A numerical algorithm for singular optimal LQ control systems

TL;DR

This paper introduces a numerical algorithm based on presymplectic constraint algorithm and singular value decomposition to solve singular linear-quadratic optimal control problems, enabling the analysis of complex systems with singular arcs.

Contribution

It presents a novel numerical method combining geometric and algebraic techniques to handle singular LQ control systems with high index and stability.

Findings

Algorithm successfully solves large singular LQ systems.

Stable behavior observed in high-index examples.

Provides a semi-explicit system construction at each step.

Abstract

A numerical algorithm to obtain the consistent conditions satisfied by singular arcs for singular linear-quadratic optimal control problems is presented. The algorithm is based on the presymplectic constraint algorithm (PCA) by Gotay-Nester \cite{Go78,Vo99} that allows to solve presymplectic hamiltonian systems and that provides a geometrical framework to the Dirac-Bergmann theory of constraints for singular Lagrangian systems \cite{Di49}. The numerical implementation of the algorithm is based on the singular value decomposition that, on each step allows to construct a semi-explicit system. Several examples and experiments are discussed, among them a family of arbitrary large singular LQ systems with index 3 and a family of examples of arbitrary large index, all of them exhibiting stable behaviour.