Min function given by integral and inequality type constraint

[Reynolds]

Greetings to you all. I need your help with the following problem



We minimize

 2a + \int_0^1 tx(t)dt \to \min
 

 1 - a - \it_t^1 x(s)ds \leq 0   a.e. t \in (0,1)
 x(t) \geq 0   a.e t \in (0,1)
 a \geq 0
 x \in L^2((0,1)).

I want to formulate dual problem and to prove that the optimal value of primary problem is 2 and the optimal valeu of the dual problem is 1.


(it looks like isoperimetric problem...)


Can you please give me some hints or show me where I can find how to solve problems like this.
Thank you !

 

[stapel]

We minimize

2a + \int_0^1 tx(t)dt \to \min
  
 1 - a - \it_t^1 x(s)ds \leq 0   a.e. t \in (0,1)
 x(t) \geq 0   a.e t \in (0,1)
 a \geq 0
 x \in L^2((0,1)).

Does the coded text mean the following?

. . . . .\(\displaystyle \mbox{Minimize: }\, 2a\, +\, \int_0^1\, (\,t\, \cdot\, x(t)\, )\, dt\)

. . . . .\(\displaystyle 1\, -\, a\, -\, \int_t^1\, (\, x(s)\, )\, ds\, \leq\, 0\, \mbox{ almost everywhere }\, t\, \in\, (0,\, 1)\)

. . . . .\(\displaystyle x(t)\, \geq\, 0\, \mbox{ almost everywhere }\, t\, \in\, (0,\, 1)\)

. . . . .\(\displaystyle a\, \geq\, 0\)

. . . . .\(\displaystyle x\, \in\, L^2 \bigg( \, (0,\, 1)\, \bigg) \)

Thank you!

 

[Reynolds]

Oh, yeah!!
Sorry I tried to find Latex tags, but I was not clever enough . Do you have any hints, ideas where I can find similar problems?
Thanx