My problem is the following. Let $t\in[0,1]$, let $u(t)\in[0,1]$ be the control funciton, and $a\in[\underline{a},\overline{a}]$ is a parameter. I know that the solution to the problem $$ \int_0^1 L(u,a,t) dt \rightarrow \max_{u\in[0,1]} $$ where $L$ is a `nice' (continuously differentiable, etc.) function, takes the following form: $u(t)=0$ for $t<t_a$ and $u(t)=1$ for $t\geq t_a$. So, it is a bang-bang solution with switching time depending on parameter $a$.
Now, the quesiton I have: is it true, that solution to $$ \int_{\underline{a}}^{\overline{a}}\left(\int_0^1 L(u,a,t) dt\right)f(a)da \rightarrow \max_{u\in[0,1]} $$ where $f(a)>0$ -- a continuous (density) funciton, takes the same form with a unique switching point?