A Stochastic Maximum Principle for Control Problems Constrained by the Stochastic Navier-Stokes Equations

We consider the control problem of the stochastic Navier-Stokes equations in multidimensional domains introduced in \cite{ocpc} restricted to noise terms defined by Q-Wiener processes. Using a stochastic maximum principle, we derive a necessary optimality condition to design the optimal control based on an adjoint equation, which is given by a backward SPDE. Moreover, we show that the optimal control satisfies a sufficient optimality condition. As a consequence, we can solve uniquely control problems constrained by the stochastic Navier-Stokes equations especially for two-dimensional as well as for three-dimensional domains.


Introduction
In this paper, we discuss an optimal control problem for the unsteady Navier-Stokes equations influenced by noise terms. Concerning fluid dynamics, noise enters the system due to structural vibration and other environmental effects. The aim is to control flow fields affected by noise, where we incorporate physical requirements, such as drag minimization, lift enhancement, mixing enhancement, turbulence minimization and stabilization, see [42] and the references therein.
In the last decades, existence and uniqueness results of solutions to the stochastic Navier-Stokes equations has been studied extensively. Unique weak solutions of the stochastic Navier-Stokes equations exist only for two-dimensional domains. In [35,43], weak solutions are considered with noise terms given by Wiener processes. Weak solutions with Lévy noise are considered in [5,15]. For three-dimensional domains, uniqueness is still an open problem and weak solutions are introduced as martingale solutions, see [4,7,16,17,37]. Another approach uses the semigroup theory leading to the definition of mild solutions. The existence and uniqueness of a mild solution over an arbitrary time interval can be obtained under certain additional assumptions, see [10,12]. In general, a unique mild solution of the stochastic Navier-Stokes equations does not exist. Thus, stopping times are required to obtain a well defined local mild solutions. For a local mild solution with additive noise given by Wiener processes, we refer to [3]. In [14,36], the stochastic Navier-Stokes equations with additive Lévy noise are considered. A generalization to multiplicative Lévy noise can be found in [2]. In [24], an existence and uniqueness result for strong pathwise solutions is given. For further definitions of solutions to the fractional stochastic Navier-Stokes equations, see [12].
The cost functional considered in this paper is motivated by common control strategies. In [28,33,39,46], the problem is formulated as a tracking type problem arising in data assimilation. Approaches that minimize the enstrophy can be found in [9,13,26,42]. In [49], the cost functional combines both strategies by introducing weights. The shortcoming of these papers is the restriction to two-dimensional domains. In [8,34], optimal control problems for the stochastic Navier-Stokes equations in three-dimensional domains are considered, where the state equation is defined as a martingale solution. Recall that the martingale solution for three-dimensional domains is not unique and thus, only existence results of the optimal control can be obtained.
To overcome these problems, we consider the control problem introduced in [2], which is a generalization of the control problems mentioned above. The solution of the stochastic Navier-Stokes equations is given by a local mild solution, which covers especially two as well as three-dimensional domains. Hence, a unique solution exists up to a certain stopping time. Since the solution as well as the stopping time depend on the control, the cost functional related to the control problem has to incorporate the stopping time resulting in a nonconvex optimization problem. This represents the main difficulty here. However, the existence and uniqueness result of the optimal control is provided in [2]. In this paper, we derive a stochastic maximum principle to obtain an explicit formula the optimal control has to satisfy. For the deterministic case in a two-dimensional domain, we refer to [29]. We calculate the Gâteaux derivative of the local mild solution to the stochastic Navier-Stokes equations, which is given by the local mild solution to the linearized stochastic Navier-Stokes equations. Therefore, we can determine the Gâteaux derivative of the cost functional and hence, the necessary optimality condition is stated as a variational inequality. This result is well known for general optimization problems of functionals, see [29,50]. To derive an explicit formula for the optimal control based on the variational inequality, a duality principle is required, which gives a relation between the linearized stochastic Navier-Stokes equations and the corresponding adjoint equation. Since the control problem is constrained by a SPDE with multiplicative noise, the adjoint equation becomes a backward SPDE. Existence and uniqueness results of mild solutions to backward SPDEs are mainly based on a martingale representation theorem, see [30]. These martingale representation theorems are only available for infinite dimensional Wiener processes and real valued Lévy processes, see [21,38,41]. Thus, we have to restrict the problem to noise terms defined by Q-Wiener processes. In general, a duality principle for SPDEs is based on an Itô product formula, which is not applicable to mild solutions of SPDEs. Here, we approximate the local mild solution of the linearized stochastic Navier-Stokes equations and the mild solution of the adjoint equation by strong formulations. As a consequence, we obtain a duality principle for the strong formulations and due to suitable convergence results, we prove that this duality principle holds for the mild solutions as well. By the variational inequality and the duality principle, we design the optimal control based on the adjoint equation. Moreover, we show that the Gâteaux derivatives and the Fréchet derivatives of the cost functional coincides up to order two. Hence, we obtain that the optimal control also satisfies a sufficient optimality condition provided in [45].
The main contribution of this paper is the derivation of a solution to the control problem introduced in [2] using a stochastic maximum principle. Thus, we are able to control the stochastic Navier-Stokes equations in multidimensional domains uniquely. As a consequence, it remains to solve a system of coupled forward and backward SPDEs.
The paper is organized as follows. In Section 2, we discuss the functional analytic background, which is standard in the literature on mild solutions to the deterministic unsteady Navier-Stokes equations. Moreover, a brief introduction to stochastic integrals subject to Q-Wiener processes is given. The existence of a unique local mild solution to the stochastic Navier-Stokes equations and some useful properties are stated in Section 3. Section 4 addresses the control problem considered in this paper, which is given by a nonconvex optimization problem. We calculate the Gâteaux derivatives as well as the Fréchet derivatives of the cost functional related to the control problem up to order two such that we can state necessary and sufficient optimality conditions. In Section 5, we introduce the adjoint equation as the mild solution of a backward SPDE. The approximation of the local mild solution to the linearized stochastic Navier-Stokes equations and the mild solution of the adjoint equation is shown in Section 6. In Section 7, we utilize a necessary optimality condition stated as a variational inequality to deduce a formula for the optimal control, which also satisfies a sufficient optimality condition.

Functional Background
Throughout this paper, let D ⊂ R n , n ≥ 2, be a bounded and connected domain with C ∞ boundary ∂D. For s ≥ 0, let H s (D) denote the usual Sobolev space and for s > 1 2 let H s 0 (D) = {y ∈ H s (D) : y = 0 on ∂D}. We introduce the following common spaces: where η denotes the unit outward normal to ∂D. The space H equipped with the inner product for every y = (y 1 , ..., y n ), z = (z 1 , ..., z n ) ∈ H becomes a Hilbert space. For all x = (x 1 , ..., x n ) ∈ D, we denote We set D j y = (D j y 1 , ..., D j y n ) for every y = (y 1 , ..., y n ) ∈ V and |j| ≤ 1. Then the space V equipped with the inner product for every y, z ∈ V becomes a Hilbert space. The norms on H and V are denoted by · H and · V , respectively. We get the orthogonal Helmholtz decomposition where ⊕ denotes the direct sum. Then there exists an orthogonal projection Π : (L 2 (D)) n → H, see [20]. Next, we define the Stokes Operator A : D(A) ⊂ H → H by Ay = −Π∆y for every y ∈ D(A), where D(A) = H 2 (D) n ∩V .
The Stokes operator A is positive, self adjoint and has a bounded inverse. Moreover, the operator −A is the infinitesimal generator of an analytic semigroup (e −At ) t≥0 such that e −At L(H) ≤ 1 for all t ≥ 0. For more details, see [19,22,23,48]. Hence, we can introduce fractional powers of the Stokes operator, see [40,47,48]. For α > 0, we define where Γ(·) denotes the gamma function. The operator A −α is linear, bounded and invertible on H. Hence, we define for all α > 0 Moreover, we set A 0 = I, where I is the identity operator on H. For α > 0, the operator A α is linear and closed on H with dense domain D(A α ) = R(A −α ), where R(A −α ) denotes the range of A −α . Next, we provide some useful properties of fractional powers to the Stokes operator.
Lemma 1 (cf. Section 2.6, [40]). Let A : D(A) ⊂ H → H be the Stokes operator. Then (i) for α, β ∈ R, we have A α+β y = A α A β y for every y ∈ D(A γ ), where γ = max{α, β, α + β}, (iv) the operator A α e −At is bounded for all t > 0 and there exist constants M α , θ > 0 such that and there exists a constant C > 0 such that for every y ∈ D(A α ) As a consequence of the previous lemma, we obtain that the space D(A α ) with α ≥ 0 equipped with the inner product y, z D(A α ) = A α y, A α z H for every y, z ∈ D(A α ) becomes a Hilbert space. In this paper, the space D(A α ) with α ∈ (0, 1) is used frequently. A concrete characterization in terms of Sobolev spaces can be found in [12,19,47]. As a direct consequence of the fact that the Stokes operator A is self adjoint, we get the following result.
Lemma 2. Let A : D(A) ⊂ H → H be the Stokes operator. Then, the operator A α is self adjoint for all α ∈ R.
Next, we define the bilinear operator B(y, z) = Π(y ·∇)z for some y, z ∈ H. If y = z, we write B(y) = B(y, y). Then we have the following properties.

Stochastic Processes and the Stochastic Integral
In this section, we give a brief introduction to stochastic integrals, where the noise term is defined as a Hilbert space valued Wiener process. For more details, see [11,21]. Throughout this paper, let (Ω, F , P) be a complete probability space endowed with a filtration (F t ) t∈[0,T ] satisfying F t = s>t F s for all t ∈ [0, T ] and F 0 contains all sets of F with P-measure 0. Let E be a separable Hilbert space. We denote by L(E) the space of linear and bounded operators defined on E. Let Q ∈ L(E) be a symmetric and nonnegative operator such that Tr Q < ∞. Then we have the following definition.
A predictable process is F t -adapted. The converse is in general not true. However, the following result is useful to conclude that a stochastic process has a predictable version.
For the remaining part of this section, let (W (t)) t∈[0,T ] be a Q-Wiener process with values in E and covariance operator Q ∈ L(E). Then there exists a unique operator Q 1/2 ∈ L(E) such that Q 1/2 • Q 1/2 = Q. We denote by L (HS) (Q 1/2 (E); H) the space of Hilbert-Schmidt operators mapping from Q 1/2 (E) into another separable Hilbert space H. Let (Φ(t)) t∈[0,T ] be a predictable process with values in L (HS) (Q 1/2 (E); H) such that The following proposition is useful when dealing with a closed operator A : D(A) ⊂ H → H.
Proposition 1 (cf. Proposition 4.15, [11]). If Φ(t)y ∈ D(A) for every y ∈ E, all t ∈ [0, T ] and P-almost surely, Next, we state a product formula for infinite dimensional stochastic processes, which we use to obtain a duality principle. The formula is an immediate consequence of the Itô formula, see [21,Theorem 2.9].
Then we have for all t ∈ [0, T ] and P-a.s.
Next, we introduce stochastic convolutions. Let (S(t)) t≥0 be a C 0 -semigroup on H. Then the stochastic convolution (I(t)) t∈[0,T ] given by is well defined for all t ∈ [0, T ] and P-almost surely. Under additional assumptions, we get the following maximal inequality.
Proposition 2 (cf. Proposition 1.3 (ii), [27]). Let the C 0 -semigroup (S(t)) t≥0 satisfy S(t) L(H) ≤ 1 for all In order to define local mild solutions to SPDEs, we need to introduce a stopped stochastic convolution. Here, we can adopt the results shown in [6,Appendix]. Let τ be a stopping time with values in [0, T ]. We consider the stopped process (I(t ∧ τ )) t∈[0,T ] , where t ∧ τ = min{t, τ }. Unfortunately, the formula is not well defined due to the fact that we integrate a process, which is not even (F t ) t∈[0,T ] adapted. To overcome this problem, we introduce a process (I τ (t)) t∈[0,T ] given by for all t ∈ [0, T ] and P-almost surely. We get the following result.
Finally, we state a martingale representation theorem for Q-Wiener processes, which we use to construct solutions of backward SPDEs. We note that the covariance operator Q ∈ L(E) is symmetric and nonnegative such that Tr Q < ∞. Hence, there exists a complete orthonormal system (e k ) k∈N in E and a bounded sequence of nonnegative real numbers (µ k ) k∈N such that Qe k = µ k e k for each k ∈ N. Then for arbitrary t ∈ [0, T ] and P-almost surely, a Q-Wiener process has the expansion where (w k (t)) t∈[0,T ] , k ∈ N, are real valued mutually independent Brownian motions. The convergence is in L 2 (Ω). Furthermore, we assume that the complete probability space (Ω, F , P) is endowed with the filtration T ] and we require that the σ-algebra F satisfies F = F T . Then we have the following martingale representation theorem.

The Stochastic Navier-Stokes Equations
In this section, we recall briefly the motivation of the stochastic Navier-Stokes equations and we state the existence and uniqueness result for the local mild solution, see [2]. Moreover, we state some useful properties.
We consider the following Navier-Stokes equations with Dirichlet boundary condition: where y(t, x, ω) ∈ R n denotes the velocity field with F 0 -measurable initial value ξ(x, ω) ∈ R n and p(t, x, ω) ∈ R describes the pressure of the fluid. The parameter ν > 0 is the viscosity parameter (for the sake of simplicity, we assume ν = 1) and f (t, x, ω, y) ∈ R n is the external random force dependent on the velocity field. Here, we assume that the external random force can be decomposed as the sum of a control term and a noise term. Using the spaces and operators introduced in Section 2.1, we obtain the stochastic Navier-Stokes equations in D(A α ): where (W (t)) t∈[0,T ] is a Q-Wiener process with values in H and covariance operator Q ∈ L(H). We introduce the ) becomes a Banach space. The set of admissible controls U is a nonempty, closed, bounded and convex subset of the Hilbert space and we have for each m ∈ N, all t ∈ [0, T ] and P-a.s.
In the previous definition, note that the stopped stochastic convolution is well defined according to Section 2.2.
The proof of the existence and uniqueness of a local mild solution to system (7) can be shown in two steps. First, we consider a modified system to get a mild solution well defined over the whole time interval [0, T ]. Then we introduce suitable stopping times such that the mild solution of the modified system and the local mild solution of system (7) coincides. Let us introduce the following system in D(A α ): where m ∈ N and π m : Then we get for every y, z ∈ D(A α ) and we have for all t ∈ [0, T ] and P-a.s.

Remark 2.
(i) It suffices to assume that the operator G satisfies a growth condition and a Lipschitz condition, see [11]. In this paper, the additional assumptions are necessary to derive the Gâteaux derivative of the local mild solution to system (7). Moreover, the adjoint operator of G is required for the adjoint equation.
(ii) In [2], it is shown that the processes (y m (t)) t∈[0,T ] and (y(t)) t∈[0,τ ) are mean square continuous. Due to the fact that E sup t∈[0,T ] y m (t) 2 D(A α ) < ∞ and the operator G is linear and bounded, we can conclude that the stochastic convolution has a continuous modification, see [11,Theorem 6.10]. Hence, the processes (y m (t)) t∈[0,T ] and (y(t)) t∈[0,τ ) have continuous modifications as well.
Next, we state some useful results. In what follows, we always assume that the parameters α ∈ (0, 1), δ ∈ [0, 1) and β ∈ [0, α] satisfy the assumptions of Theorem 1 and the stopping times (τ m ) m∈N are given by equation (12). Moreover, let the initial value ξ ∈ L 2 (Ω; D(A α )) of system (8) and system (7) be fixed. To illustrate the dependence on the control u ∈ L 2 F (Ω; L 2 ([0, T ]; D(A β ))), let us denote by (y m (t; u)) t∈[0,T ] and (y(t; u)) t∈[0,τ u ) the mild solution of system (8) and the local mild solution of system (7), respectively. Note that the stopping times (τ u m ) m∈N and τ u depend on the control as well. Whenever these processes and the stopping times are considered for fixed control, we use the notation introduced above. We have the following continuity property. For k = 2, a proof can be found in [2,Lemma 5.3].
By definition, we have for all t ∈ [0, τ u m ) and P-a.s. y(t; u) = y m (t; u). Hence, a similar result of the previous lemma holds for the local mild solution of system (7). In the following lemmas, we show some useful properties of the stopping times.
) and let the stopping time τ u m be given by (12).

A Generalized Control Problem
In this section, we introduce the cost functional and the related control problem. First, we calculate the Gâteaux derivative of the local mild solution of system (7), which is given by the local mild solution of the linearized stochastic Navier-Stokes equations. Hence, we are able to derive the Gâteaux derivatives of the cost functional and using a mean value theorem, we show that the Gâteaux derivatives and the Fréchet derivatives coincides. We introduce the cost functional J m : where m ∈ N is fixed and γ ∈ [0, α]. Moreover, the process (y(t; u)) t∈[0,τ u ) is the local mild solution of system (7) corresponding to the control u ∈ L 2 ) is a given desired velocity field. The task is to find a control u m ∈ U such that The control u m ∈ U is called an optimal control. Note that for γ = 0, the formulation coincides with a tracking problem, see [28,33,39,46]. For γ = 1 2 and y d = 0, we minimize the enstrophy, see [9,26,42]. Hence, we are dealing with a generalized cost functional, which incorporates common control problems in fluid dynamics.
Theorem 3 (Theorem 5.2, [2]). Let the functional J m be given by (13). Then there exists a unique optimal control u m ∈ U .

The Linearized Stochastic Navier-Stokes Equations
We introduce the following system in D(A α ): where v ∈ L 2 F (Ω; L 2 ([0, T ]; D(A β ))), the process (y(t)) t∈[0,τ ) is the local mild solution of system (7) and the process (W (t)) t∈[0,T ] is a Q-Wiener process with values in H and covariance operator Q ∈ L(H). The operators A, B, F, G are introduced in Section 2.1 and Section 3, respectively.
The following existence and uniqueness result holds also for a general F 0 -measurable initial value z(0) = z 0 ∈ L 2 (Ω; D(A α )). Since we prove that the local mild solution of system (14) is the Gâteaux derivative of the local mild solution of system (7), it suffices to show the result for z 0 = 0.
Similarly to Section 3, we first consider the following system in D(A α ): where the process (y m (t)) t∈[0,T ] is the mild solution of system (8) and π m : D(A α ) → D(A α ) is given by (9).
and we have for all t ∈ [0, T ] and P-a.s.
By Theorem 1, we get the existence and uniqueness of the mild solution (y m (t)) t∈[0,T ] to system (8) for fixed m ∈ N and fixed control u ∈ L 2 F (Ω; L 2 ([0, T ]; D(A β ))). Recall that the initial value ξ ∈ L 2 (Ω; D(A α )) is fixed as well. Thus, we have the following existence and uniqueness result. The proof can be obtained similarly to Theorem 1.
Due to Theorem 2, we get the existence and uniqueness of the local mild solution (y(t)) t∈[0,τ ) to system (7) for ). Thus, we have the following existence and uniqueness result, where the stopping times (τ m ) m∈N are given by equation (12). The proof can be obtained similarly to Theorem 2.
Next, we show some properties, which we use to calculate the Gâteaux derivative of the cost functional (13). Note that the mild solution of system (8) depends on the control u ∈ L 2 F (Ω; L 2 ([0, T ]; D(A β ))). Hence, the mild solution of system (15) ). Let us denote by (z m (t; u, v)) t∈[0,T ] the mild solution of system (15). Similarly, we indicate by (z(t; u, v)) t∈[0,τ u ) the local mild solution of system (14) corresponding to the controls ). Whenever these processes are considered for fixed controls, we use the notation introduced above.
Proof. Let the stochastic process (y m (t; u)) t∈[0,T ] be the mild solution of system (8) . By definition, we have for all t ∈ [T 1,m , T ] and P-a.s.
where c 2,m > 0 is a constant. By continuing the method, we obtain inequality (16).
Lemma 11. For fixed m ∈ N, let (z m (t; u, v)) t∈[0,T ] be the mild solution of system (14) corresponding to the Proof. Let the process (y m (t; u)) t∈[0,T ] be the mild solution of system (8) corresponding to the control u ∈ L 2 F (Ω; L 2 ([0, T ]; D(A β ))). To simplify the notation, we set for all t ∈ [0, T ] and P-a.s.
Lemma 12. For fixed m ∈ N, let (z m (t; u, v)) t∈[0,T ] be the mild solution of system (15) corresponding to the controls u, v ∈ L 2 F (Ω; L 2 ([0, T ]; D(A β ))). Then there exists a constant c > 0 such that for every u 1 , Proof. We define the operator B(y, z) = B(z, y)+ B(y, z) for every y, z ∈ D(A α ). Since the operator B is bilinear on D(A α ) × D(A α ), the operator B is bilinear as well and using Lemma 3, we get for every y, z ∈ D(A α ) Let (y m (t; u i )) t∈[0,T ] be the mild solution of system (8) (10), (11) and (17), Proposition 2 with k = 2 and the Cauchy-Schwarz inequality, there exist constants . Using Lemma 7 with k = 2 and Lemma 10 with k = 4, we can conclude that there exists a constant C * 3 > 0 such that nonempty and open, we denote the partial Gâteaux derivative and the partial Fréchet derivative at , respectively. First, we show that the local mild solution of system (14) is the partial Gâteaux derivative of the local mild solution to system (7) with respect to control variable. Theorem 6. Let (y(t; u)) t∈[0,τ u ) and (z(t; u, v)) t∈[0,τ u ) be the local mild solution of system (7) and system (14) corresponding to the controls u, v ∈ L 2 F (Ω; L 2 ([0, T ]; D(A β ))), respectively. Then for fixed m ∈ N, the Gâteaux derivative of y(t; u) at u ∈ L 2 F (Ω; L 2 ([0, T ]; D(A β ))) in direction v ∈ L 2 F (Ω; L 2 ([0, T ]; D(A β ))) satisfies for all t ∈ [0, τ u m ) and P-a.s.
This enables us to derive the Gâteaux derivative of the cost functional.
Using Lemma 11, the functional d G Φ 1 (u) is linear. Moreover, by Lemma 1 (v), Lemma 10 with k = 2 and the Cauchy-Schwarz inequality, there exists a constant C * > 0 such that Hence, the functional d G Φ 1 (u) is bounded. Note that the functional Φ 2 : L 2 F (Ω; L 2 ([0, T ]; D(A β ))) → R is given by the squared norm in the Hilbert Obviously, the functional d G Φ 2 (u) is linear and bounded. Using equation (27) and equation (28), the Gâteaux derivative of J m at u ∈ L 2 F (Ω; Since d G Φ 1 (u) and d G Φ 2 (u) are linear and bounded, the functional d G J m (u) is linear and bounded as well.
Recall that the set of admissible controls U is a closed, bounded and convex subset of the Hilbert space L 2 F (Ω; L 2 ([0, T ]; D(A β ))) such that 0 ∈ U . Hence, the optimal control u m ∈ U satisfies the necessary optimality condition for fixed m ∈ N and every u ∈ U . Due to Theorem 7, we get the variational inequality for fixed m ∈ N and every u ∈ U . We will use this inequality to derive an explicit formula for the optimal control u m ∈ U . For more details on necessary optimality conditions as variational inequalities, we refer to [29,50]. Next, we state the second order Gâteaux derivative of the cost functional (13). Moreover, we show that the Gâteaux derivatives and the Fréchet derivatives coincide, which will enable us to obtain a sufficient optimality condition.
Proof. The claim can be shown similarly to Theorem 7.
Corollary 3. Let the functional J m : L 2 F (Ω; L 2 ([0, T ]; D(A β ))) → R be defined by (13). Then the Fréchet derivative at u ∈ L 2 F (Ω; where the process (z(t; u, v)) t∈[0,τ u ) is the local mild solution of system (14) corresponding to the controls u, v ∈ is continuous with respect to u. Proof. Using Theorem 7, we have that the Gâteaux derivative at u ∈ L 2 F (Ω; Since u → d G J m (u) is a continuous mapping, we can conclude and by Theorem 7, the operator d F J m (u) is linear and bounded. Since is continuous as well.
Similarly to the previous corollary, we obtain that the cost functional is twice Fréchet differentiable. ) → R be defined by (13). Then the Fréchet derivative of order two at u ∈ L 2 F (Ω; where the processes (z(t; u, v i )) t∈[0,τ u ) are the local mild solution of system (14) corresponding to the controls is continuous with respect to u.

The Adjoint Equation
We will use the necessary optimality condition (30) to derive an explicit formula the optimal controls u m ∈ U has to satisfy. Therefor, we need a duality principle, which gives us a relation between the local mild solution to system (14) and the corresponding adjoint equation. Since the control problem considered in this paper is constrained by a SPDE with linear multiplicative noise, the adjoint equation is specified by a backward SPDE. For mild solutions of backward SPDEs, the existence and uniqueness result is mainly based on a martingale representation theorem, see [30].
We introduce the following backward SPDE in D(A δ ): where m ∈ N and the process (y(t)) t∈[0,τ ) is the local mild solution of system (7). The stopping times (τ m ) m∈N are defined by equation (12) and y d ∈ L 2 ([0, T ]; D(A γ )) is the given desired velocity field. The operator A and its fractional powers are introduced in Section 2. . We can rewrite this equivalently as for every h ∈ H and every z ∈ D(A α ). By the closed graph theorem, we get that the operator A α B * δ (y, ·) : H → H is linear and bounded.

Definition 7. A pair of predictable processes
and we have for all t ∈ [0, T ] and P-a.s.
To prove the existence and uniqueness of the mild solution to system (31), we need the following auxiliary results.
Furthermore, there exists a constant c * > 0 such that for all t ∈ [0, T ] E sup Proof. For δ = ε = 0, a proof can be found in [30, Lemma 2.1]. For arbitrary ε ∈ 0, 1 2 and δ ∈ [0, 1 2 − ε), one can show the result similarly using the properties of fractional powers to the operator A provided by Lemma 1.
Moreover, there exist constants C 1 , C 2 > 0 such that for each k ∈ N E sup Hence, equation (40) satisfies the assumptions of Lemma 13 and Corollary 5. Let T 1,m ∈ [0, T ). Due to inequality (35) and inequality (37), there exist constants Using inequality (36) and inequality (38), there exist constants Hence, we obtain for each k ∈ N E sup We choose T 1,m ∈ [0, T ) such that K m < 1. Thus, we can conclude that the sequence (z k is a Cauchy sequence on the interval [T 1,m , T ]. Using equation (40), we have for each k ∈ N, all t ∈ [0, T 1,m ] and P-a.s.
T is a Cauchy sequence on the interval [T 2,m , T 1,m ]. By continuing the method, we can conclude that the sequence (z By equation (39), one can easily verify that the pair of processes (z * m (t), Φ m (t)) t∈[0,T ] satisfy equation (34).
, then the restriction γ +δ < 1 2 vanishes in the previous theorem. Moreover, note that we have the additional restrictions α, δ < 1 2 . Since γ ≤ α in equation (13), we can not solve the control problem introduced in Section 4 for the special case γ = 1 2 . However, one can overcome this problem if system (7) is driven by an additive noise, i.e. the operator G does not depend on the velocity field y(t).

Approximation by a Strong Formulation
In general, a duality principle of solutions to forward and backward SPDEs can be obtained by applying an Itô product formula. This formula is not applicable to solutions in a mild sense. Here, we approximate the mild solutions of system (15) and system (31) by strong formulations. Recall that these mild solutions take values in the domain of fractional powers to the Stokes operator and hence, we need convergence results in the corresponding spaces. According to [11], one can use the Yosida approximation of the Stokes operator A. For applications regarding a duality principle, see [18,44]. This approximation holds only in the underlying Hilbert space H and thus, we can not obtain suitable convergence results. Here, we apply the approach introduced in [25,31]. The basic idea is to formulate a mild solution with values in D(A) using the resolvent operator R(λ) introduced in Section 2.1. Thus, we get convergence results in the domain of fractional power operators and the mild solutions coincide with strong solutions. Although, the convergence is only available for forward SPDEs, we are also able to show the result for the backward equation. In this section, we omit the dependence on the controls for the sake of simplicity.

The Forward Equation
Here, we give an approximation of the mild solution to system (15). We introduce the following SPDE in where m ∈ N, λ > 0 and v ∈ L 2 F (Ω; L 2 ([0, T ]; D(A β ))). The operators A, B, R(λ), F, G are introduced in Section 2.1 and Section 3, respectively. The mapping π m : D(A α ) → D(A α ) is given by (9) and the process (y m (t)) t∈[0,T ] is the mild solution of system (8). The process (W (t)) t∈[0,T ] is a Q-Wiener process with values in H and covariance operator Q ∈ L(H).
and we have for all t ∈ [0, T ] and P-a.s.
Recall that the operators R(λ) and AR(λ) are linear and bounded on H. Hence, an existence and uniqueness result of a mild solution (z m (t, λ)) t∈[0,T ] to system (41) can be obtained similarly to Theorem 4 for fixed m ∈ N and fixed λ > 0. In the following lemma, we state a strong formulation of the mild solution to system (41), which is an immediate consequence of [31, Proposition 2.3].
Az m (s, λ) + A δ R(λ)A −δ [B(R(λ)z m (s, λ), π m (y m (s))) + B(π m (y m (s)), R(λ)z m (s, λ))] ds We get the following convergence result. Proof. Let I be the identity operator on H. We define the operator B(y, z) = B(z, y) + B(y, z) for every y, z ∈ D(A α ). Since B is bilinear on D(A α ) × D(A α ), the operator B is bilinear as well and using Lemma 3, we get for every y, z ∈ D(A α ) Recall that the operator G : H → L (HS) (Q 1/2 (H); D(A α )) is linear and bounded. By definition, we find for all λ > 0, all t ∈ [0, T ] and P-a.s.

The Backward Equation
Here we give an approximation of the mild solution to system (31). We introduce the following backward SPDE in D(A 1+δ ): where m ∈ N and λ > 0. The operators A, R(λ), B * δ , G * are introduced in Section 2.1 and Section 5, respectively. The process (y(t)) t∈[0,τ ) is the local mild solution of system (7) with stopping times (τ m ) m∈N defined by (12) and y d ∈ L 2 ([0, T ]; D(A γ )) is the given desired velocity field. The process (W (t)) t∈[0,T ] is a Q-Wiener process with values in H and covariance operator Q ∈ L(H).
and we have for all t ∈ [0, T ] and P-a.s.
Recall that the operators R(λ) and AR(λ) are linear and bounded on H. Hence, an existence and uniqueness result of a mild solution (z * m (t, λ), Φ m (t, λ)) t∈[0,T ] to system (46) can be obtained similarly to Theorem 8 for fixed m ∈ N and fixed λ > 0. Moreover, we get the following result. The following lemma provides a strong formulation of the mild solution to system (46), which is an immediate consequence of [1, Theorem 3.4 and Theorem 4.1].

Design of the Optimal Controls
Based on the results provided in the previous sections, we are able to show a duality principle, which gives us a relation between the local mild solution of system (14) and the mild solution of system (31). Note that the local mild solution of system (8) depends on the control u ∈ L 2 F (Ω; L 2 ([0, T ]; D(A β ))). Hence, the mild solution of system (31) depends on the control u ∈ L 2 F (Ω; L 2 ([0, T ]; D(A β ))) as well. Let us denote this mild solution by (z * m (t; u), Φ m (t; u)) t∈[0,T ] .
Theorem 9. Assume that the processes (y(t; u)) t∈[0,τ u ) and (z(t; u, v)) t∈[0,τ u ) are the local mild solutions of system (7) and system (14) corresponding to the controls u, v ∈ L 2 F (Ω; L 2 ([0, T ]; D(A β ))), respectively. Moreover, let the pair (z * m (t; u), Φ m (t; u)) t∈[0,T ] be the mild solution of system (31) Proof. For the sake of simplicity, we omit the dependence on the controls. First, we prove the result for the approximations derived in Section 6. Let (z m (t, λ)) t∈[0,T ] be the mild solution of system (41). Using Lemma 14, We conclude that the right and the left hand side of equation (61) converges as λ → ∞ and equation (53) holds.
Based on the necessary optimality condition formulated as the variational inequality (30) and the duality principle derived in the previous theorem, we are able to deduce a formula the optimal control has to satisfy. First, we introduce a projection operator. Note that the set of admissible controls U is a closed subset of the Hilbert space L 2 F (Ω; L 2 ([0, T ]; D(A β ))). We denote by P U : L 2 F (Ω; L 2 ([0, T ]; D(A β ))) → U the projection onto U , i.e. Theorem 10. Let (z * m (t; u), Φ m (t; u)) t∈[0,T ] be the mild solution of system (31) corresponding to the control u ∈ L 2 F (Ω; L 2 ([0, T ]; D(A β ))). Then for fixed m ∈ N, the optimal control u m ∈ U satisfies for almost all t ∈ [0, T ] and P-a.s.
Proof. Using inequality (30) and Theorem 9, the optimal control u m ∈ U satisfies for every u ∈ U E τ um m 0 z * m (t; u m ), F (u(t) − u m (t)) H dt + E T 0 A β u m (t), A β (u(t) − u m (t)) H dt ≥ 0.
By Corollary 6, we have