Citation: |
[1] |
R. Abraham and J. E. Marsden, Foundations of Mechanics, AMS Chelsea Publishing, 1978.doi: 10.1090/chel/364. |
[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. |
[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. |
[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. |
[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. |
[6] |
A. M. Bloch, Nonholonomic Mechanics and Control, Springer Science & Business Media, 2003.doi: 10.1007/978-1-4939-3017-3. |
[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. |
[8] |
A. E. Bryson, Applied Optimal Control: Optimization$,$ Estimation and Control, CRC Press, 1975. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
P. Crouch, F. Silva Leite and M. Camarinha, A second order Riemannian variational problem from a Hamiltonian perspective, 1998. |
[16] |
M. P. do Carmo, Riemannian Geometry, Birkhäuser, 1992.doi: 10.1007/978-1-4757-2201-7. |
[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. |
[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. |
[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. |
[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. |
[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. |
[22] |
J. M. Lee, Introduction to Smooth Manifolds, Springer-Verlag, New York, 2003.doi: 10.1007/978-0-387-21752-9. |
[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. |
[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. |
[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. |
[26] |
J. W. Milnor, Morse Theory, Princeton University Press, 1963. |
[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. |
[28] |
S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds I, Tohoku Mathematical Journal, Second Series, 10 (1958), 338-354. |
[29] |
S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds II, Tohoku Mathematical Journal, Second Series, 14 (1962), 146-155. |
[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. |
[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. |
[32] |
J. L. Speyer and D. H. Jacobson, Primer on Optimal Control Theory, SIAM, 2010.doi: 10.1137/1.9780898718560. |
[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. |
[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. |
[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. |
[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. |