Regular and Chaotic Dynamics
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Regul. Chaotic Dyn.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Regular and Chaotic Dynamics, 2011, Volume 16, Issue 5, Pages 514–535
DOI: https://doi.org/10.1134/S156035471105008X
(Mi rcd467)
 

This article is cited in 12 scientific papers (total in 12 papers)

Optimal Control on Lie groups and Integrable Hamiltonian Systems

Velimir Jurdjevic

Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 3G3 Canada
Citations (12)
Abstract: Control theory, initially conceived in the 1950's as an engineering subject motivated by the needs of automatic control, has undergone an important mathematical transformation since then, in which its basic question, understood in a larger geometric context, led to a theory that provides distinctive and innovative insights, not only to the original problems of engineering, but also to the problems of differential geometry and mechanics.
This paper elaborates the contributions of control theory to geometry and mechanics by focusing on the class of problems which have played an important part in the evolution of integrable systems. In particular the paper identifies a large class of Hamiltonians obtained by the Maximum principle that admit isospectral representation on the Lie algebras g=pk of the form
dLλdt=[Ωλ,Lλ]Lλ=LpλLk(λ2s)A,Lpp,Lkk.
The spectral invariants associated with Lλ recover the integrability results of C.G.J. Jacobi concerning the geodesics on an ellipsoid as well as the results of C. Newmann for mechanical problem on the sphere with a quadratic potential. More significantly, this study reveals a large class of integrable systems in which these classical examples appear only as very special cases.
Keywords: Lie groups, control systems, the Maximum principle, symplectic structure, Hamiltonians, integrable systems.
Received: 02.03.2011
Accepted: 06.05.2011
Bibliographic databases:
Document Type: Article
Language: English
Citation: Velimir Jurdjevic, “Optimal Control on Lie groups and Integrable Hamiltonian Systems”, Regul. Chaotic Dyn., 16:5 (2011), 514–535
Citation in format AMSBIB
\Bibitem{Jur11}
\by Velimir Jurdjevic
\paper Optimal Control on Lie groups and Integrable Hamiltonian Systems
\jour Regul. Chaotic Dyn.
\yr 2011
\vol 16
\issue 5
\pages 514--535
\mathnet{http://mi.mathnet.ru/rcd467}
\crossref{https://doi.org/10.1134/S156035471105008X}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2844862}
\zmath{https://zbmath.org/?q=an:1309.49004}
Linking options:
  • https://www.mathnet.ru/eng/rcd467
  • https://www.mathnet.ru/eng/rcd/v16/i5/p514
  • This publication is cited in the following 12 articles:
    1. Simone Fiori, “Discrete-Time Dynamical Systems on Structured State Spaces: State-Transition Laws in Finite-Dimensional Lie Algebras”, Symmetry, 17:3 (2025), 463  crossref
    2. Pierre-Louis Giscard, Mohammadali Foroozandeh, “Exact solutions for the time-evolution of quantum spin systems under arbitrary waveforms using algebraic graph theory”, Computer Physics Communications, 282 (2023), 108561  crossref
    3. S. Fiori, L.  Del Rossi, “Minimal control effort and time Lie-group synchronisation design based on proportional-derivative control”, International Journal of Control, 95:1 (2022), 138  crossref
    4. Simone Fiori, “Manifold Calculus in System Theory and Control—Second Order Structures and Systems”, Symmetry, 14:6 (2022), 1144  crossref
    5. Foroozandeh M., Singh P., “Optimal Control of Spins By Analytical Lie Algebraic Derivatives”, Automatica, 129 (2021), 109611  crossref  mathscinet  isi  scopus
    6. Fiori S., “Extension of Pid Regulators to Dynamical Systems on Smooth Manifolds (M-Pid)”, SIAM J. Control Optim., 59:1 (2021), 78–102  crossref  mathscinet  isi  scopus
    7. Catherine E. Bartlett, Rory Biggs, Claudiu C. Remsing, “Control systems on nilpotent Lie groups of dimension ≤4: Equivalence and classification”, Differential Geometry and its Applications, 54 (2017), 282  crossref
    8. Rory Biggs, Claudiu C. Remsing, UNIPA Springer Series, Lie Groups, Differential Equations, and Geometry, 2017, 127  crossref
    9. Rory Biggs, Claudiu C. Remsing, “Control Systems on Three-Dimensional Lie Groups: Equivalence and Controllability”, J Dyn Control Syst, 20:3 (2014), 307  crossref
    10. Velimir Jurdjevic, Springer INdAM Series, 5, Geometric Control Theory and Sub-Riemannian Geometry, 2014, 219  crossref
    11. Rory Biggs, Claudiu C. Remsing, “Feedback Classification of Invariant Control Systems on Three-Dimensional Lie Groups”, IFAC Proceedings Volumes, 46:23 (2013), 506  crossref
    12. Velimir Jurdjevic, “Affine-quadratic problems on Lie groups”, Mathematical Control & Related Fields, 3:3 (2013), 347  crossref
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Statistics & downloads:
    Abstract page:137
     
      Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2025