From 16f27ab91c6e11375bfb82ff03c113f9bbb2a755 Mon Sep 17 00:00:00 2001 From: Petr Velycko Date: Fri, 16 May 2025 14:00:15 +0200 Subject: [PATCH] opravy preklepu --- linearni-rovnice-konst-koef.tex | 2 +- linearni-rovnice.tex | 6 +++--- prvni-integral.tex | 2 +- skripta.pdf | Bin 227171 -> 227208 bytes stabilita.tex | 18 +++++++++--------- 5 files changed, 14 insertions(+), 14 deletions(-) diff --git a/linearni-rovnice-konst-koef.tex b/linearni-rovnice-konst-koef.tex index 1685296..4b2fb64 100644 --- a/linearni-rovnice-konst-koef.tex +++ b/linearni-rovnice-konst-koef.tex @@ -69,7 +69,7 @@ Z obecného tvaru řešení dostáváme, že $x(t) = \Phi(t) \Phi^{-1}(t_0) x_0$ $$ x(t) = e^{(t - t_0)A}x_0 + \int_{t_0}^t e^{(t - s)A} g(s) ds. $$ \end{corollary} -Další otázka, kterou se budeme zabývat je hledání maticové exponenciály. K tomu použijeme takzvaný Jordanův kanonický tvar matice. +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 diff --git a/linearni-rovnice.tex b/linearni-rovnice.tex index ef23626..54f6fa8 100644 --- a/linearni-rovnice.tex +++ b/linearni-rovnice.tex @@ -60,9 +60,9 @@ V přednášce MA3 jste již studovali tento typ rovnic, teď se však budeme v 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 - $$ |x(t)| \leq |x(t_0)| \int_{t_0}^t \| A(s)\| |x(s)| + |g(s)| ds \overset{\begin{subarray}{c}\|A(s)\| \leq L\\|g(s)| \leq \tilde{C}\end{subarray}}{\leq} C + \int_{t_0}^t L|x(s)|C| ds \leq $$ + $$ |x(t)| \leq |x(t_0)| + \int_{t_0}^t (\| A(s)\| |x(s)| + |g(s)|) ds \overset{\begin{subarray}{c}\|A(s)\| \leq L\\|g(s)| \leq \tilde{C}\\|x(t_0)|=C\end{subarray}}{\leq} C + \int_{t_0}^t (L|x(s)| + \tilde C) ds \leq $$ Z Gronwallova lemmatu dostaneme - $$ \leq \tilde{C} + C(\beta - t_0) + \int_{t_0}^t L|x(s)| ds \implies |x(t)| \leq \underbrace{[\tilde{C} + C(\beta - t_0) e^{L(\beta - t_0}]}_{R}. $$ + $$ \leq C + \tilde{C}(\beta - t_0) + \int_{t_0}^t L|x(s)| ds \implies |x(t)| \leq \underbrace{C + \tilde C(\beta - t_0) e^{L(\beta - t_0)}}_{R}. $$ Došli jsme ke sporu s Větou \ref{thm-leaving-compact}, neboť řešení $x$ nemůže opustit kompakt $K$. \end{proof} @@ -83,7 +83,7 @@ Použijeme znalosti lineární algebry k tomu, abychom mohli formalizovat postup \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 $n$. + 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 $y = z$. Nalezli jsme $n$-prvkovou bázi, tedy prostor $\mathcal{R}_H$ má dimenzi $n$. \end{proof} \begin{definition} diff --git a/prvni-integral.tex b/prvni-integral.tex index c4ebc9f..d946f94 100644 --- a/prvni-integral.tex +++ b/prvni-integral.tex @@ -8,7 +8,7 @@ V celé této kapitole budeme uvažovat autonomní rovnici pro $f$ spojitou a lokálně lipschitzovskou. \begin{definition} - Funkci $U: \Omega \to \R$ nazveme \textit{prvním integrálem} rovnice \eqref{eq-auto}, jestliže $U \in C^1(\Omega)$ a je nekonstantní a zároveň $t \to U(x(t))$ je konstantní pro každé řešení $x$ dané rovnice v $\Omega$. + Funkci $U: \Omega \to \R$ nazveme \textit{prvním integrálem} rovnice \eqref{eq-auto}, jestliže $U \in C^1(\Omega)$ a je nekonstantní a zároveň $t \mapsto U(x(t))$ je konstantní pro každé řešení $x$ dané rovnice v $\Omega$. \end{definition} Například, máme-li rovnici $x'' + kx = 0$ (lze pomocí ní popsat kmitání pružiny s hybností $k > 0$), funkce $V(x', x) = \frac{1}{2}x'^2 + \frac{k}{2} x^2$ je 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z(~#pa0RcKp$q~LR^H6urcPMu62j73F>;KtI%<5ih7stpx)DFsP|Ft z+5>Eke%JzA;uHRB^cy~S>#n$?H&0W6T{Xo|*6#F2- z;Z(?DrR3E)@H+rOS)N)+M^UrXk;Sd9m2v9=lv5_D!_Qxnlo#>+%M%e|}Q< ztUR&r@3~wewm!n;CE^QS;qumo{Nc4r#TOOj^09_o>GCr1rM!2!N_l wfxB=I?!yCk2#??iEW>g 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{stabilní}, jestliže pro všechna $t_0 \in I$ a $\varepsilon > 0$ existuje $\delta > 0$ takové, že pro $|x_0| < \delta$ platí, že $\varphi(t, t_0, x_0)$ je definováno a splňuje $|\varphi(t, t_0, x_0)| < \varepsilon$ pro libovolné $t \geq t_0$; \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{lokální atraktor}, jestliže $\forall t_0 \in I$ existuje $\eta > 0$ tak, že pro $|x_0| < \eta$ je definován výraz $\varphi(t, t_0, x_0)$ 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{uniformně stabilní}, jestliže pro všechna $\varepsilon > 0$ existuje $\delta > 0$ takové, že pro všechna $t_0 \in I$ z $|x_0| < \delta$ plyne, že výraz $\varphi(t, t_0, x_0)$ je definován 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} @@ -17,7 +17,7 @@ Lemma \ref{lemma-sol-dist} nám teoreticky poskytuje spojitost řešící funkce 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 $g(u, t) = f(\tilde x(t) + u, t) - f(\tilde x(t), t)$. +Obecněji řečeno, 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 ekvivalentní stabilitě libovolného řešení příslušné homogenní rovnice \eqref{eq-homogenous-linear-ode}. @@ -62,7 +62,7 @@ Matice $A$ splňující $\Re \lambda < 0$ pro všechna $\lambda \in \sigma(A)$ s 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 - $$ e^{t\alpha} \| x(t) \| \leq ce^{t_0\alpha} \|x_0\| e^{c\gamma(t - t_0}. $$ + $$ e^{t\alpha} \| x(t) \| \leq ce^{t_0\alpha} \|x_0\| e^{c\gamma(t - t_0)}. $$ 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} @@ -84,7 +84,7 @@ Matice $A$ splňující $\Re \lambda < 0$ pro všechna $\lambda \in \sigma(A)$ s Použijeme seřezávací funkci $$\eta(t) = \begin{cases}1, t < \frac{\delta}{2};\\ 0, t > \delta;\\ - \textit{spojité prodloužení na }(\frac{\delta}{2}, \delta). + \text{spojité prodloužení na }(\frac{\delta}{2}, \delta). \end{cases}.$$ 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. $$\|x(t)\| \leq c\|x(t_0)\| e^{-\beta(t - t_0)}, \beta > 0. $$ @@ -93,7 +93,7 @@ Matice $A$ splňující $\Re \lambda < 0$ pro všechna $\lambda \in \sigma(A)$ s \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. + 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 $x_0$ není stabilní. \end{theorem} \begin{proof}