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Nonlinear H-Infinity Control, Hamiltonian Systems and Hamilton-Jacobi Equations by M.D.S. Aliyu

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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(x,Vx)=0,xMX,

(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 (x¯,V¯x)

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