Calculus of Variations with Inequality Constraints

[MDR123]

Hi, I am working on a calculus of variations problem and have a general question.

Specifically, I was wondering about what kind of constraint functions are possible.

I have a constraint of the form:

$$f(x)x - \int_{x_0}^x f(z) dz \leq K$$

If I had a constraint that just depends on x or a pure integral condition how to deal with it. However, it is unclear to me how to deal with a condition that depends upon both.

My other idea for an approach is to notice that the above condition is increasing if f is increasing. In addition, I have a bounded range for x. So, I know the below condition implies the above condition, but how to apply it to a calculus of variations question is unknown to me as well.

$$f(\bar{x})\bar{x} - \int_{x_0}^{\bar{x}} f(z) dz \leq K$$
$$f'(x) \geq 0$$

Thank you,
MDR123

 

[fresh_42]

You could use the fundamental theorem of calculus and write your condition as $f(x)x-F(x) = \int_{x_0}^x f'(z)z\,dz\leq K_1$. Maybe this helps.