Difference between revisions of "Aufgaben:Problem 14"
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+ | =Problem 4a= | ||
+ | Let \( T > 0,\eta \in C^{1}([0,T]), \phi \in C^{0}([0,T]) \), and assume that: | ||
+ | $$ \frac{d}{dt}\eta(t) \leqslant \eta(t)\phi(t), \forall t \in [0,T] $$ | ||
+ | Show that: | ||
+ | $$\eta(t) \leqslant \eta(0)e^{\int_{0}^{t} \phi(s) ds}, \forall t \in [0,T] $$ | ||
+ | |||
+ | ''Hint'': consider \( \xi(t) := \eta(t) e^{-\int_{0}^{t} \phi(s) ds} \) | ||
+ | |||
+ | =Solution= | ||
+ | Using chain rule: | ||
+ | $$ \frac{d}{dt} \xi(t) = \frac{d}{dt} (\eta(t) e^{-\int_{0}^{t} \phi(s) ds}) = (\frac{d}{dt} \eta(t)) e^{-\int_{0}^{t} \phi(s) ds} + \eta(t) (\frac{d}{dt} e^{-\int_{0}^{t} \phi(s) ds}) = (\frac{d}{dt} \eta(t)) e^{-\int_{0}^{t} \phi(s) ds} - \eta(t) \phi(t)e^{-\int_{0}^{t} \phi(s) ds})$$ | ||
+ | |||
+ | Substituting \( \eta(t) = \xi(t) e^{\int_{0}^{t} \phi(s) ds} \) in our assumption, then we get \( \forall t \in [0,T] \): | ||
+ | |||
+ | $$ | ||
+ | \begin{align} | ||
+ | \frac{d}{dt}\eta(t) &\leqslant \eta(t)\phi(t) \\ | ||
+ | \frac{d}{dt}(\xi(t) e^{\int_{0}^{t} \phi(s) ds}) &\leqslant \xi(t) e^{\int_{0}^{t} \phi(s) ds}\phi(t) \\ | ||
+ | (\frac{d}{dt}\xi(t)) e^{\int_{0}^{t} \phi(s) ds} + \xi(t) \frac{d}{dt}(e^{\int_{0}^{t} \phi(s) ds})&\leqslant \xi(t) e^{\int_{0}^{t} \phi(s) ds}\phi(t) \\ | ||
+ | (\frac{d}{dt}\xi(t)) e^{\int_{0}^{t} \phi(s) ds} + \xi(t) e^{\int_{0}^{t} \phi(s) ds}\phi(t)&\leqslant \xi(t) e^{\int_{0}^{t} \phi(s) ds}\phi(t) \\ | ||
+ | (\frac{d}{dt}\xi(t)) e^{\int_{0}^{t} \phi(s) ds}&\leqslant 0 \\ | ||
+ | \frac{d}{dt}\xi(t) &\leqslant 0 \\ | ||
+ | \end{align} | ||
+ | $$ | ||
+ | , where we used in the last step that \(\forall x: e^{x}>0 \). | ||
+ | |||
+ | Now integrating the result from 0 to t and since \(g(x) \leqslant h(x) \Rightarrow \int g(x) dx \leqslant \int h(x) dx \): | ||
+ | |||
+ | $$ | ||
+ | \begin{align} | ||
+ | \int_{0}^{t} \frac{d}{dt'}\xi(t') dt' = \xi(t) - \xi(0) &\leqslant 0 = \int_{0}^{t} 0 dx \\ | ||
+ | \eta(t) e^{-\int_{0}^{t} \phi(s) ds}-\eta(0) e^{-\int_{0}^{0} \phi(s) ds} &\leqslant 0 \\ | ||
+ | \eta(t) e^{-\int_{0}^{t} \phi(s) ds}-\eta(0) e^{0} &\leqslant 0 \\ | ||
+ | \eta(t) e^{-\int_{0}^{t} \phi(s) ds}&\leqslant \eta(0) \\ | ||
+ | \eta(t) &\leqslant \eta(0)e^{\int_{0}^{t} \phi(s) ds} \\ | ||
+ | \end{align} | ||
+ | $$ |
Revision as of 15:49, 26 December 2014
Problem 4a
Let \( T > 0,\eta \in C^{1}([0,T]), \phi \in C^{0}([0,T]) \), and assume that: $$ \frac{d}{dt}\eta(t) \leqslant \eta(t)\phi(t), \forall t \in [0,T] $$
Show that: $$\eta(t) \leqslant \eta(0)e^{\int_{0}^{t} \phi(s) ds}, \forall t \in [0,T] $$
Hint: consider \( \xi(t) := \eta(t) e^{-\int_{0}^{t} \phi(s) ds} \)
Solution
Using chain rule: $$ \frac{d}{dt} \xi(t) = \frac{d}{dt} (\eta(t) e^{-\int_{0}^{t} \phi(s) ds}) = (\frac{d}{dt} \eta(t)) e^{-\int_{0}^{t} \phi(s) ds} + \eta(t) (\frac{d}{dt} e^{-\int_{0}^{t} \phi(s) ds}) = (\frac{d}{dt} \eta(t)) e^{-\int_{0}^{t} \phi(s) ds} - \eta(t) \phi(t)e^{-\int_{0}^{t} \phi(s) ds})$$
Substituting \( \eta(t) = \xi(t) e^{\int_{0}^{t} \phi(s) ds} \) in our assumption, then we get \( \forall t \in [0,T] \):
$$ \begin{align} \frac{d}{dt}\eta(t) &\leqslant \eta(t)\phi(t) \\ \frac{d}{dt}(\xi(t) e^{\int_{0}^{t} \phi(s) ds}) &\leqslant \xi(t) e^{\int_{0}^{t} \phi(s) ds}\phi(t) \\ (\frac{d}{dt}\xi(t)) e^{\int_{0}^{t} \phi(s) ds} + \xi(t) \frac{d}{dt}(e^{\int_{0}^{t} \phi(s) ds})&\leqslant \xi(t) e^{\int_{0}^{t} \phi(s) ds}\phi(t) \\ (\frac{d}{dt}\xi(t)) e^{\int_{0}^{t} \phi(s) ds} + \xi(t) e^{\int_{0}^{t} \phi(s) ds}\phi(t)&\leqslant \xi(t) e^{\int_{0}^{t} \phi(s) ds}\phi(t) \\ (\frac{d}{dt}\xi(t)) e^{\int_{0}^{t} \phi(s) ds}&\leqslant 0 \\ \frac{d}{dt}\xi(t) &\leqslant 0 \\ \end{align} $$ , where we used in the last step that \(\forall x: e^{x}>0 \).
Now integrating the result from 0 to t and since \(g(x) \leqslant h(x) \Rightarrow \int g(x) dx \leqslant \int h(x) dx \):
$$ \begin{align} \int_{0}^{t} \frac{d}{dt'}\xi(t') dt' = \xi(t) - \xi(0) &\leqslant 0 = \int_{0}^{t} 0 dx \\ \eta(t) e^{-\int_{0}^{t} \phi(s) ds}-\eta(0) e^{-\int_{0}^{0} \phi(s) ds} &\leqslant 0 \\ \eta(t) e^{-\int_{0}^{t} \phi(s) ds}-\eta(0) e^{0} &\leqslant 0 \\ \eta(t) e^{-\int_{0}^{t} \phi(s) ds}&\leqslant \eta(0) \\ \eta(t) &\leqslant \eta(0)e^{\int_{0}^{t} \phi(s) ds} \\ \end{align} $$