Let us now return to the symmetric Riemannian pairs (G,K) and the Cartan decompositions. In the previous chapter we investigated the relevance of the left-invariant distributions with values in p for the structure of G and the associated quotient space G/H. In particular, we showed that the controllability assumption singled out Lie group pairs (G,K) in which the Lie algebraic conditions of Cartan took the strong form, namely,

In this chapter we will consider complementary variational problems on G defined by a positive-definite quadratic form Q(u, v) in k and an element. More precisely, we will consider the left-invariant affine distributions D(g) = {g(A + X) : X ∈ k} defined by an element A ∈ p. Each affine distribution defines a natural control problem in G,

with control functions u(t) taking values in.

We will be interested in the conditions on A that guarantee that any two points of G can be connected by a solution of (9.1), and secondly, we will be interested in the solutions of (9.1) which transfer an initial point g0 to a given terminal point g1 for which the energy functional is minimal. We will refer to this problem as the affine-quadratic problem.

In what follows, we will use ⟨u, v⟩ to denote the negative of the Killing form Kl(u, v) = Tr(ad(u) ∈ ad(v)) for any u and v in g. Since the Killing form is negative-definite on, the restriction of ⟨, ⟩ to is positive-definite, and can be used to define a bi-invariant metric on. This metric will be used as a bench mark for the affine-quadratic problems. For that reason we will express the quadratic form Q(u, v) as ⟨Q(u), v⟩ for some self-adjoint linear mapping Q on k which satisfies ⟨Q(u), u ⟩> 0 for all u = 0 in k. Then, Q = I yields the negative of the Killing form, i.e., the bi-invariant metric on.