**Gerhard-W. Weber
Department of Mathematics,
Darmstadt University of Technology, Darmstadt, Germany.**

We will study one-parameter families
of differentiable optimal control problems given by:
Here, at given times *t*the inequality constraint functions are of semi-infinite nature, the
objective functional may also be of max-type.
For each
the problem
is
equivalent to a
one-parameter family
of differentiable optimization problems. From these the
consideration of generalized critical trajectories,
such as a local minimum trajectory, comes into our investigation.
According to a concept introduced by Hettich, Jongen and Stein in optimization,
we distinguish eight types of generalized critical trajectories.
Under suitable continuity, compactness and integrability assumptions,
those problems, which exclusively have generalized critical points being
of one of these eight types, are generic.
We study normal forms and characteristic examples, locally
around these trajectories.

Moreover, we indicate the related concept of structural stability of optimal control problems due to the topological behaviour of the lower level sets under small data perturbations. Finally, we discuss the numerical consequences of our investigations for pathfollowing techniques with jumps.

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