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

09/2023

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

Friday 10th February 2023

Buchwald, S.; Ciaramella, G.; Salomon, J.

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 on 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 show that the controls obtained with the greedy algorithm on the corresponding linearized system 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 obtained. The main convergence results are obtained for the reconstruction of drift operators in linear and bilinear dynamical systems.