Admissible Directions in Optimal Control Under Uniqueness Assumptions

It is well-known that, for a mathematical programming problem involving equality and inequality constraints, the uniqueness of a Lagrange multiplier associated with a local solution implies, under certain smoothness assumptions, second order necessary optimality conditions. Those conditions hold on a set of critical directions defined by those points satisfying the constraints and for which the minimizing function and the standard Lagrangian coincide. No similar links between uniqueness of multipliers and second order conditions seem to have been established for optimal control problems. In this paper, we provide some results in this direction. In particular, we study and completely solve a natural conjecture which provides, under uniqueness assumptions, nonnegative second variations on a classical cone of admissible directions.