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I am looking for some bibliography regarding solving integral optimization problems numerically (preferably using Matlab).

I want to solve problems of the type $$ \min_{r \in A} \int_a^b (f(x,r(x),r'(x))dx $$ where $A$ is a class of admissibility for $r$, where some constraints may appear.

I am not looking for a general method to solve all these problems. In my case $f$ is usually a very simple function. I just want to see some examples solved in detail, so that I could try and adapt the methods to my case.

I would be grateful if you could point me in the right direction.

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I have deleted my answer as it appears I misread the question. I can't say that I have seen a matlab solution for the problem that you are posting here. – Dennis Jaheruddin Dec 5 '12 at 15:10

The general problem you stated belongs to the Calculus of variation. To give you a general idea of how to solve this let me adapt your notation.

We want to minimize \begin{align} \int_a^b f(x,y(x),y'(x) \,dx \text{ for } y\in A \end{align} Where $A$ is a class of function. To minimize the above integral we need to solve the Euler Lagrange Equation \begin{align} \frac{d}{dx}f_{y'} -f_y&=0 \\ \Leftrightarrow f_{y'x}+f_{y'y}y'+f_{y'y'}y''&=f_y \end{align} Which is a 2nd order differential equation for y, as f is known. This method is called indirect method.

To give you an concrete example, I think you could read this .pdf starting at chapter 7.1.1 It starts with the general theory and gives you the matlab code for an explicit example.

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