# American Institute of Mathematical Sciences

September  2016, 8(3): 257-272. doi: 10.3934/jgm.2016007

## Neighboring extremal optimal control for mechanical systems on Riemannian manifolds

 1 Department of Mathematics, University of Michigan, Ann Arbor, MI 48109 2 Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI 48109-2140, United States, United States

Received  October 2015 Revised  June 2016 Published  September 2016

In this paper, we extend neighboring extremal optimal control, which is well established for optimal control problems defined on a Euclidean space (see, e.g., [8]) to the setting of Riemannian manifolds. We further specialize the results to the case of Lie groups. An example along with simulation results is presented.
Citation: Anthony M. Bloch, Rohit Gupta, Ilya V. Kolmanovsky. Neighboring extremal optimal control for mechanical systems on Riemannian manifolds. Journal of Geometric Mechanics, 2016, 8 (3) : 257-272. doi: 10.3934/jgm.2016007
##### References:
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Silva Leite, The dynamic interpolation problem: On Riemannian manifolds, Lie groups, and symmetric spaces, Journal of Dynamical and Control Systems, 1 (1995), 177-202. doi: 10.1007/BF02254638.  Google Scholar [15] P. Crouch, F. Silva Leite and M. Camarinha, A second order Riemannian variational problem from a Hamiltonian perspective, 1998. Google Scholar [16] M. P. do Carmo, Riemannian Geometry, Birkhäuser, 1992. doi: 10.1007/978-1-4757-2201-7.  Google Scholar [17] A. L. Dontchev and W. W. Hager, Lipschitzian stability in nonlinear control and optimization, SIAM Journal on Control and Optimization, 31 (1993), 569-603. doi: 10.1137/0331026.  Google Scholar [18] A. L. Dontchev, W. W. Hager, A. B. Poore and B. Yang, Optimality, stability, and convergence in nonlinear control, Applied Mathematics and Optimization, 31 (1995), 297-326. doi: 10.1007/BF01215994.  Google Scholar [19] A. L. Dontchev and W. W. Hager, Lipschitzian stability for state constrained nonlinear optimal control, SIAM Journal on Control and Optimization, 36 (1998), 698-718. doi: 10.1137/S0363012996299314.  Google Scholar [20] R. Gupta, A. M. Bloch and I. V. Kolmanovsky, Combined homotopy and neighboring extremal optimal control, Optimal Control Applications and Methods, (2016), to appear. Google Scholar [21] R. V. Iyer, R. Holsapple and D. Doman, Optimal control problems on parallelizable Riemannian manifolds: Theory and applications, ESAIM: Control, Optimisation and Calculus of Variations, 12 (2006), 1-11. doi: 10.1051/cocv:2005026.  Google Scholar [22] J. M. Lee, Introduction to Smooth Manifolds, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21752-9.  Google Scholar [23] P. D. Loewen and H. Zheng, Generalized conjugate points for optimal control problems, Nonlinear Analysis: Theory, Methods & Applications, 22 (1994), 771-791. doi: 10.1016/0362-546X(94)90226-7.  Google Scholar [24] J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Springer Science & Business Media, 1999. doi: 10.1007/978-0-387-21792-5.  Google Scholar [25] P. M. Mereau and W. F. Powers, Conjugate point properties for linear quadratic problems, Journal of Mathematical Analysis and Applications, 55 (1976), 418-433.  Google Scholar [26] J. W. Milnor, Morse Theory, Princeton University Press, 1963.  Google Scholar [27] L. Noakes, G. Heinzinger and B. Paden, Cubic splines on curved spaces, IMA Journal of Mathematical Control and Information, 6 (1989), 465-473. doi: 10.1093/imamci/6.4.465.  Google Scholar [28] S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds I, Tohoku Mathematical Journal, Second Series, 10 (1958), 338-354.  Google Scholar [29] S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds II, Tohoku Mathematical Journal, Second Series, 14 (1962), 146-155.  Google Scholar [30] H. Schättler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples, Springer Science & Business Media, 2012. doi: 10.1007/978-1-4614-3834-2.  Google Scholar [31] F. Silva Leite, M. Camarinha and P. Crouch, Elastic curves as solutions of Riemannian and sub-Riemannian control problems, Mathematics of Control, Signals, and Systems, 13 (2000), 140-155. doi: 10.1007/PL00009863.  Google Scholar [32] J. L. Speyer and D. H. Jacobson, Primer on Optimal Control Theory, SIAM, 2010. doi: 10.1137/1.9780898718560.  Google Scholar [33] D. R. Tyner and A. D. Lewis, Geometric jacobian linearization and LQR theory, Journal of Geometric Mechanics, 2 (2010), 397-440. doi: 10.3934/jgm.2010.2.397.  Google Scholar [34] V. Zeidan and P. Zezza, The conjugate point condition for smooth control sets, Journal of Mathematical Analysis and Applications, 132 (1988), 572-589. doi: 10.1016/0022-247X(88)90085-6.  Google Scholar [35] V. Zeidan and P. Zezza, Conjugate points and optimal control: Counterexamples, IEEE Transactions on Automatic Control, 34 (1989), 254-256. doi: 10.1109/9.21115.  Google Scholar [36] V. Zeidan, The riccati equation for optimal control problems with mixed state-control constraints: Necessity and sufficiency, SIAM Journal on Control and Optimization, 32 (1994), 1297-1321. doi: 10.1137/S0363012992233640.  Google Scholar

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##### References:
 [1] R. Abraham and J. E. Marsden, Foundations of Mechanics, AMS Chelsea Publishing, 1978. doi: 10.1090/chel/364.  Google Scholar [2] A. A. Agrachev and Y. L. Sachkov, Control Theory from the Geometric Viewpoint, Springer Science & Business Media, 2004. doi: 10.1007/978-3-662-06404-7.  Google Scholar [3] C. Altafini, Reduction by group symmetry of second order variational problems on a semidirect product of Lie groups with positive definite Riemannian metric, ESAIM: Control, Optimisation and Calculus of Variations, 10 (2004), 526-548. doi: 10.1051/cocv:2004018.  Google Scholar [4] M. Barbero-Liñán, A Geometric Study of Abnormality in Optimal Control Problems for Control and Mechanical Control Systems, PhD thesis, Technical University of Catalonia, 2008. Google Scholar [5] M. Barbero-Liñán, Characterization of accessibility for affine connection control systems at some points with nonzero velocity, in Proceedings of IEEE Conference on Decision and Control and European Control Conference, 2011, 6528-6533. Google Scholar [6] A. M. Bloch, Nonholonomic Mechanics and Control, Springer Science & Business Media, 2003. doi: 10.1007/978-1-4939-3017-3.  Google Scholar [7] J. V. Breakwell and H. Yu-Chi, On the conjugate point condition for the control problem, International Journal of Engineering Science, 2 (1965), 565-579. doi: 10.1016/0020-7225(65)90037-6.  Google Scholar [8] A. E. Bryson, Applied Optimal Control: Optimization$,$ Estimation and Control, CRC Press, 1975.  Google Scholar [9] F. Bullo, Invariant Affine Connections and Controllability on Lie Groups, Technical Report Final Project Report for CIT-CDS 141a, Control and Dynamical Systems, California Institute of Technology, 1995. Google Scholar [10] F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems, Springer Science & Business Media, 2005. doi: 10.1007/978-1-4899-7276-7.  Google Scholar [11] F. Bullo and A. D. Lewis, Reduction, linearization, and stability of relative equilibria for mechanical systems on Riemannian manifolds, Acta Applicandae Mathematicae, 99 (2007), 53-95. doi: 10.1007/s10440-007-9155-5.  Google Scholar [12] J.-B. Caillau, O. Cots and J. Gergaud, Differential continuation for regular optimal control problems, Optimization Methods and Software, 27 (2012), 177-196. doi: 10.1080/10556788.2011.593625.  Google Scholar [13] N. Caroff and H. Frankowska, Conjugate points and shocks in nonlinear optimal control, Transactions of the American Mathematical Society, 348 (1996), 3133-3153. doi: 10.1090/S0002-9947-96-01577-2.  Google Scholar [14] P. Crouch and F. Silva Leite, The dynamic interpolation problem: On Riemannian manifolds, Lie groups, and symmetric spaces, Journal of Dynamical and Control Systems, 1 (1995), 177-202. doi: 10.1007/BF02254638.  Google Scholar [15] P. Crouch, F. Silva Leite and M. Camarinha, A second order Riemannian variational problem from a Hamiltonian perspective, 1998. Google Scholar [16] M. P. do Carmo, Riemannian Geometry, Birkhäuser, 1992. doi: 10.1007/978-1-4757-2201-7.  Google Scholar [17] A. L. Dontchev and W. W. Hager, Lipschitzian stability in nonlinear control and optimization, SIAM Journal on Control and Optimization, 31 (1993), 569-603. doi: 10.1137/0331026.  Google Scholar [18] A. L. Dontchev, W. W. Hager, A. B. Poore and B. Yang, Optimality, stability, and convergence in nonlinear control, Applied Mathematics and Optimization, 31 (1995), 297-326. doi: 10.1007/BF01215994.  Google Scholar [19] A. L. Dontchev and W. W. Hager, Lipschitzian stability for state constrained nonlinear optimal control, SIAM Journal on Control and Optimization, 36 (1998), 698-718. doi: 10.1137/S0363012996299314.  Google Scholar [20] R. Gupta, A. M. Bloch and I. V. Kolmanovsky, Combined homotopy and neighboring extremal optimal control, Optimal Control Applications and Methods, (2016), to appear. Google Scholar [21] R. V. Iyer, R. Holsapple and D. Doman, Optimal control problems on parallelizable Riemannian manifolds: Theory and applications, ESAIM: Control, Optimisation and Calculus of Variations, 12 (2006), 1-11. doi: 10.1051/cocv:2005026.  Google Scholar [22] J. M. Lee, Introduction to Smooth Manifolds, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21752-9.  Google Scholar [23] P. D. Loewen and H. Zheng, Generalized conjugate points for optimal control problems, Nonlinear Analysis: Theory, Methods & Applications, 22 (1994), 771-791. doi: 10.1016/0362-546X(94)90226-7.  Google Scholar [24] J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Springer Science & Business Media, 1999. doi: 10.1007/978-0-387-21792-5.  Google Scholar [25] P. M. Mereau and W. F. Powers, Conjugate point properties for linear quadratic problems, Journal of Mathematical Analysis and Applications, 55 (1976), 418-433.  Google Scholar [26] J. W. Milnor, Morse Theory, Princeton University Press, 1963.  Google Scholar [27] L. Noakes, G. Heinzinger and B. Paden, Cubic splines on curved spaces, IMA Journal of Mathematical Control and Information, 6 (1989), 465-473. doi: 10.1093/imamci/6.4.465.  Google Scholar [28] S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds I, Tohoku Mathematical Journal, Second Series, 10 (1958), 338-354.  Google Scholar [29] S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds II, Tohoku Mathematical Journal, Second Series, 14 (1962), 146-155.  Google Scholar [30] H. Schättler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples, Springer Science & Business Media, 2012. doi: 10.1007/978-1-4614-3834-2.  Google Scholar [31] F. Silva Leite, M. Camarinha and P. Crouch, Elastic curves as solutions of Riemannian and sub-Riemannian control problems, Mathematics of Control, Signals, and Systems, 13 (2000), 140-155. doi: 10.1007/PL00009863.  Google Scholar [32] J. L. Speyer and D. H. Jacobson, Primer on Optimal Control Theory, SIAM, 2010. doi: 10.1137/1.9780898718560.  Google Scholar [33] D. R. Tyner and A. D. Lewis, Geometric jacobian linearization and LQR theory, Journal of Geometric Mechanics, 2 (2010), 397-440. doi: 10.3934/jgm.2010.2.397.  Google Scholar [34] V. Zeidan and P. Zezza, The conjugate point condition for smooth control sets, Journal of Mathematical Analysis and Applications, 132 (1988), 572-589. doi: 10.1016/0022-247X(88)90085-6.  Google Scholar [35] V. Zeidan and P. Zezza, Conjugate points and optimal control: Counterexamples, IEEE Transactions on Automatic Control, 34 (1989), 254-256. doi: 10.1109/9.21115.  Google Scholar [36] V. Zeidan, The riccati equation for optimal control problems with mixed state-control constraints: Necessity and sufficiency, SIAM Journal on Control and Optimization, 32 (1994), 1297-1321. doi: 10.1137/S0363012992233640.  Google Scholar
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