Schwarz Inequality Method for Weighted Minimum Effort Terminal Control of Multidimensional Systems

Abstract

This study presents major developments in relation to problem dimensionality and formulation generality, endowing the Schwarz-inequality-based approach for finite-horizon minimum effort control with an extended degree of applicability. The extension of the Schwarz inequality method is mainly based on considering the static Lagrange multiplier that combines multiple terminal constraint relations as a vector. The minimum effort cost including a positive definite weighting function is then shown to be closely related to an associated Rayleigh quotient of the Lagrange multiplier vector. Investigation on the Rayleigh quotient form results in a simple expression for the optimal Lagrange multiplier vector, and thereby gives the closed-loop optimal control law in a time-varying state feedback form with a feedforward term. The proposed vector-based formalism provides as a direct consequence a unified means of design that encompasses several existing results developed for low-dimensional problems based on scalar-based formalism and also enables potential new developments in optimal trajectory shaping guidance.