Một số bài toán điều khiển tối ưu đối với hệ phương trình navier stokes voigt

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 encour-agement 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 COMMITTALINTHEDISSERTATION ACKNOWLEDGEMENTS i ii CONTENTS iii LISTOFSYMBOLS INTRODUCTION Chapter PRELIMINARIES AND AUXILIARY RESULTS 1.1 Function spaces 7 1.1.1 Regularities of boundaries 1.1.2 Lp and Sobolev spaces 1.1.3 Solenoidal function spaces 1.1.4 Spaces of abstract functions 11 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 1.4.2 Some auxiliary results on linearized equations 1.5 Some definitions in Convex Analysis Chapter A DISTRIBUTED OPTIMAL CONTROL PROBLEM 21 22 25 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 4.1 Setting of the problem 67 67 4.2 Solvability of the 3D Navier-Stokes-Voigt equations with nonhomogeneous boundary conditions 4.3 Existence of optimal solutions 69 75 4.4 First-order and second-order necessary optimality conditions 77 4.4.1 First-order necessary optimality conditions 77 4.4.2 Second-order necessary optimality conditions 81 4.5 Second-order sufficient optimality conditions 84 CONCLUSIONANDFUTUREWORK 88 LISTOFPUBLICATIONS 89 REFERENCES 90 iv LIST OF SYMBOLS the set of real numbers R R + R n A:=B ¯ A (:; :)X kxkX X′ ′ hx ; xiX′,X X ,! Y p L (Ω) the set of positive real numbers ndimensional Euclidean vector space A is defined by B the closure of the set A scalar product in the Hilbert space X norm of x in the space X the dual space of the space X ′ ′ duality pairing between x X and x X X is imbedded in Y the space of Lebesgue measurable functions f p Ω jf(x)j dx < +1 L 0(Ω) such that R the space of functions f L (Ω) such that ∞ L (Ω) ∞ C0 (Ω) ¯ C(Ω) > >W m,p(Ω); > > > > > R Ω f(x)dx = the space of almost everywhere bounded functions on Ω the space of infinitely differentiable functions with compact support in Ω the space of continuous functions on ¯ Ω >Hm(Ω); > < m H0 (Ω); > > > > >>Hs(Ω); > > Sobolev spaces > : s H (Γ) −m H (Ω) −s H (Γ) L (Ω) m the dual space of H (Ω) 2 L (Ω) L (Ω) L (Ω) (analogously applied for all other kinds of spaces) (:; :) the scalar product in L (Ω) ((:; :)) the scalar product in H (Ω) ((:; :))1 the scalar product in H (Ω) j:j 1 k:k the norm in H (Ω) k:k1 x y the scalar product between x; y R r ( ∂x ; ∂x ; ; ∂x n ) ry ( ∂ y r r y, div y V H; V ∂y ; ∂ ∂y ; ; ∂y ∂x1 ∂x2∂xn y ∂ ∂x1 ∂y1 ∂x1 + y2 + ∂y2 n ∂ ) ∂ ∂ + + yn ∂x2 ∂xn ∂yn + + ∂xn ∂x2 ∞ fy C0 (Ω) : div y = 0g p the closures of V in L (Ω) and H 0(Ω) the space of functions f : [0; T ] ! X such ∞ that kf(t)k X dt < the space of functions f : [0; T ] ! X such L (0; T ; X); < p < L (0; T; X) R T p that kf(:)kX is almost everywhere bounded on [0; T ] p p W (0; T ; X) C([0; T]; X) fy L (0; T ; X) : yt L (0; T ; X)g fxkg sequence of vectors xk xk converges strongly to x xk converges weakly to x the normal cone of U at the point u 1,p xk ! x xk * x NU (u) TU (u) i.e a.e the space of continuous functions from [0; T ] to X the polar cone of tangents of U at u id est (that is) almost every p page 2D 3D two-dimensional three-dimensional The proof is complete 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 , of the three-dimensional NavierStokes equations for the sake of direct numerical simulations In fact, the Navier-Stokes-Voigt system belongs to the so-called -models in fluid mechanics (see e.g [38]), but it has attractive advantages over other -models in that one does not need to impose any additional artificial boundary conditions (except the Dirichlet boundary conditions) to get the global wellposedness We also refer the interested reader to [21] for some interesting applications of Navier-Stokes-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 equa-tions, see e.g [3, 18, 19, 31, 41, 42, 60, 74] In the whole space R , 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 op-timal 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 opti-mized may not be modeled by ordinary differential equations, instead partial differential equations are used For example, heat conduction, diffusion, elec-tromagnetic waves, fluid flows can be modeled by partial differential equations In particular, optimal control of partial differential equations in fluid mechan-ics 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]) 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 au-thors, 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 prob-lems 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 Navier-Stokes 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- as follows L(gβ) L(¯g) = Z T (z; y¯ + + Z z(T ); y¯(T ) yd)dt + ! T 2 jzj + ( ; y¯ yd) dt + ! khkW yT + 3(¯g; h)W 1,2(0,T ;H1/2(Γ)) 2 jz(T )j + ( (T ); y¯(T ) yT ) (0,T ;H1/2(Γ)) + Sβ; 1,2 where Z T (z; )dt + 1 Sβ = T Z T + 2 Z0 ( ; )dt + β + (z(T ); (T )) + + 2 (z; β)dt + Z T j j dt T Z T ( β; y¯ yd)dt + Z0 j (T )j 2 (z(T ); β(T )) + 1 2 ( (T ); β(T )) + 2( β(T ); y¯(T ) yT ) + j βj dt : j β(T )j From this and the first-order necessary condition we deduce that T L(gβ) L(¯g) = Z0 + 2 jzj + ( ; y¯ yd) dt ! jz(T )j + ( (T ); y¯(T ) yT ) + khkW 1,2 (0,T ;H1/2(Γ)) From (4.35), it is easy to check that Sβ ! as ! Since L(gβ) + Sβ: L(¯g) 0, we obtain Z0 T 2 jzj + ( ; y¯ yd) dt + jz(T )j + ( (T ); y¯(T ) yT ) + khkW 1,2 (0,T ;H1/2(Γ)) (4.36) 0; h A : Now, let be defined in (4.25) By taking v = (t) in (4.25) and then integrating from to T we obtain ZT ZT ( ; y¯ yd)dt + 2( (T ); y¯(T ) yT ) = b(z; z; w)dt: This together with (4.36) imply (4.32) The proof is complete 83 4.5 Second-order sufficient optimality conditions A sufficient condition for a control to be an optimal solution is given in the following theorem (condition (4.37)) Moreover, we can prove that (4.37) even implies a W 1,2 -growth in a W 1,2 -neighborhood around the optimal solution (see (4.38)) Theorem 4.5.1 Assume that g¯ Ad Denote by y¯ the state associated to g¯ and by w the unique weak solution of system (4.27) Let h be an arbitrary function in A0 1,2 and z be the unique function in the space W (0; T ; H (Ω)) such that (z; h) satisfies equations (4.28) If g¯ satisfies the first-order necessary condition and the following assumption, in the sequel called the second-order su fficient condition: Z T q(h) := 2 jzj dt + 2jz(T )j + 3khkW Z T 1,2 (0,T ;H1/2(Γ)) b(z; z; w)dt > for every h A0nf0g; (4.37) then there exist " > and > such that L(g) L(¯g) "kg holds for all g Ad with kg g¯kW (4.38) (0,T ;H1/2(Γ)) 1,2 g¯kW 1,2(0,T ;H1/2(Γ)) In particular, this implies that g¯ is a locally optimal control Proof Let us suppose that the first-order necessary and the second-order suf-ficient 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) + k and kgk kgk g¯kW 1,2 (4.39) (0,T ;H1/2(Γ)); g¯kW 1,2(0,T ;H1/2(Γ)) < 1=k Hence, we can write gk = g¯ + khk, where in R, hk A0 and khkkW 1,2(0,T ;H1/2(Γ)) = Let zk be the unique function in the 1,2 space W (0; T ; H (Ω)) such that (zk; hk) satisfies equations (4.28) Let k W k!0 1,2 (0; T ; V ) be the unique weak solution to system (4.33) with the right-hand side of the first equation being 2B(zk; zk) Let k W 84 1,2 (0; T ; V ) be the unique weak solution of the following system 8kt + A k + 2A kt + B( k; y¯) + B(¯y; k) + kB(zk; k) + kB( k; zk) > + k > > k B( k; k) + > B( k; k) + k B( k; k) k k > > > > > > > > > k(t) = on Γ; for a.e t > [0; T ]; > > > > >k(0) > = 0: > > > > > : Since khkkW 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 fzkg in the space W 1,2 (0; T ; H (Ω)) This implies that the sequence fB(zk; zk)g is bounded in −1 1,2 the space L (0; T ; H (Ω)) and then the sequence f kg is bounded in W (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 1,2 2 solution ( k; pk) W (0; T ; V ) L (0; T ; L 0(Ω)) By applying a similar argument as in the proof of Theorem 4.4.1 we can prove that k ! in W 1,2 (4.41) (0; T ; V ) as k ! 1: From the boundedness, we can extract a subsequence of f(zk; hk)g, denoted again 1,2 ˜ (Ω)) (0;T;H by f(zk; hk)g, which weakly converges to (˜z; h) in the space W W 1,2 1/2 (0; T ; H (Γ)) Analogously as in the proof of Theorem 4.3.3 we deduce ˜ ˜ ˜ that (˜z; h) satisfies equations (4.28) We will show that h Adnf0g and q(h) 0, which contradicts (4.37) and so we get the claim Indeed, since the space W space C([0; T ]; H check that h 1,2 (0; T ; H 1/2 (Γ)) is continuously imbedded in the 1/2 (Γ)) and compactly imbedded in C([0; T ]; L (Γ)), it is easy to ˜ ˜ A0 Now, we are going to show that h 6= By assumption, g¯ satisfies the first-order necessary condition, so we have L(gk) L(¯g) = k 2 q(hk) + k Sk; (4.42) 85 where Sk = T k Z0 T Z (zk; k)dt + (zk; k)dt + k T + 2Z 2k ( k; k)dt + T T Z 2k ( k; y¯ yd)dt + Z j kj2dt k2 2 2 j kj dt kj k(T )j (zk(T ); k(T )) + k T 2 + k(zk(T ); k(T )) + Z + ( (T ); (T )) + ( (T ); y¯(T ) y ) + k k T k k 2 kj k(T )j : From (4.41) and the boundedness of sequences fzkg; f kg, we have lim Sk = 0: k→∞ It follows from (4.39) and (4.42) that 1q(h ) + S Hence We assume that ˜ k T Z k k