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 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 COMMITTAL IN THE DISSERTATION i ACKNOWLEDGEMENTS .ii CONTENTS iii 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 1.5 1.4.1 Solvability of the 3D Navier-Stokes-Voigt equations with homogeneous boundary conditions 21 1.4.2 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 88 LIST OF PUBLICATIONS 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 (., )X scalar product in the Hilbert space X ǁxǁX norm of x in the space X 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 such that Ω |f (x)| dx < +∞ ⟨ x′, x⟩ X′,X X ‹→ Y Lp(Ω) the closure of the set A p L20(Ω) the ∫ space of functions f ∈ L2(Ω) such that L∞(Ω) the space of almost everywhere bounded functions on Ω C0∞(Ω) the space of infinitely differentiable functions with compact support in Ω ¯) C(Ω ¯ the space of continuous functions on Ω Ω f (x)dx =0 Wm,p(Ω), Hm(Ω), Hm (Ω), Sobolev spaces Hs(Ω), H s (Γ) H−m(Ω) the dual space of Hm(Ω) H−s(Γ) the dual space of Hs(Γ) L2(Ω) × L2(Ω) × L2(Ω) (analogously applied for all other kinds of spaces) L2(Ω) (., ) ((., )) ((., ))1 |.| the scalar product in L2(Ω) the scalar product in H01(Ω) the scalar product in H1(Ω) the norm in L2(Ω) ǁ.ǁ1 the norm in H01(Ω) the norm in H1(Ω) x·y the scalar product between x, y ∈ Rn ∇ (∂x1 , ∂x2 , · · · , ∇y (∂x1 , ∂x2 , · · · ǁ.ǁ ∂ ∂y ∂ ∂y , ∂ ∂xn ) ∂y ∂xn ) y·∇ y1 ∂x1 + y2 ∇ · y, div y ∂x1 V {y ∈ H, V Lp(0, T ; X), < p < ∞ the closures of V in L2(Ω) and H10(Ω) the space of functions f : [0, T ] → X such ∫T that ǁf (t)ǁpX dt < ∞ L∞(0, T ; X) the space of functions f : [0, T ] → X such that ǁf (.)ǁX 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 ∂ ∂y1 + ∂x2 + · · · + yn ∂xn ∂ ∂ ∂x2 + · · · + ∂xn C∞ (Ω) : div y = ∂y2 ∂yn 0} [0, T ] to X {xk} xk → x sequence of vectors xk xk converges strongly to x xk ~ x xk converges weakly to x NU (u) i.e a.e the normal cone of U at the point u the polar cone of tangents of U at u id est (that is) almost every p 2D 3D page two-dimensional three-dimensional Q The proof is complete TU (u) 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]) 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- following system δt + νAδ + α2 Aδt + B(y¯, δ) + B(δ, y¯) + grad p2 = −2B(z, z) in H−1(Ω), for a.e t ∈ [0, T ], ∇ · δ = in Ω, for a.e t ∈ [0, T ], (4.33) δ(t) = on Γ, for a.e t ∈ [0, T ], δ(0) = 0, and ηβ is a weak solution of the following system ηβ t + νAηβ + α2 Aηβ t + B(ηβ , y¯) + B(y¯, ηβ ) + βB(z, ηβ ) + βB(ηβ , z) 2 β β + B(δ, ηβ) + H B(ηβ, δ) + 2β B(ηβ, ηβ) + grad p3 β2 z) − β B(δ, δ) in −1(Ω), for a.e t ∈ [0, T ], = − βB(z, δ) − B(δ, ∇ · η2β(t) = in2 Ω, for a.e t4 ∈ [0, T ], (4.34) ηβ(t) = on Γ, for a.e t ∈ [0, T ], ηβ(0) = By following the lines when proving the existence of weak solutions to system (4.15), we obtain that system (4.33) possesses a unique weak solution (δ, p2) ∈ W 1,2 (0, T ; V ) × L2(0, T ;0 L2(Ω)) and that for each β ∈ R, system (4.34) has exactly one weak solution (ηβ, p3) ∈ W 1,2(0, T ; V ) × L2(0, T ; L20(Ω)) Analogously as in the proof of Theorem 4.4.1, we can check that ηβ → in W 1,2(0, T ; V ) as β → (4.35) This means there exists ”the second directional derivative of the control-to-state mapping S at g¯ in the directions h, h”, which we denote by S ′′ (g¯; h, h), in the following sense S(g¯ + βh) = S(g¯) + βS ′ (g¯; h) + β ′′ S (g¯; h, h) + o(β 2), and S ′′ (g¯; h, h) = δ After some simple computations, L(gβ ) − L(g¯) can be written 84 as follows L(gβ ) − L(g¯) ∫ =β γ1 + γ3 (z, y¯ − yd )dt + γ2 z(T ), y¯(T ) − yT γ1 + β2 ! T ∫ T ! + β 2S , ǁhǁ2 β W 1,2(0,T ;H1/2(Γ)) dt + γ2 |z(T )| + (δ(T ), y¯(T ) − yT ) |z|2 + (δ, y¯ − yd ) + γ3 (g¯, h)W 1,2 (0,T ;H1/2 (Γ)) where Sβ = γ1 ∫ ∫ T T (z, δ)dt + γ1β β γ1 ∫ ∫ ∫ T |δ|2 dt ∫ γ1 T + β (δ, ηβ)dt + γ1 (ηβ , y¯ − yd )dt + β |ηβ| 2dt 2 0 γ2 + β(z(T ), δ(T )) + γ β(z(T ), ηβ (T )) + γ2 β2|δ(T )|2 γ2 γ2 + β2(δ(T ), η (T )) + γ2 (ηβ (T ), y¯(T ) − yT ) + β2 η (T )|2 | β β 2 T γ1 (z, ηβ)dt + β2 T From this and the first-order necessary condition we deduce that L(gβ ) − L(g¯) = β + γ2 γ1 ∫ T |z|2 + (δ, y¯ − yd ) dt ! |z(T )|2 + (δ(T ), y¯(T ) − y ) + T γ3 ǁhǁ2 1,2 W (0,T ;H1/2(Γ)) + β 2S β From (4.35), it is easy to check that Sβ → as β → Since L(gβ ) − L(g¯) ≥ 0, we obtain ∫ T γ γ1 |z|2 + (δ, y¯ − yd ) |z(T )| + (δ(T ), y¯(T ) − yT ) γ3 + ǁhǁ2W 1,2(0,T ;H1/2(Γ)) ≥ 0, ∀ h ∈ A dt + (4.36) Now, let τ be defined in (4.25) By taking v = δ(t) in (4.25) and then integrating from to T we obtain ∫ T ∫ T (δ, y¯ − yd )dt + γ2(δ(T ), y¯(T ) − yT ) = −2 γ1 b(z, z, w)dt 0 This together with (4.36) imply (4.32) The proof is complete 85 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 (4.38)) 1,2 -growth in a W 1,2 -neighborhood around the optimal solution (see 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 and z be the unique function in the space W 1,2(0, T ; H1(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 sufficient condition: ∫ q(h) := γ1 T |z|2 dt + γ2|z(T )|2 + γ3ǁhǁW 1,2 (0,T ;H 1/2 (Γ)) ∫ T − b(z, z, w)dt > for every h ∈ A0\{0}, (4.37) then there exist ε > and ρ > such that L(g) − L(g¯) ≥ εǁg − g¯ǁ2 1,2 (0,T ;H 1/2 (Γ)) (4.38) W holds for all g ∈ Ad with ǁg − g¯ǁ W 1,2 (0,T ;H1/2 (Γ)) ≤ 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¯) + ǁgk — g¯ǁW 1,2 (0,T ;H1/2(Γ)) , k (4.39) and ǁgk − g¯ǁ W 1,2 (0,T ;H1/2 (Γ)) < 1/k Hence, we can write gk = g¯ + βk hk , where βk → in R, hk ∈ A0 and ǁhkǁW 1,2(0,T ;H1/2(Γ)) = Let zk be the unique function in the space W 1,2 (0, T ; H1(Ω)) such that (zk, hk) satisfies equations (4.28) Let δk ∈ W 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 ηk ∈ W 1,2(0, T ; V ) be the 86 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 ) β2k B(δk, ηk) + β2k B(ηk, δk) + β2B(ηk, ηk) + grad pk = − βk B(zk, δk) + − (4.40) k 2 βk2 −1 B(δ , δ ) in H (Ω), for a.e t ∈ [0, T ], k k B(δk, zk) − ∇ · ηk(t) = in Ω, for a.e t ∈ [0, T ], ηk(t) = on Γ, for a.e t ∈ [0, T ], ηk(0) = Since ǁhkǁW βk 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 ; L2(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 by {(zk , hk )}, which weakly converges to (z˜, ˜h) in the space W 1,2(0, T ; H1(Ω)) × W 1,2(0, T ; H1/2(Γ)) Analogously as in the proof of Theorem 4.3.3 we deduce ˜ ∈ Ad\{0} and q(h ˜ ) ≤ 0, that (z˜, h˜ ) satisfies equations (4.28) We will show that 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 check that ˜h ∈ A0 Now, we are going to show that ˜h /= By assumption, g¯ satisfies the first-order necessary condition, so we have β2 L(gk ) − L(g¯) = k q(hk ) + βk Sk , 87 (4.42) where Sk = ∫ γ1 βk ∫ T T (zk, ηk)dt + (zk, δk)dt + γ1βk ∫ ∫0 T T γ1 ∫ T |δk|2 dt βk ∫ T γ1 2 (ηk , y¯ − yd )dt + β (δk, ηk)dt + γ1 |ηk| 2dt β k k 0 γ2 γ2 + β k(z k (T ), δ k (T )) + γ 2β k(z k (T ), ηk (T )) + β |δ k (T )|2 k γ2 γ2 + β2k(δ k (T ), η k (T )) + γ 2(η k(T ), y¯(T ) − y T ) + β2k|η k (T )|2 2 γ1 + From (4.41) and the boundedness of sequences {zk}, {δk}, we have lim Sk = k→∞ It follows from (4.39) and (4.42) that Hence ∫ T γ3 − q(hk) + Sk < k b(zk, zk, w)dt + 2Sk < k (4.43) We assume that ˜h = 0, then z˜ = This leads to ∫ 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 ˜= ˜ ) ≤ early assumptions Therefore, h / It remains to prove that q(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| 2dt → |z˜|2 dt 0 From the continuity of the linear operator W 1,2(0, T ; H1(Ω)) z ›→ z(T ) ∈ H1(Ω), it follows that zk(T ) ~ 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 b(zk, zk, w)dt → b(z˜, z˜, w)dt 88 T Since the unit ball is weakly compact in the space W 1,2 (0, T ; H1/2(Γ)), we get ˜ ǁ W 1,2 (0,T ;H1/2 (Γ)) ≤ From what has already been proved, we conclude that ǁh that ˜ ) ≤ lim q(h ) ≤ q(h k 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]) 89 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) 90 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) 91 REFERENCES [1] F Abergel and R Temam (1990), On some control problems in fluid mechanics, 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physical and practical significance, one only considers Navier- Stokes- Voigt equations in the case of three or