Global ISS for the Viscous Burgers' Equation With Dirichlet Boundary Disturbances

• 成果类型: Article
• 署名作者: Zheng, Jun; Zhu, Guchuan
• 署名单位: Southwest Jiaotong University; Universite de Montreal; Polytechnique Montreal
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2024.3393850
• 发表日期: 2024
• 页码: 7174-7181
• 关键词: Approximation of Lyapunov functional; Burgers' equation; Dirichlet boundary disturbance; input-to-state stability (ISS); Lyapunov method; Stampacchia's truncation

摘要

In this note, we study the global input-to-state stability (ISS) of the viscous Burgers' equation with respect to Dirichlet boundary disturbances. More specifically, under the assumption that the reaction coefficient is strictly less than pi(2), we establish first the global ISS in the spatial L-infinity-norm for the viscous Burgers' equation with Dirichlet boundary disturbances. Then, under the assumption that the reaction coefficient is nonpositive (and respectively strictly negative), we establish the global ISS in the spatial L-q-norm for the viscous Burgers' equation whenever q is an element of [2, +infinity) (and, respectively, q is an element of [1, +infinity)). The obtained results provide a positive answer to the open question raised by Mironchenko and Prieur (2019), regarding the global ISS of the viscous Burgers' equation with Dirichlet boundary disturbances. The main technique employed in this note is the so-called Stampacchia's truncation, as well as approximations of Lyapunov functionals while establishing the ISS in the spatial L-q-norm with q is an element of [1, 2), which can be seen as a generalized Lyapunov method. Numerical simulations are conducted to illustrate the obtained results.