Abstract
The Grassmann manifold\(Gr_m({\mathbb {R}}^n)\) of all m-dimensional subspaces of the n-dimensional space \({\mathbb {R}}^n\)\((m<n)\) is widely used in image analysis, statistics and optimization. Motivated by interpolation in the manifold \(Gr_2({\mathbb {R}}^4)\), we first formulate the differential equation for desired interpolation curves called Riemannian cubics in symmetric spaces by the Pontryagin maximum principle (PMP) and then narrow down to it in \(Gr_2({\mathbb {R}}^4)\). Although computation on this low-dimensional manifold may not occur heavy burden for modern machines, theoretical analysis for Riemannian cubics is very limited in references due to its highly nonlinearity. This paper focuses on presenting analytical and geometrical structures for the so-called Lie quadratics associated with Riemannian cubics. By analysing asymptotics of Lie quadratics, we find asymptotics of Riemannian cubics in \(Gr_2({\mathbb {R}}^4)\). Finally, we illustrate our results by numerical simulations.
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Notes
For abnormal case, namely, \(\lambda _0(t)\equiv 0\), there is no way to get the optimal control u from the PMP.
With a little abuse of terminologies, we use Lie quadratic instead of homogeneous Lie quadratics for simplicity in the following discussions.
The constant \(\varepsilon \) is chosen as the maximum value of the upper bound of \(\Vert V\Vert , \Vert {\dot{V}}\Vert , \Vert {\ddot{V}}\Vert \) at \({\bar{S}}\). We can assume \({\bar{S}}\) is large enough that \(\Vert V_i\Vert , \Vert {\dot{V}}_i\Vert , \Vert {\ddot{V}}_i\Vert \) get maximum value at \({\bar{S}}\); otherwise, adjust \(\varepsilon \) by enlarging \(c_{2i}\) and \(c_{5i}\).
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The authors are very grateful to the Editors and two anonymous referees for their helpful and constructive comments and suggestions, which improved the quality of the current paper greatly and made it more suitable for readers of MCSS.
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Zhang, E., Noakes, L. Optimal interpolants on Grassmann manifolds. Math. Control Signals Syst. 31, 363–383 (2019). https://doi.org/10.1007/s00498-019-0241-9
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DOI: https://doi.org/10.1007/s00498-019-0241-9