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.
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