Abstract
Discrete optimal control is a branch of mathematics which studies optimization procedures for controlled discrete-time models – that is, the optimization of a performance index associated with a discrete-time control system. This entry gives an introduction to the topic. The formulation of a general discrete optimal control problem is described, and applications to mechanical systems are discussed.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Bibliography
Bock HG, Plitt KJ (1984) A multiple shooting algorithm for direct solution of optimal control problems. In: 9th IFAC world congress, Budapest. Pergamon Press, pp 242–247
Bloch AM, Hussein I, Leok M, Sanyal AK (2009) Geometric structure-preserving optimal control of a rigid body. J Dyn Control Syst 15(3):307–330
Bloch AM, Crouch PE, Nordkvist N (2013) Continuous and discrete embedded optimal control problems and their application to the analysis of Clebsch optimal control problems and mechanical systems. J Geom Mech 5(1):1–38
Bonnans JF, Laurent-Varin J (2006) Computation of order conditions for symplectic partitioned Runge-Kutta schemes with application to optimal control. Numer Math 103:1–10
Bou-Rabee N, Marsden JE (2009) Hamilton-Pontryagin integrators on Lie groups: introduction and structure-preserving properties. Found Comput Math 9(2):197–219
Cadzow JA (1970) Discrete calculus of variations. Int J Control 11:393–407
de León M, Martí’n de Diego D, Santamaría-Merino A (2007) Discrete variational integrators and optimal control theory. Adv Comput Math 26(1–3):251–268
Hager WW (2001) Numerical analysis in optimal control. In: International series of numerical mathematics, vol 139. Birkhäuser Verlag, Basel, pp 83–93
Hairer E, Lubich C, Wanner G (2002) Geometric numerical integration, structure-preserving algorithms for ordinary differential equations’. Springer series in computational mathematics, vol 31. Springer, Berlin
Hussein I, Leok M, Sanyal A, Bloch A (2006) A discrete variational integrator for optimal control problems on SO(3)’. In: Proceedings of the 45th IEEE conference on decision and control, San Diego, pp 6636–6641
Hwang CL, Fan LT (1967) A discrete version of Pontryagin’s maximum principle. Oper Res 15:139–146
Jordan BW, Polak E (1964) Theory of a class of discrete optimal control systems. J Electron Control 17:697–711
Jiménez F, Martín de Diego D (2010) A geometric approach to Discrete mechanics for optimal control theory. In: Proceedings of the IEEE conference on decision and control, Atlanta, pp 5426–5431
Jiménez F, Kobilarov M, Martín de Diego D (2013) Discrete variational optimal control. J Nonlinear Sci 23(3):393–426
Junge O, Ober-Blöbaum S (2005) Optimal reconfiguration of formation flying satellites. In: IEEE conference on decision and control and European control conference ECC, Seville
Junge O, Marsden JE, Ober-Blöbaum S (2006) Optimal reconfiguration of formation flying spacecraft- a decentralized approach. In: IEEE conference on decision and control and European control conference ECC, San Diego, pp 5210–5215
Kobilarov M (2008) Discrete geometric motion control of autonomous vehicles. Thesis, Computer Science, University of Southern California
Kobilarov M, Marsden JE (2011) Discrete geometric optimal control on Lie groups. IEEE Trans Robot 27(4):641–655
Leok M (2004) Foundations of computational geometric mechanics, control and dynamical systems. Thesis, California Institute of Technology. Available in http://www.math.lsa.umich.edu/~mleok
Leyendecker S, Ober-Blobaum S, Marsden JE, Ortiz M (2007) Discrete mechanics and optimal control for constrained multibody dynamics. In: 6th international conference on multibody systems, nonlinear dynamics, and control, ASME international design engineering technical conferences, Las Vegas
Marrero JC, Martín de Diego D, Martínez E (2006) Discrete Lagrangian and Hamiltonian mechanics on Lie groupoids. Nonlinearity 19:1313–1348. Corrigendum: Nonlinearity 19
Marrero JC, Martín de Diego D, Stern A (2010) Lagrangian submanifolds and discrete constrained mechanics on Lie groupoids. Preprint, To appear in DCDS-A 35-1, January 2015.
Marsden JE, West M (2001) Discrete mechanics and variational integrators. Acta Numer 10: 357–514
Marsden JE, Pekarsky S, Shkoller S (1999a) Discrete Euler-Poincaré and Lie-Poisson equations. Nonlinearity 12:1647–1662
Marsden JE, Pekarsky S, Shkoller S (1999b) Symmetry reduction of discrete Lagrangian mechanics on Lie groups. J Geom Phys 36(1–2):140–151
Ober-Blöbaum S (2008) Discrete mechanics and optimal control. Ph.D. Thesis, University of Paderborn
Ober-Blöbaum S, Junge O, Marsden JE (2011) Discrete mechanics and optimal control: an analysis. ESAIM Control Optim Calc Var 17(2):322–352
Pytlak R (1999) Numerical methods for optimal control problems with state constraints. Lecture Notes in Mathematics, 1707. Springer-Verlag, Berlin, xvi+215 pp.
Wendlandt JM, Marsden JE (1997a) Mechanical integrators derived from a discrete variational principle. Phys D 106:2232–2246
Wendlandt JM, Marsden JE (1997b) Mechanical systems with symmetry, variational principles and integration algorithms. In: Alber M, Hu B, Rosenthal J (eds) Current and future directions in applied mathematics, (Notre Dame, IN, 1996), Birkhäuser Boston, Boston, MA, pp 219–261
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this entry
Cite this entry
Martin De Diego, D. (2021). Discrete Optimal Control. In: Baillieul, J., Samad, T. (eds) Encyclopedia of Systems and Control. Springer, Cham. https://doi.org/10.1007/978-3-030-44184-5_47
Download citation
DOI: https://doi.org/10.1007/978-3-030-44184-5_47
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-44183-8
Online ISBN: 978-3-030-44184-5
eBook Packages: Intelligent Technologies and RoboticsReference Module Computer Science and Engineering