Generalizations and Applications of the Lagrange Implicit Function Theorem

The Implicit function theorem due to Lagrange is generalized to enable high order implicit rate calculations of general implicit functions about a nominal solution of interest. The sensitivities thus calculated are subsequently used in determining neighboring solutions about a nominal point, or in the case of a dynamical system, a trajectory. The generalization to dynamical systems, as a special case, enables the calculation of high order time varying sensitivities and the sensitivity of the solutions of two point boundary value problems subject to system parameter and boundary condition variations. The generalizations thus realized are applied to various problems arising in trajectory optimization. It was found that useful information relating the neighboring extremal paths can be deduced from the implicit rates characterizing the behavior in significant finite neighborhoods centered along the nominal motion. The accuracy of the solutions obtained is subsequently enhanced using a Global Local Orthogonal Polynomial (GLO-MAP) weight functions developed by the first author to blend many local approximations in a continuous fashion. Example problems illustrate the wide applicability of the presented generalizations of Lagrange’s classical results to static and dynamic optimization problems.