H Hamiltonian

The implicit function theorem is a statement of the existence, continuity, and differen-tiability of a function or set of functions. The theorem is closely related to the convergence of Newton’s method for nonlinear equations, the existence and uniqueness of solutions to nonlinear differential equations, and the sensitivity of solutions to these nonlinear problems. The implicit function theorem is presented, and high order sensitivity equations are gen-erated using implicit differentiation. Once a nominal solution is known, these sensitivities are used to construct a family of neighboring solutions. Based on results presented in the paper, the implicit function approach shows great promise compared with current methods in generating families of neighboring solutions to problems in dynamics and control. Nomenclature a Open-loop plant b Control coefficien