Skip to main content

Optimal Control and Mechanics

  • Reference work entry
  • First Online:
Encyclopedia of Systems and Control
  • 134 Accesses

Abstract

There are very natural close connections between mechanics and optimal control as both involve variational problems. This is a huge subject and we just touch on some interesting connections here. A survey and history may be found in Sussman and Willems (1997). Other aspects may be found in Bloch (2003).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 899.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 549.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Bibliography

  • Agrachev AA, Sarychev AV (1996) Abnormal sub-Riemannian geodesics: Morse index and rigidity. Ann Inst H Poincaré Anal Non Linéaire 13:635–690

    MathSciNet  Google Scholar 

  • Baillieul J (1975) Some optimization problems in geometric control theory. Ph.D. thesis, Harvard University

    Google Scholar 

  • Baillieul J (1978) Geometric methods for nonlinear optimal control problems. J Optim Theory Appl 25:519–548

    MathSciNet  Google Scholar 

  • Bliss G (1930) The problem of lagrange in the calculus of variations. Am J Math 52:673–744

    MathSciNet  Google Scholar 

  • Bloch AM (2003) (with Baillieul J, Crouch PE, Marsden JE), Nonholonomic mechanics and control. Interdisciplinary applied mathematics. Springer, New York

    Google Scholar 

  • Bloch AM, Crouch PE (1994) Reduction of Euler–Lagrange problems for constrained variational problems and relation with optimal control problems. In: Proceedings of the 33rd IEEE conference on decision and control, Lake Buena Vista. IEEE, pp 2584–2590

    Google Scholar 

  • Bloch AM, Crouch PE (1996) Optimal control and geodesic flows. Syst Control Lett 28(2):65–72

    MathSciNet  Google Scholar 

  • Bloch AM, Crouch PE, Ratiu TS (1994) Sub-Riemannian optimal control problems. Fields Inst Commun AMS 3:35–48

    MathSciNet  Google Scholar 

  • Bloch AM, Crouch P, Marsden JE, Ratiu TS (2002) The symmetric representation of the rigid body equations and their discretization. Nonlinearity 15: 1309–1341

    MathSciNet  Google Scholar 

  • Bloch AM, Crouch PE, Nordkivst N, Sanyal AK (2011) Embedded geodesic problems and optimal control for matrix Lie groups. J Geom Mech 3:197–223

    MathSciNet  Google Scholar 

  • Bloch AM, Crouch PE, Nordkivst 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 5:1–38

    MathSciNet  Google Scholar 

  • Brockett RW (1973) Lie theory and control systems defined on spheres. SIAM J Appl Math 25(2):213–225

    MathSciNet  Google Scholar 

  • Brockett RW (1981) Control theory and singular Riemannian geometry. In: Hilton PJ, Young GS (eds) New directions in applied mathematics. Springer, New York, pp 11–27

    Google Scholar 

  • Brockett RW (1983) Nonlinear control theory and differential geometry. In: Proceedings of the international congress of mathematicians, Warsaw, pp 1357–1368

    Google Scholar 

  • Gay-Balmaz F, Ratiu TS (2011) Clebsch optimal control formulation in mechanics. J Geom Mech 3:41–79

    MathSciNet  Google Scholar 

  • Griffiths PA (1983) Exterior differential systems. Birkhäuser, Boston

    Google Scholar 

  • Manakov SV (1976) Note on the integration of Euler’s equations of the dynamics of an n-dimensional rigid body. Funct Anal Appl 10:328–329

    MathSciNet  Google Scholar 

  • Marsden JE, Ratiu TS (1999) Introduction to mechanics and symmetry. Texts in applied mathematics, vol 17. Springer, New York. (1st edn. 1994; 2nd edn. 1999)

    Google Scholar 

  • Montgomery R (1993) Gauge theory of the falling cat. Fields Inst Commun 1:193–218

    Google Scholar 

  • Montgomery R (1994) Abnormal minimizers. SIAM J Control Optim 32:1605–1620

    MathSciNet  Google Scholar 

  • Montgomery R (1995) A survey of singular curves in sub-Riemannian geometry. J Dyn Control Syst 1:49–90

    Google Scholar 

  • Montgomery R (2002) A tour of sub-Riemannian geometries, their geodesics and applications. Mathematical surveys and monographs, vol 91. American Mathematical Society, Providence

    Google Scholar 

  • Ratiu T (1980) The motion of the free n-dimensional rigid body. Indiana U Math J 29:609–627

    MathSciNet  Google Scholar 

  • Rund H (1966) The Hamiltonian–Jacobi theory in the calculus of variations. Krieger, New York

    Google Scholar 

  • Strichartz R (1983) Sub-Riemannian geometry. J Diff Geom 24:221–263; see also J Diff Geom 30:595–596 (1989)

    Google Scholar 

  • Strichartz RS (1987) The Campbell–Baker–Hausdorff–Dynkin formula and solutions of differential equations. J Funct Anal 72:320–345

    MathSciNet  Google Scholar 

  • Sussmann HJ (1996) A cornucopia of four-dimensional abnormal sub-Riemannian minimizers. In: Bellaïche A, Risler J-J (eds) Sub-Riemannian geometry. Progress in mathematics, vol 144. Birkhäuser, Basel, pp 341–364

    Google Scholar 

  • Sussmann HJ, Willems JC (1997) 300 years of optimal control: from the Brachystochrone to the maximum principle. IEEE Control Syst Mag 17:32–44

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Anthony Bloch .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer-Verlag London

About this entry

Cite this entry

Bloch, A. (2015). Optimal Control and Mechanics. In: Baillieul, J., Samad, T. (eds) Encyclopedia of Systems and Control. Springer, London. https://doi.org/10.1007/978-1-4471-5058-9_46

Download citation

Publish with us

Policies and ethics