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Observability Inequalities on Measurable Sets for the Stokes System and Applications
SIAM Journal on Control and Optimization ( IF 2.2 ) Pub Date : 2020-07-29 , DOI: 10.1137/18m117652x Felipe W. Chaves-Silva , Diego A. Souza , Can Zhang
SIAM Journal on Control and Optimization ( IF 2.2 ) Pub Date : 2020-07-29 , DOI: 10.1137/18m117652x Felipe W. Chaves-Silva , Diego A. Souza , Can Zhang
SIAM Journal on Control and Optimization, Volume 58, Issue 4, Page 2188-2205, January 2020.
In this paper, we establish spectral inequalities on measurable sets of positive Lebesgue measure for the Stokes operator, as well as observability inequalities on space-time measurable sets of positive measure for nonstationary Stokes system. The latter extends the result established recently by Wang and Zhang [SIAM J. Control Optim., 55 (2017), pp. 1862--1886] to the case of observations from subsets of positive measure in both time and space variables. Furthermore, we present their applications in the shape optimization problem, as well as the time optimal control problem for the Stokes system. In particular, we give a positive answer to an open question raised by Privat, Trélat, and Zuazua [Arch. Rational Mech. Anal., 216 (2015), pp. 921--981].
中文翻译:
斯托克斯系统及其应用的可测集上的可观测性不等式
SIAM控制与优化杂志,第58卷,第4期,第2188-2205页,2020年1月。
在本文中,我们为Stokes算子建立了可测量的正Lebesgue度量集上的谱不等式,并为非平稳Stokes系统建立了时空可测量的正度量集上的可观不等式。后者将Wang和Zhang [SIAM J. Control Optim。,55(2017),pp。1862--1886]最近建立的结果扩展到从时空变量中的正度量子集进行观测的情况。此外,我们介绍了它们在形状优化问题以及Stokes系统的时间最优控制问题中的应用。特别是,我们对Privat,Trélat和Zuazua [Arch。理性机械。Anal。,216(2015),921--981页]。
更新日期:2020-07-29
In this paper, we establish spectral inequalities on measurable sets of positive Lebesgue measure for the Stokes operator, as well as observability inequalities on space-time measurable sets of positive measure for nonstationary Stokes system. The latter extends the result established recently by Wang and Zhang [SIAM J. Control Optim., 55 (2017), pp. 1862--1886] to the case of observations from subsets of positive measure in both time and space variables. Furthermore, we present their applications in the shape optimization problem, as well as the time optimal control problem for the Stokes system. In particular, we give a positive answer to an open question raised by Privat, Trélat, and Zuazua [Arch. Rational Mech. Anal., 216 (2015), pp. 921--981].
中文翻译:
斯托克斯系统及其应用的可测集上的可观测性不等式
SIAM控制与优化杂志,第58卷,第4期,第2188-2205页,2020年1月。
在本文中,我们为Stokes算子建立了可测量的正Lebesgue度量集上的谱不等式,并为非平稳Stokes系统建立了时空可测量的正度量集上的可观不等式。后者将Wang和Zhang [SIAM J. Control Optim。,55(2017),pp。1862--1886]最近建立的结果扩展到从时空变量中的正度量子集进行观测的情况。此外,我们介绍了它们在形状优化问题以及Stokes系统的时间最优控制问题中的应用。特别是,我们对Privat,Trélat和Zuazua [Arch。理性机械。Anal。,216(2015),921--981页]。