20 Second Order Opthnality Conditions 20.1 Hessian In this chapter we obtain second order necessary optimality conditions for control problems. As we know, geometrically the study of optimality reduces to the study of boundary of attainable sets (see Sect. 10.2). Consider a control system q = !u(q), q E M, u E U = int U C (20.1) where the state space M is, as usual, a smooth manifold, and the space of control parameters U is open (essentially, this means that we study optimal controls that do not come to the boundary of U, although a similar theory for bang-bang controls can also be constructed). The attainable set Aqo(tI) of system (20.1) is the image of the endpoint mapping Ft, : u(·) I-t qoo ext> !a tl !u(t) dt. We say that a trajectory q(t), t E [0, t 1], is geometrically optimal for sys- tem (20.1) if it comes to the boundary of the attainable set for the terminal time t1: Necessary conditions for this inclusion are given by Pontryagin Maximum Principle. A part of the statements of PMP can be viewed as the first order optimality condition (we see this later). Now we seek for optimality conditions of the second order. Consider the problem in a general setting. Let F:U-tM be a smooth mapping, where U is an open subset in a Banach space and M is a smooth n-dimensional manifold (usually in the sequel U is the space of A. A. Agrachev et al., Control Theory from the Geometric Viewpoint © Springer-Verlag Berlin Heidelberg 2004