Initial Data Identification for the One-Dimensional Burgers Equation

• 成果类型: Article
• 署名作者: Liard, Thibault; Zuazua, Enrique
• 署名单位: Nantes Universite; Ecole Centrale de Nantes; Centre National de la Recherche Scientifique (CNRS); CNRS - Institute for Information Sciences & Technologies (INS2I); University of Erlangen Nuremberg; Autonomous University of Madrid; University of Deusto
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2021.3096921
• 发表日期: 2022
• 页码: 3098-3104
• 关键词: entropy; minimization; Mathematical model; Airplanes; Target tracking; shape; optimal control

摘要

In this article, we study the problem of identification for the 1-D Burgers equation. This problem consists in identifying the set of initial data evolving to a given target at a final time. Due to the property of nonbackward uniqueness of the Burgers equation, there may exist multiple initial data leading to the same given target. In articles Initial data identification in conservation laws and Hamilton-Jacobi equations (arXiv:1903.06448, 2019) and The inverse problem for Hamilton-Jacobi equations and semiconcave envelopes (SIAM Journal on Mathematical Analysis, vol. 52, the authors fully characterize the set of initial data leading to a given target using the classical Lax-Hopf formula. In this article, an alternative proof based only on generalized backward characteristics is given. This leads to the hope of investigating systems of conservation laws in 1-D, where the classical Lax-Hopf formula no more holds. Moreover, numerical illustrations are presented using as a target, a function optimized for minimum pressure rise in the context of sonic-boom minimization problems. All of initial data leading to this given target are constructed using a wavefront tracking algorithm.