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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].

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