Optimal Controls of 3-Dimensional Navier--Stokes Equations with State Constraints

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Previous article Next article Optimal Controls of 3-Dimensional Navier--Stokes Equations with State ConstraintsGengsheng WangGengsheng Wanghttps://doi.org/10.1137/S0363012901385769PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractThis work is concerned with the maximum principles for optimal control problems governed by 3-dimensional Navier--Stokes equations. Some types of state constraints (time variables) are considered.[1] V. Barbu, Optimal control of variational inequalities, Research Notes in Mathematics, Vol. 100, Pitman (Advanced Publishing Program), 1984iv+298 86a:49043 Google Scholar[2] Viorel Barbu, Analysis and control of nonlinear infinite‐dimensional systems, Mathematics in Science and Engineering, Vol. 190, Academic Press Inc., 1993x+476 93j:49002 Google Scholar[3] Viorel Barbu, Optimal control of Navier‐Stokes equations with periodic inputs, Nonlinear Anal., 31 (1998), 15–31 98j:49005 CrossrefISIGoogle Scholar[4] Viorel Barbu, The time optimal control of Navier‐Stokes equations, Systems Control Lett., 30 (1997), 93–100 10.1016/S0167-6911(96)00083-7 98e:49008 CrossrefISIGoogle Scholar[5] Google Scholar[6] V. Barbu and , S. Sritharan, Flow invariance preserving feedback controllers for the Navier‐Stokes equation, J. Math. Anal. Appl., 255 (2001), 281–307 2002c:93086 CrossrefISIGoogle Scholar[7] Peter Constantin and , Ciprian Foias, Navier‐Stokes equations, Chicago Lectures in Mathematics, University of Chicago Press, 1988x+190 90b:35190 CrossrefGoogle Scholar[8] H. Fattorini and , S. Sritharan, Necessary and sufficient conditions for optimal controls in viscous flow problems, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 211–251 95c:49033 CrossrefISIGoogle Scholar[9] H. Fattorini and , S. Sritharan, Optimal control problems with state constraints in fluid mechanics and combustion, Appl. Math. Optim., 38 (1998), 159–192 10.1007/s002459900087 99d:49039 CrossrefISIGoogle Scholar[10] Vladimir Arnold and , Boris Khesin, Topological methods in hydrodynamics, Applied Mathematical Sciences, Vol. 125, Springer‐Verlag, 1998xvi+374 99b:58002 CrossrefGoogle Scholar[11] A. V. Fursikov, Control problems and theorems concerning unique solvability of a mixed boundary value problem for three‐dimensional Navier–Stokes and Euler equations, Mat. Sb., 115 (1981), pp. 281–306, 320 (in Russian). mas MATSAB 0368-8666 Mat. Sb. Google Scholar[12] A. Fursikov, Optimal control problems for Navier‐Stokes system with distributed control function, SIAM, Philadelphia, PA, 1998, 109–150 99h:49005 Google Scholar[13] Kazufumi Ito and , Sungkwon Kang, A dissipative feedback control synthesis for systems arising in fluid dynamics, SIAM J. Control Optim., 32 (1994), 831–854 95b:93094 LinkISIGoogle Scholar[14] J.‐L. Lions, Optimal control of systems governed by partial differential equations. , Translated from the French by S. K. Mitter. Die Grundlehren der mathematischen Wissenschaften, Band 170, Springer‐Verlag, 1971xi+396 42:6395 CrossrefGoogle Scholar[15] J.‐L. Lions and , E. Magenes, Non‐homogeneous boundary value problems and applications. Vol. I, Springer‐Verlag, 1972xvi+357, Translated from the French by P. Kenneth; Die Grundlehren der mathematischen Wissenschaften, Band 181 50:2670 Google Scholar[16] Xun Li and , Jiong Yong, Optimal control theory for infinite‐dimensional systems, Systems & Control: Foundations & Applications, Birkhäuser Boston Inc., 1995xii+448 96k:49004 CrossrefGoogle Scholar[17] Xun Li and , Jiong Yong, Necessary conditions for optimal control of distributed parameter systems, SIAM J. Control Optim., 29 (1991), 895–908 92d:49037 LinkISIGoogle Scholar[18] Roger Temam, Navier‐Stokes equations, Studies in Mathematics and its Applications, Vol. 2, North‐Holland Publishing Co., 1979x+519, Theory and numerical analysis; With an appendix by F. Thomasset 82b:35133 Google Scholar[19] Gengsheng Wang, Optimal control of parabolic differential equations with two point boundary state constraints, SIAM J. Control Optim., 38 (2000), 1639–1654 2001g:49034 LinkISIGoogle Scholar[20] Gengsheng Wang and , Shirong Chen, Maximum principle for optimal control of some parabolic systems with two point boundary conditions, Numer. Funct. Anal. Optim., 20 (1999), 163–174 2000c:49042 CrossrefISIGoogle Scholar[21] G. Wang, , Y. Zhao and , W. Li, Some optimal control problems governed by elliptic variational inequalities with control and state constraint on the boundary, J. Optim. Theory Appl., 106 (2000), 627–655 10.1023/A:1004613630675 2001i:49044 CrossrefISIGoogle Scholar[22] Kôsaku Yosida, Functional analysis, Springer‐Verlag, 1978xii+501, Grundlehren der Mathematischen Wissenschaften, Band 123 58:17765 CrossrefGoogle ScholarKeywordsoptimal controlNavier--Stokes equationstate constraintmaximum principle Previous article Next article FiguresRelatedReferencesCited byDetails Optimal control of the 3D damped Navier-Stokes-Voigt equations with control constraintsEvolution Equations and Control Theory, Vol. 12, No. 1 Cross Ref Pontryagin’s Maximum Principle for Distributed Optimal Control of Two Dimensional Tidal Dynamics System with State Constraints of Integral Type30 May 2022 | Acta Applicandae Mathematicae, Vol. 179, No. 1 Cross Ref Optimal control problems governed by two dimensional convective Brinkman-Forchheimer equationsEvolution Equations and Control Theory, Vol. 11, No. 3 Cross Ref The Time Optimal Control of Two Dimensional Convective Brinkman–Forchheimer Equations11 February 2021 | Applied Mathematics & Optimization, Vol. 84, No. 3 Cross Ref Optimal control of a new mechanochemical model with state constraint31 March 2021 | Mathematical Methods in the Applied Sciences, Vol. 44, No. 11 Cross Ref First-order necessary conditions of optimality for the optimal control of two-dimensional convective Brinkman–Forchheimer equations with state constraints13 May 2021 | Optimization, Vol. 24 Cross Ref Maximum principle of optimal control of a Cahn–Hilliard–Navier–Stokes model with state constraints24 January 2021 | Optimal Control Applications and Methods, Vol. 42, No. 3 Cross Ref Time-Periodic Fitzhugh-Nagumo Equation and the Optimal Control Problems2 June 2021 | Chinese Annals of Mathematics, Series B, Vol. 42, No. 3 Cross Ref An optimal control problem of the 3D viscous Camassa–Holm equations26 November 2019 | Optimization, Vol. 70, No. 1 Cross Ref Second order optimality conditions for periodic optimal control problems governed by semilinear parabolic differential equations26 March 2021 | ESAIM: Control, Optimisation and Calculus of Variations, Vol. 27 Cross Ref Optimal Controls for the Stochastic Compressible Navier--Stokes EquationsHuaqiao Wang12 October 2021 | SIAM Journal on Control and Optimization, Vol. 59, No. 5AbstractPDF (615 KB)Feedback control for non-stationary 3D Navier–Stokes–Voigt equations12 June 2020 | Mathematics and Mechanics of Solids, Vol. 25, No. 12 Cross Ref Optimal Control of Time-Periodic Navier-Stokes-Voigt Equations3 July 2020 | Numerical Functional Analysis and Optimization, Vol. 41, No. 13 Cross Ref Optimal Control Problem for the Cahn–Hilliard/Allen–Cahn Equation with State Constraint12 December 2018 | Applied Mathematics & Optimization, Vol. 82, No. 2 Cross Ref Optimal bilinear control problem related to a chemo-repulsion system in 2D domains24 March 2020 | ESAIM: Control, Optimisation and Calculus of Variations, Vol. 26 Cross Ref No-gap optimality conditions for an optimal control problem with pointwise control-state constraints20 December 2017 | Applicable Analysis, Vol. 98, No. 6 Cross Ref Existence of Optimal Controls for Compressible Viscous Flow16 March 2017 | Journal of Mathematical Fluid Mechanics, Vol. 20, No. 1 Cross Ref Bibliography6 October 2017 Cross Ref Pontryagin’s Principle for Optimal Control Problem Governed by 3D Navier–Stokes Equations17 February 2017 | Journal of Optimization Theory and Applications, Vol. 173, No. 1 Cross Ref First Order Sufficient Optimality Conditions for Navier--Stokes Flow. 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Industrial and Applied MathematicsKeywordsoptimal controlNavier--Stokes equationstate constraintmaximum principleMSC codes93C0593B5093C35PDF Download Article & Publication DataArticle DOI:10.1137/S0363012901385769Article page range:pp. 583-606ISSN (print):0363-0129ISSN (online):1095-7138Publisher:Society for Industrial and Applied Mathematics