I'll try to translate certain problem about convolutional codes to more common language of ODE, hope my translation is correct, but welcome to criticize.

Consider two given functions periodic functions $(r_1(x), r_2(x))$ (with same period); 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 say $L^2$ norm (over period of $r_i$).

**Question 1** Is the problem well-defined over $R$ ? (I.e. unique solution ?) (For $D_i$ generic - i.e. their null-spaces do not intersect).

**Question 2** Is it true that solution will be periodic function ?

**Remark About question 1:** I do not put boundary conditions, that might be some reason for non-uniqueness, but I have two operators, so if null-spaces of $D_i$ do not intersect, this avoids obvious problem which may arise from adding to s(x) any functions in the null-space for both operators.

**Remark About question 2:** A colleague of mine suggests that in similar situation of convolution codes it is "well-known" that solution might not be periodic (it is my translation of his words to ODE language it might not be correct).

In the convolutional codes are related to this setup as follows. Let us discretize $x$, so instead of $s(x)$ we will have $s(n)$ and $D_i$ will act as $s(n)-> \sum_k \tilde a_k x(n-k)$. Now let us restrict values of $x(n)$ only to +1, -1 and instead of sum consider the products $x(n)-> \sum_k x(n-k)^{a_k}$. That is what convolutional codes are doing.

We have "signal" (i.e. sequence of +1,-1) $s(k)$ : encoder maps it to a pair of functions $\tilde D_1 (s(k)), \tilde D_2(s(k))$ . Due to errors from propagation via noisy channel we get $\tilde r_1(k) = \tilde D_1 (s(k))+noise , r_2(k) = \tilde D_2(s(k))+ noise $. We want to reconstruct $s(k)$ from given $\tilde r_1, \tilde r_2$.

Tail-biting is some trick how to make everything periodic, let me omit details for the moment.