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

partial differential equations in fluid mechanics is started studying in 1980s by Fursikov when heestablished several theorems about the existence of optimal solutions to some optimal co

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: 9 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 HanoiNational University of Education on ………

This thesis can be found in:

- The National Library of Vietnam;

- Library of Hanoi National University of Education

1 Literature survey and motivation

The Navier-Stokes-Voigt equations was first introduced by Oskolkov (1973) as a model ofmotion 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 theglobal well-posedness

In the past years, the existence and long-time behavior of solutions to the Voigt equations has attracted the attention of many mathematicians In bounded domains orunbounded domains satisfying the Poincaré inequality, there are many results on the existenceand long-time behavior of solutions in terms of existence of attractors for Navier-Stokes-Voigtequations, 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 Manholiand A.L Silvestre (2016) In the whole space R3, the existence and decay rates of solutionshave been studied recently (see C Zhao and H Zhu (2015), C.T Anh and P.T Trang (2016),C.J Niche (2016))

Navier-Stokes-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 ordinarydifferential equations is of interest for its applications in many fileds such as aviation andspace technology, robotics, the control of chemical processes However, in many situations,the processes to be optimized may not be modeled by ordinary differential equations, insteadpartial 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 heestablished several theorems about the existence of optimal solutions to some optimal controlproblems governed by Navier-Stokes equations.

One of the most important objectives of optimal control theory is to obtain necessary (orpossibly necessary and sufficient) conditions for the control to be an extremum Since thepioneering work of Abergel and Temam in 1990, where the first optimality conditions to theoptimal control problem for fluid flows can be found, this matter has been studied very inten-sively 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 brieflyreview 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 distributedcontrol 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 thecase of the present of state constraints, the problem was investigated by G Wang (2002) andLiu (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 theNavier-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 Fab-rie (2010) in the nonstationary case We can see also the habilitation by M Hinze (2002), thetheses by M Sandro (1997), D Wachsmuth (2006) and references therein, for other works onoptimal control of Navier-Stokes equations

As described above, the unique existence and long-time behavior of solutions to the Stokes-Voigt equations, as well as the optimal control problems for fluid flows, in particularfor Navier-Stokes equations, have been considered by many mathematicians However, to thebest of our knowledge, the optimal control of 3D Navier-Stokes-Voigt equations has not been

Navier-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 thesispresents results on some optimal control problems for this equations in the three-dimensionalspace (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 Stokes-Voigt equations, where the objective functional is of quadratic form and the distributedcontrol belongs to a non-empty, closed, convex subset,

(P2) The time optimal control problem of the nonstationary three dimensional Stokes-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 Stokes-Voigt equations, where the objective functional is of quadratic form and the boundarycontrol variable has to satisfy some compatibility conditions

3 The structure and results of the dissertation

The dissertation has four chapters and a list of references

Chapter 1 collects several basic concepts and facts on Sobolev spaces and partial differentialequations associated with solutions of Navier-Stokes-Voigt equations as well as some auxiliaryresults

Chapter 2 presents results on the distributed optimal control problem governed by Stokes-Voigt equations

Navier-Chapter 3 provides results on the time optimal control problem governed by Voigt equations

Stokes-Chapter 4 presents results on the boundary optimal control problem governed by Stokes-Voigt equations

Navier-The results obtained in Chapters 2, 3 and 4 are answers for the problems (P1), (P2), (P3),respectively.

Chapter 2 and Chapter 3 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

4 is the content of the work [CT3] in the List of Publications, which has been submitted forpublication

1 Function spaces

• L p and Sobolev spaces: L p (Ω), W m,p (Ω), H m (Ω) (m ∈ N), H s (Ω) (s ∈ R), s ≥ 0,

H s(Γ)

• Spaces of abstract functions: L p (0, T ; X), 1 ≤ p ≤ ∞, W 1,p (0, T ; X).

2 Some useful inequalities: Hölder inequality, Poincaré’s inequality, Gronwall’s inequality,

Young’s inequality with ϵ.

3 Continuous and compact imbeddings: Rellich-Kondrachov theorem, imbeddings in stract function spaces

ab-4 Operators: the trilinear form, the operator grad, the continuous linear operators andbilinear operators

5 The unique existence of solutions to the nonstationary 3D Navier-Stokes-Voigt equationswith homogeneous Dirichlet boundary conditions

6 Some auxiliary results on linearized equations: the definition of weak solutions and someproperties of weak solutions

7 Some definitions in Convex Analysis: the normal cone and the polar cone of tangents of

a convex subset in a Hilbert space

2.1 Setting of the problem

Let Ω be a bounded domain in R3 with locally Lipschitz boundary Γ and T > 0 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

We assume that

• The initial value y0 is a given function in V The desired states have to satisfy y T ∈ V

and y Q ∈ L2(Q).

• The coefficients α T , α Q 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 U ad, 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 y Q during the whole period of

time (0, T ) and closed to the desired state y T 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

min J (y, u),

subject to the state equations

Navier-is to calculate the directional derivatives of the objective functional directly instead of usingLagrange multiplier method We choose this approach because we think that our approachseems to be natural since it uses only simple ideas, such as finding an extremum of a real-valued function of one variable Our approach also leads to an explicit form of second-orderoptimality 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 usethe same approach when dealing with the two other control problems considered in Chapter

3 and 4 To prove the second-order sufficient optimality conditions, we use the contradictionarguments 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 controlbox constraints was investigated.

In three-dimensional space, due to the unique existence of the weak solution of the Stokes-Voigt equations, we do not face the difficulties that other authors have encounteredwhen studying distributed optimal control problems for Navier-Stokes equations Also for thisreason, the optimal problem we are considering shares some similarities with the optimal controlproblem for the 2D Navier-Stokes equations considered by D Wachsmuth (2006) He workedwith 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 thecone T U ad(¯ ∩ C(¯u) is a limit of a sequence of directions in the cone F U ad(¯ ∩ C(¯u) (see (2.32)

Navier-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 controlsbelong to an arbitrary non-empty, convex, closed subset, we have no way to get a similarapproximation 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 theabove 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 ∈ U ad is said to be globally optimal if

J (¯ y, ¯ u) ≤ J(y, u), ∀u ∈ U ad

(ii) A control ¯u ∈ U ad is said to be locally optimal if there exists a constant ρ > 0 such that

J (¯ y, ¯ u) ≤ J(y, u)

holds for all u ∈ U ad with ku − ¯ukL2(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 lowing a standard technique for optimal control problems

fol-Theorem 2.2.2 The optimal control problem admits a globally optimal solution ¯ u ∈ U ad with associated state ¯ y ∈ W 1,2 (0, T ; V ).

2.3 First-order necessary optimality conditions

Assume that ¯u is a locally optimal control with associated state ¯ y Let us introduce the

The first-order necessary optimality conditions are given in the theorem below

Theorem 2.3.3 Let ¯ u be a locally optimal control with associated state ¯ y Then there exists w, which is the unique weak solution of the adjoint equations (2.9) on the interval (0, T ) Moreover,

Q (w + γ ¯ u) · hdxdt ≥ 0, ∀h ∈ T U ad(¯u). (2.11)

Here, T U ad(¯u) is the polar cone of tangents of U ad at ¯ u As a special case, the variational

Q (w + γ ¯ u) · (v − ¯u) ≥ 0, ∀v ∈ U ad

is satisfied.

2.4 Second-order sufficient optimality conditions

A sufficient condition for a control to be an optimal solution is given in the theorem below(condition (2.30)) Furthermore, by this theorem we achieve quadratical growth with respect

to the L2-norm of the objective functional in a L2-neighborhood of the optimal solution (see(2.33))

Theorem 2.4.1 Let ¯ v = (¯ y, ¯ u) be the admissible pair and suppose that ¯ v satisfies together with the adjoint state w the first-order necessary optimality conditions, i.e the equations (2.9) and the inequality (2.11) Furthermore, we assume that the pair ¯ v = (¯ y, ¯ u) satisfies the following assumption, in the sequel called the second-order sufficient condition: It holds

for all h ∈ (T U ad(¯ ∩ C(¯u))\{0}, where z is the unique weak solution of the following linearized equations on the interval (0, T )

Remark 2.4.2 If we only need the directional differentiable property of the control-to-state

mapping u 7→ y to establish the first-order necessary optimality conditions, a more regular

property of this mapping is required to prove the second-order sufficient conditions - that isFréchet differentiable up to order 2 Although we did not state explicitly, we have proved thisproperty in the proof above

Remark 2.4.3 In this chapter, a distributed optimal control problem governed by 3D

Navier-Stokes-Voigt equations in the case of absence of state constraints was investigated In the case

of the presence of pointwise control-state constraints, a similar optimal control problem wasstudied recently by N H Son and T M Nguyet (2019)

of optimal solutions, and then establish the first-order necessary optimality condition and thesecond-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 > 0 be a fixed

final time We consider the minimization of the following objective functional:

T ∗ (u) := inf {t ∈ [0, T ] : ky(t) − y e k ≤ δ}

Here, y e 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