From e56b383cc0e0e0494973676c89417523856968bd Mon Sep 17 00:00:00 2001 From: Petr Velycko Date: Thu, 15 May 2025 16:26:11 +0200 Subject: [PATCH] opravy preklepu --- floquetova-teorie.tex | 2 +- jednoznacnost-reseni.tex | 4 ++-- skripta.pdf | Bin 226907 -> 226953 bytes zavislost-na-podmince.tex | 4 ++-- 4 files changed, 5 insertions(+), 5 deletions(-) diff --git a/floquetova-teorie.tex b/floquetova-teorie.tex index 774a659..0d91c44 100644 --- a/floquetova-teorie.tex +++ b/floquetova-teorie.tex @@ -79,7 +79,7 @@ Matici $C^k$ můžeme relativně snadno spočítat pomocí převodu na Jordanův (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. + (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] diff --git a/jednoznacnost-reseni.tex b/jednoznacnost-reseni.tex index b094ac2..40993d9 100644 --- a/jednoznacnost-reseni.tex +++ b/jednoznacnost-reseni.tex @@ -29,10 +29,10 @@ V této kapitole se budeme věnovat otázce jednoznačnosti řešení diferenci \end{theorem} \begin{proof} - Volme $(x_0, t_0) \in \Omega$ a dvě řešení $(x, I), (y, J)$ taková, že $y(t_0) = x(y_0) = x_0$. Vezmeme $\delta_1 > 0$ tak, aby $f$ byla lipschitzovská na $\delta_1$-okolí $(x_0, t_0)$. + Volme $(x_0, t_0) \in \Omega$ a dvě řešení $(x, I), (y, J)$ taková, že $x(t_0) = y(t_0) = x_0$. Vezmeme $\delta_1 > 0$ tak, aby $f$ byla lipschitzovská na $\delta_1$-okolí $(x_0, t_0)$. Nechť $\delta \leq \frac{1}{2L}$ je takové, že navíc $\delta < \delta_1$ a $t$ takové, aby $(x(t), t), (y(t), t) \in U(x_0, \delta) \times (t_0 - \delta, t_0 + \delta)$. Potom platí $$ \| x(t) - y(t) \| = \| x_0 + \int_{t_0}^t f(x(s), s) ds - (x_0 + \int_{t_0}^t f(y(s),s) ds ) \| \leq $$ - $$ \left| \int_{t_0}^t \| f(x(s), s) - f(y(s), s) \| ds\right| \leq \left| \int L\|x(s) - y(s)\| ds \right| \leq L\cdot\gamma\cdot\delta \leq \frac{\gamma}{2}$$ + $$ \left| \int_{t_0}^t \| f(x(s), s) - f(y(s), s) \| ds\right| \leq \left| \int_{t_0}^t L\|x(s) - y(s)\| ds \right| \leq L\cdot\gamma\cdot\delta \leq \frac{\gamma}{2}$$ pro $\gamma := \sup \| x(s) - y(s) \|$. To platí pro všechna $t$, tedy $\gamma = \sup \|x(t) - y(t)\| \leq \frac{\gamma}{2}$, z čehož plyne $\gamma = 0$, což implikuje rovnost $x(t)$ a $y(t)$. \end{proof} diff --git a/skripta.pdf b/skripta.pdf index 9f54542800e23be021268c83cb843544d8705a62..f065be7f96c0441b7274a7a3992a0617d0694bf7 100644 GIT binary patch delta 36008 zcmV)EK)}D->65eJ_>Vma%Ev{3V7P>T}y8yxskr_uc*DLVH8aUgD+d7 z!vbEz1N&gO_p&oSjHE3Cd{h<+r~iZOeP<}czp37b@sexXe==Vn{4Lx^0{jMIQe0W@+TWV*+-BCMpE7j>CehCUv z57bH-b@%6wU(-yKI?yC7SmcSw!x5D5S}C1paH4LICt!n=r~CAiK3|@qO~V_c-yy%H z-#rgjzJ>Z|n?6PI6S`iX7zvSwTN9$5fW3+~y=vNT9U!G)l4 zrg_}V0x~gM?m4%ZkUn!GPcZK%F8i4GVC9i@FVn>0-f+ji`ItnUB&ZUsRl*fGVT^|? zVDNuwIQQ6Ye>JZFjE_oyJE#Ak4ART3MPa?utb1aw>a^@u+>gie9R$GJwAXPQFlW3s zm<@j;jcJ$J%mYZ{=^z81uhYQRcR=dU0ocKJHjm8nXAVfSCFxiwVVj{|)x)M@Glpza10&l?NjrM^%7e?%77cZON8)dIADN;jnzJN46W zL7ph@^tvM2W^Tv2OaxRI&Nx1ccIny24KHk=jc$aOpTv#u3uEY_H;Tr}dL%D6$t-tt z!45UMr~m|dtMV@1T5nlbt};^kt8E=c48HQibm-RfyT%)Ff_a7=&>W z3-yMxGQtME678C`{Q(xXp}OmLR9d^IBEqqUkzuV 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Okamžitě z předpoklady vidíme, že $w(t) \leq \Phi(t)$. Zderivujeme funkci $\Phi(t)$, dostáváme + Definujeme $\Phi(t) := K + \int_{t_0}^t w(s) g(s) ds + \varepsilon$ pro $t > t_0$ (důkaz pro $t \leq t_0$ se udělá obdobně). Okamžitě z předpokladů vidíme, že $w(t) \leq \Phi(t)$. Zderivujeme funkci $\Phi(t)$, dostáváme $\Phi'(t) = w(t) g(t) \leq \Phi(t) g(t)$ což po vydělení $\Phi(t)$ (je nenulové díky přičtení $\varepsilon$) nám dává $\frac{\Phi'(t)}{\Phi(t)} \leq g(t)$, což můžeme přeintegrovat od $t_0$ do $t$, čímž dostaneme $\int_{t_0}^t \frac{\Phi'(s)}{\Phi(s)} ds \leq \int_{t_0}^t g(s) ds$. Po vyčíslení integrálů dostaneme - $ \log (\Phi(t) - \Phi(t_0)) \leq \int_{t_0}^t g(s) ds$. Jelikož $\exp$ je rostoucí funkce, můžeme psát $ \frac{\Phi(t)}{\Phi(t_0)} \leq \exp\left(\int_{t_0}^t g(s) ds\right)$. + $ \log \Phi(t) - \log \Phi(t_0) \leq \int_{t_0}^t g(s) ds$. Jelikož $\exp$ je rostoucí funkce, můžeme psát $ \frac{\Phi(t)}{\Phi(t_0)} \leq \exp\left(\int_{t_0}^t g(s) ds\right)$. Nakonec dostáváme $w(t) \leq \Phi(t) \leq (K+\varepsilon) \exp\left(\int_{t_0}^t g(s) ds\right)$. Požadované tvrzení získáme posláním $\varepsilon$ do $0$. \end{proof}