strong form of variational problem in control theory

Question

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.

Answer 1

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

$$Q_1(s)+\int_s^T Q_2(s,t) dt=0. $$