Nonlinear H-Infinity Control, Hamiltonian Systems and Hamilton-Jacobi Equations

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:

$HJI\left(x,V_x\right)=0,\text{}\text{}\text{}x\in M\subset \mathcal{X},$

(13.25)

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

Theorem 13.2.1 Assume that V ∈ 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 $(\overline{x},{\overline{V}}_x)$