Shortest-path recovery from signature with an optimal control approach
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866909267063209984 |
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| author | Rauscher, Marco Scagliotti, Alessandro Patricio, Felipe Pagginelli |
| author_facet | Rauscher, Marco Scagliotti, Alessandro Patricio, Felipe Pagginelli |
| contents | 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 $Γ$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_10619 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Shortest-path recovery from signature with an optimal control approach Rauscher, Marco Scagliotti, Alessandro Patricio, Felipe Pagginelli Optimization and Control Numerical Analysis Systems and Control 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 $Γ$-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. |
| title | Shortest-path recovery from signature with an optimal control approach |
| topic | Optimization and Control Numerical Analysis Systems and Control |
| url | https://arxiv.org/abs/2310.10619 |