Can we apply fundamental lemma of calculus of variations

Question

Let us assume that $t_0$ and $t_1$ are some constants, and $f(t)$ and $u(t)$ is a continuous mapping from $[t_0,t_1]$ to $\mathbb{R}^p$ and $\mathbb{R}^m$, respectively. Suppose that $$\int_{t_0}^{t_1}\left[\int_0^t u(\tau)^TB^Te^{A(t-\tau)}C^Td\tau\right].\left[f(t)-\int_0^tCe^{A(t-\tau)}B\hat{u}(\tau)d\tau\right] dt = 0, \text{ for all } u(\tau). $$ Here, $A \in \mathbb{R}^{n\times n},B\in \mathbb{R}^{n\times m}, C\in \mathbb{R}^{p\times n}$. I want to find the solution to $\hat{u}(t)$, so can we imply anything about $\hat{u}(t)$? I'm thinking whether the result of Fundamental lemma of calculus of variations can be used here, but I'm still new to the topic. Any help will be good.