MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION TRAN MINH NGUYET SOME OPTIMAL CONTROL PROBLEMS FOR NAVIER STOKES VOIGT EQUATIONS (MỘT SỐ BÀI TOÁN ĐIỀU KHIỂN TỐI ƯU ĐỐI VỚI HỆ[.]
MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION TRAN MINH NGUYET SOME OPTIMAL CONTROL PROBLEMS FOR NAVIER-STOKES-VOIGT EQUATIONS (MỘT SỐ BÀI TOÁN ĐIỀU KHIỂN TỐI ƯU ĐỐI VỚI HỆ PHƯƠNG TRÌNH NAVIER-STOKES-VOIGT) DOCTORAL DISSERTATION OF MATHEMATICS Hanoi - 2019 luan an MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION TRAN MINH NGUYET SOME OPTIMAL CONTROL PROBLEMS FOR NAVIER-STOKES-VOIGT EQUATIONS Speciality: Differential and Integral Equations Speciality Code: 9.46.01.03 DOCTORAL DISSERTATION OF MATHEMATICS Supervisor: PROF.DR CUNG THE ANH Hanoi - 2019 luan an COMMITTAL IN THE DISSERTATION I assure that my scientific results are new and original To my knowledge, before I published these results, there had been no such results in any scientific document I take responsibility for my research results in the dissertation The publications in common with other authors have been agreed by the co-authors when put into the dissertation December 10, 2019 Author Tran Minh Nguyet i luan an ACKNOWLEDGEMENTS This dissertation was carried out at the Department of Mathematics and Informatics, Hanoi National University of Education It was completed under the supervision of Prof.Dr Cung The Anh First and foremost, I would like to express my deep gratefulness to Prof.Dr Cung The Anh for his careful, patient and effective supervision I am very lucky to have a chance to study with him He is an excellent researcher I would like to thank Assoc.Prof.Dr Tran Dinh Ke for his help during the time I studied at Department of Mathematics and Informatics, Hanoi National University of Education I would also like to thank all the lecturers and PhD students at the seminar of Division of Mathematical Analysis for their encouragement and valuable comments A very special gratitude goes to Thang Long University for providing me the funding during the time I studied in the doctoral program Many thanks are also due to my colleagues at Division of Mathematics, Thang Long University, who always encourage me to overcome difficulties during my period of study Last but not least, I am grateful to my parents, my husband, my brother, and my beloved daughters for their love and support Hanoi, December 10, 2019 Tran Minh Nguyet ii luan an CONTENTS i ii iii COMMITTAL IN THE DISSERTATION ACKNOWLEDGEMENTS CONTENTS LIST OF SYMBOLS INTRODUCTION Chapter 1.1 PRELIMINARIES AND AUXILIARY RESULTS Function spaces 1.1.1 Regularities of boundaries 1.1.2 Lp and Sobolev spaces 1.1.3 Solenoidal function spaces 11 1.1.4 Spaces of abstract functions 12 1.1.5 Some useful inequalities 13 1.2 Continuous and compact imbeddings 14 1.3 Operators 16 1.4 The nonstationary 3D Navier-Stokes-Voigt equations 20 1.4.1 Solvability of the 3D Navier-Stokes-Voigt equations with homogeneous boundary conditions 21 1.4.2 1.5 Some auxiliary results on linearized equations 22 Some definitions in Convex Analysis 25 Chapter A DISTRIBUTED OPTIMAL CONTROL PROBLEM 26 2.1 Setting of the problem 26 2.2 Existence of optimal solutions 28 2.3 First-order necessary optimality conditions 32 2.4 Second-order sufficient optimality conditions 41 Chapter A TIME OPTIMAL CONTROL PROBLEM 47 3.1 Setting of the problem 47 3.2 Existence of optimal solutions 49 3.3 First-order necessary optimality conditions 52 3.4 Second-order sufficient optimality conditions 59 iii luan an Chapter AN OPTIMAL BOUNDARY CONTROL PROBLEM 67 4.1 Setting of the problem 67 4.2 Solvability of the 3D Navier-Stokes-Voigt equations with nonhomogeneous boundary conditions 69 4.3 Existence of optimal solutions 75 4.4 First-order and second-order necessary optimality conditions 77 4.5 4.4.1 First-order necessary optimality conditions 77 4.4.2 Second-order necessary optimality conditions 81 Second-order sufficient optimality conditions 84 CONCLUSION AND FUTURE WORK LIST OF PUBLICATIONS 88 89 REFERENCES 90 iv luan an LIST OF SYMBOLS R the set of real numbers R+ the set of positive real numbers Rn n-dimensional Euclidean vector space A := B A¯ A is defined by B the closure of the set A (., )X scalar product in the Hilbert space X kxkX norm of x in the space X X′ the dual space of the space X hx′ , xiX ′ ,X duality pairing between x′ ∈ X ′ and x ∈ X X ,→ Y X is imbedded in Y Lp (Ω) the space of Lebesgue measurable functions f R such that Ω |f (x)|p dx < +∞ L20 (Ω) the space of functions f ∈ L2 (Ω) such that R Ω L∞ (Ω) f (x)dx = the space of almost everywhere bounded functions on Ω C0∞ (Ω) the space of infinitely differentiable functions with compact support in Ω ¯ C(Ω) W m,p (Ω), H m (Ω), H m (Ω), H s (Ω), H s (Γ) ¯ the space of continuous functions on Ω Sobolev spaces H −m (Ω) the dual space of H0m (Ω) H −s (Γ) the dual space of H s (Γ) L2 (Ω) L2 (Ω) × L2 (Ω) × L2 (Ω) (analogously applied for all other kinds of spaces) luan an (., ) the scalar product in L2 (Ω) ((., )) the scalar product in H10 (Ω) ((., ))1 the scalar product in H1 (Ω) |.| the norm in L2 (Ω) k.k the norm in H10 (Ω) k.k1 the norm in H1 (Ω) x·y the scalar product between x, y ∈ Rn ∇ ( ∂x∂ , ∂x∂ , · · · , ∂x∂ n ) ∇y ∂y ∂y ∂y ( ∂x , , · · · , ∂x ) ∂x2 n y·∇ y1 ∂x∂ + y2 ∂x∂ + · · · + yn ∂x∂ n ∇ · y , div y ∂y1 ∂x1 V {y ∈ H, V the closures of V in L2 (Ω) and H10 (Ω) ∂y2 ∂yn ∂x2 + · · · + ∂xn C∞ (Ω) : div y = + 0} Lp (0, T ; X), < p < ∞ the space of functions f : [0, T ] → X such that L∞ (0, T ; X) RT kf (t)kpX dt < ∞ the space of functions f : [0, T ] → X such that kf (.)kX is almost everywhere bounded on [0, T ] W 1,p (0, T ; X) {y ∈ Lp (0, T ; X) : yt ∈ Lp (0, T ; X)} C([0, T ]; X) the space of continuous functions from [0, T ] to X {xk } sequence of vectors xk xk → x xk converges strongly to x xk * x xk converges weakly to x NU (u) the normal cone of U at the point u TU (u) the polar cone of tangents of U at u i.e id est (that is) a.e almost every p page 2D two-dimensional 3D three-dimensional The proof is complete luan an INTRODUCTION Literature survey and motivation The Navier-Stokes-Voigt (sometimes written Voight) equations was first introduced by Oskolkov in [57] as a model of motion of certain linear viscoelastic incompressible fluids This system was also proposed by Cao, Lunasin and Titi in [12] as a regularization, for small values of α, of the three-dimensional Navier-Stokes equations for the sake of direct numerical simulations In fact, the Navier-Stokes-Voigt system belongs to the so-called α-models in fluid mechanics (see e.g [38]), but it has attractive advantages over other α-models in that one does not need to impose any additional artificial boundary conditions (except the Dirichlet boundary conditions) to get the global well-posedness We also refer the interested reader to [21] for some interesting applications of NavierStokes-Voigt equations in image inpainting In the past years, the existence and long-time behavior of solutions to the Navier-Stokes-Voigt equations has attracted the attention of many mathematicians In bounded domains or unbounded domains satisfying the Poincaré inequality, there are many results on the existence and long-time behavior of solutions in terms of existence of attractors for the Navier-Stokes-Voigt equations, see e.g [3, 18, 19, 31, 41, 42, 60, 74] In the whole space R3 , the existence and decay rates of solutions have been studied recently in [4, 56, 75] The optimal control theory has been developed rapidly in the past few decades and becomes an important and separate field of applied mathematics The optimal control of ordinary differential equations is of interest for its applications in many fileds such as aviation and space technology, robotics and the control of chemical processes However, in many situations, the processes to be optimized may not be modeled by ordinary differential equations, instead partial differential equations are used For example, heat conduction, diffusion, electromagnetic waves, fluid flows can be modeled by partial differential equations In particular, optimal control of partial differential equations in fluid mechanics was first studied in 1980s by Fursikov when he established several theorems about the existence of solutions to some optimal control problems governed by Navier-Stokes equations (see [25, 26, 27]) luan an One of the most important objectives of optimal control theory is to obtain necessary (or possibly necessary and sufficient) conditions for the control to be an extremum Since the pioneering work [1] of Abergel and Temam in 1990, where the first optimality conditions to the optimal control problem for fluid flows can be found, this matter has been studied very intensively by many authors, and in various research directions such as distributed optimal control, time optimal control, boundary optimal control and sparse optimal control Let us briefly review some results on optimality conditions of optimal control problems governed by Navier-Stokes equations that is one of the most important equations in fluid mechanics For distributed control problems, this matter was studied in [23, 33, 36, 68] These works are all in the case of absence of state constraints In the case of the presence of state constraints, the problem was investigated by Wang [71] and Liu [52] The time optimal control problem of Navier-Stokes equations was investigated by Barbu in [7] and Fernandez-Cara in [24] Optimal boundary control problems of the Navier-Stokes equations have been studied by many authors, see for instance, [32, 39, 40, 61] in the stationary case, and [10, 17, 28, 29, 34, 37] in the nonstationary case One interesting result about Pontryagin’s principle for optimal control problem governed by 3D NavierStokes equations is introduced by B.T Kien, A Rösch and D Wachsmuth in [43] We can see also the habilitation [35], the theses [69], [63] and references therein, for other works on optimal control of Navier-Stokes equations As described above, the unique existence and long-time behavior of solutions to the Navier-Stokes-Voigt equations, as well as the optimal control problems for fluid flows, in particular for Navier-Stokes equations, have been considered by many mathematicians However, to the best of our knowledge, the optimal control of three-dimensional Navier-Stokes-Voigt equations has not been studied before This is our motivation to choose the topic ”Some optimal control problems for Navier-Stokes-Voigt equations” Because of the physical and practical significance, one only considers Navier-Stokes-Voigt equations in the case of three or two dimensions The thesis presents results on some optimal control problems for this equations in the three-dimensional space (the most physically meaningful case) However, all results of the thesis are still true in the two-dimensional one (with very similar statements of results and corresponding proofs) Namely, we will study the following problems: (P1) The distributed optimal control problem of the nonstationary three di- luan an ... 16 1.4 The nonstationary 3D Navier- Stokes- Voigt equations 20 1.4.1 Solvability of the 3D Navier- Stokes- Voigt equations with homogeneous boundary conditions ... interesting applications of NavierStokes -Voigt equations in image inpainting In the past years, the existence and long-time behavior of solutions to the Navier- Stokes- Voigt equations has attracted... topic ”Some optimal control problems for Navier- Stokes- Voigt equations” Because of the physical and practical significance, one only considers Navier- Stokes- Voigt equations in the case of three or