Abstract
This work presents a methodology to derive analytical functionals, with embedded linear constraints among the components of a vector (e.g., coordinates) that is a function a single variable (e.g., time). This work prepares the background necessary for the indirect solution of optimal control problems via the application of the Pontryagin Maximum Principle. The methodology presented is part of the univariate Theory of Functional Connections that has been developed to solve constrained optimization problems. To increase the clarity and practical aspects of the proposed method, the work is mostly presented via examples of applications rather than via rigorous mathematical definitions and proofs.
Highlights
The Theory of Functional Connections (TFC) is an analytical framework developed to perform functional interpolation, that is, to derive analytical functionals, called constrained expressions, describing all functions satisfying a set of assigned constraints
This paper provides a mathematical methodology to perform functional interpolation for vector’s components that are subject to a set of linear constraints
The methodology adopts the framework of the Theory of Functional Connections, a mathematical method to derive functionals that are always satisfying a set of linear constraints
Summary
The Theory of Functional Connections (TFC) is an analytical framework developed to perform functional interpolation, that is, to derive analytical functionals, called constrained expressions, describing all functions satisfying a set of assigned constraints This framework has been developed for univariate and multivariate rectangular domains and for a wide class of constraints, including points and derivatives constraints, integral constraints, linear combination of constraints, and, partially, for component constraints. TFC applied to component constraints has been initially presented in [12] to solve firstorder ODEs. the solution provided in [12] is restricted only to the cases presented. Before presenting the univariate Theory of Functional Connections for component constraints, a brief summary of univariate TFC and a summary of the initial (and incomplete) work on component constraints presented in [12] are provided in the two sections
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