Pr´ e-Publica¸ c˜ oes do Departamento de Matem´ atica Universidade de Coimbra Preprint Number 09–05 CUBIC POLYNOMIALS AND OPTIMAL CONTROL ON COMPACT LIE GROUPS L. ABRUNHEIRO, M. CAMARINHA AND J. CLEMENTE-GALLARDO Abstract: This paper analyzes the Riemannian cubic polynomials’s problem from a Hamiltonian point of view. The description of the problem on compact Lie groups is particulary explored. The state space of the second order optimal control problem considered is the tangent bundle of the Lie group which also has a group structure. The dynamics of the problem is described by a presymplectic formalism associated with the canonical symplectic form on the cotangent bundle of the tangent bundle. Using these control geometrical tools, the equivalence between the Hamiltonian approach developed here and the known variational one is verified. Moreover, the equivalence allows us to deduce two invariants along the cubic polynomials which are in involution. Keywords: Optimal control, Symplectic geometry, Riemannian geometry, Lie groups. AMS Subject Classification (2000): 70H50, 49J15, 34A26, 53B20, 34K35. 1. Introduction Riemannian cubic polynomials (RCP) can be seen as a generalization of cubic polynomials in Euclidean spaces to Riemannian manifolds. The cubic polynomials on a Riemannian manifold are the smooth solutions of a fourth order differential equation which is the Euler-Lagrange equation of a second order variational problem. This variational problem was first introduced in 1989 (see [17]) and explored from a dynamical interpolation perspective in 1995 (see [10]). Interesting points related to this subject have been developed in the last few years, namely a geometric theory surprisingly close to the Riemannian theory of geodesics (see [2, 3, 4, 7, 8, 9, 16, 18, 19]). More recently, in [3, 16, 18], the analysis of RCP from a variational point of view was carried out for locally symmetric manifolds and some invariants along these cubic polynomials were obtained. A qualitative analysis of RCP is given in [3, 16], with special attention to the case of the Lie group SO(3), where RCP corresponds to Lie quadratics on the Lie algebra. The article [3] introduces a reduction of the RCP equation for this Lie group of rotations. In [16] some results on asymptotics and symmetries of cubics are proved for Received January 23, 2009. 1