Analysis of the Velocity Tracking Control Problem for the 3D Evolutionary Navier-Stokes Equations

. The velocity tracking problem for the evolutionary Navier–Stokes equations in three dimensions is studied. The controls are of distributed type and are submitted to bound constraints. The classical cost functional is modiﬁed so that a full analysis of the control problem is possible. First and second order necessary and suﬃcient optimality conditions are proved. A fully discrete scheme based on a discontinuous (in time) Galerkin approach, combined with conforming ﬁnite element subspaces in space, is proposed and analyzed. Provided that the time and space discretization parameters, τ and h, respectively, satisfy τ ≤ Ch 2 , the L 2 (Ω T ) error estimates of order O ( h ) are proved for the diﬀerence between the locally optimal controls and their discrete approximations. Finally, combining these techniques and the approach of Casas, Herzog, and Wachsmuth [ SIAM J. Optim., 22 (2012), pp. 795–820], we extend our results to the case of L 1 (Ω T ) type functionals that allow sparse controls.

In these equations, y = (y 1 , y 2 , y 3 ) is the velocity field of the fluid, p is the pressure, ν > 0 is the viscosity, f and u represent the body forces, and y 0 denotes the initial velocity.We can control the system through the forces u.
For two-dimensional (2D) flows, Ω ⊂ R 2 , an existence and uniqueness theorem for a solution of (1.1) has been known for a long time.The study is more complicated for the 3D flows, Ω ⊂ R 3 .In this case, two different types of solutions are distinguished: weak and strong.Under minimal assumptions, the existence of weak solutions y ∈ L 2 (0, T ; H 1 0 (Ω)) ∩ C w ([0, T ], L 2 (Ω)) can be proved.However, the uniqueness is still an open problem, unless the data (f + u, y 0 ) are small enough or final time T is sufficiently small; see, for instance, Temam [21].
A strong solution y is a weak solution that additionally belongs to L 8 (0, T ; L 4 (Ω)).In the 3D case, there exists at most one strong solution of (1.1), but its existence has not been proved until now.In the 2D case, weak and strong solutions coincide, and hence we have existence and uniqueness of a solution.
In the classical tracking control problem, the cost functional involves the L 2 norm of y − y d , where y d is the given target field.In the case of 3D flows, due to the lack of uniqueness of weak solutions or of the existence of strong solutions, the analysis is very complicated.Actually, we cannot prove first and second order optimality conditions, and error estimates for the discretization of the control problem is an open issue.As a consequence, most of the studies devoted to the control problems associated to the equations (1.1) assume that Ω ⊂ R 2 [1,5,9,11,12,14,22].
Hereafter, we assume Ω ⊂ R 3 , but we do not require the data to be small because this is not a realistic assumption.In this paper, we deal with strong solutions, which allows us to carry out a complete analysis of the control problem.However, to prove the existence of an optimal control with an associated strong solution, we have to consider a convenient cost functional.Instead of setting the L 2 norm of y − y d in the cost functional as usual, we consider the functional where λ > 0, γ ≥ 0, and y Ω ∈ L 2 (Ω) to be fixed more precisely later.
The goal is to minimize the J (u, y) in a certain class of functions, where (u, y) satisfies (1.1).If y is a weak solution of (1.1) such that J (u, y) < +∞, then y is a strong solution.With this formulation we can prove the existence of an optimal control and get the first and second order optimality conditions.Moreover, following the approach of [5], we obtain the same error estimates proved there for the numerical discretization of the control problem in three dimensions.In particular, we prove estimates of order O(h) for the difference between locally optimal controls and their discrete approximations, for τ ≤ Ch 2 , when τ, h denote the time and space discretization parameters, respectively.In addition, we also show that any strict local minimum can be approximated by a sequence of local minima of the discrete optimal control problems.Estimates of order O(h) are also obtained for the state and adjoint variables, and they are optimal in terms of the regularity on the given data.The cost functional (1.2) was introduced in [4, p. 95], where existence of optimal controls and first order optimality conditions were studied for the continuous problem.It is worth noting that it plays a crucial role also in the development of error estimates when combined with the discontinuous (in time) Galerkin framework.One of the main features of discontinuous (in time) Galerkin machinery is that the discrete scheme inherits regularity properties of the corresponding continuous problem due to its heavily implicit nature.In particular, the fact that the cost functional (1.2) yields strong solutions is an important asset at the fully discrete level, since the enhanced regularity is also inherited by the discrete state and adjoint variables.As a consequence, it allows the numerical analysis of the control to state and adjoint mappings similarly to the 2D case and as in [5].
Furthermore, we also discuss the case of sparse controls.To enforce sparsity of the controls, i.e., the localization of the controls in a small region of the domain, we modify our functional in a way to include the L 1 (Ω T ) norm.It is well understood that the inclusion of the L 1 norm in the cost functional yields sparse controls (see, for instance, [6,7,13,20,23]).In [6] necessary and sufficient second order optimality conditions are derived for a semilinear elliptic control problem.Adopting the techniques of [6] in our optimal control setting for the 3D evolutionary Navier-Stokes case, we also Downloaded 03/22/16 to 193.144.185.28.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.phpprove error estimates for the difference between the locally optimal controls and their discrete approximations based on the discontinuous (in time) Galerkin framework.

The state equation. Assumptions and preliminary results.
Hereafter Ω denotes a bounded open subset in R 3 with a Lipschitz boundary Γ.We assume that either Ω is convex or Γ is of class C 1,1 .The outward unit normal vector to Γ at a point x ∈ Γ is denoted by n(x).Given 0 < T < +∞, we set Ω T = (0, T ) × Ω and Σ T = (0, T ) × Γ.As in [5], we denote the Sobolev spaces H 1 ) , and W s,p (Ω) = W s,p (Ω; R 3 ) for 1 ≤ p ≤ ∞ and s > 0. We also consider the spaces of integrable functions ), and, for a given Banach space X, L p (0, T ; X) will denote the integrable functions defined in (0, T ) and taking values in X endowed with the usual norm.Following Lions and Magenes [17,Vol. 1] we put Endowed with the standard norm, the space We introduce the usual spaces of divergence-free vector fields, Throughout this paper, we will assume that f , u ∈ L 2 (0, T ; L 2 (Ω)) and y 0 , y Ω ∈ Y.An element y ∈ W(0, T ) is said to be a weak solution of (1.1) if and the following energy inequality holds: (2.2) where • and (•, •) denote the norm and the inner product, respectively, in L 2 (Ω), and a : The existence of a weak solution is well known; see, for instance, Ladyzhenskaya [15], Lions [16], Temam [21], etc.However, the uniqueness of a weak solution is an Downloaded 03/22/16 to 193.144.185.28.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.phpopen question so far.We say that y is a strong solution of (1.1) if it is a weak solution and y ∈ L 8 (0, T ; L 4 (Ω)).It is well known that (1.1) does not have more than one strong solution.Strong solutions satisfy the energy equality instead of the energy inequality (2.2).Hence, they seem to be physically more significant than weak solutions.Unfortunately there is no existence result for strong solutions.
Once y is found from (2.1), the existence of the pressure p ∈ D (Ω T ) can be proved in such a way that (y, p) is a solution of (1.1).
We finish this section by collecting some results, whose proofs can be found in [4].
Moreover, p is unique up to the addition of a function of L 2 (0, T ).Finally, there exists an increasing function η : [0, +∞) −→ [0, +∞) depending only on Ω and ν such that where η is as in Theorem 2.1.The proof of this corollary follows from Theorem 2.1 taking g = y and e = 0. Corollary 2.3.If problem (1.1) has a strong solution for some element ū ∈ L 2 (0, T ; L 2 (Ω)), then there exists an open neighborhood A 0 of ū in L 2 (0, T ; L 2 (Ω)) such that (1.1) has a strong solution for every u ∈ A 0 .Moreover, the mapping , then z v and z v1v2 are the unique strong solutions of the following problems: , which are unique up to the addition of a function of L 2 (0, T ).Downloaded 03/22/16 to 193.144.185.28.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Proof.Here we modify the proof of [4,Cor. 4.2.2] to correct a small mistake.First, we observe that the solution y ∈ H 2,1 (Ω T ) ∩ C([0, T ], Y) of (2.3) satisfies that y ∈ L 2 (0, T ; H).This is an immediate consequence of the proof of [4,Thm. 4.2.1].Indeed, a Galerkin approach is followed there to approximate y by using a special basis {ψ j } ∞ j=1 of H 2 (Ω) ∩ Y.The approximations take the form y m = m j=1 g j (t)ψ j , and y m = m j=1 g j (t)ψ j belonging to L 2 (0, T ; H) converges weakly to y in L 2 (0, T ; H).Let us consider the space Endowed with the norm of H 2,1 (Ω T ), this is a Hilbert space.Now, we define the mapping Therefore, ∂F ∂y (y, u)z = (v, z 0 ), with (v, z 0 ) ∈ L 2 (0, T ; H) × Y, if and only if  (y, u) : H −→ L 2 (0, T ; H)×Y is an isomorphism for every (y, u) ∈ H × L 2 (0, T ; L 2 (Ω)).Therefore, if problem (1.1) has a strong solution ȳ for a given control ū, then F (ȳ, ū) = (0, 0), and applying the implicit function theorem we deduce the existence of an open neighborhood A 0 ⊂ L 2 (0, T ; L 2 (Ω)) of ū and a mapping G : A 0 −→ H of class C ∞ such that F (G(u), u) = (0, 0) for every u ∈ A 0 .The rest of the proof is immediate.
Remark 2.4.As a consequence of Corollary 2.2, we deduce that the set of controls u ∈ L 2 (0, T ; L 2 (Ω)) for which there exists a strong solution y u is open.Hereafter, this set will be denoted by A. It is known that A is dense in L 2 (0, T ; L 2 (Ω)) with Downloaded 03/22/16 to 193.144.185.28.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.phprespect to the norm of L s (0, T ; L q (Ω)) for s, q ∈ (1, +∞) such that 4 < 2 s + 3 q ; see [18].In particular we have that for any u ∈ L 2 (0, T ; L 2 (Ω)) and any ε > 0, there exists Remark 2.5.We have seen that (1.1) admits the variational formulation given in (2.1).Analogously, (2.6) and (2.7) can be formulated in a variational form as follows: Remark 2.6.The use of P H in the definition of F given in the proof of Corollary 2.3 is necessary.In principle, one could consider the mapping Then, we have that However, ∂F ∂y (y, u) is not an isomorphism.Indeed, observe that v and P H v lead to the same solution y of the above system because (v, ψ) = (P H v, ψ) for every ψ ∈ Y.This situation is avoided by introducing the projection P H in the definition of F .It is enough to observe that if u, v ∈ L 2 (0, T ; H) and (v, ψ) = (u, ψ) for every ψ ∈ Y, then u = v.The reader is referred to [21,Chap. 1] for details.

The control problem.
In this section, we define in a precise way the optimal control problem, we prove the existence of at least one solution, and we derive the first and second order optimality conditions.First, we define the set of admissible controls as follows: where −∞ ≤ α j < β j ≤ +∞ for 1 ≤ j ≤ 3, and A is defined in Remark 2. 4. In what follows, we will make the following assumption: Now, we consider the functional J : A −→ R defined by where y u = G(u) is the state associated to u, the target y d ∈ L 14 (0, T ; L 6 (Ω)), y Ω ∈ H 1 0 (Ω), γ ≥ 0, and λ > 0. It is obvious that J(u) = J (u, y u ) ∀u ∈ A. The regularity assumed for y d is needed in the proof of Theorem 3.1.Finally, we define the control problem (P) min J(u), u ∈ U ad .
As an immediate consequence of Corollary 2.3 we have the following differentiability properties of J. Theorem 3.1.The cost functional J : A −→ R is of class C ∞ , and for every u ∈ A and v ∈ L 2 (0, T ; L 2 (Ω)) we have where uniquely defined up to the addition of a function of L 2 (0, T ).
Observe that the assumption on y d and the regularity y u ∈ H 2,1 (Ω T ) imply that and hence Theorem 2.1 shows that ϕ u ∈ H 2,1 (Ω T ).
The variational formulation of (3.5) is written as follows: The next theorem establishes the existence of at least one solution for (P), as well as the first order optimality conditions satisfied by any local minimum of (P).
Theorem 3.2.Under assumption (3.1), (P) has at least one solution.Moreover, for any local solution ū, there exist ȳ, φ ∈ H Moreover, the regularity property Proof.Let us prove the existence of a solution.Since U ad is nonempty, there exists a minimizing sequence {u k } ∞ k=1 ⊂ U ad of (P).Let us set From the definition of the functional J we deduce ) and L 8 (0, T ; L 4 (Ω)), respectively.By taking subsequences, if necessary, we can assume that u k ū and y k ȳ weakly in L 2 (0, T ; L 2 (Ω)) and L 8 (0, T ; L 4 (Ω)), respectively.From (2.5), we deduce that y k ȳ weakly in H 2,1 (Ω T ).Using the compactness of the embedding H 2,1 (Ω T ) ⊂ L 8 (0, T ; L 4 (Ω)), it is easy to pass to the limit in the state equation and to deduce that ȳ is a strong solution of (1.1) with some pressure p ∈ L 2 (0, T ; H 1 (Ω)).Hence, we have that ū ∈ A. Moreover, it is immediate that ū ∈ U α,β .Therefore, ū ∈ U ad and The optimality system (3.7)-(3.9)can be proved in the standard way by using the expression of J given in (3.3).Finally, the regularity of ū is a consequence of the embedding φ which follows from (3.9).Now, we carry out the second order analysis of (P).Since this control problem is not convex, some second order conditions are required for the numerical analysis of (P).To write the second order conditions, we need to define the cone of critical directions.To this end, let us introduce the function We also deduce as usual from (3.9), for almost all (t, x) ∈ Ω T and j = 1, 2, As for the 2D flows, we have the following second order necessary and sufficient conditions; see [5].

Approximation of the control problem (P).
In this section, Ω is assumed to be convex.We consider a family of triangulations {K h } h>0 of Ω, defined in the standard way.To each element K ∈ K h , typically a tetrahedron or a hexahedron, we associate two parameters h K and K , where h K denotes the diameter of the set K and K is the diameter of the largest ball contained in K. Define the size of the mesh by h = max K∈K h h K .We also assume that the following standard regularity assumptions on the triangulation hold: (i) There exist two positive constants K and δ K such that hK K ≤ K and h hK ≤ δ K ∀K ∈ K h and ∀h > 0.
(ii) Define Ω h = ∪ K∈K h K, and let Ω h and Γ h denote its interior and its boundary, respectively.We assume that the vertices of K h placed on the boundary Γ h are points of Γ.
Since Ω is convex, from the last assumption we have that Ω h is also convex.Moreover, we assume that 0 (Ω) formed by piecewise polynomials in Ω h and vanishing in Ω \ Ω h .We make the following assumptions on these spaces. ( (A3) The subspaces Z h and Q h satisfy the following inf-sup condition: ∃c > 0 such that (4.4) inf where b : These assumptions are satisfied by the usual finite elements considered in the discretization of Navier-Stokes equations; see [10,Chap. 2].
We also consider a subspace Y h of Z h defined by and we set It is well known that, under the previous assumptions, given an element z ∈ , where C is independent of h and z.Moreover, from this estimate and an inverse inequality it is easy to prove that for the usual finite elements considered in the discretization of Navier-Stokes equations, the following estimate holds: Hence, in addition to assumptions (A1)-(A3), we will assume We proceed now with the discretization in time.Let us consider a grid of points We make the following assumption: We have that the functions of Y σ , U σ , and Q σ are piecewise constant in time.We will look for the discrete controls in the space U σ .An element of this space can be written in the form (4.7) where χ n and χ K are the characteristic functions of (t n−1 , t n ) and K, respectively.Therefore, the dimension of U σ is 3N τ N h , where N h is the number of elements in K h .
In U σ we consider the convex subset On the other hand, the elements of Y σ can be written in the form (4.8) where χ n is as above.For every discrete state y σ , we will fix y σ (t n ) = y n,h so that y σ is continuous on the left.In particular, we have y σ (T ) = y σ (t Nτ ) = y Nτ ,h .
To define the discrete control problem, we have to consider the numerical discretization of the state equation (1.1) or, equivalently, (2.1).We achieve this goal by using a discontinuous time-stepping Galerkin method, with piecewise constants in time and conforming finite element spaces in space.For any u ∈ L 2 (0, T ; L 2 (Ω)), the discrete state equation is given by (4.9) Applying Brouwer's theorem, one can easily prove the existence of at least one solution y σ ∈ Y σ of (4.9) for every u ∈ L 2 (0, T ; L 2 (Ω)).The uniqueness is a more delicate issue.For the uniqueness we need the following extra assumption: (4.12) ∃C 0 > 0 such that τ ≤ C 0 h 2 ∀σ = (τ, h).Downloaded 03/22/16 to 193.144.185.28.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.phpFrom [5,Cor. 4.9] we obtain the following result.Theorem 4.1.Under assumption (4.12), for any bounded set K⊂L 2 (0, T ; L 2 (Ω))× L 8 (0, T ; L 4 (Ω)) there exists a constant τ K > 0 such that for any u ∈ A with (u, y u ) ∈ K, (4.9) has a unique solution y σ (u) ∈ Y σ for every τ < τ K .Moreover, the sequence {y σ (u)} τ <τK is bounded in L ∞ (0, T ; H 1 0 (Ω h )).Finally, if u, v ∈ A and (u, y u ), (v, y v ) ∈ K, then there exists a constant C K > 0 such that The proof given in [5] for the 2D case is also valid for the 3D case assuming that the controls belong to A and the pair control-state belongs to K. It is enough to take into account the estimate (2.5).Now we define the discrete control problem as follows: where For instance, y Ωh can be defined as the projection of y Ω .
Now we prove a converse result.The next theorem states that any strict local minimum of (P) can be approximated by a sequence of local minima of problems (P σ ).
Proof.Since ū is a strict local minimum of (P), there exists ε > 0 such that ū is the unique solution of the control problem where ε is taken sufficiently small so that Bε (ū) ⊂ A. Along this proof, every element u σ ∈ U σ will be extended to Ω × (0, T ) by setting u σ (t, x) = ū(t, x) for almost every (t, x) ∈ (0, T ) × (Ω \ Ω h ).Now we consider the discrete control problem From the continuity and coercivity of J σ and the fact that the set of admissible points is closed, it is enough to prove that (Q σ ) has at least one admissible point to deduce the existence of a solution.To this end, we consider the L 2 (0, T ; L 2 (Ω))-projection u σ of ū.It is obvious that u σ ∈ U σ,ad for every σ and ū − u σ L 2 (0,T ;L 2 (Ω)) → 0 as σ → 0; then there exists μ 1 > 0 such that u σ ∈ Bε (ū) for every |σ| ≤ μ 1 , which shows that u σ is an admissible point of (Q σ ) for every |σ| ≤ μ 1 .Let (ū σ , ȳσ ) be a solution of (Q σ ).Then, arguing as in the proof of Theorem 4.2, we obtain that (4.15) and (4.16) hold.Furthermore, applying Theorem 4.1, we deduce the existence of 0 < μ0 ≤ μ 1 such that (4.9) has a unique solution for every ūσ with |σ| ≤ μ0 .Using (4.15) and taking μ0 sufficiently small, we have that ūσ ∈ B ε (ū).Hence, (ū σ , ȳσ ) is a local minimum of (P σ ), and J σ attains the minimum value in ( Bε (ū) ∩ U σ,ad ) × Y σ at this point.
In the rest of this section, ū will denote a local (or global) minimum of (P) with associated state and adjoint state ȳ and φ, respectively.In addition, {(ū σ , ȳσ )} |σ|≤μ0 will be a sequence of local (or global) minima of problems (P σ ) satisfying (4.15) and (4.16).The goal is to get the rate of convergence of (ū− ūσ , ȳ − ȳσ , φ− φσ ), where φσ denotes the discrete adjoint state associated to ūσ .To this end we assume that (3.19) and (4.12) hold.The first step in the derivation of the error estimates is to prove Downloaded 03/22/16 to 193.144.185.28.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.phpthat φσ is well defined and to write the first order optimality conditions satisfied by (ū σ , ȳσ ).Since ū ∈ U ad ⊂ A, we know that ȳ ∈ H 2,1 (Ω T ).Let B(ȳ) denote the unit ball centered at ȳ in H 2,1 (Ω T ).Since G : Hence, Theorem 4.1 implies the existence of τ K > 0 such that (4.9) has a unique solution for every u ∈ B ε (ū) and all τ < τ K .Now, using (4.15) we deduce the existence of σ 0 ∈ (0, τ K ) such that ūσ ∈ B ε (ū) and that ȳσ is the unique solution of (4.9) associated to ūσ for every |σ| ≤ σ 0 .Hence, for |σ| ≤ σ 0 we can define the functions Moreover, ūσ is a local (global) minimum of J σ in B ε (ū) ∩ U σ,ad .We need to prove the differentiability of J σ to get the optimality conditions satisfied by ūσ .The reader is referred to [5, sect. 4.2] for the proof of the following theorem.
Theorem 4.4.For every |σ| ≤ σ 0 the function where ϕ σ ∈ Y σ is the solution of the adjoint equation and y σ is the solution of (4.9) corresponding to u.
Finally, we have the following theorem [5,Thm. 4.16].Theorem 4.7.Suppose that (3.19) and (4.12) hold.Then, there exists a constant C > 0 independent of σ such that 5. Sparse controls.In the applications, we are frequently required to localize the controls in a small region of the domain.An interesting issue is to guess the region where the controls are more efficient.To answer this question we consider the following control problem: where with κ > 0. In this section, we assume that −∞ < α j < 0 < β j < +∞, 1 ≤ j ≤ 3. Thus, any admissible control can take the value 0 in some points.We will show that the solutions ū of (P κ ) are sparse controls, and the size of their supports can be monitored by κ.The bigger κ, the smaller the support of ū.The functional j is convex and Lipschitz.Its subdifferential is defined by By taking v = 0 and v = 2u, respectively, we get that (5.
We have the following theorem, which is analogous to Theorem 3.
Proof.The existence of a solution is proved in the same manner as in the proof of Theorem 3.2.Let us prove the optimality conditions.First, we observe that A is open and ū ∈ A; then for every u ∈ U α,β there exists ρ u > 0 such that ū + ρ(u − ū) ∈ A ∀0 < ρ < ρ u , and hence ū + ρ(u − ū) belongs to U ad .Now we use (3.3), the convexity of j, and the local optimality of ū as follows: This implies that ū is the solution of the optimization problem min u∈U α,β ΩT ( φ + λū)u dx dt + κj(u).
Corollary 5.7.Let (ū, ζ) be as in the previous theorem, and assume that (5.16) is fulfilled.Then, the following inequality holds: and 1 ≤ j ≤ 3. Now, arguing as in Corollary 5.2, we have the following result.
Corollary 5.8.Let (ū σ , ȳσ , φσ , ζσ ) satisfy the discrete optimality system for (P κσ ).Then the following relations hold for K ∈ K h and for 1 ≤ j ≤ 3 : Moreover, from the representation formula (5.22) it follows that ζσ is unique for any fixed local minimum ūσ .
We will distinguish two cases.

Concluding remarks.
In this paper, we have considered an alternative formulation of the classical tracking control problem for 3D flows, which ensures that the optimal states are strong solutions of the Navier-Stokes system.As a consequence, we have been able to carry out a complete theoretical and numerical analysis of the optimal control problem.In particular, error estimates for the numerical discretization of the same order of the 2D case have been obtained.We emphasize that our analysis is applicable without assuming any smallness assumption on the data of our problem, and it can be also used to deal with sparse control problems.
The classical formulation of the control problem uses the L 2 norm of y − y d .It is easy to prove the existence of at least one solution (ū, ȳ) for this formulation.If we make the assumption that ȳ is a strong solution of the Navier-Stokes system, then we can follow the approach described in this paper to obtain the same results.Hence, our formulation can be regarded as a way to guarantee that ȳ is indeed a strong solution.