##
**Geometric theory of extremals in optimal control problems. I: The fold and Maxwell case.**
*(English)*
Zbl 0606.49016

An absolutely continuous function (x(t),p(t)) is called an extremal if there is some measurable u(t) for which the triple (x,p,u) obeys the differential equations and maximum condition of Pontryagin’s maximum principle. Recall the latter of these:
\[
(*)\quad H(x(t),p(t),u(t))=\max [H(x(t),p(t),v):v\in U]\quad a.e.,
\]
where H is Pontryagin’s pseudo- Hamiltonian and U is the control set. This paper concerns the singularities in the set of extremal curves: these can only arise at switching points, i.e., points (x,p) where the maximizer in (*) is not unique. Fold (resp., Maxwell) points are switching points where exactly two (resp., three) control values give the maximum in (*) and certain other conditions hold. Here they are classified as hyperbolic, parabolic, or elliptic, and the qualitative behaviour of the field of extremals near (x,p) is described in each case. Infinite differentiability of all the data is assumed, and the basic techniques are those of differential geometry. There are no examples and few references.

Reviewer: P.Loewen

### MSC:

49K15 | Optimality conditions for problems involving ordinary differential equations |

49J15 | Existence theories for optimal control problems involving ordinary differential equations |

57R45 | Singularities of differentiable mappings in differential topology |

93C10 | Nonlinear systems in control theory |

93C15 | Control/observation systems governed by ordinary differential equations |