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Caustics in optimal control: An example of bifurcation when the symmetry is broken. (English) Zbl 0799.49002
Allgower, Eugene L. (ed.) et al., Exploiting symmetry in applied and numerical analysis. Proceedings of the 22nd AMS-SIAM summer seminar in applied mathematics, Colorado State University, Fort Collins, CO, USA, July 26-August 1, 1992. Providence, RI: American Mathematical Society. Lect. Appl. Math. 29, 203-212 (1993).
The paper is a short report on the author’s joint work with I. Kupka. Let \(M\) be a 3-dimensional manifold. Consider a Hamiltonian \(H\) on \(T^* M\) given by \(H(x,p)= \langle X_ 1(x),p\rangle^ 2+ \langle X_ 2(x),p\rangle^ 2\), \((x,p)\in T^* M\), where \(X_ 1\) and \(X_ 2\) are vector fields on \(M\) such that they generate (as a Lie algebra) the entire tangent space at each point. Such Hamiltonians arise in problems of Optimal Control and sub-Riemannian geometry. Fix \(x_ 0\in M\). Let \(\theta(p)\), \((p\in T^*_{x_ 0} M)\) denote the value of the solution of the Hamilton’s equation (projected on \(M\)) at time 1. The Caustic at \(x_ 0\) is the set \(\{\theta(p)\mid\text{Det}[D_ p\theta]=0\}\), i.e. caustic is where the extremals bifurcate. This paper discusses the nature of caustics for some special Hamiltonians of above form.
For the entire collection see [Zbl 0782.00045].

49J15 Existence theories for optimal control problems involving ordinary differential equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems