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:
[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, 10 (2004), 526. 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, (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.

[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. 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, (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. 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. 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. 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. 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. 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. 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. 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).

[21]

R. V. Iyer, R. Holsapple and D. Doman, Optimal control problems on parallelizable Riemannian manifolds: Theory and applications,, ESAIM: Control, 12 (2006), 1. doi: 10.1051/cocv:2005026.

[22]

J. M. Lee, Introduction to Smooth Manifolds,, Springer-Verlag, (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, 22 (1994), 771. 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.

[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. doi: 10.1093/imamci/6.4.465.

[28]

S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds I,, Tohoku Mathematical Journal, 10 (1958), 338.

[29]

S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds II,, Tohoku Mathematical Journal, 14 (1962), 146.

[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, 13 (2000), 140. 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. 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. 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. 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. doi: 10.1137/S0363012992233640.

show all references

References:
[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, 10 (2004), 526. 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, (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.

[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. 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, (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. 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. 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. 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. 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. 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. 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. 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).

[21]

R. V. Iyer, R. Holsapple and D. Doman, Optimal control problems on parallelizable Riemannian manifolds: Theory and applications,, ESAIM: Control, 12 (2006), 1. doi: 10.1051/cocv:2005026.

[22]

J. M. Lee, Introduction to Smooth Manifolds,, Springer-Verlag, (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, 22 (1994), 771. 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.

[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. doi: 10.1093/imamci/6.4.465.

[28]

S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds I,, Tohoku Mathematical Journal, 10 (1958), 338.

[29]

S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds II,, Tohoku Mathematical Journal, 14 (1962), 146.

[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, 13 (2000), 140. 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. 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. 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. 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. doi: 10.1137/S0363012992233640.

[1]

Canghua Jiang, Kok Lay Teo, Ryan Loxton, Guang-Ren Duan. A neighboring extremal solution for an optimal switched impulsive control problem. Journal of Industrial & Management Optimization, 2012, 8 (3) : 591-609. doi: 10.3934/jimo.2012.8.591

[2]

Leonardo Colombo, David Martín de Diego. Optimal control of underactuated mechanical systems with symmetries. Conference Publications, 2013, 2013 (special) : 149-158. doi: 10.3934/proc.2013.2013.149

[3]

Leonardo Colombo, David Martín de Diego. Higher-order variational problems on lie groups and optimal control applications. Journal of Geometric Mechanics, 2014, 6 (4) : 451-478. doi: 10.3934/jgm.2014.6.451

[4]

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

[5]

Cédric M. Campos, Sina Ober-Blöbaum, Emmanuel Trélat. High order variational integrators in the optimal control of mechanical systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4193-4223. doi: 10.3934/dcds.2015.35.4193

[6]

Anthony M. Bloch, Peter E. Crouch, Nikolaj Nordkvist. Continuous and discrete embedded optimal control problems and their application to the analysis of Clebsch optimal control problems and mechanical systems. Journal of Geometric Mechanics, 2013, 5 (1) : 1-38. doi: 10.3934/jgm.2013.5.1

[7]

Robert J. Kipka, Yuri S. Ledyaev. Optimal control of differential inclusions on manifolds. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4455-4475. doi: 10.3934/dcds.2015.35.4455

[8]

Leonardo Colombo, David Martín de Diego. Second-order variational problems on Lie groupoids and optimal control applications. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6023-6064. doi: 10.3934/dcds.2016064

[9]

Michael Basin, Pablo Rodriguez-Ramirez. An optimal impulsive control regulator for linear systems. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 275-282. doi: 10.3934/naco.2011.1.275

[10]

Elena Goncharova, Maxim Staritsyn. Optimal control of dynamical systems with polynomial impulses. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4367-4384. doi: 10.3934/dcds.2015.35.4367

[11]

Rein Luus. Optimal control of oscillatory systems by iterative dynamic programming. Journal of Industrial & Management Optimization, 2008, 4 (1) : 1-15. doi: 10.3934/jimo.2008.4.1

[12]

Qiying Hu, Wuyi Yue. Optimal control for resource allocation in discrete event systems. Journal of Industrial & Management Optimization, 2006, 2 (1) : 63-80. doi: 10.3934/jimo.2006.2.63

[13]

Simone Göttlich, Patrick Schindler. Optimal inflow control of production systems with finite buffers. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 107-127. doi: 10.3934/dcdsb.2015.20.107

[14]

Urszula Ledzewicz, Stanislaw Walczak. Optimal control of systems governed by some elliptic equations . Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 279-290. doi: 10.3934/dcds.1999.5.279

[15]

Luca Galbusera, Sara Pasquali, Gianni Gilioli. Stability and optimal control for some classes of tritrophic systems. Mathematical Biosciences & Engineering, 2014, 11 (2) : 257-283. doi: 10.3934/mbe.2014.11.257

[16]

Getachew K. Befekadu, Eduardo L. Pasiliao. On the hierarchical optimal control of a chain of distributed systems. Journal of Dynamics & Games, 2015, 2 (2) : 187-199. doi: 10.3934/jdg.2015.2.187

[17]

Qiying Hu, Wuyi Yue. Optimal control for discrete event systems with arbitrary control pattern . Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 535-558. doi: 10.3934/dcdsb.2006.6.535

[18]

Leonardo Colombo, Fernando Jiménez, David Martín de Diego. Variational integrators for mechanical control systems with symmetries. Journal of Computational Dynamics, 2015, 2 (2) : 193-225. doi: 10.3934/jcd.2015003

[19]

Kathrin Flasskamp, Sebastian Hage-Packhäuser, Sina Ober-Blöbaum. Symmetry exploiting control of hybrid mechanical systems. Journal of Computational Dynamics, 2015, 2 (1) : 25-50. doi: 10.3934/jcd.2015.2.25

[20]

Ellina Grigorieva, Evgenii Khailov. Optimal control of pollution stock. Conference Publications, 2011, 2011 (Special) : 578-588. doi: 10.3934/proc.2011.2011.578

2016 Impact Factor: 0.857

Metrics

  • PDF downloads (2)
  • HTML views (0)
  • Cited by (0)

[Back to Top]