13.2 A Factorization Approach for Solving the HJIE

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 [157]. In this regard, let us write HJIE (13.9) in the form:

$HJI\left(x,{V}_{x}\right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{}\text{}\text{}x\in M\subset \mathcal{X},$ |
(13.25) |

where *HJI* : *T* ^{⋆} *M* →ℜ. Then we have the following theorem.

**Theorem 13.2.1** *Assume that V ∈ C*^{2}(*M*)*, and the functions f*(.), *g*_{1}(.), *g*_{2}(.), *h*(.) *are smooth C*^{2}(*M*) *functions. Then HJI*(., .) *is continuously-differentiable in an open neighborhood N* × Ψ ⊂ *T* ^{⋆} *M of the origin. Furthermore, let* $(\overline{x},{\overline{V}}_{x})$

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