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 pre-computed solutions of interest. The sensitivities thus calculated are subsequently used in determining neighboring solutions about an existing root (for algebraic systems) or trajectory (in case of dynamical systems). The generalization to dynamical systems, as a special case, enables the calculation of high order time varying sensitivities of the solutions of boundary value problems with respect to the parameters of the system model and/or functions describing the boundary condition. 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 these implicit rates characterizing the behavior in the neighborhood of the existing solutions. The accuracy of solutions obtained is subsequently enhanced using an averaging scheme based on the 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.

[1]  Harold R. Parks,et al.  The Implicit Function Theorem , 2002 .

[2]  G. Bierman Factorization methods for discrete sequential estimation , 1977 .

[3]  J. Junkins,et al.  A Study of Six Near-Earth Asteroids , 2005 .

[4]  Maria do Rosário de Pinho,et al.  Mixed constraints in optimal control: an implicit function theorem approach , 2007, IMA J. Math. Control. Inf..

[5]  J. Junkins,et al.  Robust nonlinear least squares estimation using the Chow-Yorke homotopy method , 1984 .

[6]  J. Turner Automated Generation of High-Order Partial Derivative Models , 2003 .

[7]  George M. Siouris,et al.  Applied Optimal Control: Optimization, Estimation, and Control , 1979, IEEE Transactions on Systems, Man, and Cybernetics.

[8]  Solving Kepler’s Equation using Bézier curves , 2007 .

[9]  James D. Turner,et al.  Solution of Two-Point Boundary-Value Problems Using Lagrange Implicit Function Theorem , 2009 .

[10]  Helmut Maurer,et al.  Sensitivity Analysis for Optimal Control Problems Subject to Higher Order State Constraints , 2001, Ann. Oper. Res..

[11]  Urho A. Rauhala,et al.  Array Algebra Expansion of Matrix and Tensor Calculus: Part 1 , 2002, SIAM J. Matrix Anal. Appl..

[12]  C. Chicone Ordinary Differential Equations with Applications , 1999, Texts in Applied Mathematics.

[13]  J. Hale,et al.  Ordinary Differential Equations , 2019, Fundamentals of Numerical Mathematics for Physicists and Engineers.

[14]  John L. Junkins,et al.  Multi-Resolution Methods for Modeling and Control of Dynamical Systems , 2008 .

[15]  G. Rauwolf,et al.  Near-optimal low-thrust orbit transfers generated by a genetic algorithm , 1996 .

[16]  Srinivas R. Vadali,et al.  Fuel-Optimal, Low-Thrust, Three-Dimensional Earth-Mars Trajectories , 2001 .

[17]  R. Abraham,et al.  Manifolds, Tensor Analysis, and Applications , 1983 .

[18]  R. Battin An introduction to the mathematics and methods of astrodynamics , 1987 .

[19]  J. Junkins,et al.  Identifying Near-term Missions and Impact Keyholes for Asteroid 99942 Apophis , 2006 .

[20]  Paolo L. Gatti,et al.  Introduction to Dynamics and Control of Flexible Structures , 1996 .

[21]  Tamara G. Kolda,et al.  Categories and Subject Descriptors: G.4 [Mathematics of Computing]: Mathematical Software— , 2022 .

[22]  D. D. Mueller,et al.  Fundamentals of Astrodynamics , 1971 .

[23]  H. Maurer,et al.  Sensitivity analysis for parametric control problems with control-state constraints , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[24]  F. Markley,et al.  Kepler Equation solver , 1995 .

[25]  John L. Junkins,et al.  Re-examination of eigenvector derivatives , 1987 .

[26]  Enrique F. Castillo,et al.  Sensitivity Analysis in Calculus of Variations. Some Applications , 2008, SIAM Rev..

[27]  Urho A. Rauhala,et al.  Array Algebra Expansion of Matrix and Tensor Calculus: Part 2 , 2002, SIAM J. Matrix Anal. Appl..