Observability inequalities on measurable sets for the stokes system and applications

Abstract

In this paper, we establish spectral inequalities on measurable sets of positiveLebesgue measure for the Stokes operator, as well as observability inequalities on space-time mea-surable sets of positive measure for nonstationary Stokes system. The latter extends the resultestablished recently by Wang and Zhang [SIAM J. Control Optim.,55 (2017), pp. 1862--1886] to thecase 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 controlproblem for the Stokes system. In particular, we give a positive answer to an open question raisedby Privat, Tr\'elat, and Zuazua [Arch. Rational Mech. Anal.,216 (2015), pp. 921--981].