Pointwise second-order necessary optimality conditions for the Mayer problem with control constraints (Q2873855)

After introducing some notations and presenting some preliminary results, in this paper second-order necessary optimality conditions are proved for an arbitrary set \(U\) and any integrable optimal control at times when it belongs to the interior of \(U\). Further, the cases when \(U\) is a convex polytope and when \(U\) is of class \(C^2\) are also treated. Finally, second-order necessary optimality conditions for the Mayer problem involving endpoint constraints are proved. These results are obtained by adding the calmness assumption in order to reduce the Mayer problem with endpoint constraints to one without them, but involving a nondifferentiable cost function. For this equivalent problem pointwise conditions are derived using the variational approach for the free endpoint problems.