Pointwise second-order necessary optimality conditions and second-order sensitivity relations in optimal control

This paper is devoted to pointwise second-order necessary optimality conditions for the Mayer problem arising in optimal control theory. We first show that with every optimal trajectory it is possible to associate a solution p(⋅) of the adjoint system (as in the Pontryagin maximum principle) and a matrix solution W(⋅) of an adjoint matrix differential equation that satisfy a second-order transversality condition and a second-order maximality condition. These conditions seem to be a natural second-order extension of the maximum principle. We then prove a Jacobson like necessary optimality condition for general control systems and measurable optimal controls that may be only “partially singular” and may take values on the boundary of control constraints. Finally we investigate the second-order sensitivity relations along optimal trajectories involving both p(⋅) and W(⋅).