From 180b81eca1c709f578978b075c2a83c983f68ab4 Mon Sep 17 00:00:00 2001 From: Petr Velycko Date: Thu, 15 May 2025 11:33:34 +0200 Subject: [PATCH] dukaz vety 9.6 + opravy --- ljapunovske-funkce.tex | 36 ++++++++++++++++++++++++++++++++++++ skripta.pdf | Bin 207797 -> 213068 bytes skripta.tex | 3 +++ stabilita.tex | 4 +++- 4 files changed, 42 insertions(+), 1 deletion(-) diff --git a/ljapunovske-funkce.tex b/ljapunovske-funkce.tex index 54bc004..60c05df 100644 --- a/ljapunovske-funkce.tex +++ b/ljapunovske-funkce.tex @@ -89,6 +89,42 @@ Na závěr si uvedeme větu, která nám poskytuje několik možných charakteri \end{enumerate} \end{theorem} +\begin{proof} + Ekvivalence mezi body (i), (ii) a (iii) byla již dokázána ve Větách \ref{thm-stable-hurwitz}, \ref{thm-stable-fundamental} (iv) a větách o tvaru maticové exponenciály. + + Dokážeme, že z (iii) plyne (iv). Definujeme $B := \int_0^{\infty} e^{tA^T} e^{tA} dt$. Platí $e^{tA^T} = (e^{tA})^T, (A^T)^k = (A^k)^T$, proto + $$e^{tA^T} = \sum_{k=1}^\infty \frac{1}{k!} (A^T)^k = \left(\sum_{k=1}^\infty \frac{1}{k!}A^k\right)^T. $$ + Dále pro $\alpha > 0$ z vlastnosti (iii) můžeme psát $\|e^{tA^T}\| = \|e^{tA}\| \leq ce^{-t\alpha}$, + proto + $$ \| e^{tA^T} e^{tA} \| \leq ce^{-\alpha t} \cdot ce^{-\alpha t} = c^2 e^{-2\alpha t}, $$ + tedy integrál v definici matice $B$ je dobře definovaný. + + Ukážeme, že $B$ je symetrická a pozitivně definitní. Platí + \begin{align*} + B^T &= \left( \int_0^\infty e^{tA^T} e^{tA} dt \right)^T = \int_0^\infty (e^{tA^T}e^{tA})^T dt = \\ + &= \int_0^\infty (e^{tA})^T (e^{tA^T})^T dt = \int_0^\infty e^{tA^T}e^{tA} dt = B. + \end{align*} + Dále $B$ splňuje + \begin{align*} + \inner*{Bx, x} &= \inner*{\int_0^\infty e^{tA^T}e^{tA} dt x, x} = \int_0^\infty \inner*{e^{tA^T}e^{tA}x, x} dt \\ + &= \int_0^\infty \inner*{e^{tA}x, e^{tA}x} dt = \int_0^\infty \|e^{tA}x\|^2 dt > 0, + \end{align*} + kde kladnost tohoto výrazu plyne z toho, že $\|e^{tA}x\|$ je nenulové pro $x \neq 0$. + + Zbývá dokázat, že $B$ splňuje rovnost $A^T B + BA = -I$. Skutečně, můžeme psát + \begin{align*} + A^T B &= A^T \int_0^\infty e^{tA^T}e^{tA}dt = \int_0^\infty \odv*{e^{tA^T}}{t}e^{tA} dt \overset{\text{per partes}}=\left[ e^{tA^T} e^{tA} \right]^\infty_0 \\ &- \int_0^\infty e^{tA^T} \odv*{e^{tA}}{t} dt = 0 - I - \int_0^\infty e^{tA^T}e^{tA}dt A = -I - BA. + \end{align*} + + Pro důkaz implikace (iv) $\implies$ (i), ukážeme, že $V(x, t) := \inner*{Bx, x}$ je ljapunovská funkce splňující podmínku z Věty \ref{thm-asymptotic-ljapunov}. Jelikož chceme, aby platilo $\omega(x) \leq V(x, t) \leq \lambda(x)$, stačí volit $\lambda(x) = \omega(x) = \inner*{Bx, x}$. Nalezneme vhodnou funkci $\eta$ tak, aby $\odv*{V(x(t), t)}{t} \leq -\eta(x)$. Díky vlastnostem skalárního součinu platí + \begin{align*} + \odv*{\inner*{Bx(t), x(t)}}{t} &= \inner*{Bx'(t), x(t)} + \inner*{Bx(t), x'(t)} = \\ + &= \inner*{BAx(t), x(t)} + \inner*{Bx(t), Ax(t)} = \\ + &= \inner*{(BA + A^TB)x(t), x(t)} = -\|x(t)\|^2. + \end{align*} + Stačí tedy volit $\eta(x) = \|x\|^2$, a tedy dle výše zmíněné Věty \ref{thm-asymptotic-ljapunov} je naše úloha asymptoticky stabilní. +\end{proof} + Poslední rovnosti se běžně říká \textit{Ljapunovova rovnice}. Dále z bodu (iv) plyne, že $V(x) = x\cdot Bx$ je ljapunovskou funkcí rovnice $x' = Ax$, pomocí níž můžeme sepsat alternativní důkaz věty o linearizované stabilitě (Věta \ref{thm-linearized-stability}). \hfill \textit{konec 11. přednášky (9.5.2025)} diff --git a/skripta.pdf b/skripta.pdf index a9d49ab14cc808a5b5e4738248d9a91afc6471c1..a31b916ff79a64ceee7cf906456c1991a879bf03 100644 GIT binary patch delta 44111 zcmY(KQ+p*0u&iU-wrzW2+qP}3U}D?0Cw4NitqE3a+t}aP&$-!m{R^t=t?p`FLzoUi z{4+KT;1Uvob8~mKFmZtM-ncT*OQ4WS3p~+UHfyhy-5(YVca<}~3i->N=>zt_0z7KE zWg#ck+`zpVbUMlHkTMt7TZrpI!gSJ#erkp*>=}3r0-VPX1Gx}Bc=m^wcwg$^k964F{+6+@%_w@0!d&@cEYj6US*=EH+J;sLcz65x`dezQRRv8>Vzr4o) zj6x3|F6KdYF?N5kE9lb%;yOB2>n}~Sf}c=n`ku0}Vj4BO24r16%?ud>_}?(cnp0-h z8%Jd8?0LF#mK^L$FC=YU7I`r@{R7NvzLEZMxR*{$Dk#ra?!_1{=3YQ&`|L~MKZ;HXc zjl!clqW_W*CHYuZ*W1#Sk(al-l4w#=R4F2KfS%_I8X05UC+G zr%#FT+VeQSzMj&*dUs==k1`KUXAHE8R&{sAwr$=}J*bIgw=B;^e397D&rd%SIuM*R zJO5y5AYfZ!MZi6DnFZR=fAHvMiPa$}9;0u%nK_5)FH-wGc+w3(LO+>OcyBIjvhLxg 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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 $[t_0, \infty)$; @@ -32,10 +33,11 @@ V případě řešení lineární rovnice \eqref{eq-linear-ode}, tj. $x' = A(t)x \end{theorem} \begin{theorem} + \label{thm-stable-hurwitz} 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 $\lambda \in \sigma(A)$. + \item (uniformě) asymptoticky stabilní, právě když $\Re \lambda < 0$ pro všechna vlastní čísla $\lambda \in \sigma(A)$. \end{enumerate} \end{theorem}