Stabilization of 3D Navier–Stokes Equations to Trajectories by Finite-Dimensional RHC

Local exponential stabilization of the three-dimensional Navier–Stokes system to a given reference trajectory via receding horizon control (RHC) is investigated. The RHC enters as the linear combinations of a finite number of actuators. The actuators are spatial functions and can be chosen in particular as indicator functions whose supports cover only a part of the spatial domain.

To be more precise, we show that there exists an r > 0 such that for every initial function y 0 satisfying y 0 −ŷ 0 H 1 0 (Ω;R 3 ) ≤ r , and the receding horizon state y rh corresponding to the RHC u rh (y 0 ) ∈ L 2 ((0, ∞); R N ), it holds where the positive constants c V and ζ are independent of y 0 .
One efficient approach for the stabilization of a class of continuous-time infinitedimensional controlled systems is receding horizon framework, see e.g., [1][2][3][4][5] and the references therein. In this approach, a stabilizing RHC is constructed through the concatenation of a sequence of finite horizon open-loop optimal controls on overlapping temporal intervals covering [0, ∞). These optimal control problems are computed according to a performance index function which enhances the desirable properties and structures of the control. Here, for every T ∈ (0, ∞], we consider the following performance index function for β > 0 and initial pair (t 0 , y 0 ) ∈ R + × H 1 (Ω; R 3 ), where | · | 2 stands for the 2norm. The receding horizon framework bridges to a certain degree the gap between closed-loop control and open-loop control. The main issue is then to justify the stability of RHC. Depending on the structure of the underlying problem, this is usually done, by techniques involving the design of appropriate sequences of temporal intervals, using an adequate concatenation scheme, or adding terminal costs and\or constraints to the finite horizon problems. Due to the structure of the receding horizon framework, the resulting control acts as a feedback mechanism.
Considering the performance index function defined in (1.3), the stabilization of the control system (1.1) towards the trajectoryŷ of (1.2) can be also reformulated as the following infinite horizon optimal control problem In connection to the infinite horizon problem O P o ∞ (y 0 ), the receding horizon framework delivers approximations to the solution of this problem which are considered suboptimal solutions.
We continue our investigations on the receding horizon framework for infinite horizon optimal control problems governed by partial differential equations, that we initiated in [2] for autonomous systems and, recently extended in [3] for time-varying infinite-dimensional linear systems. In this framework, the exponential stability and suboptimality of RHC are obtained by generating an appropriate sequence of overlapping temporal intervals and applying a suitable concatenation scheme. There is no need for terminal costs or terminal constraints imposed on the open-loop problems. Previously, this framework was investigated for finite-dimensional autonomous systems in e.g, [6,7] and for discrete-time autonomous systems in e.g, [8,9].
In the receding horizon approach, we choose a sampling time δ > 0 and an appropriate prediction horizon T > δ. Then, we define sampling instances t k := kδ for k = 0 . . . . At every sampling instance t k , an open-loop optimal control problem is solved over a finite prediction horizon (t k , t k + T ). Then the optimal control is applied to steer the system from time t k with the initial state y rh (t k ) until time t k+1 := t k + δ at which point, a new measurement of state is assumed to be available. The process is repeated starting from the new measured state: we obtain a new optimal control and a new predicted state trajectory by shifting the prediction horizon forward in time. The sampling time δ is the time period between two sample instances. Throughout, we denote the receding horizon state-and control variables by y rh (·) and u rh (·), respectively. Also, (y * T (·; t 0 , y 0 ), u * T (·; t 0 , y 0 )) stands for the optimal state and control of the optimal control problem with finite time horizon T , and initial function y 0 at initial time t 0 . The receding horizon framework is summarized in Algorithm 1.
While most of the literature on feedback stabilization is concerned with the stabilization of the Navier-Stokes equations to the steady-state, similarly to [14], we are interested in the stabilization towards a given reference trajectory by means of finite- in Ω, dimensional controls. In this case, it is needed to derive the stabilizability results for a nonautonomous system, which requires different techniques compared to the case of autonomous systems, see e.g, [14,Introduction] for more details. We also refer the readers to [29][30][31] for more recent results concerning the stabilizability of the nonautonomous parabolic-like differential equations by finite-dimensional controls.

Contributions
This manuscript deals with the analysis of the RHC for the Navier-Stokes equations. Besides the fact that RHC has not been investigated for these equations before, the present paper also contains novelties compared to our recent investigations [1][2][3] on the analysis of RHC for infinite-dimensional systems: (i) Here our objective is to stabilize the system around a given reference trajectory. In this case, depending on the regularity of the given reference trajectory, in order to study the stability and wellposedness of RHC, we consider the translated controlled system. This system, obtained by subtracting (1.1) from (1.2), is a system of nonlinear time-varying equations and, thus, we are concerned with the local stabilizability of a nonautonomous system. (ii) For the three-dimensional Navier-Stokes equations, in order to ensure the uniqueness, we need to work with the so-called strong variational solutions. For this purpose, the stabilizability of the controlled system is investigated with respect to the H 1 -norm.
In this matter, in order to establish the exponential stability of RHC, we need to derive an observability type inequality with respect to the H 1 -norm. (iii) For the threedimensional Navier-Stokes equations, the existence of the global strong solution is only guaranteed for small initial data and forcing terms. Therefore, an extra effort needs to be made to guarantee both well-posedness of the open-loop subproblems and the smallness of the H 1 -norm of the states at sampling instances t k during the concatenation process within the receding horizon framework. Furthermore, compared to the work [14] dealing with the stabilization of Navier-Stokes equations to reference trajectories, the present work differs not only in the fact that we employ and investigate the receding horizon framework for stabilization, but also that our theory allows the stabilization to less regular reference trajectories, and as actuators, we can use the indicator functions which are more practical in applications in comparison to the eigenfunctions of the Stokes operator.

Organization of the Paper
The rest of the paper is organized as follows: In Sect. 2, we introduce the notions and functional spaces used in the theory of the three-dimensional Navier-Stokes equations. Then, we review some preliminaries about the well-posedness and regularity of the solution to the translated system, which is obtained by subtracting (1.1) from (1.2). Based on four key properties, Sect. 3 deals with the stability and suboptimality of RHC. In Sect. 4, we investigate the local stabilizability of (1.1) around a given trajectorŷ y by finitely many controllers. Further, sufficient conditions on the set of actuators are given, for which the stabilizability results hold. Then in Sect. 5, first the validity of the four properties given in Sect. 3 is established. Then the main results i.e., the local exponential stabilizability of the receding horizon state towards a given target trajectory and the suboptimality of RHC are proven. Finally, to improve the readability of the paper, we provide proofs to some of results from Sects. 2-4 only in the appendix.

Functional Spaces and Translated Systems
We write R + for the set of non-negative real numbers. For a Banach space X , we denote by · X the associated norm, by X the associated dual space, and by ·, · X ,X the dual paring between X and X . In the case that X is a Hilbert space, we use the scalar product (·, ·) X . Further, L(X , Y ) denotes the space of continuous linear operators from X to Y with the usual operator norm · L(X,Y ) . In case X = Y , we write L(X ) := L(X , X ) instead. Let X and Y be Banach spaces, then for any open interval (t 0 , t 1 ) ⊂ R + we define where the derivative ∂ t is taken in the sense of distributions. This space is endowed with the norm We frequently use open intervals of the form I T (t 0 ) := (t 0 , t 0 +T ) ⊂ R + with t 0 ∈ R + and T ∈ R + ∪ {∞}. Then, we denote [t 0 , t 0 + T ] and [t 0 , ∞), by I T (t 0 ) and I ∞ (t 0 ), respectively.
Let Ω ⊂ R 3 be an open, bounded, and connected set with smooth boundary ∂Ω. Throughout, for simplicity, we use the notations L p := L p (Ω; where n is the unit outward normal vector on ∂Ω. The spaces H and V are the closure of the space D with respect to the L 2 -and H 1 0 -norms, respectively. It is wellknown that with a densely compact embedding, and as a consequence, we recall from e.g., [32] that for an open interval (t 0 , t 1 ) ⊂ R + it holds  In order to define the weak variational form of the Navier-Stokes equations, we introduce the continuous bilinear form B : It is wellknown from e.g., [33,Lemma 1.3.], that for b it holds that b(y, v, v) = 0 and b(y, v, w) = −b(y, w, v).
(2.2) Moreover, using standard Sobolev's embeddings, we can obtain that where c is a generic constant depending on Ω.
We denote the nonlinear term in the Navier-Stokes equations by N (y) := B(y, y).
For any givenŷ ∈ V , we define the linear operator For specifying the regularity of the reference trajectory, we need to introduce the following Banach space where the subscript w stands for the weak measureability, see e.g., [34, Sects. 5.0 and 9.1]. For any given t 0 ∈ R + , T ∈ R + ∪ {∞}, and a fix σ > 6 5 , we consider the spaces W t 0 ,T and V t 0 ,T for the measurable vector functions y = (y 1 , y 1 , where L ∞ div := {y ∈ L ∞ : div y = 0 in Ω}. Further, for given t 0 ∈ R + , T ∈ R + ∪ {∞}, and λ ≥ 0, we will use the Banach These spaces will be used within the contraction mapping theorem for the existence results for the nonlinear system of equations. Using the Leray projection and the notations introduced above, (1.1) and (1.2) can equivalently be written as and respectively. Setting v := y −ŷ, v 0 := y 0 −ŷ 0 , and subtracting (2.4) from (2.5), we come up with the following system of time-varying nonlinear differential equations This system is called the translated system. Our control objective can now be expressed, equivalently, as the local exponential stabilization of the nonlinear time-varying system (2.6) to zero with respect to V -norm by means of RHC.

Local Existence and Estimates
In this section, we are concerned with the well-posedness and regularity of the system of nonlinear time-varying Eq. (2.6). Letŷ be the solution to (2.5) for a pair (ŷ 0 ,f). Then for every initial function v 0 and forcing term f, we consider the auxiliary nonlinear system and for λ ≥ 0 the auxiliary linear system Throughout the paper, we impose the following regularity condition for the reference trajectoryŷ.
Assumption 1 Let (ŷ,p) be a global smooth solution to (1.2), for which it holds with constantsˆ > 0, σ > 6 5 , and R > 0, that Our stabilizability result is based on the concatenation of exact controllability controls on a family of finite intervals covering [0, ∞). Here we used the exact controllability result given in [25,Proposition 1.] and, thus, the regularity condition (RA) is motivated by the one given in [25, p. 3].
Remark 1 Due to regularity condition (RA), for every (t 0 , T ) ∈ R 2 + the quantities ŷ W t 0 ,T and ŷ V t 0 ,T are bounded by constants depending only onˆ , T , and R. Lemma 1 Let ν > 0 and λ ≥ 0 be given. Then, for every

Proof
The proof follows by using the standard arguments given in e.g. [33,Chapter 3] and estimates (2.3). Thus, we omit the proof here.
In the next proposition, we investigate the existence of the nonlinear system (2.7) for small pairs of initial functions v 0 and forcing functions f.

Proposition 1 For every given T > 0, there exists r = r (T ) > 0 such that for every
Moreover, for this solution we have the following estimates where the constant c 4 depends onŷ and T , and the constant K depends on T ,ŷ, v 0 , and f.

Proof
The proof is given in Appendix 1.
In the next Lemma, we establish an observability inequality which is essential for the exponential stability of RHC.
Proof The proof follows by energy estimates and it is given in Appendix 2.

Stability of RHC
This section is devoted to investigating the stability of RHC. For simplicity in presentation, we use the notations B := [Π 1 , . . . , Π N ] for the set of actuators , we consider the following nonlinear time-varying controlled system Further, due to Proposition 1, for every given T > 0, there exists a radius r c : For the sake of convenience in presentation, we proceed the stability analysis of RHC with a general class of incremental functions which contains the one associated to (1.3) as a spacial case (See Remark 2). In this matter, for defining the optimal control problems associated to the receding horizon framework, we consider an incremental functions : where the number α > 0 is independent of (t, v, u). For every interval length T > 0, initial state v 0 ∈ V , and initial time t 0 , we use frequently the finite horizon optimal control problems of the form Then, the reeding horizon algorithm for dealing with the infinite horizon problem is given in Algorithm 2.

Remark 2
It is easy to check that by setting 2) holds for α := min{1, β} and Algorithm 1 can be equivalently expressed by Algorithm 2.
To be more precise, due to (3.3) and using the fact that v = y −ŷ, it can be easily verified that the finite horizon optimal control problems defined on the same temporal interval in both of Algorithms 1 and 2 are equivalent. Thus, both of these algorithms deliver the same RHC u rh and approximations for the value functions. Hence, we restrict ourselves here to investigate the stability and suboptimality of RHCs obtained by Algorithm 2.

Algorithm 2 Receding Horizon Algorithm for the Translated Equation
Require: Let the prediction horizon T , the sampling time δ < T , and the initial point v 0 ∈ V be given. Then we proceed through the following steps: 4: Go to step 2.

Definition 1 For any
Similarly, for every (T , t 0 , v 0 ) ∈ R 2 + × V , the finite horizon value function V T : In order to show the exponential stability and suboptimality of RHC obtained by Algorithm 2, we need to verify the following properties for C S( P1: There exists a radius r s such that for every positive number T , V T is globally decrescent on B r s (0) with respect to the V -norm. That is, there exists a continuous, non-decreasing, and bounded function γ : Since, in Algorithm 2, the solution of O P ∞ (v 0 ) is approximated by solving a sequence of the finite horizon open-loop optimal controls, we need a priori to guarantee that any of these optimal control problems in Step 2 of Algorithm 2 is well-posed.
For any given T > 0, there exists a radius r e = r e (T ) such that for every (t 0 , v 0 ) ∈ R + × B r e (0) and u satisfying we have the following properties: with a positive constantc 1 =c 1 (T ). P3: Every finite horizon optimal control problem of the form O P T (t 0 , v 0 ), over the set of all control u satisfying (3.5), admits a solution. P4: For every δ with 0 < δ ≤ T , there exists a constantc 2 The estimate (3.7) will be used to derive the exponential stability of RHC. The validity of Properties P1-P4 will be addressed in Sect. 5. In particular, the justification of Property P1 is based on the stabilizability of (2.6) by finitely many controllers. This result will be investigated in Sect. 4.
For the sake of simplicity, throughout this section, we use the notation * .
Therefore, O P T (t 0 , v 0 ) can be considered as an unconstrained problem.

Lemma 3 If P1-P3 hold and T > δ > 0, then there exists a neighbourhood
and t 0 +T Proof First observe that due to (3.8) in Remark 3, we have for Thus, using (3.4) and (3.6), we have for everyt Choosing Now, we come to the verification of (3.9) for v 0 ∈ B d 1 (0). Due to Bellman's optimality principle, we have for every t * ∈ [δ, T ] that (3.12) where v u in the above equality is the solution to C S(t * − δ, t 0 + δ, v * T (t 0 + δ; t 0 , v 0 )) for any u ∈ L 2 ((t 0 + δ, t 0 + t * ); R N ) and in the last inequality, (3.4) and (3.11) were used.

Lemma 4 Suppose that P1-P3 hold, and for given
we have the following estimates and t 0 +T Proof The proof is similar to the one given in [3,Lemma 2.4]. The only difference lies on the fact that here the estimates are with respect to the V -norm instead of H -norm. This requires that v * T (·; v 0 , t 0 ) ∈ C(I T (t 0 ); V ) which is true according to Property P2.

Proof
The proof is given in Appendix 3.
Theorem 1 (Suboptimality and exponential stability) Suppose that P1-P4 hold and let a sampling time δ > 0 be given. Then there exist numbers T * > δ and α ∈ (0, 1), such that for every fixed prediction horizon T ≥ T * and every v 0 ∈ B d 2 (0) with d 2 (T ) > 0, the receding horizon control u rh obtained from Algorithm 2 satisfies the suboptimality inequality 17) and the exponential stability inequality where T ≥ T * , d 1 is defined as in Lemma 3, andc 2 =c 2 (δ, T ) is given in Property P4. For the moment, we will show by induction that for every integer k ≥ 1, the following conditions hold: and Induction base (k = 1): Since d 2 ≤ d 1 , the assumptions of Proposition 2 are applicable and we have where α and ζ have been defined in Proposition 2.
Induction step: We assume that (3.19)-(3.21) hold for k = k with k ∈ N, we will show that (3.19)-(3.21) are also satisfied for (3.22) and  (3.24) and Moreover, due to the induction hypothesis ((3.19) for k = k ), P1 is applicable and by using (3.8) in Remark 3, we can write (3.26) Hence, for initial pair (t k , v rh (t k )), Property P4 is also applicable and we can write that (3.27) Hence v rh (t k +1 ) ∈ B d 1 (0). From this together with (3.24) and (3.25), we can conclude the induction step and, thus, (3.19)-(3.21) hold for any k ∈ N 0 . Now, taking the limit k → ∞ in (3.20), we find which concludes (3.17). Now we turn to inequality (3.18). Using (3.27) and setting c V =c 2 γ (T ) Moreover, for every t > 0 there exists a k ∈ N such that t ∈ [t k , t k+1 ]. Using (3.6), (3.26), and (3.28), we have for t and therefore, by setting c V :=c 1 c V (1+ γ (T ) α )η −1 , we are finished with the verification of (3.18) and the proof is complete. Remark 4 if we had α = 1, the inequality (3.17) would imply the optimality of the RHC u rh . Since γ (T ) is bounded and δ is fixed, it follows from (7.10) that lim T →∞ α(T ) = 1. This means that, RHC is asymptotically optimal.
We introduce a set of actuators Further, without loss of generality, we assume that U ω is linearly independent.
To prove the local stabilizability of C S(∞, t 0 , v 0 ), first, we study the stabilizability of this controlled system without the nonlinear term N . Then using the perturbation theory, we extend the result to the local stabilizability for the original controlled system with the nonlinearity. In this matter, we consider the following linear system where q ∈ L 2 (I ∞ (t 0 ); L 2 ) stands for the control input and P N : L 2 → spanÛ ⊂ L 2 is the orthogonal projection onto the span ofÛ. A stabilizing control q for (4.3) is constructed through concatenation of a sequence of controls on equidistant finite horizon intervals covering [t 0 , ∞). These controls are associated to the null controllability problems introduced in the next lemma.

Lemma 5
Suppose that λ ≥ 0, and a nonempty open set ω ⊂ Ω be given. Further, assume that for the reference trajectoryŷ regularity condition (RA) holds. Then for every T > 0, and (t 0 , v 0 ) ∈ R + × H , the following system is null controllable with a constant c ob = c ob (T , λ,ŷ). That is, there exists a control η * ∈ L 2 (I T (t 0 ), L 2 (ω)) satisfying

5)
whose associated state at time t 0 + T is equal to zero.
Proof The proof is inspired by those given in [29,Theorem 2.10] and [31] with the deferences that here we deal with a system of equations and for the Navier-Stokes system, we need to deal with the Leray projection. For the sake of completeness, we give the proof in Appendix 4. In For both the examples, we investigate the actuators component-wise. In this matter, for the sequence of scalar valued spatial functions {φ j } M j=1 ⊂ L 2 (Ω), we consider the functions j ∈ L 2 defined by (4.12) Then we can definê 13) where N = M 3 . For this setting, the orthogonal projection P N : L 2 → spanÛ will have the form where P M : L 2 (Ω) → span({φ j } M j=1 ) stands for the orthogonal projection from L 2 (Ω) onto span({φ j } M j=1 ). Therefore, due to (4.11) and (4.14), condition (COAC) holds provided that This means that, we only need to verify condition (COAC) component-wise. In each example, we choose {φ j } M j=1 in (4.12), and in such a way that (4.15) holds. In this case, for the correspondingÛ defined in (4.13) and the chosen , Proposition 3 is applicable and U ω defined in (4.2) is the desirable set of actuators. 1, 2, 3, . . . } is a complete system of eigenfunctions associated to the negative of Laplacian −Δ, which is defined on the domain ω with homogeneous Dirichlet boundary conditions. We may also assume that these eigenfunctions are ordered with respect to the increasing sequence of the eigenvalues 0 < λ 1 ≤ λ 2 ≤ · · · with lim i→∞ λ i = ∞. Moreover, let ∈ C 2 (Ω) satisfying (4.1) be given. For instance, can be chosen to be a bump function.

Example 1 (Laplacian Eigenfunctions) Suppose that {φ
By defining the orthonormal projection P ω M : ) and setting φ j := E 0φ j for j = 1, . . . , M with the extension-to-zero operator E 0 : L 2 (ω) → L 2 (Ω), we obtain for every w ∈ L 2 (Ω) and v ∈ H 1 0 (Ω) that (4.16) Therefore, for these choices ofÛ defined by (4.12)-(4.13) and , condition (COAC) holds due to (4.15) if for a large enough N = M 3 the following inequality holds For ω of the form (4.10), due to the asymptotic behaviour λ M ≥ DM and |ω| := and |B| denoting the volume of the unit ball in R 3 , we obtain the following estimate on the number of required actuators Example 2 (Piecewise constant functions) Here we set := χ ω for (4.10), where χ ω : Ω → {0, 1} is the characteristic function defined on ω. Then we consider the uniform partitioning of ω to a family of sub-rectangles. For every i ∈ {1, 2, 3}, the interval Then the set of actuators is defined by setting φ i := Since μ M → 0 ( equivalently μ N → 0) as d i → 0 for each i ∈ {1, 2, 3}, due to (4.15) it can be shown that for this choice ofÛ defined in (4.12)-(4.13), condition (COAC) is satisfied provided that the partitions are fine enough so that

Consequently, condition (COAC) is satisfied provided that
where d min := min 1≤i≤3 d i andĪ := max 1≤i≤3Īi . Then the inequality M 2 I 6 ≥ c 6 Π Υ 3 π 6 is sufficient for (COAC) and we have the following lower bound on the number of actuators Due to Proposition 3, system (4.3) is globally stabilizable and the control q(v 0 ) ∈ L 2 (I ∞ (t 0 ); L 2 ) can be taken as a bounded function of an initial function v 0 ∈ H . Relying on this, in the following Proposition, we derive a stabilizing feedback law of the form This feedback law enters the linear system (4.3) in the place of P N q(t).
In the next theorem, we show the local exponential stabilizability of the nonlinear system relying on the results of Proposition 4.  = cK(ŷ, ,Û , λ, ν), for which the first statement in Proposition 4 holds. Further, there exists a radius r s such that for every pair is well-posed and its solution satisfies
Proof The proof is provided in Appendix 5.

Main Result
In this section, we present the main result of the paper, i.e, the local exponential stability of the RHC obtained by Algorithm 1, or equivalently, by Algorithm 2 for the setting (3.3). Beforehand, we need to verify Properties P1-P4 for the incremental function defined in (3.3). Clearly, in this case satisfies (3.2) with α := min{1, β}. Proof Due to Theorem 2, there exist a uniformly bounded family of continuous oper-atorsK λ (t) : H → span U ω and a radius r s such that for every pair (t 0 , v 0 ) ∈ R + × B r s (0) the feedback lawK λ v is exponentially stabilizing. By defining the linear isomorphism I : span U ω → R N , and settingũ(v 0 ) = (ũ 1 , . . . ,ũ N ) t := IK λ v, we obtain that Using (4.21) and (5.1), we can infer that where constant c I is related to I. Due to the definition of V T and using (4.21) and (5.2), we can write for any given Hence, by setting γ (T ) := In the next proposition, we investigate Properties P2-P4.
Proposition 6 (Verification of P2-P4) Suppose that the incremental function is defined as in (3.3) and let T > 0 be given. Then there exists a ball B r e (0) ⊂ V with radius r e = r e (T ), such that for every (t 0 , v 0 ) ∈ R + × B r e (0) and u satisfying
Proof First we deal with the verification of P2. Due to Proposition 1, for every given T > 0, there exists a radius r = r (T ) such that for every  Using this fact together with (5.8), we find that Further, for the terms in the last line of (5.11), we obtain (5.12) and where c > 0 is a generic constant depending on Ω, and in both of (5.12) and (5.13) we have used Agmon's inequality [37,Lemma 13.2] z L ∞ ≤ c ag z with c ag depending on Ω. Due to the fact that the embedding L 2 (I T (t 0 ); D(A)) ∩ is compact (see e.g., [38]) and the terms v n and, thus, v * ∈ W (I T (t 0 ); D(A), H ) is the solution corresponding to the control u * .
Since v n → v * strongly in L 2 (I T (t 0 ); V ) and u n u * in L 2 (I T (t 0 ); R N ) we have and, as a consequence, the pair (v * , u * ) is optimal. We are finished with the justification of P3.
Finally, due to (5.3) and (5.6), Lemma 2 is applicable. Using estimate (2.15) for any δ with 0 < δ ≤ T , we obtain  (5.14) and the exponentially stable estimate where ζ and c V depend on U ω , δ, and T , but are independent of y 0 .

Conclusions
To sum up, we have established the local stabilizability of the three-dimensional Navier-Stokes system towards a given trajectory satisfying suitable regularity conditions via finite-dimensional RHC. This RHC enters as time depending linear combinations of a finite number of actuators. Our theory allows us to employ the indicator functions whose supports cover a part of the domain as actuators.
In this paper, we confined ourselves to the three-dimensional Navier-Stokes equations. Obviously, all the results remain also valid for the two-dimensional Navier-Stokes equations. We believe that for the two-dimensional case the local exponential stabilizability of the RHC can even be proven for the weakly variational solution, as a consequence, with respect to the H -norm.
with c depending only on Ω and ν. Then, using (7.9) and Gronwall's Lemma for the interval I δ (t 0 ), we obtain where c depends also on T . Moreover, due to (2.12) and (2.13), we have where in the last inequality we have use the fact that ŷ 2 V t 0 ,δ ≤ĉ for a constantĉ =ĉ(δ, R,ˆ ) independent of t 0 (see Remark 1). Hence, by setting c 5  Therefore, due to the definition of wλ, using a similar argument as in the proof of Proposition 3, and (7.27), we conclude that