Given smooth functions $Q\_1(t), Q\_2(s,y)$ which are both bounded above and below, consider the following variational problem for $T,t>0$ and $T>t$:


$\int\_0^T \delta w(s) Q\_1(s) \ ds + \int\_0^T \int\_0^t \delta w(s) Q(s,t) \ ds \ dt = 0$, for all smooth and bounded variations $\delta w$ which need not be zero at the end points,


what is the strong form of this variational problem?


I would like to find an equation of the form $K(Q\_1(t), Q\_2(s,t) = 0$, $\forall s,t$. is this possible, and if so, how can it be done.



strong form of variational problem in control theory
Note that the second integral is the integral over the triangle $\newcommand{\bR}{\mathbb{R}}$


$$ \Delta\_T=\big\{ (s,t)\in\bR^2;\;\;0\leq s\leq t\leq T\big\}. $$
Fubini's theorem shows that for any integrable function $f: \Delta\_T\to\bR$ we have


$$\int\_0^T \left(\int\_0^t f(s,t) ds\right)dt=\int\_{\Delta\_T} f(s,t) dsdt=\int\_0^T\left(\int\_s^T f(s,t) dt)\right) ds. $$


If we let $f(s,t):= \delta w(s) Q\_2(s,t)$ and we deduce


$$\int\_0^T\left(\int\_0^t \delta w(s) Q\_2(s,t) ds\right) dt=\int\_0^T\delta w(s)\left(\int\_s^T Q\_2(s,t) dt\right) ds $$


This shows that the "strong" form you seek is


$$Q\_1(s)+\int\_s^T Q\_2(s,t) dt=0. $$


