In this section, we discuss a factorization approach that may yield exact global solutions of the HJIE for the class of affine nonlinear systems. We begin with a discussion of sufficiency conditions for the existence of exact solutions to the HJIE (13.9) which are provided by the Implicit-function Theorem . In this regard, let us write HJIE (13.9) in the form:
where HJI : T ⋆ M →ℜ. Then we have the following theorem.
Theorem 13.2.1 Assume that V ∈ C2(M), and the functions f(.), g1(.), g2(.), h(.) are smooth C2(M) functions. Then HJI(., .) is continuously-differentiable in an open neighborhood N × Ψ ⊂ T ⋆ M of the origin. Furthermore, let