American Institute of Mathematical Sciences

June  2011, 3(2): 197-223. doi: 10.3934/jgm.2011.3.197

Embedded geodesic problems and optimal control for matrix Lie groups

 1 Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, United States 2 Department of Electrical Engineering, University of Hawai‘i at Mānoa, Honolulu, HI 96822, United States 3 Department of Mechanical Engineering, University of Hawai‘i at Mānoa, Honolulu, HI 96822, United States 4 Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 88003, United States

Received  July 2009 Revised  June 2011 Published  July 2011

This paper is devoted to a detailed analysis of the geodesic problem on matrix Lie groups, with left invariant metric, by examining representations of embeddings of geodesic flows in suitable vector spaces. We show how these representations generate extremals for optimal control problems. In particular we discuss in detail the symmetric representation of the so-called $n$-dimensional rigid body equation and its relation to the more classical Euler description. We detail invariant manifolds of these flows on which we are able to define a strict notion of equivalence between representations, and identify naturally induced symplectic structures.
Citation: Anthony M. Bloch, Peter E. Crouch, Nikolaj Nordkvist, Amit K. Sanyal. Embedded geodesic problems and optimal control for matrix Lie groups. Journal of Geometric Mechanics, 2011, 3 (2) : 197-223. doi: 10.3934/jgm.2011.3.197
References:

show all references

References:
 [1] François Gay-Balmaz, Tudor S. Ratiu. Clebsch optimal control formulation in mechanics. Journal of Geometric Mechanics, 2011, 3 (1) : 41-79. doi: 10.3934/jgm.2011.3.41 [2] Sujit Nair, Naomi Ehrich Leonard. Stable synchronization of rigid body networks. Networks & Heterogeneous Media, 2007, 2 (4) : 597-626. doi: 10.3934/nhm.2007.2.597 [3] Zahra Al Helal, Volker Rehbock, Ryan Loxton. Modelling and optimal control of blood glucose levels in the human body. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1149-1164. doi: 10.3934/jimo.2015.11.1149 [4] Sebastián Ferrer, Francisco J. Molero. Andoyer's variables and phases in the free rigid body. Journal of Geometric Mechanics, 2014, 6 (1) : 25-37. doi: 10.3934/jgm.2014.6.25 [5] Kai Koike. Wall effect on the motion of a rigid body immersed in a free molecular flow. Kinetic & Related Models, 2018, 11 (3) : 441-467. doi: 10.3934/krm.2018020 [6] Giancarlo Benettin, Anna Maria Cherubini, Francesco Fassò. Regular and chaotic motions of the fast rotating rigid body: a numerical study. Discrete & Continuous Dynamical Systems - B, 2002, 2 (4) : 521-540. doi: 10.3934/dcdsb.2002.2.521 [7] Giancarlo Benettin, Massimiliano Guzzo, Anatoly Neishtadt. A new problem of adiabatic invariance related to the rigid body dynamics. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 959-975. doi: 10.3934/dcds.2008.21.959 [8] Arnab Roy, Takéo Takahashi. Local null controllability of a rigid body moving into a Boussinesq flow. Mathematical Control & Related Fields, 2019, 9 (4) : 793-836. doi: 10.3934/mcrf.2019050 [9] Alex Bombrun, Jean-Baptiste Pomet. Asymptotic behavior of time optimal orbital transfer for low thrust 2-body control system. Conference Publications, 2007, 2007 (Special) : 122-129. doi: 10.3934/proc.2007.2007.122 [10] Urszula Ledzewicz, Helen Moore. Optimal control applied to a generalized Michaelis-Menten model of CML therapy. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 331-346. doi: 10.3934/dcdsb.2018022 [11] Matthias Hieber, Miho Murata. The $L^p$-approach to the fluid-rigid body interaction problem for compressible fluids. Evolution Equations & Control Theory, 2015, 4 (1) : 69-87. doi: 10.3934/eect.2015.4.69 [12] Paul Deuring, Stanislav Kračmar, Šárka Nečasová. A leading term for the velocity of stationary viscous incompressible flow around a rigid body performing a rotation and a translation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1389-1409. doi: 10.3934/dcds.2017057 [13] Bernard Bonnard, Olivier Cots, Nataliya Shcherbakova. The Serret-Andoyer Riemannian metric and Euler-Poinsot rigid body motion. Mathematical Control & Related Fields, 2013, 3 (3) : 287-302. doi: 10.3934/mcrf.2013.3.287 [14] Šárka Nečasová, Joerg Wolf. On the existence of global strong solutions to the equations modeling a motion of a rigid body around a viscous fluid. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1539-1562. doi: 10.3934/dcds.2016.36.1539 [15] Marshall Hampton, Anders Nedergaard Jensen. Finiteness of relative equilibria in the planar generalized $N$-body problem with fixed subconfigurations. Journal of Geometric Mechanics, 2015, 7 (1) : 35-42. doi: 10.3934/jgm.2015.7.35 [16] Jean-Baptiste Caillau, Bilel Daoud, Joseph Gergaud. Discrete and differential homotopy in circular restricted three-body control. Conference Publications, 2011, 2011 (Special) : 229-239. doi: 10.3934/proc.2011.2011.229 [17] Thalya Burden, Jon Ernstberger, K. Renee Fister. Optimal control applied to immunotherapy. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 135-146. doi: 10.3934/dcdsb.2004.4.135 [18] Ellina Grigorieva, Evgenii Khailov. Optimal control of pollution stock. Conference Publications, 2011, 2011 (Special) : 578-588. doi: 10.3934/proc.2011.2011.578 [19] Hang-Chin Lai, Jin-Chirng Lee, Shuh-Jye Chern. A variational problem and optimal control. Journal of Industrial & Management Optimization, 2011, 7 (4) : 967-975. doi: 10.3934/jimo.2011.7.967 [20] Qun Lin, Ryan Loxton, Kok Lay Teo. The control parameterization method for nonlinear optimal control: A survey. Journal of Industrial & Management Optimization, 2014, 10 (1) : 275-309. doi: 10.3934/jimo.2014.10.275

2018 Impact Factor: 0.525