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We discuss the use of Dirac structures to obtain a better understanding of the geometry of a class of optimal control problems and their reduction by symmetries. In particular we will show how to extend the reduction of Dirac structures recently proposed by Yoshimura and Marsden [Yo09] to describe the reduction of a class of optimal control problems with a Lie group of symmetry. We will prove that, as in the case of reduction of implicit Hamiltonian or Lagrangian systems, the reduction of the variational principle and the reduction of the Dirac structure describing the Pontryagin Maximum Principle first order differential conditions coincide. Moreover they will also reproduce E. Mart\'inez Lie algebroids reduction approach [Mr04] to optimal control systems with symmetry. The geodesic subriemannian problem considered as an optimal control problem on the Heisenberg group, will be discussed as a simple example illustrating these results.

For $\alpha\in(0,\pi/2)$, let $(\Sigma)_\alpha$ be the control system $\dot{x}=(F+uG)x$, where $x$ belongs to the two-dimensional unit sphere $S^2$, $u\in [-1,1]$ and $F,G$ are $3\times3$ skew-symmetric matrices generating rotations with perpendicular axes of respective length $\cos(\alpha)$ and $\sin(\alpha)$. In this paper, we study the time optimal synthesis (TOS) from the north pole $(0,0,1)^T$ associated to $(\Sigma)_\alpha$, as the parameter $\alpha$ tends to zero. We first prove that the TOS is characterized by a ``two-snakes'' configuration on the whole $S^2$, except for a neighborhood $U_\alpha$ of the south pole $(0,0,-1)^T$ of diameter at most $\O(\alpha)$. We next show that, inside $U_\alpha$, the TOS depends on the relationship between $r(\alpha):=\pi/2\alpha-[\pi/2\alpha]$ and$\alpha$. More precisely, we characterize three main relationships, by considering sequences $(\alpha_k)_{k\geq 0}$ satisfying $(a)$$r(\alpha_k)=\bar{r}$; $(b)$ $r(\alpha_k)=C\alpha_k$ and $(c)$ $r(\alpha_k)=0$, where $\bar{r}\in (0,1)$ and $C>0$. In each case, we describe the TOS and provide, after a suitable rescaling, the limiting behavior, as $\alpha$ tends to zero, of the corresponding TOS inside $U_\alpha$., Comment: 28 pages, 10 figures