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.

翻译：本文致力于基于文献[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]所提策略的贪婪重构算法的开发与收敛性分析。这些方法能够设计控制函数序列，从而简化非线性动力系统中未知算子的识别。原始贪婪重构算法策略基于重构过程的离线/在线分解，并通过先验选择的线性独立矩阵集获得未知算子的ansatz。在先前工作[S. Buchwald, G. Ciaramella and J. Salomon, SIAM J. Control Optim., 59(6), pp. 4511-4537]中，线性识别问题已获得收敛性结果。本文处理更具一般性的非线性系统情况。具体而言，我们基于线性化系统引入一种新型贪婪算法，随后证明该算法获得的控制量能够使经典高斯-牛顿法在在线非线性识别问题中实现局部收敛。进一步将这一结果拓展至非线性系统获得的控制量，并同样证明其局部收敛性。主要收敛性结果针对具有线性和双线性控制结构的动力系统中漂移算子的重构问题。