Một số bài toán điều khiển tối ưu đối với hệ phương trình Navier - Stokes - Voigt

t))dt+ γ(u¯,mk)W 1,2(0,T ;L2(Ω)) < tk k − tk 2 q(mk)− tkAk. (3.53) 65 From the convergence of the sequences S(mk),Σ(mk) and the weak convergence of the sequence {mk} in the space W 1,2(0, T ;L2(Ω)) we deduce that the sequence {q(mk)} is bounded. Passing to the limit in (3.53) we have∫ T 0 (w(t), m˜(t))dt+ γ(u¯, m˜)W 1,2(0,T ;L2(Ω)) ≤ 0. This and (3.51) imply that m˜ ∈ C(u¯). So we obtain m˜ ∈ TUad(u¯)∩C(u¯). It remains to show that q(m˜) ≤ 0 and m˜ 6= 0. Since mk ∈ FUad(u¯) ⊂ TUad(u¯), the first-order necessary condition implies that∫ T 0 (w(t),mk(t))dt+ γ(u¯,mk)W 1,2(0,T ;L2(Ω)) ≥ 0. This together with (3.52) lead to q(mk) < 2( 1 k − Ak). Setting q∗(m) = S2(m) + 2TΣ(m). Since S(mk),Σ(mk) converge to S(m˜),Σ(m˜) respectively, we get q∗(mk)→ q∗(m˜) as k →∞. Moreover, q(mk) = q ∗(mk) + γ < 2( 1 k − Ak), so we obtain m˜ 6= 0 by the assumption γ > 0. Since mk ⇀ m˜ in W 1,2(0, T ;L2(Q)) and ‖mk‖W 1,2(0,T ;L2(Q)) = 1, we have ‖m˜‖W 1,2(0,T ;L2(Q)) ≤ 1. Therefore, q(m˜) ≤ limk→∞ q(mk) ≤ 0, and the theorem is proved. Conclusion of Chapter 3 In this chapter, we have studied a time optimal control problem for 3D Navier- Stokes-Voigt equations, where the control belongs to an arbitrary non-empty, convex, closed subset in the space W 1,2(0, T ;L2(Ω)). We have achieved the fol- lowing results: 1) Existence of globally optimal solutions (Theorem 3.2.3); 2) The first-order necessary optimality condition (Theorem 3.3.4); 3) The second-order sufficient optimality condition (Theorem 3.4.1). These are the first results on time optimal control of Navier-Stokes-Voigt equa- tions. Moreover, we derive both necessary and sufficient conditions instead of only necessary conditions, compare to a close result on time optimal control for the variable density Navier-Stokes equations (see [24]). 66 Chapter 4 AN OPTIMAL BOUNDARY CONTROL PROBLEM In this chapter, we consider an optimal boundary control problem for the 3D Navier-Stokes-Voigt equations in bounded domains. First, we prove a new result on the existence and uniqueness of solutions to the Navier-Stokes-Voigt equations with nonhomogeneous Dirichlet boundary conditions. Then, we show the existence of an optimal solution, the first-order and second-order necessary optimality conditions, and the second-order sufficient optimality conditions. The second-order optimality conditions obtained appear in a new form and seem to be sharp in the sense that the gap between them is minimal. The content of this chapter is based on the work [CT3] in the List of Publi- cations, which has been submitted and has not been published. 4.1 Setting of the problem Let Ω be a bounded domain in R3 with C1 boundary Γ. Denote by Q the time-space cylinder Ω × (0, T ) and by n the unit outer normal to Γ. We study the following optimal boundary control problem: Minimize the cost functional L(g) = γ1 2 ∫ T 0 ∫ Ω |y − yd|2dxdt+ γ2 2 ∫ Ω |y(T )− yT |2dx + γ3 2 ∫ T 0 (‖g‖2H1/2(Γ) + ‖gt‖2H1/2(Γ))dt, where g is the boundary control variable and the state variable y is a weak solution to the following 3D Navier-Stokes-Voigt equations on the interval (0, T ) yt − ν∆y − α2∆yt + (y · ∇)y +∇p = 0 in Q ∇ · y = 0 in Q, y = g on Γ× (0, T ), y(0) = y0 in Ω. (4.1) To study the above optimal boundary control problem, we assume that 67 • The initial velocity y0 is a given function in H1(Ω), which satisfies ∇· y0 = 0 and ∫ Γ y0 · nds = 0; • The functions yd, yT are given desired states that belong to the spaces L2(Q) and L2(Ω), respectively; • The coefficients γ1, γ2 are non-negative real numbers, where at least one of them is positive to get a non-trial objective functional. The coefficient γ3, which measures the cost of the control, is a positive real number; • The boundary control g belongs to the set of admissible controls Ad, which will be specified later in (4.2). Finding g to minimize L(g) means that one want to find a boundary control that satisfies a numerous purposes: the corresponding state is closed to the desired state yd during the whole period of time (0, T ) and closed to the desired state yT at final time T , and the cost is low (expressed through the point that the norm of g is small). In general, boundary controls add more difficulties than distributed controls. From the viewpoint of analysis, the choice an appropriate analytic and function spaces is neither unique nor obvious. From the practical perspective the influence of boundary forces being much weaker than that of body forces, it is more challenging to reach the design objective. In the past years, optimal boundary control problems of the Navier-Stokes equations have been studied by many authors, see for instance, [32, 39, 40, 61, 10, 17, 28, 29, 34, 37]. Some of these papers only treat questions concerning the existence of optimal solutions and the derivation of optimality systems from which optimal controls and states may be deduced. Others present formal derivations of optimality systems, define algorithms for the approximation of solutions of these systems, and the results of numerical experiments. We also refer the readers to [8, 9, 11, 16] for recent works on optimal boundary control of the 2D Boussinesq system, the MHD system and the 2D simplified Ericksen-Leslie system. We now describe precisely our optimal boundary control problem. (PL): Find the control g ∈ Ad that minimizes the cost function L(g) = γ1 2 ‖y − yd‖2L2(Q) + γ2 2 |y(T )− yT |2 + γ3 2 ‖g‖2W 1,2(0,T ;H1/2(Γ)) subject to equations (4.1). 68 Here, Ad is the set of admissible boundary controls defined by Ad = { g ∈ W 1,2(0, T ;H1/2(Γ)) : g(0) = y0 on Γ and ∫ Γ g · nds = 0 for a.e. t ∈ [0, T ] } . (4.2) From the early assumptions that y0 ∈ H1(Ω) and ∫ Γ y0 · nds = 0, the function g defined by g(t) = y0|Γ for every t ∈ [0, T ] belongs to Ad. Hence, Ad 6= ∅. Using a new result on the unique solvability of the 3D Navier-Stokes-Voigt equations with time-dependent nonhomogeneous Dirichlet boundary conditions (see Theorem 4.2.2 below) and following the general lines of the approach in [34], we then prove the existence of optimal solutions and derive the first-order necessary optimality condition. Next, we develop the ideas in a natural way to get the second-order necessary optimality condition. The approach that we use here to derive the second-order necessary optimality condition is a bit different from the usual one for optimal control problems, which often mentions about the C2 continuity property of the control-to-state mapping and Lagrange function. We don’t need to prove that the control-to-state mapping belongs to class C2, but only prove that it has directional derivatives up to order 2, which is sufficient to obtain the second-order necessary optimality condition. Hence, this approach can be applied to the cases when the control-to-state doesn’t belong to class C2. The second-order sufficient optimality condition is proved by using the contradiction argument (see Theorem 4.5.1 below) as that were used in Chapter 2 and 3. It is worthy noticing that our second-order optimality conditions appear in a new form and seem to be sharp in the sense that the sufficient condition is very close to the associated necessary one. It is also emphasized that our choice of boundary controls is optimal in the sense that it is necessary to ensure the well-posedness of the state equations. 4.2 Solvability of the 3D Navier-Stokes-Voigt equations with nonhomogeneous boundary conditions In this section, we are going to study the existence and uniqueness of solutions to the nonstationary 3D Navier-Stokes-Voigt equations with nonhomogeneous 69 Dirichlet boundary conditions yt − ν∆y − α2∆yt + (y · ∇)y +∇p = 0, x ∈ Ω, t > 0, ∇ · y = 0, x ∈ Ω, t > 0, y(x, t) = g(x, t), x ∈ Γ, t > 0, y(x, 0) = y0(x), x ∈ Ω. (4.3) Here, y = y(x, t) = (y1(x, t), y2(x, t), y3(x, t)) is the unknown velocity, y0 = y0(x) is the initial velocity, p = p(x, t) is the unknown pressure, g = g(x, t) is a given vector function defined for x ∈ Γ and t > 0, ν > 0 is the kinematic viscosity coefficient and α 6= 0 is the length-scale parameter characterizing the elasticity of the fluid. First, we recall a result on the existence and uniqueness of solutions to the Stokes equations. Lemma 4.2.1. [66, Chapter I, Section 2.4, 2.5] Let Ω be an open bounded set of class C1 in R3. Let there be given f ∈ H−1(Ω), g ∈ L2(Ω), φ ∈ H1/2(Γ) such that∫ Ω gdx = ∫ Γ φ · n ds. Then there exists a unique weak solution (u, p) ∈ H1(Ω)× L20(Ω) of the following Stokes problem  −ν∆u+ grad p = f in Ω, ∇ · u = g in Ω, u = φ on Γ. Moreover, there exists a constant C depending only on ν,Ω such that ‖u‖1 + ‖p‖L2(Ω) ≤ C (‖f‖H−1(Ω) + ‖g‖L2(Ω) + ‖φ‖H1/2(Γ)). The existence and uniqueness of solutions to the nonhomogeneous boundary value problem for the Navier-Stokes-Voigt equations is stated in the theorem below. Theorem 4.2.2. Let Ω be a bounded open domain of class C1 in R3. Assume that y0 ∈ H1(Ω) and g ∈ W 1,2(0, T ;H1/2(Γ)) satisfy the following conditions ∇ · y0 = 0, y0 = g(0) on Γ,∫ Γ g · nds = 0 for a.e. t ∈ [0, T ]. 70 Then, system (4.3) possesses a unique weak solution (y, p) ∈ W 1,2(0, T ;H1(Ω)) × L2(0, T ;L20(Ω)) in the following sense yt + νAy + α 2Ayt +B(y, y) + grad p = 0 in H−1(Ω), ∇ · y(t) = 0 in Ω, y(t) = g(t) on Γ, y(0) = y0, (4.4) for a.e. t ∈ [0, T ]. Here, A, B, grad are the operators defined in Section 1.3 and yt can be seen as an element in the space H−1(Ω) by 〈yt, v〉H−1(Ω),H10(Ω) := (yt, v), for v ∈ H10(Ω). Moreover, if ‖y0‖, ‖g‖W 1,2(0,T ;H1/2(Γ)) ≤M, (4.5) then there exists a constant C = C(M) such that ‖y‖W 1,2(0,T ;H1(Ω)) ≤ C. (4.6) Remark 4.2.3. Since each function g ∈ W 1,2(0, T ;H1/2(Γ)) is equal to a function f ∈ C([0, T ];H1/2(Γ)) (except a set of measure zero on [0, T ]), so we can consider that the compatibility condition holds at every point in [0, T ] if we identify each function g ∈ W 1,2(0, T ;H1/2(Γ)) with a function f ∈ C([0, T ];H1/2(Γ)). Proof. For each t ∈ [0, T ], we consider the following problem νAu(t) + grad q(t) = 0 in H−1(Ω), ∇ · u(t) = 0 in Ω, u(t) = g(t) on Γ. (4.7) By Proposition 1.2.4, we know that the space W 1,2(0, T ;H1/2(Γ)) is continuously imbedded in the space C([0, T ];H1/2(Γ)), so for each t ∈ [0, T ], g(t) ∈ H1/2(Γ). Therefore, it follows from Lemma 4.2.1 that problem (4.7) has a unique solution u(t) ∈ H1(Ω) and q(t) ∈ L20(Ω) satisfying ‖u(t)‖1 + ‖q(t)‖L2(Ω) ≤ C‖g(t)‖H1/2(Γ). (4.8) Now, we show that q ∈ L2(0, T ;L20(Ω)) and u ∈ W 1,2(0, T ;H1(Ω)). Since g ∈ W 1,2(0, T ;H1/2(Γ)), we have g ∈ L2(0, T ;H1/2(Γ)), and then we get from (4.8) that q ∈ L2(0, T ;L20(Ω)). We only need to prove that u ∈ W 1,2(0, T ;H1(Ω)). 71 First, we assume g ∈ C1([0, T ];H1/2(Γ)). We will prove that u ∈ C1([0, T ];H1(Ω)) and u satisfies the following estimates ‖u(t)‖1 ≤ C‖g(t)‖H1/2(Γ), (4.9) ‖ut(t)‖1 ≤ C‖gt(t)‖H1/2(Γ), (4.10) for every t ∈ [0, T ]. Indeed, inequality (4.9) comes from (4.8), so we only have to prove that the function u : [0, T ] → H1(Ω) is continuously differentiable and satisfies (4.10). Let t0 be a fixed point in [0, T ] and t be any point in [0, T ], t 6= t0. The problem  νAϕ+ grad q0 = 0 in H−1(Ω), ∇ · ϕ = 0 in Ω, ϕ = gt(t0) on Γ, (4.11) has a unique solution ϕ ∈ H1(Ω), q0 ∈ L20(Ω). Set φ(t) = u(t)− u(t0) t− t0 − ϕ, q1(t) = q(t)− q(t0) t− t0 − q0. We have φ(t), q1(t) is the unique solution of the following system νAφ(t) + grad q1(t) = 0 in H−1(Ω), ∇ · φ(t) = 0 in Ω, φ(t) = g(t)− g(t0) t− t0 − gt(t0) on Γ. Hence, we get∥∥∥∥u(t)− u(t0)t− t0 − ϕ ∥∥∥∥ 1 = ‖φ(t)‖1 ≤ C ∥∥∥∥g(t)− g(t0)t− t0 − gt(t0) ∥∥∥∥ H1/2(Γ) . Letting t→ t0 in this inequality we get that u is differentiable at t0 and ut(t0) = ϕ. Then, applying estimate (4.8) to problem (4.11) we get (4.10). The continuity of ut can be achieved from the continuity of gt. Next, let g be an arbitrary function in the space W 1,2(0, T ;H1(Γ)). From (1.6) we know that C1([0, T ];H1/2(Γ)) is dense in W 1,2(0, T ;H1/2(Γ)). Hence, there exists a sequence gn ∈ C1([0, T ];H1/2(Γ)) converging to g in the space W 1,2(0, T ;H1/2(Γ)). Denote by un the unique solution to problem (4.7) with the right-hand side of the third equation being gn. We have un ∈ C1([0, T ];H1(Ω)) and ‖un‖W 1,2(0,T ;H1(Ω)) ≤ C‖gn‖W 1,2(0,T ;H1/2(Γ)), (4.12) 72 thanks to estimates (4.9) and (4.10). Since gn → g in W 1,2(0, T ;H1(Γ)) we can easily get from (4.8) that un → u in L2(0, T ;H1(Ω)). (4.13) It follows from (4.12) that the sequence {un} is bounded in W 1,2(0, T ;H1(Ω)). Hence, we can extract a subsequence denoted again by {un} that weakly con- verges to some w in W 1,2(0, T ;H1(Ω)). This together with (4.13) imply that w ≡ u, so u ∈ W 1,2(0, T ;H1(Ω)). Now, let assumption (4.5) be satisfied, we can assume that ‖gn‖W 1,2(0,T ;H1/2(Γ)) ≤M + 1 for every n. This together with (4.12) give us the following estimate ‖un‖W 1,2(0,T ;H1(Ω)) ≤ C.(M + 1). Since un ⇀ u in W 1,2(0, T ;H1(Ω)), we get ‖u‖W 1,2(0,T ;H1(Ω)) ≤ C.(M + 1). (4.14) For the rest of the proof, we use C to denote several constants that may depend on M . Now, for u ∈ W 1,2(0, T ;H1(Ω)) given as above, we will show that the following system λt + νAλ+ α 2Aλt +B(λ, λ) +B(u, λ) +B(λ, u) + grad q˜ = −ut + α2Aut −B(u, u) in H−1(Ω), for a.e. t ∈ [0, T ], ∇ · λ(t) = 0 in Ω, for a.e. t ∈ [0, T ], λ(t) = 0 on Γ, for a.e. t ∈ [0, T ], λ(0) = y0 − u(0) (4.15) has a unique weak solution λ ∈ W 1,2(0, T ;V ) and q˜ ∈ L2(0, T ;L20(Ω)). Indeed, set f = −ut − α2Aut −B(u, u), then f ∈ L2(0, T ;H−1(Ω)) and by (4.14) we have ‖f‖L2(0,T ;H−1(Ω)) ≤ C. First, we prove that the following system λt + νAλ+ α 2Aλt +B(λ, λ) +B(u, λ) +B(λ, u) = f in V ′ for a.e. t ∈ [0, T ], ∇ · λ(t) = 0 in Ω, for a.e. t ∈ [0, T ], λ(t) = 0 on Γ, for a.e. t ∈ [0, T ], λ(0) = y0 − u(0) (4.16) 73 has a unique weak solution λ ∈ W 1,2(0, T ;V ). The proof is standard by using the Galerkin method, so we only present here some priori estimates. Multiplying the first equation in (4.16) by λ(s) we get 1 2 d dt (|λ(s)|2 + α2‖λ(s)‖2) + ν‖λ(s)‖2 = −b(λ(s), u(s), λ(s)) + 〈f(s), λ(s)〉V ′,V . (4.17) The right-hand side can be estimated by∣∣〈f(s), λ(s)〉V ′,V ∣∣ ≤ ‖f(s)‖V ′‖λ(s)‖ ≤ ν 2 ‖λ(s)‖2 + C‖f(s)‖2V ′ , |b(λ(s), u(s), λ(s))| ≤ C|λ(s)|1/2‖λ(s)‖3/2 ≤ ν 2 ‖λ(s)‖2 + C|λ(s)|2. From these estimates and (4.17), it follows that d dt (|λ(s)|2 + α2‖λ(s)‖2) ≤ C|λ(s)|2 + C‖f(s)‖2V ′ ≤ C(|λ(s)|2 + α2‖λ(s)‖2) + C‖f(s)‖2V ′ . Applying Gronwall’s inequality we obtain |λ(s)|2 + α2‖λ(s)‖2 ≤ (|λ(0)|2 + ‖λ(0)‖2)eCs + C ∫ s 0 eC(s−θ)‖f(θ)‖2V ′dθ. This implies that λ ∈ L∞(0, T ;V ) and ‖λ‖L∞(0,T ;V ) ≤ C. (4.18) Next, multiplying the first equation in (4.16) by λt, then integrating from 0 to T yields∫ T 0 (|λt(s)|2 + α2‖λt(s)‖2)ds+ ν ∫ T 0 ((λ(s), λt(s)))ds = − ∫ T 0 b(λ(s), λ(s), λt(s))ds− ∫ T 0 b(u(s), λ(s), λt(s))ds − ∫ T 0 b(λ(s), u(s), λt(s))ds+ ∫ T 0 〈f(s), λt(s)〉V ′,V ds. (4.19) From (4.14) and (4.18), we can find an upper bound for the right-hand side of (4.19) by applying (1.10), that is, α2 2 ∫ T 0 ‖λt(s)‖2ds+ C ∫ T 0 ‖f(s)‖2V ′ds+ C. 74 Then, it follows from (4.19) that∫ T 0 ‖λt(s)‖2ds ≤ C‖λ(0)‖2 + C ∫ T 0 ‖f(s)‖2V ′ds+ C ≤ C‖λ(0)‖2 + C ∫ T 0 ‖f(s)‖2H−1(Ω)ds+ C ≤ C. (4.20) This implies λt ∈ L2(0, T ;V ), and by (4.18), we have ‖λ‖W 1,2(0,T ;H1(Ω)) ≤ C. Set ψ = λt + νAλ+ α 2Aλt +B(λ, λ) +B(u, λ) +B(λ, u)− f. It is clear that ψ ∈ L2(0, T ;H−1(Ω)). Moreover, 〈ψ(t), v〉H−1(Ω),H10(Ω) = 0 ∀v ∈ V, for a.e. t ∈ [0, T ]. From Proposition 1.3.2, there exists a unique q˜ ∈ L2(0, T ;L20(Ω)) such that grad q˜ = ψ for a.e. t ∈ [0, T ]. We have shown that (λ, q˜) satisfies (4.15). Set y := u + λ, p := q + q˜, we get that y ∈ W 1,2(0, T ;H1(Ω)) and p ∈ L2(0, T ;L20(Ω)) is the unique solution to system (4.1). Estimate (4.6) follows from (4.14), (4.18) and (4.20). 4.3 Existence of optimal solutions We start with some definitions of optimal solutions. Definition 4.3.1. A control g¯ ∈ Ad is said to be globally optimal if L(g¯) ≤ L(g), ∀g ∈ Ad. Definition 4.3.2. A control g¯ ∈ Ad is said to be locally optimal if there exists a constant ρ > 0 such that L(g¯) ≤ L(g) holds for every g ∈ Ad with ‖g − g¯‖W 1,2(0,T ;H1/2(Γ)) ≤ ρ. Theorem 4.3.3. The problem (PL) has at least one globally optimal solution. Proof. For every control g in Ad, by Theorem 4.2.2, there exists a unique weak solution to system (4.1). We see that L(g) ≥ 0 for every g ∈ Ad. Hence, there exists the infimum of L over all admissible controls 0 ≤ L¯ := inf g∈Ad L(g) <∞. (4.21) 75 Moreover, there is a sequence {gn} of admissible controls such that L(gn) → L¯ as n→∞. Denote by (yn, pn) the unique weak solution of system (4.1) with the right-hand side of the third equation being gn. From the convergence we see that the set {L(gn)} is bounded. This implies that the set {gn} is bounded in W 1,2(0, T ;H1/2(Γ)). From (4.6) we deduce that the sequence {yn} is bounded in W 1,2(0, T ;H1(Ω)). From these boundedness, we can extract a subsequence, which is denoted again by {(yn, gn)}, converging weakly in W 1,2(0, T ;H1(Ω))×W 1,2(0, T ;H1/2(Γ)) to some limit (y¯, g¯). We will show that g¯ is admissible, i.e. g¯ ∈ Ad, and y¯ is the state associated to g¯. To show g¯ ∈ Ad we have to prove that g¯(0) = y0 on Γ, (4.22)∫ Γ g¯ · nds = 0 for a.e. t ∈ [0, T ]. (4.23) Since gn(0) = y0 on Γ, (4.22) comes directly from the continuity of the imbedding W 1,2(0, T ;H1/2(Γ)) ↪→ C([0, T ];H1/2(Γ)) and of the map C([0, T ];H1/2(Γ))→ H1/2(Γ), g 7→ g(s). Here, s is a fixed given point in [0, T ]. From these continuities, we also obtain that for every t ∈ [0, T ], gn(t) ⇀ g¯(t) in H1/2(Γ) as n → ∞. Moreover, H1/2(Γ) is compactly imbedded in L2(Γ), so we have gn(t) → g¯(t) in L2(Γ). This gives (4.23). Now, to prove that y¯ is the associated state to g¯, we will show the existence of a function p¯ ∈ L2(0, T ;L20(Ω)) such that (y¯, p¯, g¯) satisfies equations (4.4). Since yn ⇀ y¯ in W 1,2(0, T ;H1(Ω)), it implies that ynt+νAyn+α2Aynt+B(yn, yn) weakly converges to y¯t+ νAy¯+α2Ay¯t+B(y¯, y¯) in L2(0, T ;H−1(Ω)), and so in L2(0, T ;V ′). Since 〈ynt(s) + νAyn(s) + α2Aynt(s) +B(yn(s), yn(s)), v(t)〉L2(0,T ;V ′),L2(0,T ;V ) = − ∫ T 0 〈grad pn(t), v(t)〉H−1(Ω),H10(Ω) = 0 for every v ∈ L2(0, T ;V ), we get ynt(s) + νAyn(s) + α 2Aynt(s) +B(yn(s), yn(s)) = 0 in L2(0, T ;V ′). Hence y¯t(s) + νAy¯(s) + α 2Ay¯t(s) +B(y¯(s), y¯(s)) = 0 in L2(0, T ;V ′), 76 then we have 〈y¯t(s) + νAy¯(s) + α2Ay¯t(s) +B(y¯(s), y¯(s)), v〉 = 0 ∀v ∈ V, a.e. s ∈ [0, T ]. This ensures the unique existence of a function p¯ ∈ L2(0, T ;L20(Ω)) such that (y¯, p¯) satisfies the first equation of (4.4), thanks to Proposition 1.3.2. Since (yn, gn) weakly converges to (y¯, g¯) in W 1,2(0, T ;H1(Ω))×W 1,2(0, T ;H1/2(Γ)), we can easily check that (y¯, g¯) satisfies the three remain equations in (4.4). Finally, it remains to show that L(g¯) = L¯. From (4.21) we have L¯ ≤ L(g¯), so we only need to prove L¯ ≥ L(g¯). Indeed, it is clear that the function J : W 1,2(0, T ;H1(Ω))×W 1,2(0, T ;H1/2(Γ))→ R defined by J(y, g) = γ1 2 ‖y − yd‖2L2(Q) + γ2 2 |y(T )− yT |2 + γ3 2 ‖g‖2W 1,2(0,T ;H1/2(Γ)), is sequentially weakly lower semicontinuous, so we have lim inf n→∞ J(yn, gn) ≥ J(y¯, g¯). This implies that L¯ ≥ L(g¯). The proof is complete. 4.4 First-order and second-order necessary optimality condi- tions 4.4.1 First-order necessary optimality conditions Set A0 = { h ∈ W 1,2(0, T ;H1/2(Γ)) : h(0) = 0 on Γ and ∫ Γ h · nds = 0 for a.e. t ∈ [0, T ] } . We see that A0 is a subspace of the the space W 1,2(0, T ;H1/2(Γ)) and is exactly the normal cone and the polar cone of tangents of Ad at an arbitrary point g ∈ Ad. The following theorem gives the first-order necessary condition for a control to be an optimal solution. 77 Theorem 4.4.1. If g¯ ∈ Ad is an optimal solution to problem (PL) then g¯ satisfies the following condition γ3 ∫ T 0 (g¯, h)H1/2(Γ)dt+ γ3 ∫ T 0 (g¯t, ht)H1/2(Γ)dt − ∫ T 0 〈τ(t), h(t)〉H−1/2(Γ),H1/2(Γ)dt− 〈pi, h(T )〉H−1/2(Γ),H1/2(Γ) = 0, ∀h ∈ A0, (4.24) where τ ∈ L2(0, T ;H−1/2(Γ)), pi ∈ H−1/2(Γ) are defined by 〈τ(s), h〉H−1/2(Γ),H1/2(Γ) := −(wt(s), v) + ν(∇w(s),∇v)− α2(∇wt(s),∇v) −B(y¯(s), w(s), v) +B(v, y¯(s), w(s)) + (σ(s),∇ · v)− γ1(y¯(s)− yd(s), v), for evey h ∈ H1/2(Γ), for a.e. s ∈ [0, T ], (4.25) 〈pi, h〉H−1/2(Γ),H1/2(Γ) = (w(T ), v) + α2(∇w(T ),∇v) + (κ,∇ · v) − γ2(y¯(T )− yT , v), for every h ∈ H1/2(Γ). (4.26) In (4.25) and (4.26), v is an element in H1(Ω) such that v|Γ = h and (w, σ, κ) ∈ W 1,2(0, T ;V )×L2(0, T ;L20(Ω))×L20(Ω) is the unique weak solution of the following adjoint system −wt + νAw − α2Awt −B(y¯, w) + B˜(y¯, w) + gradσ = γ1(y¯ − yd) in H−1(Ω) for a.e. t ∈ [0, T ], ∇ · w(t) = 0 in Ω, for a.e. t ∈ [0, T ], w(t) = 0 on Γ, for a.e. t ∈ [0, T ], w(T ) + α2Aw(T ) + gradκ = γ2(y¯(T )− yT ) in H−1(Ω). (4.27) Here, 〈B˜(y¯, w), v〉H−1(Ω),H10(Ω) := B(v, y¯, w). Remark 4.4.2. The value of the right-hand sides in (4.25) and (4.26) is inde- pendent of the v chosen. Indeed, denote by F (v) the right-hand side of (4.25). If there are two functions v1, v2 ∈ H1(Ω) such that v1|Γ = v2|Γ = h then v1 − v2 ∈ H10(Ω). By taking v1 − v2 as a test function for the first equation of (4.27) we get F (v1) = F (v2). A similar argument can be applied to get the conclusion for (4.26). Proof. Assume that g¯ ∈ Ad is an optimal solution to the problem (PL) and y¯ is the state associated to g¯. Let h be any fixed element of A0. Set gβ = g¯ + βh, we have gβ ∈ Ad for every β ∈ R. Denote by yβ the state associated to gβ. We can 78 wite yβ = y¯ + βz + βηβ, where z is a weak solution of the following system zt + νAz + α 2Azt +B(y¯, z) +B(z, y¯) + grad p1 = 0 in H−1(Ω) for a.e. t ∈ [0, T ], ∇ · z(t) = 0 in Ω, for a.e. t ∈ [0, T ], z(t) = h(t) on Γ, for a.e. t ∈ [0, T ], z(0) = 0, (4.28) and ηβ is a weak solution of the following system ηβt + νAηβ + α 2Aηβt +B(y¯, ηβ) +B(ηβ, y¯) + βB(z, ηβ) + βB(ηβ, z) +βB(ηβ, ηβ) + grad p2 = −βB(z, z) in H−1(Ω) for a.e. t ∈ [0, T ], ∇ · ηβ(t) = 0 in Ω, for a.e. t ∈ [0, T ], ηβ(t) = 0 on Γ, for a.e. t ∈ [0, T ], ηβ(0) = 0. (4.29) Using the same arguments as in the proof of Theorem 4.2.2, we can prove that system (4.28) possesses a unique weak solution (z, p1) ∈ W 1,2(0, T ;H1(Ω))× L2(0, T ;L20(Ω)). By following the lines when proving the existence of weak so- lutions to system (4.15), we obtain that for each β ∈ R system (4.29) has exactly one weak solution (ηβ, p2) ∈ W 1,2(0, T ;V ) × L2(0, T ;L20(Ω)). In fact, ηβ ∈ W 1,2(0, T ;V ) is the unique weak solution of the following system ηβt + νAηβ + α 2Aηβt +B(y¯, ηβ) +B(ηβ, y¯) + βB(z, ηβ) + βB(ηβ, z) +βB(ηβ, ηβ) = −βB(z, z) in V ′, for a.e. t ∈ [0, T ], ∇ · ηβ(t) = 0 in Ω, for a.e. t ∈ [0, T ], ηβ(t) = 0 on Γ, for a.e. t ∈ [0, T ], ηβ(0) = 0. Since y¯, z ∈ W 1,2(0, T ;H1(Ω)) and W 1,2(0, T ;H1(Ω)) is continuously imbedded in C([0, T ];H1(Ω)) we get that y¯, z ∈ C([0, T ];H1(Ω)). From this result and (1.10), (1.11), we get that ηβ → 0 in W 1,2(0, T ;V ) as β → 0, by a similar argument as in the proof of Theorem 2.3.3. This means that the control-to-state mapping S : g 7→ y has the directional derivative at g¯ in the direction h, i.e. S ′(g¯;h), and S ′(g¯;h) = z. 79 Now, we will establish a necessary optimality condition. By hypothesis, L(gβ)− L(g¯) ≥ 0. On the other hand, we have L(gβ)− L(g¯) = β ( γ1 ∫ T 0 (z, y¯ − yd)dt+ γ2(z(T ), y¯(T )− yT ) + γ3(g¯, h)W 1,2(0,T ;H1/2(Γ)) ) + βZβ, (4.30) where Zβ = γ1 2 β ∫ T 0 |z + ηβ|2dt+ γ1 ∫ T 0 (ηβ, y¯ − yd)dt+ γ2 2 β|z(T ) + ηβ(T )|2 + γ2(ηβ(T ), y¯(T )− yT ) + γ3 2 β‖h‖2W 1,2(0,T ;H1/2(Γ)). Since ηβ → 0 in W 1,2(0, T ;V ) as β → 0, it is easy to check that Zβ → 0 as β → 0. Dividing both sides of (4.30) by β then letting β → 0+ and β → 0− we get γ1 ∫ T 0 (z, y¯ − yd)dt+ γ2(z(T ), y¯(T )− yT ) + γ3(g¯, h)W 1,2(0,T ;H1/2(Γ)) = 0. (4.31) Now, assume that (w, σ, κ) is a weak solution of system (4.27). Then, by the definition of operator grad, we imply that w is a weak solution of the following system  −wt + νAw − α2Awt −B(y¯, w) + B˜(y¯, w) = γ1(y¯ − yd) in V ′, for a.e. t ∈ [0, T ], ∇ · w(t) = 0 in Ω, for a.e. t ∈ [0.T ], w(t) = 0 on Γ, for a.e. t ∈ [0, T ], w(T ) + α2Aw(T ) = γ2(y¯(T )− yT ) in V ′. From the proof of Theorem 2.3.3, we know that there exists a unique weak solution w ∈ W 1,2(0, T ;V ) of the above system. Analogously as in the proof of the unique existence of solutions to system (4.15), we get the unique existence of functions σ ∈ L2(0, T ;L20(Ω)) and κ ∈ L20(Ω), which together with w above satisfy equations (4.27). We have proved that system (4.27) possesses a unique weak solution (w, σ, κ) ∈ W 1,2(0, T ;V )× L2(0, T ;L20(Ω))× L20(Ω). Next, we prove that τ , which is defined in (4.25), belongs to L2(0, T ;H−1/2(Γ)). Let h be an element in the space H1/2(Γ). By the trace theorem, there exists v ∈ H1(Ω) such that v|Γ = h and ‖v‖H1(Ω) ≤ C‖h‖H1/2(Γ). 80 We use this function v in the definition of τ in (4.25). Since w ∈ W 1,2(0, T ;V ), y¯ ∈ W 1,2(0, T ;H1(Ω)), σ ∈ L2(0, T ;L20(Ω)), yd ∈ L2(Q), we can easily check that τ ∈ L2(0, T ;H−1/2(Γ)). By a similar argument, we can show that pi, which is defined by (4.26), belongs to the space H−1/2(Γ). Now, taking v = z(t) in (4.25) and then integrating from 0 to T we have γ1 ∫ T 0 (y¯ − yd, z)dt+ γ2(y¯(T )− yT , z(T )) = − ∫ T 0 〈τ(t), h(t)〉H−1/2(Γ),H1/2(Γ)dt− 〈pi, h(T )〉H−1/2(Γ),H1/2(Γ). From this and (4.31) we get (4.24). 4.4.2 Second-order necessary optimality conditions The following theorem gives the second-order necessary optimality condition for the problem (PL) (condition (4.32) below). Theorem 4.4.3. Assume that g¯ ∈ Ad is an optimal solution to the problem (PL). Denote by y¯ the state associated to g¯ and by w the adjoint state, i.e. the unique weak solution of system (4.27). Let h be 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). Set q(h) := γ1 ∫ T 0 |z|2dt+ γ2|z(T )|2 + γ3‖h‖2W 1,2(0,T ;H1/2(Γ)) − 2 ∫ T 0 b(z, z, w)dt. Then we have q(h) ≥ 0, ∀h ∈ A0. (4.32) Proof. Let h be any fixed element of A0. Set gβ = g¯ + βh, we have gβ ∈ Ad for every β ∈ R. Denote by yβ the state associated to gβ. We can wite yβ = y¯ + βz + β2 2 δ + β2ηβ, where z is the unique weak solution of system (4.28), δ is a weak solution of the 81 following system δt + νAδ + α 2Aδt +B(y¯, δ) +B(δ, y¯) + grad p2 = −2B(z, z) in H−1(Ω), for a.e. t ∈ [0, T ], ∇ · δ = 0 in Ω, for a.e. t ∈ [0, T ], δ(t) = 0 on Γ, for a.e. t ∈ [0, T ], δ(0) = 0, (4.33) and ηβ is a weak solution of the following system ηβt + νAηβ + α 2Aηβt +B(ηβ, y¯) +B(y¯, ηβ) + βB(z, ηβ) + βB(ηβ, z) + β2 2 B(δ, ηβ) + β2 2 B(ηβ, δ) + β 2B(ηβ, ηβ) + grad p3 = −β 2 B(z, δ)− β 2 B(δ, z)− β 2 4 B(δ, δ) in H−1(Ω), for a.e. t ∈ [0, T ], ∇ · ηβ(t) = 0 in Ω, for a.e. t ∈ [0, T ], ηβ(t) = 0 on Γ, for a.e. t ∈ [0, T ], ηβ(0) = 0. (4.34) 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 ;L20(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 → 0 in W 1,2(0, T ;V ) as β → 0. (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 2 2 S ′′(g¯;h, h) + o(β2), and S ′′(g¯;h, h) = δ. After some simple computations, L(gβ)−L(g¯) can be written 82 as follows L(gβ)− L(g¯) = β ( γ1 ∫ T 0 (z, y¯ − yd)dt+ γ2 ( z(T ), y¯(T )− yT ) + γ3(g¯, h)W 1,2(0,T ;H1/2(Γ)) ) + β2 ( γ1 2 ∫ T 0 ( |z|2 + (δ, y¯ − yd) ) dt+ γ2 2 ( |z(T )|2 + (δ(T ), y¯(T )− yT ) ) + γ3 2 ‖h‖2W 1,2(0,T ;H1/2(Γ)) ) + β2Sβ, where Sβ = γ1 2 β ∫ T 0 (z, δ)dt+ γ1β ∫ T 0 (z, ηβ)dt+ γ1 8 β2 ∫ T 0 |δ|2dt + γ1 2 β2 ∫ T 0 (δ, ηβ)dt+ γ1 ∫ T 0 (ηβ, y¯ − yd)dt+ γ1 2 β2 ∫ T 0 |ηβ|2dt + γ2 2 β(z(T ), δ(T )) + γ2β(z(T ), ηβ(T )) + γ2 8 β2|δ(T )|2 + γ2 2 β2(δ(T ), ηβ(T )) + γ2(ηβ(T ), y¯(T )− yT ) + γ2 2 β2|ηβ(T )|2. From this and the first-order necessary condition we deduce that L(gβ)− L(g¯) = β2 ( γ1 2 ∫ T 0 ( |z|2 + (δ, y¯ − yd) ) dt + γ2 2 ( |z(T )|2 + (δ(T ), y¯(T )− yT ) ) + γ3 2 ‖h‖2W 1,2(0,T ;H1/2(Γ)) ) + β2Sβ. From (4.35), it is easy to check that Sβ → 0 as β → 0. Since L(gβ) − L(g¯) ≥ 0, we obtain γ1 2 ∫ T 0 ( |z|2 + (δ, y¯ − yd) ) dt+ γ2 2 ( |z(T )|2 + (δ(T ), y¯(T )− yT ) ) + γ3 2 ‖h‖2W 1,2(0,T ;H1/2(Γ)) ≥ 0, ∀h ∈ A0. (4.36) Now, let τ be defined in (4.25). By taking v = δ(t) in (4.25) and then integrating from 0 to T we obtain γ1 ∫ T 0 (δ, y¯ − yd)dt+ γ2(δ(T ), y¯(T )− yT ) = −2 ∫ T 0 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 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 0 |z|2dt+ γ2|z(T )|2 + γ3‖h‖2W 1,2(0,T ;H1/2(Γ)) − 2 ∫ T 0 b(z, z, w)dt > 0 for every h ∈ A0\{0}, (4.37) then there exist ε > 0 and ρ > 0 such that L(g)− L(g¯) ≥ ε‖g − g¯‖2W 1,2(0,T ;H1/2(Γ)) (4.38) holds for all g ∈ Ad with ‖g − g¯‖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 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¯) + 1 k ‖gk − g¯‖2W 1,2(0,T ;H1/2(Γ)), (4.39) and ‖gk − g¯‖W 1,2(0,T ;H1/2(Γ)) < 1/k. Hence, we can write gk = g¯ + βkhk, where βk → 0 in R, hk ∈ A0 and ‖hk‖W 1,2(0,T ;H1/2(Γ)) = 1. 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 84 unique weak solution of the following system ηkt + νAηk + α 2Aηkt +B(ηk, y¯) +B(y¯, ηk) + βkB(zk, ηk) + βkB(ηk, zk) + β2k 2 B(δk, ηk) + β2k 2 B(ηk, δk) + β 2 kB(ηk, ηk) + grad pk = − βk 2 B(zk, δk) −βk 2 B(δk, zk)− β2k 4 B(δk, δk) in H−1(Ω), for a.e. t ∈ [0, T ], ∇ · ηk(t) = 0 in Ω, for a.e. t ∈ [0, T ], ηk(t) = 0 on Γ, for a.e. t ∈ [0, T ], ηk(0) = 0. (4.40) 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 → 0 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 that (z˜, h˜) satisfies equations (4.28). We will show that h˜ ∈ Ad\{0} and q(h˜) ≤ 0, 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˜ 6= 0. By assumption, g¯ satisfies the first-order necessary condition, so we have L(gk)− L(g¯) = β2k 2 q(hk) + β 2 kSk, (4.42) 85 where Sk = γ1 2 βk ∫ T 0 (zk, δk)dt+ γ1βk ∫ T 0 (zk, ηk)dt+ γ1 8 β2k ∫ T 0 |δk|2dt + γ1 2 β2k ∫ T 0 (δk, ηk)dt+ γ1 ∫ T 0 (ηk, y¯ − yd)dt+ γ1 2 β2k ∫ T 0 |ηk|2dt + γ2 2 βk(zk(T ), δk(T )) + γ2βk(zk(T ), ηk(T )) + γ2 8 β2k|δk(T )|2 + γ2 2 β2k(δk(T ), ηk(T )) + γ2(ηk(T ), y¯(T )− yT ) + γ2 2 β2k|ηk(T )|2. From (4.41) and the boundedness of sequences {zk}, {δk}, we have lim k→∞ Sk = 0. It follows from (4.39) and (4.42) that 1 2 q(hk) + Sk < 1 k . Hence γ3 − 2 ∫ T 0 b(zk, zk, w)dt+ 2Sk < 2 k . (4.43) We assume that h˜ = 0, then z˜ = 0. This leads to∫ T 0 b(zk, zk, w)dt→ 0 as k →∞, by Lemma 1.3.3. We thus get from (4.43) that γ3 ≤ 0, which contradicts the early assumptions. Therefore, h˜ 6= 0. It remains to prove that q(h˜) ≤ 0. Indeed, the space W 1,2(0, T ;H1(Ω)) is compactly imbedded in L2(0, T ;L2(Ω)), so we have ∫ T 0 |zk|2dt→ ∫ T 0 |z˜|2dt. From the continuity of the linear operator W 1,2(0, T ;H1(Ω)) 3 z 7→ 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 0 b(zk, zk, w)dt→ ∫ T 0 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)) ≤ 1. From what has already been proved, we conclude that q(h˜) ≤ lim k→∞ q(hk) ≤ 0. This ends the proof. Conclusion of Chapter 4 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 nonhomo- geneous 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 Navier- Stokes-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 Navier- Stokes equations (see [34]). 87 CONCLUSION AND FUTURE WORK Conclusion In this thesis, a number of optimal control problems governed by three- dimensional 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: 1. Existence of optimal solutions, the first-order necessary optimality condi- tion and the second-order sufficient optimality condition for a distributed optimal control problem and a time optimal control problem. 2. Existence of optimal solutions, the first-order necessary optimality condi- tion, 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: 1. Numerical approximations for the above optimal control problems (see the survey article [13] for related results on Navier-Stokes equations). 2. Optimal control of Navier-Stokes-Voigt equations with bang-bang controls (see [14] for results on 2D Navier-Stokes equations). 3. Optimal control of Navier-Stokes-Voigt equations with measure valued con- trols (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. 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