Globally convergent homotopies for discrete-time optimal control
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866929618770984960 |
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| author | Esterhuizen, Willem Flaßkamp, Kathrin Hoffmann, Matthias Worthmann, Karl |
| author_facet | Esterhuizen, Willem Flaßkamp, Kathrin Hoffmann, Matthias Worthmann, Karl |
| contents | Homotopy methods are attractive due to their capability of solving difficult optimisation and optimal control problems. The underlying idea is to construct a homotopy, which may be considered as a continuous (zero) curve between the difficult original problem and a related, comparatively easy one. Then, the solution of the easier one is continuously perturbed along the zero curve towards the sought-after solution of the original problem. We propose a methodology for the systematic construction of such zero curves for discrete-time optimal control problems drawing upon the theory of globally convergent homotopies for nonlinear programs. The proposed framework ensures that for almost every initial guess at a solution there exists a suitable homotopy path that is, in addition, numerically convenient to track. We demonstrate the results by solving optimal path planning problems for a linear system and the nonlinear nonholonomic car (Dubins' vehicle). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_07852 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Globally convergent homotopies for discrete-time optimal control Esterhuizen, Willem Flaßkamp, Kathrin Hoffmann, Matthias Worthmann, Karl Optimization and Control Systems and Control 49K15, 49M99, 90C30, 93B40, 93C55 Homotopy methods are attractive due to their capability of solving difficult optimisation and optimal control problems. The underlying idea is to construct a homotopy, which may be considered as a continuous (zero) curve between the difficult original problem and a related, comparatively easy one. Then, the solution of the easier one is continuously perturbed along the zero curve towards the sought-after solution of the original problem. We propose a methodology for the systematic construction of such zero curves for discrete-time optimal control problems drawing upon the theory of globally convergent homotopies for nonlinear programs. The proposed framework ensures that for almost every initial guess at a solution there exists a suitable homotopy path that is, in addition, numerically convenient to track. We demonstrate the results by solving optimal path planning problems for a linear system and the nonlinear nonholonomic car (Dubins' vehicle). |
| title | Globally convergent homotopies for discrete-time optimal control |
| topic | Optimization and Control Systems and Control 49K15, 49M99, 90C30, 93B40, 93C55 |
| url | https://arxiv.org/abs/2306.07852 |