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Extra info for Advanced Topics in Control Systems Theory: Lecture Notes from FAP 2005
The distribution D spanned by F1 , F2 is a contact distribution deﬁned as the kernel of the 1-form α = dz + (xdy − ydx). A sub-Riemannian metric is associated to a metric of the form g = a(q)dx2 + 2b(q)dxdy + c(q)dy 2 . By choosing suitable coordinates, the smooth functions a, b and c can be normalized to a = c = 1 and b = 0. The case g = dx2 + dy 2 is called the Heisenberg case or the sub-Riemannian ﬂat contact case. Heisenberg sub-Riemannian geometry and the Dido problem. We observe that the previous problem can be written x˙ = u1 , y˙ = u2 , z˙ = xy ˙ − yx, ˙ and T 0 (x˙ 2 + y˙ 2 )1/2 dt → min in order that: (i) The length of a curve t → (x(t), y(t), z(t)) is the length of the projection in the xy-plane.
Let H (t, z) be a smooth Hamiltonian vector ﬁeld whose integral curves are the extremals of an optimal control problem with ﬁxed time T and initial manifold M0 . The time tf is a focal time along the BC-extremal z if there is a Jacobi ﬁeld J such that J(0) is in Tz(0) M0⊥ and J is vertical at tf . Both concepts ﬁt in the same geometric framework: a one parameter family of Lagrangian manifolds obtained by transporting the initial submanifold with the ﬂow. The Jacobi ﬁelds span the tangent spaces of the Lagrangian manifolds computed along the reference extremal.
Properties of the model. In this model, we have gathered in one normal form all the information required to evaluate the endpoint mapping (and thus the accessibility set) up to second-order relevant terms. The adjoint covector is oriented by the condition H0 ≥ 0 and normalized to p = (1, 0, . . , 0) The linearized system along the reference trajectory is a constant linear system in Brunovsky normal form. Indeed, x˙ 1 = 1 + q(x1 , x2 , . . , xn ) x˙ 2 = x3 .. x˙ n = u with q(x1 , x2 , . . , xn ) = n i,j=2 aij xi xj .