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In this paper, we consider the signature-to-path reconstruction problem fromthe control theoretic perspective. Namely, we design an optimal control problemwhose solution leads to the minimal-length path that generates a givensignature. In order to do that, we minimize a cost functional consisting of twocompeting terms, i.e., a weighted final-time cost combined with the $L^2$-normsquared of the controls. Moreover, we can show that, by taking the limit toinfinity of the parameter that tunes the final-time cost, the problem$\Gamma$-converges to the problem of finding a sub-Riemannian geodesicconnecting two signatures. Finally, we provide an alternative reformulation ofthe latter problem, which is particularly suitable for the numericalimplementation.

Shortest-path recovery from signature with an optimal control approach