American Institute of Mathematical Sciences

doi: 10.3934/jgm.2022003
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Riemannian cubics close to geodesics at the boundaries

 1 University of Coimbra, CMUC, Department of Mathematics, Coimbra, Portugal 2 University of Coimbra, Department of Mathematics, and Institute of Systems and Robotics, DEEC-UC, Coimbra, Portugal 3 Arlington College of Engineering, University of Texas at Arlington, Arlington, Texas, USA

* Corresponding author: Margarida Camarinha

Received  June 2021 Revised  October 2021 Early access February 2022

In this paper we investigate the existence and uniqueness of Riemannian cubics under boundary conditions on position and velocity. We restrict the study to cubics close to geodesics at the boundaries. In other words, we consider the boundary data in a neighborhood of geodesic boundary data. We define a map that generalizes the Riemannian exponential, the biexponential. This map is used to establish the correspondence between initial and boundary data. We also emphasize the relation between biconjugate points and bi-Jacobi fields along cubics by means of the biexponential map.

Citation: Margarida Camarinha, Fátima Silva Leite, Peter Crouch. Riemannian cubics close to geodesics at the boundaries. Journal of Geometric Mechanics, doi: 10.3934/jgm.2022003
References:
 [1] L. Abrunheiro, M. Camarinha and J. Clemente-Gallardo, Cubic polynomials on Lie groups: Reduction of the Hamiltonian system, J. Phys. A: Math. Theor., 44 (2011), 355203, 16 pp. doi: 10.1088/1751-8113/44/35/355203. [2] J. Arroyo, O. J. Garay and J. J. Mencía, Unit speed stationary points of the acceleration, J. Math. Phys., 49 (2008), 013508, 16 pp. doi: 10.1063/1.2830433. [3] P. Balseiro, T. J. Stuchi, A. Cabrera and J. Koiller, About simple variational splines from the Hamiltonian viewpoint, J. Geom. Mech., 9 (2017), 257-290.  doi: 10.3934/jgm.2017011. [4] E. Batzies, K. Hüper, L. Machado and F. Silva Leite, Geometric mean and geodesic regression on Grassmannians, Linear Algebra Appl., 466 (2015), 83-101.  doi: 10.1016/j.laa.2014.10.003. [5] A. Bloch, L. Colombo, R. Gupta and D. M. de Diego, A geometric approach to the optimal control of nonholonomic mechanical systems, Analysis and Geometry in Control Theory and its Applications 35–64, Springer INdAM Ser., 11, Springer, Cham, 2015. doi: 10.1007/978-3-319-06917-3_2. [6] A. M. Bloch and P. E. Crouch, Optimal control, optimization, and analytical mechanics, Mathematical Control Theory, 268–321, Springer, New York, 1999. doi: 10.1007/978-1-4612-1416-8_8. [7] A. Bloch, M. Camarinha and L. J. Colombo, Dynamic interpolation for obstacle avoidance on Riemannian manifolds, Internat. J. Control, 94 (2021), 588-600.  doi: 10.1080/00207179.2019.1603400. [8] A. Bloch, M. Camarinha and L. Colombo, Variational point-obstacle avoidance on Riemannian manifolds, Math. Control Signals Systems, 33 (2021), 109-121.  doi: 10.1007/s00498-021-00276-0. [9] A. Bloch, L. Colombo and F. Jiménez, The variational discretization of the constrained higher-order Lagrange-Poincaré equations, Discrete Contin. Dyn. Syst., 39 (2019), 309-344.  doi: 10.3934/dcds.2019013. [10] A. M. Bloch, R. Gupta and I. V. Kolmanovsky, Neighboring extremal optimal control for mechanical systems on Riemannian manifolds, J. Geom. Mech., 8 (2016), 257-272.  doi: 10.3934/jgm.2016007. [11] G. Bogfjellmo, K. Modin and O. Verdier, A numerical algorithm for C2-splines on symmetric spaces, SIAM J. Numer. Anal., 56 (2018), 2623-2647.  doi: 10.1137/17M1123353. [12] R. Caddeo, S. Montaldo, C. Oniciuc and P. Piu, The Euler-Lagrange method for biharmonic curves, Mediterr. J. Math., 3 (2006), 449-465.  doi: 10.1007/s00009-006-0090-x. [13] M. Camarinha, The Geometry of Cubic Polynomials in Riemannian Manifolds, Ph.D thesis, University of Coimbra, 1996. Available from: http://hdl.handle.net/10316/1954. [14] M. Camarinha, F. Silva Leite and P. Crouch, On the geometry of Riemannian cubic polynomials, Differential Geom. Appl., 15 (2001), 107-135.  doi: 10.1016/S0926-2245(01)00054-7. [15] M. Camarinha, F. Silva Leite and P. Crouch, Existence and uniqueness for Riemannian cubics with boundary conditions, in Lecture Notes in Electrical Engineering, 695 (2021), LNEE, 322–331. doi: 10.1007/978-3-030-58653-9_31. [16] L. Colombo, S. Ferraro and D. Martín de Diego, Geometric integrators for higher-order variational systems and their application to optimal control, J. Nonlinear Sci., 26 (2016), 1615-1650.  doi: 10.1007/s00332-016-9314-9. [17] P. Crouch and F. Silva Leite, The dynamic interpolation problem on Riemannian manifolds, Lie groups and symmetric spaces, J. Dynam. Control Systems, 1 (1995), 177-202.  doi: 10.1007/BF02254638. [18] P. Crouch, F. Silva Leite and M. Camarinha, Hamiltonian structure of generalized cubic polynomials, IFAC Proceedings Volumes, 33 (2000), 13-18.  doi: 10.1016/S1474-6670(17)35541-6. [19] S. Gabriel and J. Kajiya, Spline interpolation in curved space, in SIGGRAPH 85 Course Notes for State of the Art in Image Synthesis, (1985), 1–14. [20] F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu and F.-X. Vialard, Invariant higher-order variational problems Ⅱ, J. Nonlinear Sci., 22 (2012), 553-597.  doi: 10.1007/s00332-012-9137-2. [21] R. Giambò, F. Giannoni and P. Piccione, Optimal control on Riemannian manifolds by interpolation, Math. Control Signals Systems, 16 (2004), 278-296.  doi: 10.1007/s00498-003-0139-3. [22] R. Gupta, A. M. Bloch and I. V. Kolmanovsky, Combined homotopy and neighboring extremal optimal control, Optimal Control Appl. Methods, 38 (2017), 459-469.  doi: 10.1002/oca.2253. [23] B. Heeren, M. Rumpf and B. Wirth, Variational time discretization of Riemannian splines, IMA J. Numer. Anal., 39 (2019), 61-104.  doi: 10.1093/imanum/drx077. [24] I. I. Hussein and A. M. Bloch, Dynamic coverage optimal control for multiple spacecraft interferometric imaging, J. Dyn. Control Syst., 13 (2007), 69-93.  doi: 10.1007/s10883-006-9004-2. [25] G. Y. Jiang, $2$-harmonic maps and their first and second variational formulas, Chinese Ann. Math. Ser. A, 7 (1986), 389-402. [26] L. Machado, F. Silva Leite and K. Krakowski, Higher-order smoothing splines versus least squares problems on Riemannian manifolds, J. Dyn. Control Syst., 16 (2010), 121-148.  doi: 10.1007/s10883-010-9080-1. [27] L. Noakes, Null cubics and Lie quadratics, J. Math. Phys., 44 (2003), 1436-1448.  doi: 10.1063/1.1537461. [28] L. Noakes, Approximating near-geodesic natural cubic splines, Commun. Math. Sci., 12 (2014), 1409-1425.  doi: 10.4310/CMS.2014.v12.n8.a2. [29] L. Noakes, G. Heinzinger and B. Paden, Cubic splines on curved spaces, IMA J. Math. Control Inform., 6 (1989), 465-473.  doi: 10.1093/imamci/6.4.465. [30] L. Noakes and T. Popiel, Quadratures and cubics in SO(3) and SO(1, 2), IMA J. Math. Control Inform., 23 (2006), 463-473.  doi: 10.1093/imamci/dni069. [31] L. Noakes and T. S. Ratiu, Bi-Jacobi fields and Riemannian cubics for left-invariant SO(3), Commun. Math. Sci., 14 (2016), 55-68.  doi: 10.4310/CMS.2016.v14.n1.a3. [32] T. Popiel, Higher order geodesics in Lie groups, Math. Control Signals Systems, 19 (2007), 235-253.  doi: 10.1007/s00498-007-0012-x. [33] P. Schrader, Existence of variationally defined curves with higher order elliptic Lagrangians, Nonlinear Anal., 115 (2015), 1-11.  doi: 10.1016/j.na.2014.11.016. [34] A. Trouvé and F.-X. Vialard, Shape splines and stochastic shape evolutions: A second order point of view. (English summary), Quart. Appl. Math., 70 (2012), 219-251.  doi: 10.1090/S0033-569X-2012-01250-4.

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References:
 [1] L. Abrunheiro, M. Camarinha and J. Clemente-Gallardo, Cubic polynomials on Lie groups: Reduction of the Hamiltonian system, J. Phys. A: Math. Theor., 44 (2011), 355203, 16 pp. doi: 10.1088/1751-8113/44/35/355203. [2] J. Arroyo, O. J. Garay and J. J. Mencía, Unit speed stationary points of the acceleration, J. Math. Phys., 49 (2008), 013508, 16 pp. doi: 10.1063/1.2830433. [3] P. Balseiro, T. J. Stuchi, A. Cabrera and J. Koiller, About simple variational splines from the Hamiltonian viewpoint, J. Geom. Mech., 9 (2017), 257-290.  doi: 10.3934/jgm.2017011. [4] E. Batzies, K. Hüper, L. Machado and F. Silva Leite, Geometric mean and geodesic regression on Grassmannians, Linear Algebra Appl., 466 (2015), 83-101.  doi: 10.1016/j.laa.2014.10.003. [5] A. Bloch, L. Colombo, R. Gupta and D. M. de Diego, A geometric approach to the optimal control of nonholonomic mechanical systems, Analysis and Geometry in Control Theory and its Applications 35–64, Springer INdAM Ser., 11, Springer, Cham, 2015. doi: 10.1007/978-3-319-06917-3_2. [6] A. M. Bloch and P. E. Crouch, Optimal control, optimization, and analytical mechanics, Mathematical Control Theory, 268–321, Springer, New York, 1999. doi: 10.1007/978-1-4612-1416-8_8. [7] A. Bloch, M. Camarinha and L. J. Colombo, Dynamic interpolation for obstacle avoidance on Riemannian manifolds, Internat. J. Control, 94 (2021), 588-600.  doi: 10.1080/00207179.2019.1603400. [8] A. Bloch, M. Camarinha and L. Colombo, Variational point-obstacle avoidance on Riemannian manifolds, Math. Control Signals Systems, 33 (2021), 109-121.  doi: 10.1007/s00498-021-00276-0. [9] A. Bloch, L. Colombo and F. Jiménez, The variational discretization of the constrained higher-order Lagrange-Poincaré equations, Discrete Contin. Dyn. Syst., 39 (2019), 309-344.  doi: 10.3934/dcds.2019013. [10] A. M. Bloch, R. Gupta and I. V. Kolmanovsky, Neighboring extremal optimal control for mechanical systems on Riemannian manifolds, J. Geom. Mech., 8 (2016), 257-272.  doi: 10.3934/jgm.2016007. [11] G. Bogfjellmo, K. Modin and O. Verdier, A numerical algorithm for C2-splines on symmetric spaces, SIAM J. Numer. Anal., 56 (2018), 2623-2647.  doi: 10.1137/17M1123353. [12] R. Caddeo, S. Montaldo, C. Oniciuc and P. Piu, The Euler-Lagrange method for biharmonic curves, Mediterr. J. Math., 3 (2006), 449-465.  doi: 10.1007/s00009-006-0090-x. [13] M. Camarinha, The Geometry of Cubic Polynomials in Riemannian Manifolds, Ph.D thesis, University of Coimbra, 1996. Available from: http://hdl.handle.net/10316/1954. [14] M. Camarinha, F. Silva Leite and P. Crouch, On the geometry of Riemannian cubic polynomials, Differential Geom. Appl., 15 (2001), 107-135.  doi: 10.1016/S0926-2245(01)00054-7. [15] M. Camarinha, F. Silva Leite and P. Crouch, Existence and uniqueness for Riemannian cubics with boundary conditions, in Lecture Notes in Electrical Engineering, 695 (2021), LNEE, 322–331. doi: 10.1007/978-3-030-58653-9_31. [16] L. Colombo, S. Ferraro and D. Martín de Diego, Geometric integrators for higher-order variational systems and their application to optimal control, J. Nonlinear Sci., 26 (2016), 1615-1650.  doi: 10.1007/s00332-016-9314-9. [17] P. Crouch and F. Silva Leite, The dynamic interpolation problem on Riemannian manifolds, Lie groups and symmetric spaces, J. Dynam. Control Systems, 1 (1995), 177-202.  doi: 10.1007/BF02254638. [18] P. Crouch, F. Silva Leite and M. Camarinha, Hamiltonian structure of generalized cubic polynomials, IFAC Proceedings Volumes, 33 (2000), 13-18.  doi: 10.1016/S1474-6670(17)35541-6. [19] S. Gabriel and J. Kajiya, Spline interpolation in curved space, in SIGGRAPH 85 Course Notes for State of the Art in Image Synthesis, (1985), 1–14. [20] F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu and F.-X. Vialard, Invariant higher-order variational problems Ⅱ, J. Nonlinear Sci., 22 (2012), 553-597.  doi: 10.1007/s00332-012-9137-2. [21] R. Giambò, F. Giannoni and P. Piccione, Optimal control on Riemannian manifolds by interpolation, Math. Control Signals Systems, 16 (2004), 278-296.  doi: 10.1007/s00498-003-0139-3. [22] R. Gupta, A. M. Bloch and I. V. Kolmanovsky, Combined homotopy and neighboring extremal optimal control, Optimal Control Appl. Methods, 38 (2017), 459-469.  doi: 10.1002/oca.2253. [23] B. Heeren, M. Rumpf and B. Wirth, Variational time discretization of Riemannian splines, IMA J. Numer. Anal., 39 (2019), 61-104.  doi: 10.1093/imanum/drx077. [24] I. I. Hussein and A. M. Bloch, Dynamic coverage optimal control for multiple spacecraft interferometric imaging, J. Dyn. Control Syst., 13 (2007), 69-93.  doi: 10.1007/s10883-006-9004-2. [25] G. Y. Jiang, $2$-harmonic maps and their first and second variational formulas, Chinese Ann. Math. Ser. A, 7 (1986), 389-402. [26] L. Machado, F. Silva Leite and K. Krakowski, Higher-order smoothing splines versus least squares problems on Riemannian manifolds, J. Dyn. Control Syst., 16 (2010), 121-148.  doi: 10.1007/s10883-010-9080-1. [27] L. Noakes, Null cubics and Lie quadratics, J. Math. Phys., 44 (2003), 1436-1448.  doi: 10.1063/1.1537461. [28] L. Noakes, Approximating near-geodesic natural cubic splines, Commun. Math. Sci., 12 (2014), 1409-1425.  doi: 10.4310/CMS.2014.v12.n8.a2. [29] L. Noakes, G. Heinzinger and B. Paden, Cubic splines on curved spaces, IMA J. Math. Control Inform., 6 (1989), 465-473.  doi: 10.1093/imamci/6.4.465. [30] L. Noakes and T. Popiel, Quadratures and cubics in SO(3) and SO(1, 2), IMA J. Math. Control Inform., 23 (2006), 463-473.  doi: 10.1093/imamci/dni069. [31] L. Noakes and T. S. Ratiu, Bi-Jacobi fields and Riemannian cubics for left-invariant SO(3), Commun. Math. Sci., 14 (2016), 55-68.  doi: 10.4310/CMS.2016.v14.n1.a3. [32] T. Popiel, Higher order geodesics in Lie groups, Math. Control Signals Systems, 19 (2007), 235-253.  doi: 10.1007/s00498-007-0012-x. [33] P. Schrader, Existence of variationally defined curves with higher order elliptic Lagrangians, Nonlinear Anal., 115 (2015), 1-11.  doi: 10.1016/j.na.2014.11.016. [34] A. Trouvé and F.-X. Vialard, Shape splines and stochastic shape evolutions: A second order point of view. (English summary), Quart. Appl. Math., 70 (2012), 219-251.  doi: 10.1090/S0033-569X-2012-01250-4.
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