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=p⊕k of the form
dLλdt=[Ωλ,Lλ]Lλ=Lp−λLk−(λ2−s)A,Lp∈p,Lk∈k.
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.
\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
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This publication is cited in the following 12 articles:
Simone Fiori, “Discrete-Time Dynamical Systems on Structured State Spaces: State-Transition Laws in Finite-Dimensional Lie Algebras”, Symmetry, 17:3 (2025), 463
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
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
Simone Fiori, “Manifold Calculus in System Theory and Control—Second Order Structures and Systems”, Symmetry, 14:6 (2022), 1144
Foroozandeh M., Singh P., “Optimal Control of Spins By Analytical Lie Algebraic Derivatives”, Automatica, 129 (2021), 109611
Fiori S., “Extension of Pid Regulators to Dynamical Systems on Smooth Manifolds (M-Pid)”, SIAM J. Control Optim., 59:1 (2021), 78–102
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
Rory Biggs, Claudiu C. Remsing, UNIPA Springer Series, Lie Groups, Differential Equations, and Geometry, 2017, 127
Rory Biggs, Claudiu C. Remsing, “Control Systems on Three-Dimensional Lie Groups: Equivalence and Controllability”, J Dyn Control Syst, 20:3 (2014), 307
Velimir Jurdjevic, Springer INdAM Series, 5, Geometric Control Theory and Sub-Riemannian Geometry, 2014, 219
Rory Biggs, Claudiu C. Remsing, “Feedback Classification of Invariant Control Systems on Three-Dimensional Lie Groups”, IFAC Proceedings Volumes, 46:23 (2013), 506
Velimir Jurdjevic, “Affine-quadratic problems on Lie groups”, Mathematical Control & Related Fields, 3:3 (2013), 347