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I want to find one method or approach or idea which compute following statement: $$ \sup_{t \in [0,1]} \left( \inf_{X \in C^1([0,1])} \left\| \frac{dX(t)}{dt} - A(t)X(t) - F(t) \right\| \right) $$ Thanks alot.

I want to find another idea for solving:

$$\displaystyle\min\int_0^1\left(\left\|\frac{dX(t)}{dt}-A(t)X(t)-F(t)\right\|\mathrm{d}t\right) ,X(0)=X_0,X(t)\in K\subseteq\mathbb{R}^n\;,$$

where $K$ is a closed interval.

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    Could you provide some more explanation? That equation is incomprehensible.2011-12-27
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    Tried to fix up the equation. Rehana, please use LaTeX formatting in the future, as described [here](http://meta.math.stackexchange.com/questions/480/math-markup-diagrams-etc-pointers-please/484#484).2011-12-27
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    Thanks alot my friend for your attention, I edit my question by title' essential problem for homework".2011-12-27
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    We need more things to be able to answer your question correctly. What is $X$'s image? $X : [0,1] \to ???$ If you're studying analysis, this is one of the most important things to notice... and what is $A$ and $F$? then we can start talking.2011-12-27
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    Dear Henning Makholm, A and F are not fixed.2011-12-27
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    @Rehana, then what are they? They are not bound anywhere in your expression.2011-12-27
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    Dear Patrick, X(.)∈c^ı [0,ı] that is, X(.):[0,1]→[0,1] is continues and first order differential.2011-12-27
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    @Rehana, in C¹(A), the A usually defines only the _domain_, whereas the codomain is left implicit. Also: You should use `@` instead of "Dear" -- that will make the software automatically alert those you speak to about your comment.2011-12-27
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    Please don't write \underset{X \in C^1([0,1])}{\inf}. I change this to \inf_{X \in C^1([0,1])}. In "display" mode, that causes the subscript to be directly under the "inf". In "inline" mode, the subscript will appear below and to the right of "inf". The whole point of having operator names like \inf, \det, \gcd, etc., instead of writing \mathrm{inf} or the like is that the standard conventions are built in. That also includes things like a space between "inf" and "A" when you write \inf A, and non-italicization of "inf".2011-12-27
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    @MichaelHardy Fyi your comment may not be seen by the author of said TeX code since you did not explicitly address him via @ name.2011-12-27
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    +1 to Hardy's comment; so that it stays on top, @BillDubuque :) BTW, I have been thinking the OP is always notified of the comments under his posts.2011-12-27
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    @Michael Hardy : Thanks for pointing that out. I was the one that used the underset... I didn't know this worked here. I usually don't do it when I TeX but a lot of things I'm used to do don't work here so I tend to code as explicitly as possible..2011-12-27
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    @Rehana, are you the same user as [KaySid](http://math.stackexchange.com/users/21800/kaysid)? It looks like you're both using the same broken math-formula-to-unicode paster.2011-12-27
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    @Kannappan The author of said TeX code was not the OP so he will not be pinged without explicit mention.2011-12-27
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    Can I ask someone who understands the question to edit the title into something more informative than "essential problem", please? The best I can come up with is "supremum of something involving a derivative," but surely someone can do better than that.2011-12-28

2 Answers 2

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Since $X$ is allowed depend on $t$, you can just choose $X$ to be $u\mapsto (u-t)F(t)$. Then $X(t)=0$ takes care of the $A(t)X(t)$ term, and $-F(t)$ is cancelled out by the derivative of $X$.

Therefore the $\inf$ is $0$ for every $t$, and thus the result of the entire expression is $0$, too.

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    @Rehana, I'm not your dear friend, I'm a random stranger on the internet. Also, you have some funky unicode characters there. I have no idea what the mathematical meaning of U+2592 `MEDIUM SHADE` is even supposed to be. Please stick to LaTeX.2011-12-27
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    ,Thanks , It is good idea, would you mind to give me another idea which is similar to sup inf for solve (dX(t))/dt=A(t)X(t)+F(t) by min∫_0^1▒‖dX(t)/dt-A(t)X(t)-F(t) ‖dt X(0)=X_0 X(t)∈K⊆R^(n ),K is close interval〗2011-12-27
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We can do the following. Let $X(t) = Y(t) \, \mathrm{exp}\left( \left( \int_0^t A(z) \, dz \right)t \right)$ that I will write (for short) $Y \, e^{t \int A }$. Then the differential equation $$ \frac{dX}{dt} - A(t)X(t) - F(t) = 0 $$ becomes $$ \frac{d(Y \, e^{t \int A })}{dt} - A Y \, e^{t \int A } - F(t) = Y'e^{t \int A } + AY \, e^{t \int A } -A Y \, e^{t \int A } - F(t) = Y' \, e^{t \int A } - F(t) = 0 $$ which means that $$ X(t) = Y(t) \, e^{t \int A }= e^{t \int_0^t A} \int_0^t e^{-u \int_0^u A} F(u) \, du. $$ Therefore your supremum is zero whenever this function is in $C^1([0,1])$. When $A$ and $F$ are continuous we can see that this is the case.

Hope that helps,

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    @Rehana, it would be so much easier if you used latex to model your questions/comments.2011-12-27
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    @DimitriSurinx, sorry, I don't have Latex, this software was damaged on my notebook.2011-12-27
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    @Patrick, Thanks , It is good idea, would you mind to give me another idea which is similar to sup inf for solve (dX(t))/dt=A(t)X(t)+F(t) by min∫_0^1▒‖dX(t)/dt-A(t)X(t)-F(t) ‖dt X(0)=X_0 X(t)∈K⊆R^(n ),K is close interval〗2011-12-27
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    @Rehana You don't have to have LaTeX installed on your machine in order to use it on this website. See [the help page](http://meta.math.stackexchange.com/questions/107/faq-for-math-stackexchange/117#117). If Javascript doesn't work on your computer then that's a more serious problem.2011-12-27
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    @Patrick, why did you say supremum is zero for above function?2011-12-29