $h(x)$ nulová a nakonec pro $\|x\| \in [\frac{\delta}{2}, \delta]$ platí $\|h(x)\| \leq \|g(x)\|$. Tato porušená rovnice již splňuje předpoklad Lemmatu \ref{lemma-sol-eq-est}. Aplikací tohoto lemmatu dostáváme odhad na řešení této porušené rovnice. + $$\|x(t)\| \leq c\|x(t_0)\| e^{-\beta(t - t_0)}, \beta > 0. $$ + Vezmeme $x(t_0)$ dost malé, potom díky předchozímu odhadu funkce $x$ zůstane v $B(0, \frac{\delta}{2})$ pro všechna $t \geq 0$, tedy $h(x) = g(x)$, z čehož nakonec dostáváme, že $x$ řeší i původní rovnici $x' = Ax + g(x)$. Tím jsme dokázali, že řešení začínající blízko nuly zkolabují v nekonečnu, což je právě definice asymptotické stability. +\end{proof} + +\begin{theorem}[o linearizované nestabilitě] + \label{thm-linearized-instability} + Je dána rovnice $x' = f(x)$. Nechť $f(x_0) = 0$ a $f(x)$ je $C^1$ na okolí $x_0$ a nechť existuje vlastní číslo $\lambda \in \sigma(A)$ takové, že $\Re\lambda > 0$, kde $A = \nabla f(x_0)$. Potom $x_0$ je (uniformně) asymptoticky stabilní. +\end{theorem} + +\begin{proof} + Idea důkazu: nejdříve si matici $A$ převedeme do Jordanova kanonického tvaru, poté si ukážeme, že řešení ``se drží" nestabilního směru. Nakonec důkaz formálně dokončíme pomocí věty o opuštění kompaktu (Věta \ref{thm-leaving-compact}). + + Bez újmy na obecnosti uvažujme $x_0 = 0$. Nechť tedy $A = VJV^{-1}$ je převod matice $A$ do Jordanova tvaru. Přenásobíme matici $J$ zprava maticí $H = \diag(\eta, \eta^2,\dots,\eta^n)$ a zleva maticí $H^{-1} = \diag(\eta^{-1}, \dots, \eta^{-1})$. Dostáváme matici v Jordanově kanonickém tvaru, kde místo jedniček máme $\eta$. Potom $A = (VH) \tilde J (VH)^{-1}$ + Můžeme psát + $$ x' = Ax + g(x) = (VH)\tilde J (VH)^{-1}x + g(x). $$ + Nechť $y := (VH)^{-1}x$, potom + $$ y' = (VH)^{-1}x' = \tilde Jy + (VH)^{-1}g(VHy). $$ + Označme $\tilde g(y) := g(VHy)$. Dostali jsme tedy rovnici $y' = \tilde Jy + \tilde g(y)$. Platí $\tilde g(y) = o(\|y\|)$ (plyne z toho, že $g(x) = o(\|x\|)$). + + Mějme vektor $y = (y_1, \dots, y_n)$. Nechť $z = (z_1, \dots, z_j)$ jsou nestabilní směry a $w = (w_{j+1}, \dots, w_n)$ jsou stabilní a centrální směry. Potom nechť $\varphi(t) := \sum_{i=1}^j |z_i|^2$ a $\psi(t) := \sum_{i=j+1}^n |w_i|^2$. Můžeme psát $\varphi(t) = \sum_{i=1}^j z_i \bar z_i$ a analogicky pro $w_i$. Implicitně zde předpokládáme závislost na čase. Dále definujeme $Z := \{ (z, w) \in \R^n: \varphi(t) \geq \psi(t) \}$. + + Nechť $\lambda_0 := \min\{ \Re\lambda; \lambda \in \sigma(A), \Re\lambda > 0\}$. Vezmeme $\eta = \frac{\lambda_0}{6}$ a $\delta > 0$ takové, aby $\|\tilde g(y)\| \leq \eta \|y\|$ pro všechna $y \in B_\delta$. Dále definujeme $K := Z \cap \overline{B_\delta}$. + + Uvažujme řešení $y(t) = (z(t), w(t))$, pro které budeme odhadovat. Ukážeme, že platí, že pokud $\varphi(t_1) - \psi(t_1) > 0$, potom $\varphi(t) - \psi(t) > 0$ pro všechna $t > t_1$, dokud jsme v $B_\delta$. Ukážeme, že $(\varphi(t) - \psi(t))' > 0$. + + Můžeme psát + \begin{align*} + \varphi'(t) &= \odv*{\sum_{i=1}^j z_i \bar z_i}{t} = \sum_{i=1}^j z_i'(t) \bar z_i(t) + z_i(t) \bar z_i'(t) = \sum_{i=1}^j 2\Re z_i z_i'\\ + &= 2\Re \left( \sum_{i=1}^j z_i \lambda_i z_i + \left(\sum_{i=1}^j z_i \mu z_{i+1} \right) + \sum_{i=1}^j z_i \tilde g_i(z, w) \right). + \end{align*} + Z toho máme, že + \begin{align*} + \varphi'(t) &= \sum_{i=1}^j 2\Re \bar \lambda_i |z_i|^2 + 2\Re \sum_{i=1}^j z_i \mu \bar z_{i+1} + 2\Re \sum_{i=1}^j z_i \overline{\tilde g_i(z, w)}\\ + &\geq 2\lambda_0 \sum_{i=1}^j |z_i|^2 - \left|2\Re \sum_{i=1}^j z_i \mu \bar z_{i+1}\right| - \left| 2\Re \sum_{i=1}^j z_i \overline{\tilde g_i(z, w)}\right|. + \end{align*} + Dále platí (AG-nerovnost) + $$ \left| \sum_{i=1}^j \eta z_i \bar z_{i+1} \right| \leq \sum_{j=1}^i \eta \frac{1}{2} (|z_i|^2 + |z_{i+1}|^2) \leq \eta \frac{1}{2} (\varphi(t) + \varphi(t)) = \eta \varphi(t) $$ + a také + $$ \left| \sum_{i=1}^j z_i \overline{\tilde g_i(z, w)} \right| \leq \frac{1}{2} \sum_{i=1}^j (|z_i|^2\eta + \frac{1}{\eta}|\tilde g_i (z, w)|^2) = \frac{1}{2} \eta \varphi(t) + \eta \varphi(t), $$ + neboť $\sum |g_i|^2 = |g(y)|^2 < \eta^2 |y|^2 \leq 2\eta^2 |z|^2 = 2\eta^2 \varphi(t)$. + + Celkově tedy máme, že $\varphi'(t) \geq (2\lambda_0 - 2\eta -3\eta) \varphi(t) \geq 7\eta \varphi(t)$, jelikož $\lambda_0 > 3\eta$. Obdobně budeme postupovat s odhadem pro $\psi'(t)$. Můžeme psát + \begin{align*} + \psi'(t) &= 2\Re \sum_{i=j+1}^n w_i \bar w_i' = 2\Re \sum_{i=j+1}^n w_i \overline{(\lambda_i w_i + \eta w_{i+1} + \tilde g_i(z, w))}\\ + &= 2\Re \left(\lambda_i \sum_{i=j+1}^n |w_i|^2 + \sum_{i=j+1}^n w_i \eta \bar w_{i+1} + \sum_{i=j+1}^n w_i \overline{\tilde g_i(z, w)} \right)\\ + &\leq 2\eta \frac{1}{2}\left(\sum_{i=j+1}^n \left(|w_i|^2 + |w_{i+1}|^2\right) + \sum_{i=j+1}^n \left(|w_i|^2\eta + \frac{1}{\eta}|\bar g_i^2(z, w)|\right)\right)\\ + &\leq 2\eta \psi(t) + \eta \psi(t) + 2\eta \varphi(t) \leq 5\eta \varphi(t), + \end{align*} + přičemž v poslední nerovnosti jsme využili faktu, že $\psi(t) < \varphi(t)$. Z těchto dvou právě dokázaných nerovností již plyne $\varphi'(t) - \psi'(t) \geq 2\eta\varphi(t) > 0$. + + Z nerovnosti pro $\varphi'(t)$ platí $\frac{\varphi'(t)}{\varphi(t)} \geq 7 \eta$. Zintegrováním dostaneme následující nerovnost pro funkci $\varphi$: + $$ \varphi(t) \geq e^{7\eta (t - t_0)} \varphi(t_0). $$ + + Nechť $y(0) = (z(0), w(0)) \in K$. Potom existuje $T > 0$ takové, že platí nerovnost $e^{7\eta(T - t_0)} \| \varphi_z(t_0) \|^2 > \delta^2$. Uvažujme kompakt $C := K \times [t_0, T + \varepsilon]$. Naše řešení $y$ tento kompakt opustí. Buď $t_1 = \inf \{ t \geq t_0; (y(t), t) \in C\}$. + + Pak $t_1 < T + \varepsilon$. Skutečně, nechť $t_1 \geq T + \varepsilon$, potom $\|y(t_1)\|^2 \geq \varphi_z(t_1) \geq e^{7\eta (t_1 - t_0)} \varphi_z(t_0) > \delta^2$. To je spor s předpokladem, že $t_1$ je infimum. Jistě platí $y(t_0) \in \delta K$. Funkce $\varphi - \psi$ je rostoucí, tedy $\varphi(t_0) - \psi(t_0) \geq \varphi(0) - \psi(0)$. Z toho již plyne, že $|y(t_0)| = \delta$. Dostali jsme, že řešení $y$ opustí otevřenou kouli o poloměru $\delta$, což je přesně to, co jsme chtěli dokázat v této větě. +\end{proof} + +\hfill \textit{konec 9. přednášky (25.4.2025)}