Chaves-Silva, Felipe W.; Souza, Diego A.; Zhang, Can Observability inequalities on measurable sets for the Stokes system and applications. (English) Zbl 1446.49029 SIAM J. Control Optim. 58, No. 4, 2188-2205 (2020). Summary: 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 G. Wang and C. Zhang [SIAM J. Control Optim. 55, No. 3, 1862–1886 (2017; Zbl 1365.93058)] 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 Y. Privat et al. [Arch. Ration. Mech. Anal. 216, No. 3, 921–981 (2015; Zbl 1319.35272)]. Cited in 1 Document MSC: 49Q10 Optimization of shapes other than minimal surfaces 76D55 Flow control and optimization for incompressible viscous fluids 93B07 Observability 49R05 Variational methods for eigenvalues of operators 28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets 28A75 Length, area, volume, other geometric measure theory Keywords:spectral inequality; observability inequality; Stokes system; shape optimization; time optimal control Citations:Zbl 1365.93058; Zbl 1319.35272 PDFBibTeX XMLCite \textit{F. W. Chaves-Silva} et al., SIAM J. Control Optim. 58, No. 4, 2188--2205 (2020; Zbl 1446.49029) Full Text: DOI arXiv References: [1] J. Apraiz and L. Escauriaza, Null-control and measurable sets. ESAIM: COCV, 19 (2013), 239-254. · Zbl 1262.35118 [2] J. Apraiz, L. Escauriaza, G. Wang, and C. Zhang, Observability inequalities and measurable sets, J. Eur. Math. Soc., 16 (2014), pp. 2433-2475. · Zbl 1302.93040 [3] O. Bodart, M. González-Burgos, and R. Pérez-García, Existence of insensitizing controls for a semilinear heat equation with a superlinear nonlinearity, Comm. Partial Differential Equations, 29 (2004), pp. 1017-1050. · Zbl 1067.93035 [4] N. Burq and N. Tzvetkov, Random data Cauchy theory for supercritical wave equations. I. Local theory, Invent. Math., 173 (2008), pp. 449-475. · Zbl 1156.35062 [5] F. W. Chaves-Silva and G. Lebeau, Spectral inequality and optimal cost of controllability for the Stokes system, ESAIM: COCV, 22 (2016), pp. 1137-1162. · Zbl 1357.35178 [6] J. M. Coron and S. Guerrero, Null controllability of the N-dimensional Stokes system with N-1 scalar controls, J. Differential Equations, 246 (2009), pp. 2908-2921. · Zbl 1172.35042 [7] L. Escauriaza, S. Montaner, and C. Zhang, Observation from measurable sets for parabolic analytic evolutions and applications, J. Math. Pures Appl., 104 (2015), pp. 837-867. · Zbl 1328.35070 [8] L. Escauriaza, S. Montaner, and C. Zhang, Analyticity of solutions to parabolic evolutions and applications, SIAM J. Math. Anal., 49 (2017), pp. 4064-4092. · Zbl 1377.35048 [9] H. Fattorini, Infinite Dimensional Linear Control Systems; the Time Optimal and Norm Optimal Problems, North-Holland Math. Stud. 201, Elsevier, Amsterdam, 2005. · Zbl 1135.93001 [10] E. Fernández-Cara, S. Guerrero, O. Yu. Imanuvilov, and J. P. Puel, Local exact controllability of the Navier-Stokes system, J. Math. Pures Appl., 83 (2004), pp. 1501-1542. · Zbl 1267.93020 [11] A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series 34, Research Institute of Mathematics, Seoul National University, Seoul, 1996. · Zbl 0862.49004 [12] O. Yu. Imanuvilov, J. P. Puel, and M. Yamamoto, Carleman estimates for parabolic equations with nonhomogeneous boundary conditions, Chin. Ann. Math. Ser. B, 30 (2009), pp. 333-378. · Zbl 1184.35087 [13] C. Kahane, On the spatial analyticity of solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 33 (1969), pp. 386-405. · Zbl 0186.16801 [14] K. Masuda, On the analyticity and the unique continuation theorem for solutions of the Navier-Stokes equation, Proc. Japan Acad., 43 (1967), pp. 827-832. · Zbl 0204.26901 [15] S. Micu, I. Roventa, and M. Tucsnak, Time optimal boundary controls for the heat equation, J. Funct. Anal., 263 (2012), pp. 25-49. · Zbl 1243.49005 [16] C. B. Morrey and L. Nirenberg, On the analyticity of the solutions of linear elliptic systems of partial differential equations, Commun. Pur. Appl. Math., X (1957), pp. 271-290. · Zbl 0082.09402 [17] Y. Privat, E. Trélat, and E. Zuazua, Optimal shape and location of sensors for parabolic equations with random initial data, Arch. Ration. Mech. Anal., 216 (2015), pp. 921-981. · Zbl 1319.35272 [18] K. D. Phung and G. Wang, An observability estimate for parabolic equations from a general measurable set in time and its applications, J. Eur. Math. Soc., 15 (2013), pp. 681-703. · Zbl 1258.93037 [19] L. Robbiano, Fonction de coût et contrôle des solutions des équations hyperboliques, Asympt. Analysis, 10 (1995), pp. 95-115. · Zbl 0882.35015 [20] S. Vessella, A continuous dependence result in the analytic continuation problem, Forum Math., 11 (1999), pp. 695-703. · Zbl 0933.35192 [21] G. Wang, \(L^\infty \)-Null controllability for the heat equation and its consequences for the time optimal control problem, SIAM J. Control Optim., 47 (2008), pp. 1701-1720. · Zbl 1165.93016 [22] G. Wang and C. Zhang, Observability estimate from measurable sets in time for some evolution equations, SIAM J. Control Optim., 55 (2017), pp. 1862-1886. · Zbl 1365.93058 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.