GENERIC PROPERTIES OF CONJUGATE POINTS IN OPTIMAL CONTROL PROBLEMS

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 ψ ∈ C⁴(Rⁿ), 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.