  The first part of the paper studies a class of optimal control problems in
Bolza form, where the dynamics is linear w.r.t.~the control function. A
necessary condition is derived, for the optimality of a trajectory which starts
at a conjugate point. The second part is concerned with a classical problem in
the Calculus of Variations, with free terminal point. For a generic terminal
cost $\psi\in \C^4(\mathbb{R}^n)$, applying the previous necessary condition we
show that the set of conjugate points is contained in the image of an
$(n-2)$-dimensional manifold, and has locally bounded $(n-2)$-dimensional
Hausdorff measure.
