From db835c8aa5b951e088a81fc4535a004412d1b1a1 Mon Sep 17 00:00:00 2001 From: Petr Velycko Date: Thu, 15 May 2025 12:53:20 +0200 Subject: [PATCH] prednaska 16.5.2025 --- floquetova-teorie.tex | 96 ++++++++++++++++++++++++++++++++++++++++++ skripta.pdf | Bin 213068 -> 226707 bytes skripta.tex | 3 ++ 3 files changed, 99 insertions(+) create mode 100644 floquetova-teorie.tex diff --git a/floquetova-teorie.tex b/floquetova-teorie.tex new file mode 100644 index 0000000..29b9e05 --- /dev/null +++ b/floquetova-teorie.tex @@ -0,0 +1,96 @@ +\section{Floquetova teorie} + +V této kapitole se budeme zabývat lineární rovnicí +\begin{equation} + \label{eq-linear-periodic} + x' = A(t)x + b(t) +\end{equation} +a příslušnou homogenní rovnicí +\begin{equation} + \label{eq-linhom-periodic} + x' = A(t)x +\end{equation} +pro $T$-periodické funkce $A \in C(\R, \R^{n\times n})$ a $b \in C(\R, \R^n)$. Budeme zkoumat periodicitu a stabilitu řešení této rovnice. + +\begin{observation} + \begin{enumerate}[(i)] + \item Pro každou počáteční podmínku $x_0 \in \R^n$ existuje právě jedno maximální řešení obou těchto rovnic a to je definované na celém $\R$. + \item Buď $x$ řešení \eqref{eq-linear-periodic} a definujeme $y(t) := x(t + T)$. Potom $y$ je také řešení dané rovnice. Skutečně, platí + $$y'(t) = x'(t + T) = A(t+T)x(t+T) + b(t+T) = A(t)y(t) + b(t).$$ + \item Řešení $x$ je $T$-periodické právě tehdy, když $x(T) = x(0)$. Implikace $\implies$ je triviální, pro důkaz té opačné můžeme definovat $y(t) := x(t + T)$, což je díky již dokázanému také řešení a $y(0) = x(T) = x(0)$. Z jednoznačnosti řešení se tyto dvě řešení musí rovnat, tedy máme $x(t) = y(t) = x(t + T)$ pro všechna $t$. + \item Nechť $\Phi$ je fundamentální matice \eqref{eq-linhom-periodic} taková, že $\Phi(0) = I$. Potom řešení $x$ rovnice \eqref{eq-linhom-periodic} je periodické právě tehdy když $\Phi(T)x_0 = x_0$, jinými slovy, $x_0$ je vlastní vektor matice $\Phi(T)$ příslušný vlastnímu číslu $1$. + \end{enumerate} +\end{observation} + +\begin{definition} + Nechť $\Phi$ je fundamentální matice \eqref{eq-linhom-periodic} taková, že $\Phi(0) = I$. Pak matici $C = \Phi(T)$ nazveme \textit{maticí monodromie}. +\end{definition} + +Dle předchozího pozorování není těžké zpozorovat, že počet periodických řešení naší úlohy úzce souvisí s dimenzí jádra matice $C - I$, tedy násobností vlastního čísla $1$ matice $C$. + +\begin{lemma}[Maticový logaritmus] + \label{lemma-matrix-log} + Nechť $A \in \R^{n\times n}$ je regulární. Pak existuje matice $B \in \C^{n\times n}$ splňující $e^B = A$. Tato matice $B$ nemusí být určena jednoznačně. +\end{lemma} + +Ústřední větou celé Floquetovy teorie je následující věta, která nám umožňuje jistým způsobem charakterizovat chování fundamentální matice pro rovnici \eqref{eq-linhom-periodic}. + +\begin{theorem}[Floquet] + \label{thm-floquet} + Existuje $Q \in C(\R, \R^{n\times n})$ $T$-periodická, $Q(t)$ regulární pro všechna $t \in \R$ a $B \in \C^{n\times n}$ tak, že + $$ \Phi(t) = Q(t)e^{tB} \quad\forall t \in \R. $$ + Zde $\Phi(t)$ je fundamentální matice úlohy \eqref{eq-linhom-periodic} taková, že $\Phi(0) = I$. +\end{theorem} + +\begin{proof} + Nechť $C = \Phi(T)$ je matice monodromie. Z Lemmatu \ref{lemma-matrix-log} nalezneme komplexní matici $\tilde B \in \C^{n \times n}$ takovou, že $e^{\tilde B} = C$ a položíme $B = \frac{1}{T}\tilde B$ (tedy $e^{T\cdot B} = C$ z vlastností exponenciály). + + Definujeme $Q(t) := \Phi(t)e^{-tB}$. Tato matice je regulární, spojitá a splňuje $\Phi(t) = Q(t)e^{tB}$. Dokážeme, že je $T$-periodická. K tomu budeme potřebovat funkci $\Psi(t) := \Phi(t + T)$. Platí + $$ \Psi'(t) = \Phi'(t + T) = A(t + T)\Phi(t + T) = A(t)\Phi(t), $$ + tedy $\Psi$ je taky fundamentální matice a platí $\Psi(0) = \Phi(t) = C$. Zároveň také $\Phi(t)C$ je fundamentální maticí \eqref{eq-linhom-periodic} a $\Phi(0)C = C$. Z jednoznačnosti dostáváme, že $\Psi(t) = \Phi(t) C$ pro všechna $t \in \R$. Nakonec + $$ Q(t + T) = \Phi(t + T)e^{-(t + T)B} = \Psi(t) e^{-TB}e^{-tB} = \Phi(t) C C^{-1} e^{-tB} = Q(t), $$ + čímž je důkaz ukončen. +\end{proof} + +Floquetova transformace $y(t) = Q^{-1}(t)x(t)$ převádí řešení \eqref{eq-linhom-periodic} na řešení úlohy $y' = By$, což je rovnice s konstantními koeficienty. Platí +$$ y(t) = Q^{-1}(t)\Phi(t)x_0 = Q^{-1}(t)Q(t) e^{tB}x_0 = e^{tB}x_0. $$ +Položíme-li $t = kT + s$, kde $k \in \Z$ a $s \in [0, T)$, pak +\begin{align*} + x(t) &= \Phi(t)x_0 = Q(t)e^{tB}x_0 = Q(s) e^{(kT+s)B}x_0 = \\ + &= Q(s)e^{sB}(e^{TB})^k x_0 = \Phi(s)C^k x_0. +\end{align*} +Matici $C^k$ můžeme relativně snadno spočítat pomocí převodu na Jordanův kanonický tvar, což v praxi často výrazně zjednodušuje výpočty. + +\begin{corollary} + Pokud pro spektrum matice monodromie platí $\sigma(C) \in \{z \in \C: |z| < 1\}$, pak máme asymptotickou stabilitu. +\end{corollary} + +\begin{theorem}[Existence periodických řešení] + Nechť matice $A(t)$ je spojitá a $T$-periodická. Potom je ekvivalentní: + \begin{enumerate}[(i)] + \item rovnice \eqref{eq-linear-periodic} (nehomogenní) má právě jedno $T$-periodické řešení pro každou $b \in C(\R, \R^n)$ $T$-periodickou; + \item rovnice \eqref{eq-linhom-periodic} (homogenní) má pouze triviální $T$-periodické řešení; + \item $1 \not\in \sigma(C)$. + \end{enumerate} +\end{theorem} + +\begin{proof} + (ii) $\equiv$ (ii): $x$ je $T$-periodické řešení právě tehdy když $x(T) = x(0)$, coz nastane právě tehdy když $Cx_0 = x_0$. + + (i) $\implies$ (ii): Předpokládejme, že $x$ je $T$-periodické řešení \eqref{eq-linear-periodic} a pro spor předpokládejme, že $y$ je netriviální $T$-periodické řešení homogenní rovnice, tedy $x + y$ je $T$-periodické řešení \eqref{eq-linear-periodic}, ale $x + y \neq x$, což je spor s jednoznačností řešení nehomogenní úlohy. + + (iii) $\implies$ (i): Řešení nehomogenní úlohy můžeme explicitně zapsat ve tvaru $x(t) = \Phi(t)x_0 + \int_0^t \Phi(t) \Phi^{-1}(s) b(s) ds$. Dokážeme, že toto je jediné $T$-periodické řešení úlohy \eqref{eq-linear-periodic}. Můžeme psát $x(T) = Cx_0 + y$, kde $y = \int_{0^T}\Phi(T)\Phi^{-1}(s)b(s) ds$, tedy $x(T) = x_0$ právě tehdy, když $-y = (C - I)x_0$. Dle předpokladu v (iii) je matice $C - I$ regulární, tedy existuje právě jedno $x_0$ splňující danou podmínku, proto $T$-periodické řešení \eqref{eq-linear-periodic} je určeno jednoznačně. +\end{proof} + +\begin{corollary}[Stabilita pro periodické lineární rovnice] + Nulové řešení \eqref{eq-linhom-periodic} je (asymptoticky) stabilní, právě když nulové řešení $y' = By$ je (asymptoticky) stabilní. +\end{corollary} + +\begin{proof} + Z Floquetovy věty (Věta \ref{thm-floquet}) a poznámky o Floquetově transformaci víme, že $x(t) = Q(t)e^{tB}x_0 = Q(t) y(t)$. Potom + $$ \|x(t)\| \leq \|Q(t)\|\|y(t)\| \leq \max_{t \in [0, T]} \|Q(t)\| \|y(t)\| = K \|y(t)\|$$ + a obdobně + $$ \|y(t)\| \leq \|Q^{-1}(t)\|\|x(t)\| \leq \tilde K \|x(t)\|.$$ +\end{proof} + +\hfill \textit{konec 12. přednášky (16.5.2025)} diff --git a/skripta.pdf b/skripta.pdf index a31b916ff79a64ceee7cf906456c1991a879bf03..ca4b4d242f6bff3da9b6314338b95f0bc0c955dc 100644 GIT binary patch delta 70305 zcmZs?LvSt(uqGVawr$(C?VOk=cJjuyZQIF-ZQHhO%y*||F+Zw$(bd(9UOwt6fB2+J zc>iW#;WbbeChjz25D;nZ&dDg($DYd;3a3+UjQ6rh)9uoKf%rxf(Eojw_6yX2>c^hgK$Uh*DrCw zKbavrPDV!Buq6eBpI_s&NaO2MT*kYEusjF;9&#E!hfh}?uGpqkXqem21WOB{+lXor=p6HzO z1I;N#Xk%mUYVHN_FFgmls#=h*h81GLBvIP;-1;)?5S7si28}_wliK( zww$!>q3GUoe4WcD#tgWzN{I}3%y|$I=hs6KcxoDaf5`9fI@gnMUC%pvliJP+6DrNU zWUQ&ArnX+ML74zA@2@p~^)z3V5rrsgc%i^?ei*Oiph^L7(sa6f*{>vFZ{w5c*gwqH zVlXJ;K~iITJoAeTDIcN9hPIAg2kjRmQ#SQNxATE4E_`wU#yKanOe9#ekJ)Q|a8~G6 zl?}?0Qdwlso8IyqBq3sycJ|ov^cOhV0Ug~I-7-A0I5L?CG&WW~e)KiZ^qUjp-^*#X 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