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 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 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 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 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 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 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 x norm of x in the space X X X′ x′ , x the dual space of the space X X ′ ,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 such that L20 (Ω) |f (x)|p dx < +∞ the space of functions f ∈ L2 (Ω) such that Ω L∞ (Ω) 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) (., ) the scalar product in L2 (Ω) ((., )) the scalar product in H10 (Ω) ((., ))1 the scalar product in H1 (Ω) |.| the norm in L2 (Ω) the norm in H10 (Ω) 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) T f (t) p X dt for every h ∈ A0 \{0}, (4.37) then there exist ε > and ρ > such that L(g) − L(¯ g ) ≥ ε g − g¯ holds for all g ∈ Ad with g − g¯ W 1,2 (0,T ;H1/2 (Γ)) W 1,2 (0,T ;H1/2 (Γ)) (4.38) ≤ ρ In particular, this implies that g¯ is a locally optimal control Proof Let us suppose that the first-order necessary and the second-order sufficient conditions are satisfied, whereas (4.38) does not hold Then for every k ∈ Z+ , there exists a sequence of admissible controls gk ∈ Ad such that L(gk ) < L(¯ g) + and gk − g¯ W 1,2 (0,T ;H1/2 (Γ)) δk ∈ W 1,2 (0, T ; H1 (Ω)) W 1,2 (0, T ; V ) W 1,2 (0,T ;H1/2 (Γ)) , (4.39) < 1/k Hence, we can write gk = g¯ + βk hk , where βk → in R, hk ∈ A0 and hk in the space gk − g¯ k W 1,2 (0,T ;H1/2 (Γ)) = Let zk be the unique function such that (zk , hk ) satisfies equations (4.28) Let be the unique weak solution to system (4.33) with the right- hand side of the first equation being −2B(zk , zk ) Let ηk ∈ W 1,2 (0, T ; V ) be the 84 unique weak solution of the following system    ηk t + νAηk + α2 Aηk t + B(ηk , y¯) + B(¯ y , ηk ) + βk B(zk , ηk ) + βk B(ηk , zk )     βk2 βk2 βk   , η ) + + B(δ B(ηk , δk ) + βk2 B(ηk , ηk ) + grad pk = − B(zk , δk )  k k  2    βk2 βk  − B(δk , zk ) − B(δk , δk ) in H−1 (Ω), for a.e t ∈ [0, T ],    ∇ · ηk (t) = in Ω, for a.e t ∈ [0, T ],       η (t) = on Γ, for a.e t ∈ [0, T ],   k   η (0) = (4.40) k Since hk W 1,2 (0,T ;H1/2 (Γ)) = 1, we can slightly modify the arguments used in the proof of Theorem 4.2.2 to get the boundedness of the sequence {zk } in the space W 1,2 (0, T ; H1 (Ω)) This implies that the sequence {B(zk , zk )} is bounded in the space L2 (0, T ; H−1 (Ω)) and then the sequence {δk } is bounded in W 1,2 (0, T ; V ) Analogously as in the proof of the unique existence of weak solutions to system (4.15), we obtain that for each k system (4.40) has exactly one weak solution (ηk , pk ) ∈ W 1,2 (0, T ; V ) × L2 (0, T ; L20 (Ω)) By applying a similar argument as in the proof of Theorem 4.4.1 we can prove that ηk → in W 1,2 (0, T ; V ) as k → ∞ (4.41) From the boundedness, we can extract a subsequence of {(zk , hk )}, denoted again ˜ in the space W 1,2 (0, T ; H1 (Ω)) × by {(zk , hk )}, which weakly converges to (˜ z , h) W 1,2 (0, T ; H1/2 (Γ)) Analogously as in the proof of Theorem 4.3.3 we deduce ˜ satisfies equations (4.28) We will show that h ˜ ∈ Ad \{0} and q(h) ˜ ≤ 0, that (˜ z , h) which contradicts (4.37) and so we get the claim Indeed, since the space W 1,2 (0, T ; H1/2 (Γ)) is continuously imbedded in the space C([0, T ]; H1/2 (Γ)) and compactly imbedded in C([0, T ]; L2 (Γ)), it is easy to ˜ ∈ A0 Now, we are going to show that h ˜ = By assumption, g¯ check that h satisfies the first-order necessary condition, so we have βk2 L(gk ) − L(¯ g) = q(hk ) + βk2 Sk , 85 (4.42) where γ1 Sk = βk T T (zk , δk )dt + γ1 βk 0 T γ1 (zk , ηk )dt + βk2 T T |δk |2 dt T γ1 γ1 + βk2 (δk , ηk )dt + γ1 (ηk , y¯ − yd )dt + βk2 |ηk |2 dt 2 0 γ2 γ2 + βk (zk (T ), δk (T )) + γ2 βk (zk (T ), ηk (T )) + βk |δk (T )|2 γ2 γ2 + βk (δk (T ), ηk (T )) + γ2 (ηk (T ), y¯(T ) − yT ) + βk2 |ηk (T )|2 2 From (4.41) and the boundedness of sequences {zk }, {δk }, we have lim Sk = k→∞ It follows from (4.39) and (4.42) that 1 q(hk ) + Sk < k Hence T γ3 − b(zk , zk , w)dt + 2Sk < k (4.43) ˜ = 0, then z˜ = This leads to We assume that h T b(zk , zk , w)dt → as k → ∞, by Lemma 1.3.3 We thus get from (4.43) that γ3 ≤ 0, which contradicts the ˜ = It remains to prove that q(h) ˜ ≤ early assumptions Therefore, h Indeed, the space W 1,2 (0, T ; H1 (Ω)) is compactly imbedded in L2 (0, T ; L2 (Ω)), so we have T T |zk |2 dt → |˜ z |2 dt From the continuity of the linear operator W 1,2 (0, T ; H1 (Ω)) it follows that zk (T ) z → z(T ) ∈ H1 (Ω), z˜(T ) in the space H1 (Ω) In addition, H1 (Ω) is compactly imbedded in L2 (Ω), so we get |zk (T )| → |˜ z (T )| By Lemma 1.3.3, T T b(zk , zk , w)dt → b(˜ z , z˜, w)dt 86 Since the unit ball is weakly compact in the space W 1,2 (0, T ; H1/2 (Γ)), we get ˜ that h W 1,2 (0,T ;H1/2 (Γ)) ≤ From what has already been proved, we conclude that ˜ ≤ lim q(hk ) ≤ q(h) k→∞ This ends the proof Conclusion of Chapter In this chapter, we have studied an optimal boundary control problem for 3D Navier-Stokes-Voigt equations, where the objective functional has a quadratic form and the control variable has to satisfy some compatibility conditions We have achieved the following results: 1) Unique solvablility of the 3D Navier-Stokes-Voigt equations with nonhomogeneous Dirichlet boundary conditions (Theorem 4.2.2); 2) Existence of globally optimal solutions (Theorem 4.3.3); 3) The first-order necessary optimality condition (Theorem 4.4.1); 4) The second-order necessary optimality condition (Theorem 4.4.3); 5) The second-order sufficient optimality condition (Theorem 4.5.1) These are the first results on the unique exsistion of solutions to the NavierStokes-Voigt equations with nonhomogeneous Dirichlet boundary conditions, as well as on boundary optimal control of Navier-Stokes-Voigt equations Moreover, we derive both necessary and sufficient conditions instead of only necessary conditions, compare to a close result on boundary optimal control for NavierStokes equations (see [34]) 87 CONCLUSION AND FUTURE WORK Conclusion In this thesis, a number of optimal control problems governed by threedimensional Navier-Stokes-Voigt equations have been investigated The main contributions of this thesis are to prove the existence of optimal solutions and to derive the optimality conditions, namely: Existence of optimal solutions, the first-order necessary optimality condition and the second-order sufficient optimality condition for a distributed optimal control problem and a time optimal control problem Existence of optimal solutions, the first-order necessary optimality condition, the second-order necessary optimality condition and the second-order sufficient optimality condition for an optimal boundary control problem The results obtained in the thesis are meaningful contributions to the theory of 3D Navier-Stokes-Voigt equations as well as optimal control of partial differential equations in fluid mechanics Future Work Some suggestions for potential future work are proposed below: Numerical approximations for the above optimal control problems (see the survey article [13] for related results on Navier-Stokes equations) Optimal control of Navier-Stokes-Voigt equations with bang-bang controls (see [14] for results on 2D Navier-Stokes equations) Optimal control of Navier-Stokes-Voigt equations with measure valued controls (see [15] for a very recent result in this direction) 88 LIST OF PUBLICATIONS Published papers [CT1] C.T Anh and T.M Nguyet, Optimal control of the instationary three dimensional Navier-Stokes-Voigt equations, Numer Funct Anal Optim 37 (2016), 415–439 (SCIE) [CT2] C.T Anh and T.M Nguyet, Time optimal control of the unsteady 3D Navier-Stokes-Voigt equations, Appl Math Optim 79 (2019), 397–426 (SCI) Submitted papers [CT3] C.T Anh and T.M Nguyet, Optimal boundary control of the 3D Navier-Stokes-Voigt equations, submitted to Optimization (2019) 89 REFERENCES [1] F Abergel and R Temam (1990), On some control problems in fluid mechanics, Theoret Comput Fluid Dynam 1, 303–325 [2] R.A Adams (1975), Sobolev Spaces, Academic Press, New York San Francisco London [3] C.T Anh and P.T Trang (2013), Pull-back attractors for three-dimensional Navier-Stokes-Voigt equations in some unbounded domains, Proc Royal Soc Edinburgh Sect A 143, 223–251 [4] C.T Anh and P.T Trang (2016), Decay rate of solutions to the 3D NavierStokes-Voigt equations in H 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(2013), Time optimal controls of the FitzHughNagumo equation with internal control, J Dyn Control Syst 19, 483–501 96 ...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... 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 CONTENTS i ... co-authors when put into the dissertation December 10, 2019 Author Tran Minh Nguyet i ACKNOWLEDGEMENTS This dissertation was carried out at the Department of Mathematics and Informatics, Hanoi