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Let G be a Lie group. In order to study optimal control problems on a homogeneous space G/H, we identify its cotangent bundle T * G/H as a subbundle of the cotangent bundle of G. Next, this identification is used to describe the Hamiltonian lifting of vector fields on G/H induced by elements in the Lie algebra g of G. As an application, we consider a bilinear control system Σ in R 2 whose matrices generate sl(2). Through the Pontryagin maximum principle, we analyze the time-optimal problem for the angle system PΣ defined by the projection of Σ onto the projective line P 1. We compute some examples, and in particular we show that the bang-bang principle does not need to be true.

Let Σ be a bilinear control system on R 2 whose matrices generate the Lie algebra sl(2) of the Lie group Sl(2) : the group of order two real matrices with determinant 1. In this work we focus on the extremals of a quadratic cost optimal problem for the angle system PΣ defined by the projection of Σ onto the real projective line P 1. It has been proved in [2] that through the Cartan-Killing form the cotangent bundle of P 1 can be identified with a cone C in sl(2). Via the Pontryagin Maximum Principle, we explicitly show the extremals by using the mentioned identification and the special form of the trajectories associated with the lifting of vector fields on PΣ. We analyze both: the controllable case and when the system bf P Σ give rise to control sets. Some examples are shown.

In this paper the theory of jets based on Weil's near points is applied to Lie equations and pseudogroups. Linear systems of partial differential equations are interpreted, in a canonical way, as distributions on the fibre bundles of invertible jets invariant under translations. We prove the two fundamental theorems for Lie equations and generalize the results of Rodrigues; a geometric correspondence between linear and nonlinear Lie equations is given, and the symbols of a linear Lie equation and its prolongations are canonically identified with the symbols of their attached nonlinear equations. From this fact we deduce that a linear Lie equation verifies the conditions of Goldsmichmidt's criterion on formal integrability if and only if its attached nonlinear Lie equation satisfies them locally. Finally, we define the Cartan 1-form on the fibre bundle of invertible jets and give a global form to the equivalence between the Lie and Cartan definitions of continuous groups. ᮊ