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At least if everything is sufficiently smooth, $$\frac{\delta }{\delta u(s,t)} \int dy\ \phi \left( \int dz\ J(z,y,u(z,y)) \right) = \phi^{\prime } \left( \int dz\ J(z,t,u(z,t)) \right) \frac{\partial J}{\partial u} (s,t,u(s,t))$$
Too long to comment. I assume that $x$ and $u$ are within the range $[-1,1]$ for $f$ to be well-defined. Suppose $x=\sin(\theta)$, $0\leq \theta \leq \pi/2$. Let $u=\sin(\phi)$. In that case, the optimization problem is: $$\min_{\phi}~~\cos(\phi) + \frac{1}{2}(\sin(\phi)-\sin(\theta))^2.$$ Differentiating the cost function yields:  -\sin(\phi) + \sin(\...