Gauss-Newton oriented greedy algorithms for the reconstruction of operators in nonlinear dynamics

TL;DR

This paper develops and analyzes greedy algorithms based on Gauss-Newton methods for reconstructing unknown operators in nonlinear dynamical systems, extending previous linear results to nonlinear cases with convergence guarantees.

Contribution

Introduces a new greedy algorithm based on linearized systems that ensures local convergence of Gauss-Newton methods for nonlinear operator reconstruction.

Findings

Controls lead to local convergence of Gauss-Newton method

Extension of convergence results to nonlinear systems

Applicable to drift operators in linear and bilinear control systems

Abstract

This paper is devoted to the development and convergence analysis of greedy reconstruction algorithms based on the strategy presented in [Y. Maday and J. Salomon, Joint Proceedings of the 48th IEEE Conference on Decision and Control and the 28th Chinese Control Conference, 2009, pp. 375--379]. These procedures allow the design of a sequence of control functions that ease the identification of unknown operators in nonlinear dynamical systems. The original strategy of greedy reconstruction algorithms is based on an offline/online decomposition of the reconstruction process and an ansatz for the unknown operator obtained by an a priori chosen set of linearly independent matrices. In the previous work [S. Buchwald, G. Ciaramella and J. Salomon, SIAM J. Control Optim., 59(6), pp. 4511-4537], convergence results were obtained in the case of linear identification problems. We tackle here the more general case of nonlinear systems. More precisely, we introduce a new greedy algorithm based on the linearized system. Then, we show that the controls obtained with this new algorithm lead to the local convergence of the classical Gauss-Newton method applied to the online nonlinear identification problem. We then extend this result to the controls obtained on nonlinear systems where a local convergence result is also proved. The main convergence results are obtained for the reconstruction of drift operators in dynamical systems with linear and bilinear control structures.