Robust control of a Cahn-Hilliard-Navier-Stokes model

We study in this article a class of robust control problems associated with a coupled 
Cahn-Hilliard-Navier-Stokes model in a two dimensional bounded 
domain. The model consists of the Navier-Stokes equations for the 
velocity, coupled with the Cahn-Hilliard model for the 
order (phase) parameter. We prove the existence and uniqueness of solutions and we 
derive a first-order necessary optimality condition for these robust control problems.


1.
Introduction. It is well accepted that the incompressible Navier-Stokes equation governs the motions of single-phase fluids such as air or water. On the other hand, we are faced with the difficult problem of understanding the motion of binary fluid mixtures, that is fluids composed by either two phases of the same chemical species or phases of different composition. Diffuse interface models are well-known tools to describe the dynamics of complex (e.g., binary) fluids, [15]. For instance, this approach is used in [5] to describe cavitation phenomena in a flowing liquid. The model consists of the NS equation coupled with the phase-field system, cf., eg. [9,15,14,16]. In the isothermal compressible case, the existence of a global weak solution is proved in [12]. In the incompressible isothermal case, neglecting chemical reactions and other forces, the model reduces to an evolution system which governs the fluid velocity v and the order parameter φ. This system can be written as a NS equation coupled with a convective Allen-Cahn equation, [15]. The associated initial and boundary value problem was studied in [15] in which the authors proved that the system generated a strongly continuous semigroup on a suitable phase space which possesses a global attractor. They also established the existence of an exponential attractor. This entails that the global attractor has a finite fractal dimension, which is estimated in [15] in terms of some model parameters. The dynamic of simple single-phase fluids has been widely investigated although some important issues remain unresolved, [28]. In the case of binary fluids, the analysis is even more complicate and the mathematical studied is still at it infancy as noted in [15]. As noted in [14], the mathematical analysis of binary fluid flows is far from being well understood. For instance, the spinodal decomposition under shear consists of a two-stage evolution of a homogeneous initial mixture: a phase separation stage in which some macroscopic patterns appear, then a shear stage in 2076 T. TACHIM MEDJO which these patters organize themselves into parallel layers (see, e.g. [25] for experimental snapshots). This model has to take into account the chemical interactions between the two phases at the interface, achieved using a Cahn-Hilliard approach, as well as the hydrodynamic properties of the mixture (e.g., in the shear case), for which a Navier-Stokes equations with surface tension terms acting at the interface are needed. When the two fluids have the same constant density, the temperature differences are negligible and the diffuse interface between the two phases has a small but non-zero thickness, a well-known model is the so-called "Model H" (cf. [19,17]). This is a system of equations where an incompressible Navier-Stokes equation for the (mean) velocity v is coupled with a convective Cahn-Hilliard equation for the order parameter φ, which represents the relative concentration of one of the fluids.
The necessary conditions for optimal control problems governed by fluid mechanic models such as the NS systems have been studied by several authors (see for instance [31,32,30,29,21,22,23,3]). In [29], the authors studied a Pontryagin's maximum principle for optimal control problems (with a state constraint) governed by the 3D NS equations. In order to overcome the problem associated with the state constraint, the authors first defined a new penalty functional depending on a small parameter with which they approximated the original problem with a family of optimal control problems P without state constraint. A Pontryagin's maximum principle is derived for the approximate problem P and the limit as goes to 0 yields an optimality condition for the original control problem with a state constraint. For control problems associated to the Cahn-Hilliard-Navier-Stokes (CH-NS) systems, not that much have been done. In [24], the author studied a Pontryagin's maximum principle for optimal control problems (with a state constraint) governed by a 2D CH-NS model. Following the work of [29], he derived an optimality condition for the control problem. In [13], the authors studied some distributed optimal control problem associated with a CH-NS system. They proved the existence of a solution and derived a first-order optimality condition. Similar results are obtained in [18], where the authors studied an optimal boundary control problem associated with a time-discrete CH-NS system.
As described in [4], robust control theory, which generalizes optimal control theory, can be represented as a differential game between an engineer seeking the "best" control which stabilizes the flow perturbation with limited control effort and, simultaneously, nature seeking the "maximally malevolent" disturbance which destabilizes the flow perturbation with limited disturbance magnitude. In [4], the authors present a general framework for robust control problem in fluid mechanics. Given a fairly general cost functional J = J (ψ, φ), the authors in [4] proved the existence of a saddle point (ψ,φ), which maximizes J with respect to the disturbance ψ and minimizes J with respect to the control f 2 , subject to the Navier-Stokes equations. In this article, we study some robust control problems associated with the CH-NS systems. We prove the existence and uniqueness of solutions using the framework given in [4]. Let us note that the coupling between the Navier-Stokes and the Cahn-Hilliard systems introduces in the system a highly nonlinear coupling term that makes the analysis of these control problems more involved.
The article is divided as follows. In the next section we present the CH-NS model and its mathematical setting. We recall from [8,6,7,15,14] some existence and uniqueness results as well as some a priori estimates on the solution. We also present some estimates for the linearized system. The third section presents the robust control framework as well as the main results. We first study the linear problem and then, we consider the full nonlinear case. For each of these problems, we prove the existence and uniqueness of solution and we derive an associated firstorder optimality condition. Finally, some a priori estimates on the adjoint systems are given in the Appendix.

2.
The CH-NS model and its mathematical setting.
2.1. Governing equations. In this article, we consider a model of homogeneous incompressible two-phase flow. More precisely, we assume that the domain M of the fluid is a bounded domain in 2 . Then, we consider the system in M × (0, +∞).
In (1), the unknown functions are the velocity v = (v 1 , v 2 ) of the fluid, its pressure p and the order (phase) parameter φ. The external volume force Q is given. The quantity µ is the variational derivative of the following free energy functional where, e.g., F (r) = r 0 f (ζ)dζ. Here, the constants ν 1 > 0, ν 3 > 0 and K > 0 correspond to the kinematic viscosity of the fluid, the mobility constant and the capillarity (stress) coefficient respectively. Here ν 2 , α > 0 are two physical parameters describing the interaction between the two phases. In particular, ν 2 is related with the thickness of the interface separating the two fluids.
A typical example of potential F is that of logarithmic type. However, this potential is often replaced by a polynomial approximation of the type F (r) = γ 1 r 4 − γ 2 r 2 , γ 1 , γ 2 being positive constants. As noted in [14], (1) 1 can be replaced by The stress tensor ∇φ ⊕ ∇φ is considered the main contribution modeling capillary forces due to surface tension at the interface between the two phases of the fluid.
Regarding the boundary conditions for these models, we assume that the boundary conditions for φ are the natural no-flux condition where ∂M is the boundary of M and η is the outward normal to ∂M. These conditions ensure the mass conservation. Note that (4) implies that From (5), we deduce the conservation of the following quantity

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where |M| stands for the Lebesgue measure of M. More precisely, we have Concerning the boundary condition for v, we assume the Dirichlet (no-slip) boundary condition v = 0, on ∂M × (0, ∞).
(8) Therefore we assume that there is no relative motion at the fluid-solid interface.
The initial condition is given by 2.2. Mathematical setting. We first recall from [14] a weak formulation of (1), (4), (8)- (9). Hereafter, we assume that the domain M is bounded with a smooth boundary ∂M (e.g., of class C 3 ). We also assume that f ∈ C 4 ( ) satisfies where c f is some positive constant and m ∈ [3, +∞) is fixed.
Hereafter, if X is a Banach space, we will denote by X * the dual space of X. To simplify the notations, the duality paring between X and X * will be denoted ·, · and the norm in X * will be denoted · X * .
We set We denote by H 1 and V 1 the closure of V 1 in (L 2 (M)) 2 and (H 1 0 (M)) 2 respectively. The scalar product in H 1 is denoted by (·, ·) L 2 and the associated norm by | · | L 2 . Moreover, the space V 1 is endowed with the scalar product We now define the operator A 0 by where P 1 is the Leray-Helmotz projector in L 2 (M) onto H 1 . Then, A 0 is a selfadjoint positive unbounded operator in H 1 which is associated with the scalar product defined above. Furthermore, A −1 0 is a compact linear operator on H 1 and |A 0 · | L 2 is a norm on D(A 0 ) that is equivalent to the H 2 −norm.
We introduce the linear nonnegative unbounded operator on L 2 (M) and we endow D(A N ) with the norm |A N · | L 2 + | · | L 2 , which is equivalent to the H 2 −norm. We also define the linear positive unbounded operator on the Hilbert space L 2 0 (M) of the L 2 − functions with null mean Note that B −1 N is a compact linear operator on L 2 0 (M). More generally, we can define B s N for any s ∈ , noting that |B We introduce the bilinear operators B 0 , B 1 (and their associated trilinear forms b 0 , b 1 ) as well as the coupling mapping R 0 , which are defined from D( respectively. More precisely, we set Note that R 0 (µ, φ) = Pµ∇φ. We recall that B 0 , B 1 and R 0 satisfy the following estimates We recall that (due to the mass conservation) we have Thus, up to a shift of the order parameter field, we can always assume that the mean of φ is zero a the initial time and, therefore it will remain zero for all positive times. Hereafter, we assume that We set The space Y is a complete metric space with respect to the norm We define the Hilbert space V by endowed with the scalar products whose associated norm is Finally we set Throughout this article, we shall denote by c i , K i , K several positive constants that depend on the data (v 0 , φ 0 ) and Q. We will also denote by c a generic positive constant that depends on the domain M. To simplify the notations, we set (without loss of generality) ν 1 = ν 2 = ν 3 = α = K = 1.
is called a weak solution to (24) and (9) is called a strong solution on the time interval [0, T ] if in addition to (25), it satisfies The weak formulation of (24) was proposed and studied in [8,6,7,15,14] (see also [2,1,10]), where the existence and uniqueness results for weak and strong solutions were proved.
2.3. Some a priori estimates. In this part, we first derive some a priori estimates on the solution to (24), (9).
Note that if (v, φ) is a smooth solution to (1), by taking the scalar product in H 1 of (1) 1 with v, then taking the scalar product in L 2 (M) of (1) 3 with µ, we derive that d dt Proposition 1. For (v 0 , φ 0 ) ∈ Y and f ∈ C 1 ( ). The system (24), (9) has a unique weak solution (v, φ)(t). Moreover, the following estimate holds: where hereafter, Q 0 will denote a monotone non-decreasing function independent of the time t and the initial conditions. Proof. The existence and uniqueness of weak solutions is proved in [8,6,7]. To derive (28), we proceed as in [14], (see Proposition 3.1 an Lemma 3.3 in [14]). To derive (28), we proceed follows. We take the scalar product in L 2 (M) of (24) 3 with 2φ. Adding the resulting equation to (27) gives where and C e = 2C F |M| > 0, |M| > 0 being the Lebesgue measure of M, and C F > 0 is a constant large enough to ensure that E(t) is nonnegative. Note that F is bounded from below by a constant. With this choice of C e , we can find C f > 0 such that It follows from (29) that which gives and (28) 1 follows from (33) and (31). Using (24) 1 , we can check (using well known regularity result) that and (28) 2 follows. Finally, it is clear that (28) 3 follows from (28) 1 , (28) 2 and (14)- (16). (24), (9) has a unique strong solution (v, φ)(t). Moreover the following estimate holds: If (24) corresponding to the initial condition u i 0 and the forcing Q i , then we have

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Furthermore if (w 0 , ψ 0 ) ∈ V, then the solution (w, ψ) to (58) satisfies the regularity ) ∩ C(0, T ; V) and the following estimates hold: , using the known regularity results for parabolic equations such as the linearized Navier-Stokes system (see e.g., [28]), we can rigorously prove the existence and uniqueness of solution to (58) or (57). To derive (59), we multiplying (58) 1 by w. Then we take the duality of (58) 2 and (58) 3 with A Nμ − ξA N ψ and A N ψ respectively, where ξ > 0 is small enough and will be selected later. We derive that d dt To prove (60) 1 , we take the inner product of (58) 1 with A 0 w in L 2 (M). Then we take the inner product in L 2 (M) of (58) 2 and (58) 3 with B 2 Nμ − ξB 2 N ψ and B 2 N ψ respectively, where ξ > 0 will be selected later. We derive that where It follows from (70)-(77) that (with 0 < ξ < 1 small enough) where , and (60) 1 follows from the Gronwall lemma. Note that G(t) ∈ L 1 (0, T ).
It is easy to check that (60) 2 follows from (60) 1 and the properties of the operators B 0 , B 1 and R 0 given in (14)-(16). 3. Robust control framework. Let us first introduce some notations. We define the linear operator A : V → V * by for where ·, · denotes the duality pairing between V and V * or between V i and V * i , i = 1, 2. We also define the following trilinear functional and associated operator for

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We introduce the bilinear operator R : for Finally we set for To simplify the notations, we will also set Then we can check that With the above notations, if we set u = (v, φ), u 0 = (v 0 , φ 0 ) and Q ≡ (Q, 0), then we can rewrite (24) as: Let u = (v, φ) be the strong solution to (84) given by Proposition 2. If we set ω = (w, ψ), ω 0 = (w 0 , ψ 0 ) and g = (g 1 , g 2 ), then the linearized system (58) can be rewritten as: We will also consider the following adjoint system for which a result on the existence and uniqueness of solutions is given in Proposition 14 in the Appendix.
For the control framework, in the spirit of the non-cooperative game discussed in [4], the interior forcing Q is decomposed into a disturbance f 1 ∈ L 2 (0, T ; H 1 ) and a control f 2 ∈ L 2 (0, T ; H 1 ). Thus we write where B 1 and B 2 are given bounded operators on L 2 (M). The cost functional J used in this section is defined by where the flow u is related to the disturbance f 1 and the control f 2 through the CH-NS equations and the control parameters l and γ are given. The linear operators C 1 and C 2 are bounded or unbounded operators on Y satisfying with α 0 ≥ 0, β ≥ 0 and α 0 + β > 0. Some cases of particular interest are: , ⇒ regulation of turbulent kinetic energy; • C 1 u = (d 1 ∇ × v, 0), C 2 u = 0, for u = (v, φ), ⇒ regulation of the square of the vorticity.
The goal is to find a disturbancef 1 and a controlf 2 such that (f 1 ,f 2 ) is a saddle point of the functional J .
3.1. The linear problem. In this subsection, the flow u is related to the disturbance f 1 and the control f 2 through the linearized CH-NS which models small deviations of the flow perturbation u = (v, φ) from the desired target flow U = (υ, Φ). The regularity required is given by Proposition 4. For u 0 ∈ V, the system (90) has a unique solution u ∈ C((0, T ; V)∩ L 2 (0, T ; D(A)). The and it follows that u ∈ L ∞ (0, T ; V) ∩ L 2 (0, T ; D(A)).
Proof. The proof of the existence and uniqueness of solution is given in Proposition 3. The rest of the proposition is proved as Proposition 9 given below for the full CH-NS model.
In this section, we consider the following robust control problem: To find (f 1 ,f 2 ) ∈ (L 2 (0, T ; H 1 )) 2 such that subject to the linearized CH-NS (90), where X = L 2 (0, T ; H 1 ). Here the cost function is given by (88) where the flow u is related to the disturbance f 1 and the control f 2 through the system (90). The proof of the existence of a solution (f 1 ,f 2 ) to the robust control problem for the linear case is based on the following result.
Theorem 3.1 (Existence of solution of the robust control problem in the linear case). For γ > γ 0 = γ 0 (M, T, u 0 , U ) and l > 0, the robust control problem (93) has a unique solution (f 1 ,f 2 ) that satisfies Proof. The proof follows directly from Propositions 5 and 6.

3.2.
Characterization of the saddle point. In this subsection, we derive in a classical manner the adjoint equation associated with the robust control problem (93). We consider the following systems.
where u is given by (90) and hereafter, if M is a bounded operator defined from V into V , M * will denote the adjoint operator defined by Proposition 7. Let u ,ũ and u be the solutions of (103), (104) and (90) respectively. Then whereũ is given by (104).

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Proof. We can easily check that (109) and 3.3. The nonlinear problem. In this subsection, we consider the nonlinear case, that is the case where the flow u is related to the disturbance f 1 and the control f 2 through the following full CH-NS Hereafter, for u 0 ∈ V, we will denote by u(f 1 , f 2 ) ≡ u ∈ L ∞ (0, T ; V)∩L 2 (0, T ; D(A)) the unique solution to (111) corresponding to (f 1 , f 2 ) ∈ (L 2 (0, T ; H 1 )) 2 .

Now from (38), we derive that
i.e., Let where From (14)- (16), we also have It follows that We derive from Proposition 3 that Therefore U defined by (112) is the Frêchet derivative of U at (f 1 , f 2 ).

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It follows that (125) Therefore Let X and Y be two non-empty, closed, convex and bounded subsets of L 2 (0, T ; H 1 ). Let K > 0 such that We consider the following robust control problem: To Here the cost function is given by where the flow u is related to the disturbance f 1 and the control f 2 through the system (111). Hereafter c 0 , c 1 denote numerical coefficients depending only on M, T, u 0 , X and Y and whose value may be different in each inequality. The proof of the existence of a solution (f 1 ,f 2 ) to the robust control problem for the nonlinear case if based on the following result.
Proposition 10. Let J be a functional defined on X × Y, where X and Y are non-empty, closed, bounded, convex sets. If J satisfies: is convex lower semi-continuous, then the functional J has at least one saddle point Proof. The proof is given in [11].
Proof. 1) Since the norm is lower semi-continuous, the mapping f 1 −→ J (f 1 , f 2 ) is upper semi-continuous. To prove the concavity, it is enough to show that for every f 1 , f 1 , f 2 , h(r) = J (f 1 + rf 1 , f 2 ) is concave with respect to r, near r = 0, i.e.
where u satisfies From Proposition 3, we have where From the properties (14)-(16) of the operators B, R and E, we have where c 2 = c 2 ( u V ) > 0. It follows that which gives (see (132)) Therefore, we have the following estimate
To prove Condition 2, we first note that the map f 2 −→ J (f 1 , f 2 ) is upper semicontinuous since the norm is lower semi-continuous. For the convexity, it is enough to prove that g(r) = J (f 1 , f 2 + rf 2 ) is convex w.r.t. r near r = 0, i.e., g (0) > 0.
Note that where Following similar estimates as in the proof of Condition 1, we arrive at for l 2 > l 2 1 = D 2 + D 4 . The strict convexity of the map f 2 −→ J (f 1 , f 2 ) follows. From Propositions 10 and 11, the following results hold true.
Proposition 12. Assume that X and Y are non-empty, closed, bounded, convex subsets of L 2 (0, T ; H 1 ) and that l and γ are large enough. Then there exists a unique saddle point (f 1 ,f 2 ) ∈ X × Y and an associated flow u(f 1 ,f 2 ), such that 3.4. Identification of the gradients. We now state the main result of this section.
Theorem 3.2. Assume that X and Y are non-empty, closed, bounded, convex subset of L 2 (0, T ; L 2 (M)) and that l and γ are large enough. The robust control problem (128) has a unique solution (f 1 ,f 2 ) ∈ X × Y. Moreover, the gradients of the cost functional J are given by DJ whereũ is found from the solution (u,ũ) of the following coupled systems du dt Proof. The gradients DJ Df1 and DJ Df2 are determined as in the linear case by solving the system (150)-(151). The uniqueness of solutions is proved using the strict concavity and the strict convexity of the functional J .

3.5.
Application to data assimilation. It is well known that chaotic problems, such as weather system, are highly susceptible to the small disturbance present in all physical systems. A classical control problem arising in meteorology and oceanography, in relation with data assimilation, is the adjustment of initial conditions in order to obtain a flow that agrees with a desired target flow (i.e., the observations). Given a set of measurements of some actual flow ϑ on [0, T ], the problem is to determine a "best" estimate as to the initial state of the model u that leads to the observed system behavior, while simultaneously forcing the model with a small component of the worst-case disturbance which perturbs u away from the observed system behavior ϑ, [20,26,27]. Define = u−ϑ as the difference between the estimated flow u and the observed flow ϑ. The cost function considered for this problem is given by The measurements of the actual flow C 1 ϑ and C 2 ϑ(T ) are assumed to be given. The linear operators C 1 and C 2 are bounded or unbounded operators on Y satisfying (89). For C 1 = 0 and C 2 = d 1 I, where I is the identity operator, the goal is to match the potential vorticity of the estimated flow with that of the observed flow at the end time T. More details on the functional J are given in [4]. The results given in this subsection are generalizations of those given in the previous subsection, therefore we will omit the details of the proofs. Let X and Y be two non-empty, closed, convex and bounded subsets of L 2 (0, T ; H 1 ) and V respectively. Let K > 0 such that Here we assume that B 2 is a bounded mapping from H 1 into V. We consider the following robust control problem: To As for the body force control problem (128), we have the following results.
Proposition 13. Assume that X and Y are non-empty, closed, bounded, convex subset of L 2 (0, T ; L 2 (M)) and V respectively and that l and γ are large enough.
The robust control problem (154) has a unique solution (f 1 ,f 2 ) ∈ X × Y. Moreover, the gradients of the cost functional J are given by whereũ is found from the solution (u,ũ) of the following coupled systems Proof. The proof is similar to that of Theorem 3.2.
4. Appendix 1. In this part, we study the existence and uniqueness of solutions to the adjoint systems. Let u = (v, φ) be the strong solution to (84) given by Proposition 2 with u 0 ∈ V. We consider the following adjoint system: where the unknown is ω = (w, ψ) and ω 0 = (w 0 , ψ 0 ) is given. The existence and uniqueness of solution to (158) is given in [24].

Appendix 2:
An intuitive introduction to robust control theory. We recall from [4] some motivations for studying robust control problems such as the ones considred in this article. Consider the present problem as a differential game between an engineer seeking the best control f 2 which stabilizes the flow perturbation with limited control effort and, simultaneously, nature seeking the maximally malevolent disturbance f 1 which destabilizes the flow perturbation with limited disturbance magnitude. The parameter γ 2 factors into such a competition as a weighting on the magnitude of the disturbance which nature can afford to offer, in a manner analogous to the parameter l 2 , which is a weighting on the magnitude of the control which the engineer can afford to offer.
The parameter l 2 may be interpreted as the price of the control to the engineer. The l → ∞ limit corresponds to prohibitively expensive control, and results in f 2 → 0 in the minimization with respect to f 2 for the present problem. Reduced values of l increase the cost functional less upon the application of a control f 2 . A non-zero control results whenever the control f 2 can affect the flow perturbation (v, φ) in such a way that the net cost functional J is reduced.
The parameter γ 2 may be interpreted as the price of the disturbance to nature. The γ → ∞ limit results in f 1 → 0 in the maximization with respect to f 1 , leading to the optimal control formulation of [3] for f 2 alone. Reduced values of γ decrease the cost functional less upon the application of a disturbance. A non-zero disturbance results whenever the disturbance can affect the flow perturbation u in such a way that the net cost functional J is increased.
Solving for the control f 2 which is effective even in the presence of a disturbance f 1 which maximally spoils the control objective is a way of achieving system robustness. A control which works even in the presence of the malevolent disturbance f 1 will also be robust to a wide class of other possible disturbances. Put another way, the introduction of the worst-case disturbance in the robust approach is a means of detuning the optimal controls. It results in a set of controls which may have somewhat degraded performance when no disturbances are present. However, much greater system robustness (i.e., better performance) is attained in cases for which unknown disturbances are present in the system, and thus the approach is relevant for applications in physical systems, in which unpredictable disturbances are ubiquitous. In the present systems, for γ < γ 0 for some critical value γ 0 (an upper bound of which is established in this paper), the non-cooperative game is not known to have a finite solution; essentially, the malevolent disturbance wins. The control f 2 corresponding to γ = γ 0 results in a stable system even when nature is on the brink of making the system unstable. However, the control determined with γ = γ 0 is not always the most suitable, as it may result in a very large control magnitude and degraded performance in response to disturbances with structure more benign than the worst-case scenario. In the implementation, variation of l and γ provides the flexibility in the control design which is necessary to achieve the desired trade-offs between Gaussian and worst-case disturbance response and the control magnitude required.