The question is pair to MO117505 and translates some problem on error-correction codes to similar problem about differential operators. (See also If “force” is periodic does it imply “velocity” is periodic ?).

**Setup** Consider two given periodic functions $(r_1(x), r_2(x))$ (with same period "T"); two linear differential operators $ D_1 (s(x)) = (\sum_i a_i s^{(i)}(x) )$ ,
and similar $D_2$ with some other coefficients (both coefficients are (say) constant, may be I also need something like positive definite - not sure)

**Optimization problem:** Find $(s(x) )$, such that
$||D_1(s) - r_1||^2 + ||D_2(s) - r_2||^2 -> min.$ Where $||.||$ is the $L^2$-norm on the intervals specified below.

**Boundary conditions** Let us fix boundary condition at zero: $b_0 = s(0), b_1=s'(0) ...$.

**Question 1** Consider optimization problem on interval $[0, Y]$. Does the result of optimization depends on the boundary
conditions for large x>>T ? I mean take $Y=NT$ very large comparing with period "T" and look on $s(x)$ for $x$ very large, I would expect
the difference between $s_{boundary1}(x)-s_{boundary2}(x)$ will be small, is it true ?

**Question 2** Is true that function s(x) tends to be periodic for large "x"? I.e. $|s(x+T)-s(x)|-> 0 $ for $x-> \infty $

**Question 3** Look on s(x) for $x \in$ interval $[(N-1)T, NT]$ is it true that for $N-> \infty $ we get same $s(x)$ as if
we solve the same optimization problem without boundary conditions ?

PS It might be that Question 3 is not clearly formulated.

Let me try again.

**Question 3b** Assume the answer on question 2 is positive i.e. s(x) does not depend on initial conditions for large x. Is it true in this case that the solution of the same optimization problem WITHOUT boundary conditions will be the same as s(x) for large x ?