Second order analysis of control-affine problems with scalar state   constraint
  In this article we establish new second order necessary and sufficient optimality conditions for a class of control-affine problems with a scalar control and a scalar state constraint. These optimality conditions extend to the constrained state framework the Goh transform, which is the classical tool for obtaining an extension of the Legendre condition. 
M. Soledad Aronna and Fr\'ed\'eric Bonnans and Bean San Goh
math.OC math.NA