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MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION TRAN MINH NGUYET SOME OPTIMAL CONTROL PROBLEMS FOR NAVIER-STOKES-VOIGT EQUATIONS SUMMARY OF DOCTORAL THESIS IN MATHEMATICS Speciality: Differential and Integral Equations Speciality Code: 46 01 03 HA NOI, 2019 This dissertation has been written at Hanoi National University of Education Supervisor: Prof Dr Cung The Anh Referee 1: Prof D.Sc Vu Ngoc Phat Institute of Mathematics - VAST Referee 2: Assoc Prof Dr Nguyen Sinh Bay Thuong Mai University Referee 3: Assoc Prof Dr Tran Dinh Ke Hanoi National University of Education The thesis shall be defended at the University level Thesis Assessment Council at Hanoi National University of Education on ……… This thesis can be found in: - The National Library of Vietnam; - Library of Hanoi National University of Education INTRODUCTION Literature survey and motivation The Navier-Stokes-Voigt equations was first introduced by Oskolkov (1973) as a model of motion of certain linear viscoelastic incompressible fluids This system was also proposed by Cao, Lunasin and Titi (2006) as a regularization, for small values of α, of the 3D 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 Holst, Lunasin and Tsogtgerel (2010)), but it has attractive advantage over other α-models in that one does not need to impose any additional artificial boundary condition (besides the Dirichlet boundary conditions) to get the global well-posedness In the past years, the existence and long-time behavior of solutions to the Navier-StokesVoigt 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 Navier-Stokes-Voigt equations, see e.g V.K Kalantarov and E.S Titi (2009), G Yue and C.K Zhong (2011), J García-Luengo, P Marín-Rubio and J Real (2012), Y Qin, X Yang and X Liu (2012), C.T Anh and P.T Trang (2013), M Conti Zalati and C.G Gal (2015), P.D Damázio, P Manholi and A.L Silvestre (2016) In the whole space R3 , the existence and decay rates of solutions have been studied recently (see C Zhao and H Zhu (2015), C.T Anh and P.T Trang (2016), C.J Niche (2016)) The optimal control theory has been developed rapidly in the past few decades and become 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, the control of chemical processes However, in many situations, the processes to be optimized may not be modeled by ordinary differential equations, instead partial 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 is started studying in 1980s by Fursikov when he established several theorems about the existence of optimal solutions to some optimal control problems governed by Navier-Stokes equations 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 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 NavierStokes equations that is one of the most important equations in fluid mechanics For distributed control problems, this matter was studied by H.O Fattorini and S Sritharan (1994), M D Gunzburger and S Manservisi (1999), M Hinze and K Kunisch (2001), F Tröltzsch and D Wachsmuth (2006) These works are all in the case of absence of state constraints In the case of the present of state constraints, the problem was investigated by G Wang (2002) and Liu (2010) The time optimal control problem of Navier-Stokes equations was investigated by Barbu (1997) and E Fernandez-Cara (2012) Optimal boundary control problems of the Navier-Stokes equations have been studied by many authors, see for instance, M.D Gunzburger, L.S Hou and Th.P Svobodny (1991), J.C De Los Reyes and K Kunisch (2005), C John and D Wachsmuth (2009), M Holst, E Lunasin and G Tsogtgerel (2010) in the stationary case, and M Berggren (1998), A.V Fursikov, M.D Gunzburger and L.S Hou (1998, 2005), M.D Gunzburger and S Manservisi (2000), M Hinze and K Kunisch (2004), M Colin and P Fabrie (2010) in the nonstationary case We can see also the habilitation by M Hinze (2002), the theses by M Sandro (1997), D Wachsmuth (2006) 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 NavierStokes-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 3D 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 dimensional NavierStokes-Voigt equations, where the objective functional is of quadratic form and the distributed control belongs to a non-empty, closed, convex subset, (P2) The time optimal control problem of the nonstationary three dimensional NavierStokes-Voigt equations, where the set of admissible controls is an arbitrary non-empty, closed, convex subset, (P3) The boundary optimal control problem of the nonstationary three dimensional NavierStokes-Voigt equations, where the objective functional is of quadratic form and the boundary control variable has to satisfy some compatibility conditions Objectives The objectives of this dissertation are to prove the existence of optimal solutions and to give the necessary and sufficient optimality conditions for problems (P1), (P2), (P3), namely, (i) to show the existence of optimal solutions and to establish the first-order necessary and the second-order sufficient optimality conditions for problems (P1) (ii) to prove the existence of optimal solutions and to derive the first-order necessary and the second-order sufficient optimality conditions for problems (P2) (iii) to get the existence of optimal solutions and to give the first-order and second-order necessary, the second-order sufficient optimality conditions for problems (P3) The structure and results of the dissertation The dissertation has four chapters and a list of references Chapter collects several basic concepts and facts on Sobolev spaces and partial differential equations associated with solutions of Navier-Stokes-Voigt equations as well as some auxiliary results Chapter presents results on the distributed optimal control problem governed by NavierStokes-Voigt equations Chapter provides results on the time optimal control problem governed by Navier-StokesVoigt equations Chapter presents results on the boundary optimal control problem governed by NavierStokes-Voigt equations The results obtained in Chapters 2, and are answers for the problems (P1), (P2), (P3), respectively Chapter and Chapter are based on the content of the papers [CT1], [CT2] in the List of Publications which were published in the journals Numerical Functional Analysis and Optimization and Applied Mathematics and Optimization, respectively The results of Chapter is the content of the work [CT3] in the List of Publications, which has been submitted for publication Chapter PRELIMINARIES AND AUXILIARY RESULTS In this chapter, we review some basic concepts and results on function spaces, imbeddings, operators, Navier-Stokes-Voigt equations and present some auxiliary results on linearized equations Function spaces • Lp and Sobolev spaces: Lp (Ω), W m,p (Ω), H m (Ω) (m ∈ N), H s (Ω) (s ∈ R), s ≥ 0, H s (Γ) • Spaces of abstract functions: Lp (0, T ; X), ≤ p ≤ ∞, W 1,p (0, T ; X) Some useful inequalities: Hölder inequality, Poincaré’s inequality, Gronwall’s inequality, Young’s inequality with ϵ Continuous and compact imbeddings: Rellich-Kondrachov theorem, imbeddings in abstract function spaces Operators: the trilinear form, the operator grad, the continuous linear operators and bilinear operators The unique existence of solutions to the nonstationary 3D Navier-Stokes-Voigt equations with homogeneous Dirichlet boundary conditions Some auxiliary results on linearized equations: the definition of weak solutions and some properties of weak solutions Some definitions in Convex Analysis: the normal cone and the polar cone of tangents of a convex subset in a Hilbert space Chapter A DISTRIBUTED OPTIMAL CONTROL PROBLEM In this chapter, we will study an optimal control problem with quadratic objective functional for the 3D Navier-Stokes-Voigt equations in bounded domains Here, the control variable plays the role of the force field in the equations We will show the existence of optimal solutions, the first-order necessary optimality conditions and the second-order sufficient optimality conditions The content of this chapter is based on the work [CT1] in the List of Publications 2.1 Setting of the problem Let Ω be a bounded domain in R3 with locally Lipschitz boundary Γ and T > be a fixed final time Denote by Q the cylinder Ω × (0, T ) In this chapter, we consider the minimization of the following quadratic objective functional αT J(y, u) = ∫ αQ |y(x, T ) − yT (x)| dx + Ω ∫∫ |y(x, t) − yQ (x, t)|2 dxdt Q γ + ∫∫ |u(x, t)|2 dxdt Q Here the free variables - state y and control u - have to fulfill the following 3D Navier-StokesVoigt equations yt − ν∆y − α2 ∆yt + (y · ∇)y + ∇p ∇·y = u, x ∈ Ω, t > 0, = 0, x ∈ Ω, t > 0, y(x, t) = 0, x ∈ Γ, t > 0, y(x, 0) = y0 (x), x ∈ Ω We assume that • The initial value y0 is a given function in V The desired states have to satisfy yT ∈ V and yQ ∈ L2 (Q) • The coefficients αT , αQ are non-negative real numbers, where at least one of them is positive to get a non-trivial objective functional The regularization parameter γ, which measures the cost of the control, is also a positive number • The set of admissible controls, denoted by Uad , is non-empty, convex, closed in L2 (Q) Finding u to minimize J(y, u) means that one want to find a control that satisfies a numerous purposes: the corresponding state is closed to the desired state yQ 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 u is small) We can reformulate the given optimal control problem as follows: Find J(y, u), subject to the state equations yt + νAy + α2 Ayt + B(y, y) = u in L2 (0, T ; V ′ ), y(0) = y0 in V, and the control constraint u ∈ Uad We will exploit the methods and ideas used for the optimal control problem of NavierStokes equations to ours While the existence of the optimal solution is derived in a classical way, we choose a slightly different approach when establishing optimality conditions, which is to calculate the directional derivatives of the objective functional directly instead of using Lagrange multiplier method We choose this approach because we think that our approach seems to be natural since it uses only simple ideas, such as finding an extremum of a realvalued function of one variable Our approach also leads to an explicit form of second-order optimality conditions, which doesn’t perform through second derivatives of Lagrange function as usual, although after some calculations the values of the two forms are equal We also use the same approach when dealing with the two other control problems considered in Chapter and To prove the second-order sufficient optimality conditions, we use the contradiction arguments as that were used by D Wachsmuth (2006) for 2D Navier-Stokes equations, where an optimal control problem of this system with a quadratic objective functional and control box constraints was investigated In three-dimensional space, due to the unique existence of the weak solution of the NavierStokes-Voigt equations, we not face the difficulties that other authors have encountered when studying distributed optimal control problems for Navier-Stokes equations Also for this reason, the optimal problem we are considering shares some similarities with the optimal control problem for the 2D Navier-Stokes equations considered by D Wachsmuth (2006) He worked with box constraint, but we choose the set of admissible controls to be an arbitrary non-empty, convex, closed set When dealing with box constraints, one can prove that each direction in the cone TUad (¯ u) ∩ C(¯ u) is a limit of a sequence of directions in the cone FUad (¯ u) ∩ C(¯ u) (see (2.32) for the definition of C(¯ u)) This is the essential point that D Wachsmuth used to establish the second-order optimality condition However, because of the choice that admissible controls belong to an arbitrary non-empty, convex, closed subset, we have no way to get a similar approximation when the optimal control u¯ does not belong to the interior of the admissible control set Therefore, we can not establish the second-order necessary optimality conditions However, if we choose the admissible control set as D Wachsmuth does, we can also use the above approximation to obtain a similar conditions 2.2 Existence of optimal solutions First, we give some definitions of solutions to the optimal control problem above Definition 2.2.1 (i) A control u¯ ∈ Uad is said to be globally optimal if J(¯ y , u¯) ≤ J(y, u), ∀u ∈ Uad (ii) A control u¯ ∈ Uad is said to be locally optimal if there exists a constant ρ > such that J(¯ y , u¯) ≤ J(y, u) holds for all u ∈ Uad with u − u¯ L2 (Q) ≤ ρ Here, y¯ and y denote the states associated with u¯ and u, respectively The following theorem shows the existence of an optimal solution, which is proved by following a standard technique for optimal control problems Theorem 2.2.2 The optimal control problem admits a globally optimal solution u¯ ∈ Uad with associated state y¯ ∈ W 1,2 (0, T ; V ) Chapter TIME OPTIMAL CONTROL PROBLEM In this chapter, we will consider an optimal control problem for the 3D Navier-Stokes-Voigt equations in bounded domains, where the time needed to reach a desired state plays an essential role and the control variable is the force field in the equations We will first prove the existence of optimal solutions, and then establish the first-order necessary optimality condition and the second-order sufficient optimality conditions The content of this chapter is based on article [CT2] in the List of Publications 3.1 Setting of the problem Let Ω be a bounded domain in R3 with locally Lipschitz boundary Γ and T > be a fixed final time We consider the minimization of the following objective functional: γ J(u) = T ∗ (u) + u 2 W 1,2 (0,T ;L2 (Ω)) , (3.1) where T ∗ (u) := inf {t ∈ [0, T ] : y(t) − ye ≤ δ} Here, ye is the desired state, δ, γ are given constants and y is the state associated to control u, i.e y is the unique weak solution of the following Navier-Stokes-Voigt equations on the interval 11 (0, T ) yt − ν∆y − α2 ∆yt + (y · ∇)y + ∇p ∇·y = u, x ∈ Ω, t > 0, = 0, x ∈ Ω, t > 0, y(x, t) = 0, x ∈ Γ, t > 0, y(x, 0) = y0 (x), x ∈ Ω (3.2) The functional (3.1) provides a balance for ”being near ye as soon as possible” and ”using a small control u”, and we consider the problem of minimizing the time needed to reach the desired state This problem is called a time optimal control problem for the Navier-Stokes-Voigt equations (3.2) The time optimal control problem of differential equations was studied first for ordinary differential equations (J.P LaSalle (1960)) Then such problem was investigated in the context of partial differential equations (PDEs), see e.g V Barbu (1993, 1997), G Wang (2004), D.K Phung, G Wang and X Zhang (2007), E Fernandez-Cara (2012), S Micu (2012), K Kunisch and D Wachsmuth (2013), J Zheng and Y Wang (2013), K Kunisch and L Wang (2013, 2016), S Micu and L.E Temereanca (2014), D.K Phung, L Wang and C Zhang (2014) In these works, existence of optimal solutions and/or first-order necessary optimality conditions were given A usual assumption imposed is to assume the existence of an admissible control such that the corresponding value of the objective functional is finite This is a really difficult question, even in the finite-dimensional case, because it is related to type of controllability of the equations with some kind of control constraints On the other hand, as far as we know, there are very few results on sufficient optimality conditions for the time optimal control problem of PDEs, see for instance H.O Fattorini (2005), G Wang and E Zuazua (2012), K Kunisch and L Wang (2013) We now describe our problem precisely: Let Uad be a non-empty, convex, closed subset in the space W 1,2 (0, T ; L2 (Ω)) Find Min J(u) u∈Uad It is emphasized that taking for the set of admissible controls Uad a subset of W 1,2 (0, T ; L2 (Ω)) is not a common choice and our results are independent of the choice of the norm of the space W 1,2 (0, T ; L2 (Ω)) which is used in the cost function The reasons for our choice of the set of admissible controls will be explained later In order to study the above time optimal problem, we assume that: • The initial value y0 and the desired state ye are given functions in V ; 12 • The coefficient γ, which measures the cost of the control, is a positive number; • The given constant δ is also a positive number; • y0 − ye > δ (it turns to the trivial case if y0 − ye ≤ δ, so we add this assumption) To prove the existence of an optimal solution and establish the first-order necessary optimality conditions, we follow the general lines of the approach by E Fernandez-Cara (2012) However, regarding to these issues, the main difference between E Fernandez-Cara’s and our work is the choice of controls Here our controls are more regular, that is, they belong to W 1,2 (0, T ; L2 (Ω)) instead of L2 (0, T ; L2 (Ω)) as in E Fernandez-Cara’s paper This choice is motivated by two reasons First, it ensures that the state y¯ associated to the optimal solution u¯ is continuously differentiable in time, which turns to be an assumption in Theorem 5.6 of E Fernandez-Cara’s, and our approach for deriving a necessary optimality condition (more precisely, for differentiating T ∗ (¯ u)) cannot be used if y¯ is not continuously differentiable Then, we also need the differentiablity of y¯ to establish the second-order sufficient optimality conditions It is noticed that there is another method (based on a time-reparameterization) for the derivation of optimality conditions For this method, time-optimal control problems are reparameterized on a fixed time interval In this manner, the function T ∗ (u) appears as a parameter of the dynamical system, and one can derive an optimality condition for the time t, see for example K Kunisch, K Pieper and A Rund (2016) To prove the second-order sufficient optimality conditions (see Theorem 3.4.1 below), we use the contradiction arguments as that used in Chapter 3.2 Existence of optimal solutions The existence of a solution to the optimal control problem is stated in the following theorem Theorem 3.2.3 Assume that there exists a control u ∈ Uad such that J(u) < +∞ Then, the optimal control problem has at least one global solution Remark 3.2.5 (i) The assumption of this theorem ensures the existence of an admissible ¯ e , δ) in a finite period of time As control that steers the initial value y0 to the ball B(y we have said above, the existence of such a control is still an open question (ii) The proof follows a classical technique and is very similar to the proof of Theorem 2.2.2 The only remarkable point here is checking of the sequentially weakly lower semicontinuous property of the function T ∗ 13 3.3 First-order necessary optimality conditions Assume that the assumptions of Theorem 3.2.3 are satisfied and let u¯ be a solution to the given time optimal control problem Denote by y¯ the state associated to u¯ Set T = T ∗ (¯ u) Furthermore, we assume that (3.6) < T < T, (( )) y¯(T ) − ye , y¯t (T ) < Remark 3.3.2 If u¯ is an optimal solution then (( (3.7) y¯(T ) − ye , y¯t (T ) )) ≤ In condition (3.7), we only assume that the inequality is strict Consider the following adjoint equations −wt − ν∆w + α2 ∆wt − (¯ y · ∇)w + (∇¯ y )T w + ∇p ∇·w = 0, x ∈ Ω, t > 0, = 0, x ∈ Ω, t > 0, (3.8) = 0, x ∈ Γ, t > 0, w(x, t) w(T ) − α2 ∆w(T ) + ∇q = −∆w0 , where w0 is the element of V defined as follows w0 = − T (¯ y (T ) − ye ) ((¯ yt (T ), y¯(T ) − ye )) With the aid of the adjoint state w, we can now give the first-order necessary optimality condition (condition (3.11) below) Theorem 3.3.4 Let u¯, y¯, T be defined as above Assume that the assumptions of Theorem 3.2.3 and conditions (3.6), (3.7) are satisfied Then, the system (3.8) possesses a unique weak solution w Moreover, we have ∫ T (w(t), m(t))dt + γ(¯ u, m)W 1,2 (0,T ;L2 (Ω)) ≥ 0, ∀m ∈ TUad (¯ u) (3.11) In the sequel, the condition (3.11) is called first-order necessary optimality condition As a special case, the variational inequality ∫ T (w(t), v(t) − u¯(t))dt + γ(¯ u, v − u¯)W 1,2 (0,T ;L2 (Ω)) ≥ 0, is satisfied 14 ∀v ∈ Uad , 3.4 Second-order sufficient optimality conditions Let u¯ be some admissible control and y¯ be the state associated to u¯ Set T = T ∗ (¯ u) Denote by w the adjoint state, i.e w ∈ W 1,2 (0, T ; V ) is the unique weak solution of the system (3.8) Set ∫ { C(¯ u) := m ∈ W 1,2 (0, T ; L (Ω)) : T } (w(t), m(t))dt + γ(¯ u, m)W 1,2 (0,T ;L2 (Ω)) = Let m be in W 1,2 (0, T ; L2 (Ω)) Denote by z the unique weak solution of the linearized equations zt − ν∆z − α2 ∆zt + (z · ∇)¯ y + (¯ y · ∇)z + ∇q = m, x ∈ Ω, t > 0, ∇·z = 0, x ∈ Ω, t > 0, z(x, t) = 0, x ∈ Γ, t > 0, z(x, 0) = 0, x ∈ Ω Set S(m) = − (( Σ(m) = − − T ∫ ((z(T ), y¯(T ) − ye )) , ((¯ yt (T ), y¯(T ) − ye )) y¯tt (T )S (m) + zt (T )S(m), y¯(T ) − ye ((¯ yt (T ), y¯(T ) − ye )) T b(z(t), z(t), w(t))dt − )) y¯t (T )S(m) + z(T ) 2((¯ yt (T ), y¯(T ) − ye )) The following theorem gives a sufficient optimality condition for the optimal control problem (condition (3.31)) Furthermore, we can prove that this condition implies a quadratical growth with respect to the W 1,2 -norm in a W 1,2 -neighborhood of the optimal solution (see (3.32)) Theorem 3.4.1 Assume that T ∈ (0, T ), ((¯ yt (T ), y¯(T ) − ye )) < 0, and the map t → y¯(t) is twice Fréchet differentiable at T We assume furthermore that the two following assumptions are satisfied: (i) The first-order necessary optimality condition: ∫ T (w(t), m(t))dt + γ(¯ u, m)W 1,2 (0,T ;L2 (Ω)) ≥ 0, ∀m ∈ TUad (¯ u); (ii) It holds S (m) + 2T Σ(m) + γ m ( ) for every m ∈ TUad (¯ u) ∩ C(¯ u) \{0} 15 W 1,2 (0,T ;L2 (Ω)) > 0, (3.31) Then, there exist ε > and ρ > such that the inequality J(u) ≥ J(¯ u) + ε u − u¯ holds for every u ∈ Uad with u − u¯ W 1,2 (0,T ;L2 (Ω)) W 1,2 (0,T ;L2 (Ω)) ≤ ρ, which deduces that u¯ is a locally optimal control The condition (ii) is called second-order sufficient optimality condition 16 (3.32) Chapter OPTIMAL BOUNDARY CONTROL PROBLEM In this chapter, we consider an optimal boundary control problem for the 3D Navier-StokesVoigt 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 Dircihlet 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 condition 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 Publications, which has been submitted and has not been published 4.1 Setting of the problem Let Ω be a bounded domain in R3 with C boundary Γ We study the following optimal boundary control problem: (PL ) Minimize the cost functional γ1 L(g) = ∫ T ∫ γ2 |y − yd | dxdt + Ω ∫ γ3 |y(T ) − yT | dx + Ω ∫ T ( g H1/2 (Γ) + gt H1/2 (Γ) ) dt, where g is the boundary control variable and the state variable y is a weak solution to the 17 following 3D Navier-Stokes-Voigt equations on the interval (0, T ) yt − ν∆y − α2 ∆yt + (y · ∇)y + ∇p = in Q ∇ · y = in Q, y=g y(0) = y on Γ × (0, T ), in Ω To study the above optimal boundary control problem, we assume that • The initial velocity y0 is a given function in H1 (Ω), which satisfies ∇·y0 = 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 defined as follows ∫ { } 1,2 1/2 Ad = g ∈ W (0, T ; H (Γ)) : g(0) = y0 on Γ and g·nds = for a.e t ∈ [0, T ] Γ 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, M.D Gunzburger, L.S Hou and Th.P Svobodny (1991), M Berggren (1998), L.S Hou and S.S Ravindran (1998), A.V Fursikov, M.D Gunzburger and L.S Hou (1998, 2005), M Hinze (2002), J.C De Los Reyes and K Kunisch (2005), C John and D Wachsmuth (2009), M Colin and P Fabrie (2010), M Holst, E Lunasin and G Tsogtgerel (2010) 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 18 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 G Bärwolff and M Hinze (2006, 2008), G Bornia, M Gunzburger and S Manservisi (2013), C Cavaterra, E Rocca and H Wu (2017) for recent works on optimal boundary control of the 2D Boussinesq system, the MHD system and the 2D simplified Ericksen-Leslie system 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) and following the general lines of the approach used in M.D Gunzburger and S Manservisi (2000), 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 C 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 C , 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 C 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 and 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 Dirichlet boundary conditions yt − ν∆y − α2 ∆yt + (y · ∇)y + ∇p = 0, x ∈ Ω, t > 0, ∇·y = 0, x ∈ Ω, t > 0, (4.3) y(x, t) = g(x, t), x ∈ Γ, t > 0, y(x, 0) = y0 (x), x ∈ Ω 19 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 > is the kinematic viscosity coefficient and α = is the length-scale parameter characterizing the elasticity of the fluid 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 C 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, g · nds = for a.e t ∈ [0, T ] Γ 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 + α2 Ayt + B(y, y) + grad p ∇ · y(t) = in H−1 (Ω), = in Ω, y(t) = g(t) on Γ, y(0) = y0 , for a.e t ∈ [0, T ] Here, 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, then there exists a constant C = C(M ) such that y W 1,2 (0,T ;H1 (Ω)) ≤ C 4.3 Existence of optimal solutions Theorem 4.3.3 The problem (PL ) has at least one globally optimal solution 4.4 First-order and second-order necessary optimality conditions 4.4.1 First-order necessary optimality conditions Set 20 ∫ { 1,2 1/2 A0 = h ∈ W (0, T ; H (Γ)) : h(0) = on Γ and h · nds = } for a.e t ∈ [0, T ] Γ 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 Theorem 4.4.1 If g¯ ∈ Ad is an optimal solution to the optimal problem (PL ) then g¯ satisfies the following condition ∫ ∫ T γ3 T (¯ g , h)H1/2 (Γ) dt + γ3 (¯ gt , ht )H1/2 (Γ) dt ∫ T − τ (t), h(t) H−1/2 (Γ),H1/2 (Γ) dt − π, h(T ) H−1/2 (Γ),H1/2 (Γ) = 0, ∀h ∈ A0 , where τ ∈ L2 (0, T ; H−1/2 (Γ)), π ∈ 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) π, 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 ˜ y , w) + grad σ = γ1 (¯ −wt + νAw − α2 Awt − B(¯ y , w) + B(¯ y − yd ) in H−1 (Ω) for a.e t ∈ [0, T ], ∇ · w(t) = in Ω, for a.e t ∈ [0, T ], w(t) = on Γ, for a.e t ∈ [0, T ], w(T ) + α2 Aw(T ) + grad κ = γ2 (¯ y (T ) − yT ) in H−1 (Ω) ˜ y , w), v Here, B(¯ H−1 (Ω),H10 (Ω) (4.27) := B(v, y¯, w) Remark 4.4.2 The value of the right-hand sides in (4.25) and (4.26) is independent of the v chosen 21 4.4.2 Second-order necessary optimality condition 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 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 the following equations zt + νAz + α2 Azt + B(¯ y , z) + B(z, y¯) + grad p1 ∇ · z(t) = in H−1 (Ω) for a.e t ∈ [0, T ], = 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) Set ∫ ∫ T |z| dt + γ2 |z(T )| + γ3 h q(h) := γ1 2 W 1,2 (0,T ;H1/2 (Γ)) −2 T b(z, z, w)dt Then we have q(h) ≥ 0, ∀ h ∈ A0 (4.32) 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 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 the equations (4.28) If g¯ satisfies the first-order necessary condition and the following assumption, in the sequel called the second-order sufficient condition: ∫ T |z|2 dt + γ2 |z(T )|2 + γ3 h q(h) := γ1 W 1,2 (0,T ;H1/2 (Γ)) ∫ T b(z, z, w)dt > for every h ∈ A0 \{0}, (4.37) −2 then there exist ε > and ρ > such that L(g) − L(¯ g ) ≥ ε g − g¯ 22 W 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 23 CONCLUSION AND FUTURE WORK Conclusion In this thesis, a number of optimal control problems governed by 3D Navier-Stokes-Voigt equations have been investigated The main contributions of this thesis are to prove the existence of optimal solutions and the 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 secondorder 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 NavierStokes-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 Optimal control of Navier-Stokes-Voigt equations with bang-bang controls Optimal control of Navier-Stokes-Voigt equations with measure valued controls 24 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 NavierStokes-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-StokesVoigt equations, submitted to Optimization (2019) ... control of Navier- Stokes- Voigt equations with bang-bang controls Optimal control of Navier- Stokes- Voigt equations with measure valued controls 24 LIST OF PUBLICATIONS Published papers [CT1] C.T Anh. .. 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 NavierStokes -Voigt equations,... optimal control problem governed by Navier- StokesVoigt equations Chapter presents results on the boundary optimal control problem governed by NavierStokes -Voigt equations The results obtained
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