Optimal control with integral constraint

Question

I am faced with the following optimization problem: $$\max_{x(v_1,v_2)}\int_0^1\int_0^1f_1(v_1,v_2)x(v_1,v_2)\,dv_1\,dv_2,$$ subject to: $$\int_0^1\int_0^1f_2(v_1,v_2)x(v_1,v_2) \, dv_1 \, dv_2\geq 0,\; x(v_1,v_2)\in[0,1].$$ $f_1$ and $f_2$ are some known functions of 2 variables (affine) and $x(v_1,v_2)$ is an unknown function of the 2 variables with values between 0 and 1, not necessarily differentiable or continuous. Can you help me out? Any ideas or links to appropriate literature would be appreciated :)

Answer 1

As I wrote in the comment, the solution for the problem in which the inequality constraint is binding follows relatively standard theory of the so-called isoperimetric problems. An interested reader can find out more in, for instance, this book by Kamien and Schwartz. Basically, one uses Lagrange multiplier $\lambda$ for the constraint and gets that $x(v_1,v_2)=1$ for $f_1-\lambda f_2\geq 0$ and zero otherwise. One can then use the (binding) constraint to retrieve the exact value of $\lambda$.