Second-order sufficient conditions for an extremum in optimal control

Sufficient quadratic optimality conditions for a weak and a strong minimum are stated in an optimal control problem on a fixed time interval with mixed state-control constraints, under the assumption that the gradients of all active mixed constraints with respect to control are linearly independent. The conditions. are stated for the cases of both continuous and discontinuous controls and guarantee in each case a lower bound of the cost function increase at the reference point. They are formulated in terms of an accessory problem with quadratic form, which must be positive-definite on the so-called critical cone. In the case of discontinuous control the quadratic form has some new,terms related to the control discontinuity.