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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].

##### MSC:
 49J15 Existence theories for optimal control problems involving ordinary differential equations 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
##### Keywords:
sub-Riemannian geometry; Caustic; Hamiltonians