Optimal Control on Lie groups and Integrable Hamiltonian Systems

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 $\mathfrak{g} = \mathfrak{p} ⊕ \mathfrak{k}$ of the form
$$\frac{dL_\lambda}{dt} = [\Omega_\lambda, L_\lambda] L_\lambda = L_{\mathfrak{p}} − \lambda L_{\mathfrak{k}} − (\lambda^2 − s)A, \quad L_{\mathfrak{p}} \in \mathfrak{p}, \quad L_{\mathfrak{k}} \in \mathfrak{k}.$$
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.