Second order conditions for optimality and local controllability of   discrete-time systems
  We study local controllability and optimal control problems for invertible discrete-time control systems. We present second order necessary conditions for optimality and sufficient conditions for local controllability. The conditions are stated in geometric terms, using vector fields naturally associated to the system. The Hessian of the optimal problem is computed in terms of Lie brackets of vector fields of the system. 
M. Barbero-Li\~n\'an, B. Jakubczyk
math.OC