Parameter estimation and control by penalized multiple shooting

Multiple shooting methods have been studied and applied to parameter estimation and optimal control problems, governed by ODEs during the past decades, and also to problems governed by PDEs less frequently. In this work we explore numerically a multiple shooting method via augmented Lagrangian and penalized models, to estimate parameter values of systems modelled by ODEs. Unlike most authors, who prefer algorithms like Gauss- Newton or SQP to solve the associated optimization problems, we apply the BFGS algorithm with inexact line-search, and a variant of a dual ascent method, along with the adjoint equation method to compute derivatives or gradients. The proposed method, with the mentioned ingredients, is simple and efficient. It estimates accurately parameters of the Lorentz equations in chaotic regime. The same multiple shooting approach can also be applied to optimal control problems, particularly to simultaneously control the transition between equilibrium states and the stabilization around an unstable equilibria of a model that describes the dynamics of a Josephson Junction Array, a quantum interference device used in superconductivity.

Keywords

multiple shooting, parameter estimation, optimal control, augmented Lagrangian, quasi-Newton method

2010 Mathematics Subject Classification

Primary 65K10, 65L09, 90C26. Secondary 34A55, 34H05, 49M15, 49M37.

Full Text (PDF format)

The first-named author was supported by UAM-I and Conacyt-México.

The second-named author was supported by UJAT.

The third-named author was supported by UAM-I and Conacyt-México.

Received 27 July 2023

Accepted 11 March 2024

Published 5 April 2024