  In this paper, we consider the signature-to-path reconstruction problem from
the control theoretic perspective. Namely, we design an optimal control problem
whose solution leads to the minimal-length path that generates a given
signature. In order to do that, we minimize a cost functional consisting of two
competing terms, i.e., a weighted final-time cost combined with the $L^2$-norm
squared of the controls. Moreover, we can show that, by taking the limit to
infinity of the parameter that tunes the final-time cost, the problem
$\Gamma$-converges to the problem of finding a sub-Riemannian geodesic
connecting two signatures. Finally, we provide an alternative reformulation of
the latter problem, which is particularly suitable for the numerical
implementation.
