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Methods of control theory in nonholonomic geometry. (English) Zbl 0848.93012
Chatterji, S. D. (ed.), Proceedings of the international congress of mathematicians, ICM ’94, August 3-11, 1994, Zürich, Switzerland. Vol. II. Basel: Birkhäuser. 1473-1483 (1995).
Let $$M$$ be a manifold and $$V\subseteq TM$$ a control system. The assumption that $$V$$ is either a distribution or the intersection of it with a fibering by ellipsoids is made. Let $$\Omega_{q_0}$$ be the set of admissible trajectories with respect to $$V$$, such that $$q(0)= q_0$$. Also let $$\Omega_{q_0 q_1}$$ be the subset of trajectories such that $$q(0)= q_0$$, $$q(t)= q_1$$. A suitable differentiable structure on $$\Omega_{q_0}$$ is defined in terms of uniform convergence of trajectories and their derivatives. Let $$f_t: \Omega_{q_0}\to M$$ be the endpoint mapping $$f_t(q(\cdot))= q(t)$$. A trajectory $$q(\cdot)$$ is called a critical point if $$\lambda D f_t(q(\cdot))= 0$$ for some $$\lambda\in T^*_{q(t)} M$$, $$\lambda\neq 0$$.
This article explores, without proofs but giving appropriate references, several interesting results related to critical levels of $$f_t$$, in important cases, for instance, when $$M$$ is a Lie group or a contact manifold. This is done by using the symplectic structure of $$T^* M$$ and Hamilton’s equations $$\dot\lambda= \vec v_t^*(\lambda)$$, where, for each admissible vector field $$v_t$$, $$v^*(\lambda)= \langle \lambda, v\rangle$$ is the Hamiltonian. Nonvanishing solutions to these equations are called extremals. For instance when $$M$$ is a semisimple Lie group, the homology $$Hu(\Omega_{q_0 q_1}\backslash \{q^\ell\})$$ is calculated.
For the entire collection see [Zbl 0829.00015].

##### MSC:
 93B29 Differential-geometric methods in systems theory (MSC2000) 58A30 Vector distributions (subbundles of the tangent bundles) 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems