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Regul. Chaotic Dyn., 2011, Volume 16, Issue 5, Pages 514–535 (Mi rcd467)  

Optimal Control on Lie groups and Integrable Hamiltonian Systems

Velimir Jurdjevic

Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 3G3 Canada

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 $\frak g=\frak p\oplus\frak k$ of the form
$$ \frac{dL_\lambda}{dt} = [\Omega_\lambda,L_\lambda]L_\lambda=L_{\frak p}- \lambda L_{\frak k}-(\lambda^2-s)A,\quad L_{\frak p}\in \frak p,\quad L_{\frak k}\in \frak k. $$
The spectral invariants associated with $L_\lambda$ 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

DOI: https://doi.org/10.1134/S156035471105008X


Bibliographic databases:

MSC: 49J15, 53D05, 93B27, 74B20
Received: 02.03.2011
Accepted:06.05.2011
Language:

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://www.ams.org/mathscinet-getitem?mr=2844862}
\zmath{https://zbmath.org/?q=an:1309.49004}


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