I was wondering what are the correct FOCs for this problem: $\max_a \int_{c_1}^{d_1} \int_{c_2}^{d_1} f(a(\theta_1,\theta_2)) d\theta_1 d\theta_2$ with constraint $\frac{\partial a(\theta_1,\theta_2) }{\partial \theta_1}\geq 0$ for all $\theta_1 \in [c_1,d_1],\theta_2 \in [c_2,d_2]$
Some context though don't think is necessary : This is relevant to part of proof of a game theory research problem. So $\theta_1,\theta_2$ are types of agents. If there was no $\theta_2$ the problem would've been close to optimal control problems but now it seems more complex as it is two dimensional. My guess is that even the two dimensional one should be formulatable as optimal control with infinite number of state variables but I'm not sure how to formally attack the problem from that side. Obviously I've tried optimal control techniques.