Local controllability to trajectories for non-homogeneous 2-d incompressible Navier-Stokes equations
aa r X i v : . [ m a t h . A P ] J u l Local controllability to trajectories for non-homogeneous2-d incompressible Navier-Stokes equations ∗ Mehdi Badra † , Sylvain Ervedoza ‡ , and Sergio Guerrero § October 23, 2018
Abstract
The goal of this article is to show a local exact controllability to smooth ( C ) trajecto-ries for the 2-d density dependent incompressible Navier-Stokes equations. Our controlla-bility result requires some geometric condition on the flow of the target trajectory, whichis remanent from the transport equation satisfied by the density. The proof of this resultuses a fixed point argument in suitable spaces adapted to a Carleman weight function thatfollows the flow of the target trajectory. Our result requires the proof of new Carlemanestimates for heat and Stokes equations. Key words.
Non-homogeneous Navier-Stokes equations, Local exact controllability totrajectories.
AMS subject classifications.
The goal of this article is to discuss the local exact controllability property for the 2-dnon-homogeneous Navier Stokes equations.
Setting and main results.
Let Ω be a smooth bounded domain of R , T > , T ) × Ω by Ω T . Let us consider a trajectory ( σ, y ) of the non-homogeneousNavier-Stokes equations: ∂ t σ + div( σ y ) = f σ in Ω T ,σ∂ t y + σ ( y · ∇ ) y − ν ∆ y + ∇ q = f y in Ω T , div y = 0 in Ω T , ( σ (0) , y (0)) = ( σ , y ) in Ω . (1.1)Here, ν > f σ , f y ) are assumed to beknown.We will focus on the local exact controllability problem around the trajectory ( σ, y ) witha control exerted on the boundary (0 , T ) × ∂ Ω: Given ( σ , y ) close to the initial data ∗ This work is partially supported by the Agence Nationale de la Recherche (ANR, France), Project CISIFSnumber NT09-437023. † Laboratoire LMA, UMR CNRS 5142, Universit´e de Pau et des Pays de l’Adour, F-64013 Pau Cedex, France.E-mail: [email protected] ‡ Corresponding author. Institut de Math´ematiques de Toulouse ; UMR5219 ; Universit´e de Toulouse ;CNRS ; UPS IMT, F-31062 Toulouse Cedex 9, France. E-mail: [email protected] .Address: Institut de Math´ematiques de Toulouse, Universit´e Paul Sabatier, 118 route de Narbonne, 31062Toulouse Cedex 9. Ph.: +33561557654 § Universit´e Pierre et Marie Curie, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005 Paris, France.E-mail: [email protected] σ , y ), find control functions ( h σ , h y ) on (0 , T ) × ∂ Ω such that the solution ( σ, y ) of ∂ t σ + div( σ y ) = f σ in Ω T ,σ∂ t y + σ ( y · ∇ ) y − ν ∆ y + ∇ q = f y in Ω T , div y = 0 in Ω T , ( σ (0) , y (0)) = ( σ + ρ , y + u ) , in Ω , (1.2)with the boundary conditions: σ = σ + h σ for ( t, x ) ∈ (0 , T ) × ∂ Ω , with y ( t, x ) · n ( x ) < , (1.3) y = y + h y on (0 , T ) × ∂ Ω , (1.4)satisfies ( σ ( T ) , y ( T )) = ( σ ( T ) , y ( T )) . (1.5)Our goal is to present a positive answer to this control problem under suitable assumptionson the target trajectory ( σ, y ), and in particular one of hyperbolic nature on the flowcorresponding to y . Besides, our strategy will yield a control acting on some suitablesubsets of the boundary which correspond, roughly speaking, to the complement of thepart of the boundary in which the scalar product of the target velocity y with the normalvector n is positive for all time t ∈ [0 , T ].Going further requires some notations. We denote by L (Ω), L ∞ (Ω), H r (Ω), H r (Ω) etcfor r ≥
0, the usual Lebesgue and Sobolev spaces of scalar functions, and we write in boldthe spaces of vector-valued functions: L (Ω) = ( L (Ω)) , H r (Ω) = ( H r (Ω)) , etc. Wealso define V (Ω) : def = { v ∈ H (Ω) | div v = 0 in Ω } . In the following, we will always assume that the target velocity y belongs to C (Ω T ). It canthus be extended into a C ([0 , T ] × R ) function, still denoted the same for simplicity butnot necessarily divergence free outside Ω T . This allows to define the flow X = X ( t, τ, x )associated to that velocity y : ∀ ( t, τ, x ) ∈ [0 , T ] × R , ∂ t X ( t, τ, x ) = y ( t, X ( t, τ, x )) , X ( τ, τ, x ) = x. (1.6)Thus we define the outgoing subset of Ω for the flow X as follows:Ω T out : def = (cid:8) x ∈ Ω | ∃ t ∈ (0 , T ) s.t. X ( t, , x ) ∈ R \ Ω (cid:9) . (1.7)One of our main assumptions is the following one:Ω = Ω T out . (1.8)Note that this assumption does not depend on the extension y on [0 , T ] × R and is intrinsic.This assumption is of hyperbolic nature as it requires the time T to be large enough toguarantee that all the particles that were in Ω at time t = 0 have been transported bythe flow outside Ω in a time strictly smaller than T . Of course, this is remanent from thedensity equation (1.2) (1) in which the density is transported along the flow correspondingto the velocity of the fluid.As we said, we will not require the control to be supported on the whole boundary (0 , T ) × ∂ Ω, but only on some part of it (0 , T ) × Γ c where Γ c = ∂ Ω \ Γ and Γ (the part withoutcontrol) is an open subset of ∂ Ω satisfying the following conditions:(i) . Γ has a finite number of connected components,(ii) . sup [0 ,T ] × Γ y · n > . (1.9)Note that the above condition garantees the existence of γ > y ( t, x ) · n ( x ) ≥ γ for all ( t, x ) ∈ (0 , T ) × Γ .Our main result states as follows: heorem 1.1. Let Ω be a smooth bounded domain of R . Assume that the target trajectory ( σ, y ) solution of (1.1) satisfies ( σ, y ) ∈ C ([0 , T ] × Ω) × C ([0 , T ] × Ω) and inf [0 ,T ] × Ω σ > . (1.10) Assume that the condition (1.8) is satisfied for the time T .Then there exists ε > such that for all ( ρ , u ) ∈ L ∞ (Ω) × V (Ω) satisfying k ρ k L ∞ (Ω) + k u k H (Ω) ≤ ε, (1.11) there exists a controlled trajectory ( σ, y ) ∈ L ∞ (Ω T ) × H (0 , T ; L (Ω)) ∩ L (0 , T ; H (Ω)) solution of (1.2) – (1.4) satisfying the control requirement (1.5) .Besides, if Γ denotes an open subset of the boundary satisfying (1.9) , we may furtherimpose y = y on (0 , T ) × Γ . In particular, in that case, no boundary condition is imposedon the density on Γ . Actually, we will only prove Theorem 1.1 when Γ = ∅ . When Γ = ∅ , Theorem 1.1can be proved more easily following the same lines, as the extensions arguments we willperform can be handled much more easily. Strategy of the proof.
The proof of Theorem 1.1 is based on a technical fixed-pointprocedure, and we briefly explain below its general strategy.Setting ρ : def = σ − σ, u : def = y − y , (1.12)and f ( ρ, u ) : def = − ρ ( ∂ t u + (( y + u ) · ∇ ) u + ( u · ∇ ) y ) − σ ( u · ∇ ) u − ρ ( ∂ t y + ( y · ∇ ) y ) , (1.13)equations (1.2)–(1.5) rewrite ∂ t ρ + ( y + u ) · ∇ ρ = − u · ∇ σ in Ω T ,σ∂ t u + σ ( y · ∇ ) u + σ ( u · ∇ ) y − ν ∆ u + ∇ p = f ( ρ, u ) in Ω T , div u = 0 in Ω T , ( ρ (0) , u (0)) = ( ρ , u ) in Ω , (1.14)with the boundary conditions u = on (0 , T ) × Γ , (1.15)and with the requirement ( ρ ( T ) , u ( T )) = (0 , ) in Ω . (1.16)To construct a solution of (1.14)–(1.16), the strategy consists in finding a fixed-point tosome mapping F ( ρ , u ) : b u u defined in such a way that u = F ( ρ , u ) ( b u ) is a suitablesolution of: ∂ t ρ + ( y + b u ) · ∇ ρ = − b u · ∇ σ in Ω T ,σ∂ t u + σ ( y · ∇ ) u + σ ( u · ∇ ) y − ν ∆ u + ∇ p = f ( ρ, b u ) in Ω T , div u = 0 in Ω T , u = on (0 , T ) × Γ , ( ρ (0) , u (0)) = ( ρ , u ) in Ω , ( ρ ( T ) , u ( T )) = (0 , ) in Ω . (1.17)The mapping F ( ρ , u ) is defined in two steps. First, for a given b u , we define F ( b u , ρ ) : def = ρ, where ρ will be constructed as a suitable solution of the following control problem forthe equation of the density: ∂ t ρ + ( y + b u ) · ∇ ρ = − b u · ∇ σ in Ω T ,ρ (0) = ρ in Ω ,ρ ( T ) = 0 in Ω . (1.18) hen, we define F ( f , u ) : def = u , where u is a suitable solution of the following controlproblem for the equation of the velocity: σ∂ t u + σ ( y · ∇ ) u + σ ( u · ∇ ) y − ν ∆ u + ∇ p = f in Ω T , div u = 0 in Ω T , u = on (0 , T ) × Γ , u (0) = u in Ω , u ( T ) = in Ω . (1.19)The mapping F ( ρ , u ) is then defined as follows: F ( ρ , u ) ( b u ) : def = u , where ρ = F ( b u , ρ ) , and u = F ( f ( ρ, b u ) , u ) . (1.20)Hence our strategy decouples the control problem (1.2)–(1.5) into two control problems,(1.18) for the equation of the density, and (1.19) for the equation of the velocity, each ofwhich having different behaviors.Indeed, on one hand, the control problem (1.19) is of parabolic type, and it will behandled by using global Carleman estimates following the general approach of Fursikovand Imanuvilov [14] for the heat equations: in the case of Navier-Stokes equations, thisapproach has already been successfully implemented in the works [17, 12].On the other hand, the control problem (1.18) involves a transport equation. This canbe easily controlled provided the time T > F ( ρ , u ) to map some convex setinto itself. In order to do this, we should be able to get estimates on the above controlproblems in spaces that behave suitably with respect to both of them. In particular,this will lead us to introduce Carleman weights that follow the dynamics of the transportequation, that is weight functions which are transported by the flow. This strategy thenfollows the one recently developed in [9] for deriving local exact controllability results forthe 1d compressible Navier-Stokes equations around constant non-vanishing velocities.Actually, the Carleman estimates we develop in this article also present the featureof not vanishing at time t = 0. This allows us to construct a solution ( ρ, u ) of (1.14)without using any property of the Cauchy problem for the non-homogeneous Navier-Stokesequations. Related references and comments.
To our knowledge, control properties for non-homogeneous Navier-Stokes equations have only been studied in [10], which proves severaloptimal control results in that context for various cost functions.For the homogeneous Navier-Stokes equations, the density is assumed to be constantand thus the equations reduce to the equations on the velocity. In that case, several localexact controllability results have been established in [17, 12] based on parabolic Carlemanestimates, see e.g. [14, 11]. Later on, several different strategies have been proposed,see for instance [13, 16, 19]. We also point out that these results also use the Carlemanestimate derived in [18] for non-homogeneous elliptic problems in order to handle thepressure term.But our problem also involves some transport phenomenon, and therefore also sharessome features of the thermoelasticity equations [1], the viscoelasticity models [21, 5], andthe compressible Navier-Stokes equations [9]. Our approach is actually close to the onedeveloped in [9]. Though, the divergence free condition in the model we consider hererequires a specific treatment.In this article, we will not use any result on the Cauchy problem for (1.2), as ourstrategy will automatically construct a trajectory ( σ, y ) solving the equations (1.2). How-ever, several results are available in the literature. We refer to the work [10] for severalresults and comments on the Cauchy problem for the non-homogeneous incompressibleNavier-Stokes equations and to the references therein.Let us also note that we will need a precise understanding of the transport equationwhen transported by a flow entering the domain. More precisely, we will use in an essentialway the compactness result in [3, Theorem 4], obtained as a consequence of [2]. e also underline that Theorem 1.1 does not state the uniqueness of the controlledtrajectory ( σ, y ). This is due to the lack of regularity for the density σ which only belongsto L ∞ (Ω T ), see [8] for the uncontrolled case. Another limitation of this control result isthat it is only valid for 2-d geometry. This restriction comes from the treatment of thevelocity equation. For that, we prove a new Carleman inequality for the Stokes equationswhich hardly relies on the use of the stream function of the velocity, see Section 2.Finally note that our result also allows the use of non-trivial trajectories. For instance,if Γ = ∅ and ( σ, y ) = (1 , ), one may consider the trajectory ( σ ∗ ( t ) , y ∗ ( t )) = (1 , η ( t/T ) U )for constant vector fields U and η = η ( t ) ∈ [0 ,
1] a bump function taking value 0 at t = 0and t = 1 and with η = 1 on [1 / , / σ ∗ ( t ) , y ∗ ( t )) = (1 , ) at time t = 0and at time t = T . But for T > U , ( σ ∗ ( t ) , y ∗ ( t )) satisfies (1.8) and all theassumptions of Theorem 1.1, while whatever the time T > σ ( t ) , y ( t )) =(1 , ) clearly does not satisfy (1.8). This suggests that the geometric condition (1.8) maybe avoided in some cases using “return method” type ideas, see e.g. [6, 7]. Outline.
This article is organized as follows. Section 2 explains how to solve thecontrol problem (1.19) by the use of Carleman estimates for the Stokes operator. Section3 shows how to construct a controlled density satisfying (1.18) and to derive weightedestimates on it. Section 4 then focuses on the proof of Theorem 1.1 by putting togetherthe arguments developed in Sections 2 and 3. Finally, the Appendix gives the detailedproofs of some technical results.
This section is dedicated to the construction of a solution of (1.19).
In order to solve the control problem (1.19), we will consider (1.19) in an extended domain O as follows: O is a smooth bounded domain of R satisfyingΩ ⊂ O , ∂ O is of class C , ∂ O ∩ ∂ Ω ⊃ Γ . (2.1)We then extend ( σ, y ) on [0 , T ] × O , still denoted the same for simplicity, such that( σ, y ) ∈ C ([0 , T ] × O ) × C ([0 , T ] × O ) and inf [0 ,T ] ×O σ ( t, x ) > . (2.2)Remark that this is possible due to the assumption (1.10). As u ∈ V (Ω), extending itby zero outside Ω, we get an extension, still denoted the same, such that u ∈ H ( O ) and div u = 0 in O . (2.3)By also extending f by zero outside Ω and setting O T = (0 , T ) × O , Γ T = (0 , T ) × ∂ O wethen consider the following system σ ( ∂ t u + ( y · ∇ ) u + ( u · ∇ ) y ) − ν ∆ u + ∇ p = f + h O\ Ω in O T , div u = 0 in O T , u = on Γ T , u (0) = u in O . (2.4)Here, 1 O\ Ω is the characteristic function of O \
Ω and h ∈ L ( O T ) is a control function.Note that the presence of 1 O\ Ω in (2.4) implies that the action of the control is supportedin O \
Ω.We thus intend to solve the following control problem: Given u ∈ H ( O ) satisfying (2.3)and a source term f in some suitable space, find a control function h ∈ L ( O T ) such thatthe solution u of (2.4) satisfies u ( T ) = 0 in O . (2.5)Indeed, if we are able to solve this control problem, the restriction of the solution u to Ωwould yield a solution of the control problem (1.19). In order to solve the control problem − ∂ t ( σ v ) − D ( σ v ) y − σ v div y − ν ∆ v + ∇ p = g in O T , div v = 0 in O T , v = on Γ T , (2.6)where D v := ∇ v + t ∇ v is the symmetrized gradient.To state our result precisely, let us introduce the weight functions we will use in theCarleman estimate. We assume that we have a function ˜ ψ = ˜ ψ ( t, x ) ∈ C ( O T ) such that˜ ψ : def = ˜ ψ ( t, x ) such that ∀ ( t, x ) ∈ O T , ˜ ψ ( t, x ) ∈ [0 , , ∀ ( t, x ) ∈ Γ T , ∂ n ˜ ψ ( t, x ) ≤ , ∀ t ∈ [0 , T ] , ˜ ψ ( t ) | ∂ O is constant , ∀ t ∈ [0 , T ] , inf O ˜ ψ ( t, · ) = ˜ ψ ( t ) | ∂ O . (2.7)We also assume the existence of two open subsets ˜ ω T ⋐ ω T of [0 , T ] × ( O \
Ω) (here and inthe following, the symbol ⋐ means that there exists a compact set K T of [0 , T ] × ( O \
Ω)such that ˜ ω T ⊂ K T ⊂ ω T ) and a constant α > O T \ ˜ ω T {|∇ ˜ ψ |} ≥ α > . (2.8)For m ≥
1, we set ψ ( t, x ) : def = ˜ ψ ( t, x ) + 6 m. (2.9)We then set T > T > T ≤ / T + 2 T < T and choose a weightfunction in time θ m,µ ( t ) depending on the parameters m ≥ µ ≥ θ m,µ : def = θ m,µ ( t ) such that ∀ t ∈ [0 , T ] , θ m,µ ( t ) = 1 + (cid:18) − tT (cid:19) µ , ∀ t ∈ [ T , T − T ] , θ m,µ ( t ) = 1 , ∀ t ∈ [ T − T , T ) , θ m,µ ( t ) = 1( T − t ) m ,θ m,µ is increasing on [ T − T , T − T ] ,θ m,µ ∈ C ([0 , T )) . (2.10)For simplicity of notations in the following we omit the dependence on m and µ and wesimply write θ : def = θ m,µ . We will then take the following weight functions ϕ = ϕ ( t, x ) and ξ = ξ ( t, x ): ϕ ( t, x ) : def = θ ( t ) (cid:16) λe λ ( m +1) − exp( λψ ( t, x )) (cid:17) , ξ ( t, x ) : def = θ ( t ) exp( λψ ( t, x )) , (2.11)where s, λ are positive parameters with s ≥ λ ≥ µ is chosen as µ = sλ e λ (6 m − , (2.12)which is always bigger than 2, thus being compatible with the condition θ ∈ C ([0 , T ]).Note that the weight functions ϕ and ξ , depend on s, λ, m , and should rather be denotedby ϕ s,λ,m , resp. ξ s,λ,m , but we drop these indexes for simplicity of notations.Remark that, due to the definition of ψ in (2.9) and the conditions (2.7), we have, for all λ ≥ t, x ) ∈ O T , 34 θ ( t ) λe λ ( m +1) ≤ ϕ ( t, x ) ≤ θ ( t ) λe λ ( m +1) . (2.13)Finally, we introduce b ϕ ( t ) : def = min x ∈O ϕ ( t, x ) , ϕ ∗ ( t ) : def = max x ∈O ϕ ( t, x ) = ϕ | ∂ O ( t ) , (2.14) b ξ ( t ) : def = max x ∈O ξ ( t, x ) , ξ ∗ ( t ) : def = min x ∈O ξ ( t, x ) = ξ | ∂ O ( t ) . (2.15)Using these weight functions, we prove the following Carleman estimate for the Stokessystem (2.6): heorem 2.1. Assume that O is a smooth bounded domain extending Ω as in (2.1) , let ω , ˜ ω be two subdomains of O\ Ω such that ˜ ω ⋐ ω and set ω T = [0 , T ] × ω and ˜ ω T = [0 , T ] × ˜ ω .Let ˜ ψ as in (2.7) – (2.8) and ψ, θ, ϕ, ξ as in (2.9) – (2.10) – (2.11) .Then, for m ≥ , there exist some constants s ≥ , λ ≥ and C > such that for allsmooth solution v of (2.6) with source term g ∈ L ( O T ) , for all s ≥ s and λ ≥ λ , s / λ − / Z O ( ξ ∗ ) − /m | v (0 , · ) | e − sϕ ∗ (0) + sλ Z Z O T ξ | v | e − sϕ + s − Z Z O T ξ |∇ v | e − sϕ + s / λ − / Z T ( ξ ∗ ) − /m e − sϕ ∗ k v k H ( O ) ≤ C (cid:18) s / λ Z Z ω T b ξ | v | e sϕ ∗ − s b ϕ + s / λ − / Z Z O T ( ξ ) − /m | g | e − sϕ (cid:19) . (2.16)The proof of Theorem 2.1 is done in Sections 2.2 and 2.3. We are first going to prove aslightly improved version of the Carleman estimates (2.16) for solutions v of the simplifiedversion of the adjoint problem (2.6): − σ∂ t v − ν ∆ v + ∇ p = g in O T , div v = 0 in O T , v = 0 on Γ T . (2.17)Our approach then consists first in taking the curl of the equation (2.17) and consider theequation of w = rot v : − σ∂ t w − ν ∆ w = rot g + ∂ t v · ∇ ⊥ σ in O T . (2.18)Thus, in Section 2.2, we derive estimates on w solution of (2.18) in terms of the right handside of the equation of (2.18) and the boundary terms. It turns out that the boundaryconditions and source terms strongly depend on v itself. Hence in Section 2.3, we explainhow to estimate v in terms of w by using the stream function ζ associated to u , which isgiven by∆ ζ ( t ) = w ( t ) in O T and ζ ( t ) = c i ( t ) on [0 , T ] × γ i for i = 1 , . . . , K, (2.19)where { γ i , i = 1 , . . . , K } is the family of connected components of ∂ O and c i ( t ), i =1 , . . . , K are some constants characterizing ζ ( t ) which are chosen such that, for someLipschitz subdomain b ω of O\ Ω satisfying ˜ ω ⋐ b ω ⋐ ω , Z b ω ζ ( t ) = 0 . (2.20)Among the new features of the Carleman estimate of Theorem 2.1 with respect tothose in the literature, let us point out the following facts: • The weight function in time θ m,µ in (2.10) does not blow up as the time t goes to 0.However, our proof requires a strong convexity property close to t = 0, tuned by thechoice of the parameter µ in (2.10) as a suitable function of the parameters s and λ ,see (2.12). • The weight function ψ depends on both the time and space variables. As we shallexplain, this is not a big issue as long as we guarantee that for all t ∈ [0 , T ], ψ ( t )is constant on the boundary ∂ O , thus allowing to apply the Carleman inequality of[18] for elliptic equations.Based on Theorem 2.1, following standard duality arguments, we prove the followingcontrol result: Theorem 2.2.
Within the setting and assumptions of Theorem 2.1, there exists a constant
C > such that for all s ≥ s and λ ≥ λ , if u verifies (2.3) and f ∈ L ( O T ) satisfies Z Z O T ξ − | f | e sϕ < ∞ , (2.21) here exists a control function h ∈ L ( O T ) supported in ω T and a controlled trajectory u ∈ L ( O T ) such that u solves the control problem (2.4) – (2.5) and ( u , h ) satisfies theestimate k e sϕ ∗ u k L ( H ) ∩ H ( L ) + s / λ / Z Z O T ξ /m − | u | e sϕ + s − / Z Z ω T b ξ − | h | e s b ϕ − sϕ ∗ ≤ C (cid:18)Z Z O T ξ − | f | e sϕ + e sϕ ∗ (0 , · ) k u k H ( O ) (cid:19) . (2.22)The proof of Theorem 2.2 is given in Section 2.4. The goal of this section is to show the following estimate:
Theorem 2.3.
Let c ω T be an open subset of O T satisfying ˜ ω T ⋐ c ω T and let ˜ ψ as in (2.7) – (2.8) and ψ, θ, ϕ, ξ as in (2.9) – (2.10) – (2.11) .For all M > , there exist constants C > , s and λ such that for all s ≥ s and λ ≥ λ ,for all smooth functions w in O T , such that − σ∂ t w − ν ∆ w = a w + A · ∇ w + g + n X i =1 b i ∂ i g i + b n +1 ∂ t g n +1 in O T , with a ∈ L ∞ ( O T ) , A ∈ L ∞ (0 , T ; W , ∞ ( O )) , g , g i ∈ L ( O T ) , and coefficients b i ∈ L ∞ (0 , T ; W , ∞ ( O )) , b n +1 ∈ W , ∞ (0 , T ; L ∞ ( O )) satisfying k a k L ∞ ( O T ) + k A k L ∞ (0 ,T ; W , ∞ ( O )) + n X i =1 k b i k L ∞ (0 ,T ; W , ∞ ( O )) + k b n +1 k W , ∞ (0 ,T ; L ∞ ( O )) ≤ M, (2.23) we have s λ Z Z O T ξ | w | e − sϕ ≤ C Z Z O T | g | e − sϕ + Cs λ Z Z O T ξ (cid:0) n X i =1 | g i | (cid:1) e − sϕ + Cs λ Z Z O T ξ | g n +1 | e − sϕ + Cs λ Z Γ T ξ | w | e − sϕ + Cs λ Z Z d ω T ξ | w | e − sϕ . (2.24)The proof of Theorem 2.3 is long and is divided in three steps:1. a Carleman estimate for the heat equation with homogeneous boundary conditionsand source terms in L ( O T ); see Theorem 2.4;2. energy estimates on controlled trajectories of a heat equation with a source term in L ( O T ); see Theorem 2.5;3. a duality argument.This proof is inspired by the ones in [20], see also [11]. Below, we only state Theorems2.4–2.5, whose proofs are postponed to the appendix. Proof of Theorem 2.3.
As said above, the proof is done in three steps. An L -Carleman estimate. The first result is the following L -Carleman estimatefor the heat equation: heorem 2.4. Assume the setting of Theorem 2.3. For all m ≥ , there exist constants C > , s ≥ and λ ≥ such that for all smooth functions z on O T satisfying z = 0 on Γ T , for all s ≥ s , λ ≥ λ , we have Z O |∇ z (0) | e − sϕ (0) + s λ e λ (6 m +1) Z O | z (0) | e − sϕ (0) + sλ Z Z O T ξ |∇ z | e − sϕ + s λ Z Z O T ξ | z | e − sϕ ≤ C Z Z O T | ( − σ∂ t − ν ∆) z | e − sϕ + C s λ Z Z d ω T ξ | z | e − sϕ . (2.25)The proof of Theorem 2.4 is given in Section A.1. It is rather classical except for theweight function ϕ , which does not blow up as t → ψ whichdepends on both time and space variables. This introduces in the proof of Theorem 2.4several new technical issues, though our proof follows the lines of [14]. Estimates on a control problem.
We then analyze the following control problem:for f ∈ L ( O T ), find a control function h ∈ L ( c ω T ) such that the solution y of ∂ t ( σy ) − ν ∆ y = f + h d ω T , in O T ,y = 0 , on Γ T ,y (0 , · ) = 0 , in O , (2.26)solves the control problem: y ( T, · ) = 0 , in O . (2.27)We claim the following result: Theorem 2.5.
Assume the setting of Theorem 2.3. For all m ≥ , there exist positiveconstants C > , s ≥ and λ ≥ such that for all s ≥ s and λ ≥ λ , for all f satisfying Z Z O T ξ − | f | e sϕ < ∞ , (2.28) there exists a solution ( Y, H ) of the control problem (2.26) – (2.27) which furthermore sat-isfies the following estimate: s λ Z Z O T | Y | e sϕ + Z Z d ω T ξ − | H | e sϕ + sλ Z Z O T ξ − |∇ Y | e sϕ +1 s Z Z O T ξ − ( | ∂ t Y | + | ∆ Y | ) e sϕ + λ Z Γ T ξ − | ∂ n Y | e sϕ ≤ C Z Z O T ξ − | f | e sϕ . (2.29)The proof of Theorem 2.5 is given in Section A.2. Again, the proof is rather clas-sical and is based on the duality between the Carleman estimates, which are weightedobservability estimates, and controllability, and then on energy estimates. Note how-ever that these energy estimates have to be derived using the weight functions defined in(2.7)–(2.11), and this introduces some novelties in the computations. A duality argument.
The proof of Theorem 2.3 then relies upon the estimate (2.29)on the solution (
Y, H ) of the control problem (2.26)–(2.27) for f = ξ we − sϕ . Indeed, if( Y, H ) solves (2.26)–(2.27) for some f satisfying (2.28), multiplying the equation satisfiedby Y by w , we obtain Z Z O T w ( f + H d ω T ) + Z Γ T wν∂ n Y = Z Z O T ( a wY − w div ( A Y ) + g Y − n X i =1 g i ∂ i ( b i Y ) − g n +1 ∂ t ( b n +1 Y )) . (2.30)In particular, as f = ξ we − sϕ satisfies Z Z O T ξ − | f | e sϕ = Z Z O T ξ | w | e − sϕ , ccording to (2.29) we can construct ( Y, H ) solution of ∂ t ( σY ) − ν ∆ Y = ξ we − sϕ + H d ω T , in O T ,Y = 0 , on Γ T ,Y (0 , · ) = 0 , in O ,Y ( T, · ) = 0 , in O , (2.31)for which we have the estimate: s λ Z Z O T | Y | e sϕ + Z Z d ω T ξ − | H | e sϕ + sλ Z Z O T ξ − |∇ Y | e sϕ +1 s Z Z O T ξ − ( | ∂ t Y | + |∇ Y | ) e sϕ + λ Z Γ T ξ − | ∂ n Y | e sϕ ≤ C Z Z O T ξ | w | e − sϕ . (2.32)Using then the identity (2.30), we infer Z Z O T ξ | w | e − sϕ ≤ C (cid:18) sλ Z Z O T ξ | w | e − sϕ (cid:19) / (cid:18) sλ Z Z O T ξ − ( | Y | + |∇ Y | ) e sϕ (cid:19) / + C (cid:18) s λ Z Z O T | g | e − sϕ (cid:19) / (cid:18) s λ Z Z O T | Y | e sϕ (cid:19) / + C sλ Z Z O T ξ (cid:0) n X i =1 | g i | (cid:1) e − sϕ ! / (cid:18) sλ Z Z O T ξ − ( | Y | + |∇ Y | ) e sϕ (cid:19) / + C (cid:18) s Z Z O T ξ | g t | e − sϕ (cid:19) / (cid:18) s Z Z O T ξ − ( | Y | + | ∂ t Y | ) e sϕ (cid:19) / + C (cid:18) λ Z Γ T ξ | w | e − sϕ (cid:19) / (cid:18) λ Z Γ T ξ − | ∂ n Y | e sϕ (cid:19) / + C (cid:18)Z Z d ω T ξ | w | e − sϕ (cid:19) / (cid:18)Z Z d ω T ξ − | H | e sϕ (cid:19) / , which immediately yields the claimed result by (2.32). This section aims at proving Theorem 2.1. This will be done in two steps.We first prove the following Carleman estimate for v solution of (2.17): Theorem 2.6.
Within the setting and assumptions of Theorem 2.1, for any m ≥ , thereexist some constants s ≥ , λ ≥ and C > such that for all solution v of (2.17) withsource term g ∈ L ( O T ) , for all s ≥ s and λ ≥ λ , s / λ − / Z O ( ξ ∗ ) − /m | v (0 , · ) | e − sϕ ∗ (0) + sλ Z Z O T ξ | v | e − sϕ + s / λ − / Z T ( ξ ∗ ) − /m e − sϕ ∗ k v k H ( O ) + Z Z O T ξ | rot v | e − sϕ + s − Z Z O T ξ |∇ v | e − sϕ ≤ C (cid:18) s / λ Z Z ω T b ξ | v | e sϕ ∗ − s b ϕ + s − λ − Z Z O T ξ | g | e − sϕ + s / λ − / Z T ( ξ ∗ ) − /m e − sϕ ∗ k g k H − ( O ) + s − / λ − / Z Z O T ( ξ ∗ ) − /m | g | e − sϕ ∗ (cid:19) . (2.33)The proof of Theorem 2.6 is done below in Section 2.3.1. In Section 2.3.2 we thenexplain how Theorem 2.6 implies Theorem 2.1. .3.1 Proof of Theorem 2.6 Let v be a solution of (2.17) with source term g . As w = rot v satisfies (2.18), theCarleman estimate (2.24) applies to w : for all s ≥ s and λ ≥ λ , Z Z O T ξ | w | e − sϕ ≤ C (cid:18)Z Z d ω T ξ | w | e − sϕ + s Z Z O T ξ | v | e − sϕ + λ − Z Γ T ξ | w | e − sϕ + s − λ − Z Z O T ξ | g | e − sϕ (cid:19) . (2.34)Here and in the following b ω T = [0 , T ] × b ω where b ω is a Lipschitz subdomain O\ Ω suchthat e ω ⋐ b ω ⋐ ω . Note in particular that e ω T ⋐ b ω T ⋐ ω T .Next, because v is divergence free we also have, for all t ∈ (0 , T ), − ∆ v ( t ) = rot w ( t ) in O , v ( t ) = 0 on ∂ O . (2.35)Thus, using elliptic Carleman estimates with source term in H − ( O ) with weight e − sϕ ( t, · ) and integrating in time, see [18], we immediately get s − Z Z O T ξ |∇ v | e − sϕ + sλ Z Z O T ξ | v | e − sϕ ≤ C (cid:18)Z Z O T ξ | w | e − sϕ + sλ Z Z d ω T ξ | v | e − sϕ (cid:19) . (2.36)Combined with (2.34), and using the fact that w = rot v is bounded by ∂ n v on Γ T (recallthat v = 0 on Γ T ) and that ξ ∗ = ξ and ϕ ∗ = ϕ on (0 , T ) × ∂ O , we immediately have thatfor some s > λ >
1, for all s ≥ s and λ ≥ λ , s − Z Z O T ξ |∇ v | e − sϕ + Z Z O T ξ | w | e − sϕ + sλ Z Z O T ξ | v | e − sϕ ≤ C (cid:18)Z Z d ω T ξ | w | e − sϕ + sλ Z Z d ω T ξ | v | e − sϕ + λ − Z Γ T ( ξ ∗ ) | ∂ n v | e − sϕ ∗ + s − λ − Z Z O T ξ | g | e − sϕ (cid:19) . (2.37)We then introduce the stream function ζ associated to v , i.e. v = ∇ ⊥ ζ , which can becomputed explicitly as the solution of (2.19) for some constants c i ( t ) due to the dimension N = 2, see e.g. [15, Corollary 3.1]. Note that, by adding a constant to ζ if necessary,without loss of generality we can assume that (2.20) is also satisfied. Applying the ellipticCarleman estimate to the equation (2.19) (see e.g. [14]), we obtain that s λ Z Z O T ξ | ζ | e − sϕ + sλ Z Z O T ξ |∇ ζ | e − sϕ ≤ C (cid:18)Z Z O T ξ | w | e − sϕ + s λ Z Z d ω T ξ | ζ | e − sϕ (cid:19) . (2.38)Note that the Carleman estimate of [14] is obtained for homogeneous Dirichlet boundaryconditions. But it is easily seen that it remains true for a boundary data whose tangentialderivative at the boundary vanishes, which is the case for ζ .Of course, estimate (2.34) requires an observation term in ζ in c ω T . But Poincar´eWirtinger inequality and condition (2.20) implies, for all t ∈ [0 , T ], Z b ω | ζ ( t, · ) | ≤ C Z b ω |∇ ζ ( t, · ) | = Z b ω | rot ζ ( t, · ) | = Z b ω | v ( t, · ) | , and in particular: Z Z d ω T ξ | ζ | e − sϕ ≤ C Z Z d ω T b ξ | v | e − s b ϕ . (2.39) et us stress the fact that the 2-d assumption is also used at this stage since (2.39) relieson the identity |∇ ζ ( t, · ) | = | rot ζ ( t, · ) | .Next, we use (2.38) to derive suitable weighted energy estimates for v , hence for ∂ n v on the boundary ∂ O . But since we do not have any estimate on the pressure in the Stokesequation (2.17), we are reduced to derive energy estimates for v with weight functionsindependent of x . Estimates in L (0 , T ; H ( O )) . We set ( v a , p a ) : def = θ ( t )( v , p ) with θ ( t ) : def = s / λ − / ( ξ ∗ ) − /m e − sϕ ∗ ( t ) . Using ∂ t ϕ ∗ ≤ Cλ ( ξ ∗ ) /m in O T . (2.40)and explicit computations, we get θ ′ ≥ − Cs / λ / ( ξ ∗ ) e − sϕ ∗ ( t ) . (2.41)The pair ( v a , p a ) satisfies − σ∂ t v a − ν ∆ v a + ∇ p a = θ g − σθ ′ v in O T , div v a = 0 in O T , v a = 0 on Γ T , v a ( T ) = 0 in O . (2.42)We want to obtain an estimate of the L ( H )-norm of v a , so we multiply the partialdifferential equation in (2.42) by v a , we integrate in O T and we integrate by parts. Thisyields:12 k p σ (0 , · ) v a (0 , · ) k L ( O ) + ν k v a k L (0 ,T ; H ( O )) = Z Z O T θ g · v a − Z Z O T σθ ′ v · v a − Z Z O T ∂ t σ | v a | . (2.43)First, we remark that (cid:12)(cid:12)(cid:12)(cid:12)Z Z O T θ g · v a (cid:12)(cid:12)(cid:12)(cid:12) ≤ ν Z Z O T |∇ v a | + C Z T | θ | k g k H − ( O ) . (2.44)We then focus on the second term of (2.43) and use (2.41) − Z Z O T σθ ′ v · v a ≤ Cs / λ / Z Z O T ( ξ ∗ ) − /m v · ∇ ⊥ ζe − sϕ ∗ ( t ) = − Cs / λ / Z Z O T ( ξ ∗ ) − /m rot v ζe − sϕ ∗ ( t ) ≤ Cs / λ / Z Z O T ( ξ ∗ ) | ζ | e − sϕ ∗ ( t ) + νs / λ − / Z Z O T ( ξ ∗ ) − /m |∇ v | e − sϕ ∗ ( t ) ≤ Cs / λ / Z Z O T ( ξ ∗ ) | ζ | e − sϕ ∗ ( t ) + ν Z Z O T |∇ v a | . The last term can be handled similarly: (cid:12)(cid:12)(cid:12)(cid:12) Z Z O T ∂ t σ | v a | (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cs / λ − / Z Z O T ( ξ ∗ ) − /m | v | e − sϕ ∗ ≤ Cs / λ / Z Z O T ( ξ ∗ ) | ζ | e − sϕ ∗ ( t ) + ν Z Z O T |∇ v a | . lugging these three last estimates in (2.43), we obtain k v a (0 , · ) k L ( O ) + k v a k L (0 ,T ; H ( O )) ≤ C (cid:18) s / λ / Z Z O T ( ξ ∗ ) | ζ | e − sϕ ∗ + k θ g k L (0 ,T ; H − ( O )) (cid:19) . (2.45) Estimate in L (0 , T ; H ( O )) . Let us now set ( v b , p b ) : def = θ ( t )( v , p ) with θ ( t ) : def = s − / λ − / ( ξ ∗ ) / − / (2 m ) e − sϕ ∗ ( t ) , for which explicit computations yield: θ ′ ≥ − Cs / λ / ( ξ ∗ ) − m e − sϕ ∗ (2.46)This pair ( v b , p b ) satisfies − σ∂ t v b − ∆ v b + ∇ p b = θ g − σθ ′ v in O T , div v b = 0 in O T , v b = 0 on Γ T , v b ( T ) = 0 in O . (2.47)We then multiply the partial differential equation in (2.47) by ( − ∆ v b + ∇ p b ) /σ , we inte-grate in O T and we integrate by parts:12 Z O |∇ v b (0 , · ) | + Z Z O T σ |− ∆ v b + ∇ p b | = Z Z O T θ σ g ( − ∆ v b + ∇ p b ) − Z Z O T θ θ ′ |∇ v | . (2.48)Using (2.2) we can estimate the first term as follows: (cid:12)(cid:12)(cid:12)(cid:12)Z Z O T θ σ g ( − ∆ v b + ∇ p b ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Z O T σ | − ∆ v b + ∇ p b | + C k θ g k L ( O T ) . (2.49)For the second term, remark that by (2.46), we have θ θ ′ ≥ − Cs / λ − / ( ξ ∗ ) − /m e − sϕ ∗ = − Cθ , thus yielding − Z Z O T θ θ ′ |∇ v | ≤ C k θ v k L (0 ,T ; H ( O )) = C k v a k L (0 ,T ; H ( O )) . Therefore, using the above estimate and (2.49) into (2.48), we obtain k v b k L (0 ,T ; H ( O )) ≤ C Z Z O T | − ∆ v b + ∇ p b | ≤ C (cid:0) k θ g k L ( O T ) + k v a k L (0 ,T ; H ( O )) (cid:1) , (2.50)where we have used the classical H -estimate for the stationary Stokes system, see e.g.[4, Theorem IV.5.8]. Global Estimate on v and its normal derivative.
Since v = 0 on Γ T , classicalestimates yield k ∂ n v ( t, · ) k L ( ∂ O ) ≤ C (cid:16) k v ( t, · ) k H ( O ) k v ( t, · ) k H ( O ) + k v ( t, · ) k H ( O ) (cid:17) , and in particular, using the fact that θ ( t ) ≤ θ ( t ) for all t ∈ (0 , T ), (cid:13)(cid:13)(cid:13)(cid:13) λ − / ( ξ ∗ ) − m ∂ n v e − sϕ ∗ ( t, · ) (cid:13)(cid:13)(cid:13)(cid:13) L ( ∂ O ) ≤ C (cid:16) k θ v ( t, · ) k H ( O ) k θ v ( t, · ) k H ( O ) + k θ v ( t, · ) k H ( O ) (cid:17) . utting together (2.45) and (2.50) with this last estimate, using (2.38) and (2.39) toestimate the term in ζ and taking into account that m ≥
5, we deduce that k v a (0 , · ) k L ( O ) + k θ v k L (0 ,T ; H ( O )) + k θ v k L (0 ,T ; H ( O )) + λ − (cid:13)(cid:13)(cid:13) ( ξ ∗ ) / ∂ n v e − sϕ ∗ (cid:13)(cid:13)(cid:13) L (Γ T ) ≤ C (cid:18) s − / λ − / Z Z O T ξ | w | e − sϕ + s / λ / Z Z d ω T b ξ | v | e − s b ϕ + k θ g k L (0 ,T ; H − ( O )) + k θ g k L ( O T ) (cid:17) . (2.51) Elimination of the boundary term.
We come back to the Carleman inequality(2.37) and we combine it with (2.51): for s large enough, k v a (0 , · ) k L ( O ) + k θ v k L (0 ,T ; H ( O )) + k θ v k L (0 ,T ; H ( O )) s − Z Z O T ξ |∇ v | e − sϕ + Z Z O T ξ | w | e − sϕ + sλ Z Z O T ξ | v | e − sϕ ≤ C (cid:18)Z Z d ω T ξ | w | e − sϕ + s / λ Z Z d ω T b ξ | v | e − s b ϕ + k θ g k L (0 ,T ; H − ( O )) + k θ g k L ( O T ) + s − λ − Z Z O T ξ | g | e − sϕ (cid:19) . (2.52) Removing the observation on w . We now estimate the local term in | w | . Forthis purpose, we recall that c ω T = [0 , T ] × b ω ⋐ ω T = [0 , T ] × ω and we consider a positivefunction χ ∈ C ( O ) such that χ = 1 in b ω, χ = 0 in O \ ω. Using
Z Z b ω T ξ | w | e − sϕ ≤ Z Z b ω T b ξ | w | e − s b ϕ , (2.53)we are reduced to estimate the right hand side of (2.53): Z Z b ω T b ξ | w | e − s b ϕ ≤ Z Z ω T χ b ξ | w | e − s b ϕ ≤ Z Z ω T χ b ξ |∇ v | e − s b ϕ = − Z Z ω T χ b ξ ∆ v v e − s b ϕ + 12 Z Z ω T ∆ χ b ξ | v | e − s b ϕ . ≤ εs − / λ − / Z Z O T ( ξ ∗ ) − /m | ∆ v | e − sϕ ∗ + C ε s / λ / Z Z ω T ( ξ ∗ ) − /m b ξ | v | e sϕ ∗ − s b ϕ , where the last estimate follows from Young’s identity and where ε > ε small enough and recalling the definitionof θ , we get in particular s / λ − / Z O ( ξ ∗ ) − /m | v (0 , · ) | e − sϕ ∗ + s / λ − / Z T ( ξ ∗ ) − /m e − sϕ ∗ k v k H ( O ) s − Z Z O T ξ |∇ v | e − sϕ + sλ Z Z O T ξ | v | e − sϕ + Z Z O T ξ | rot v | e − sϕ ≤ C (cid:18) s / λ Z Z ω T b ξ | v | e sϕ ∗ − s b ϕ + s − λ − Z Z O T ξ | g | e − sϕ + s / λ − / Z T ( ξ ∗ ) − /m e − sϕ ∗ k g k H − ( O ) + s − / λ − / Z Z O T ( ξ ∗ ) − /m | g | e − sϕ ∗ (cid:19) . (2.54)This concludes the proof of Theorem 2.6. .3.2 Proof of Theorem 2.1 Let v be a smooth solution of (2.6) with source term g . Then v is a solution of (2.17)with source term ˜ g = g + ∂ t σ v + D ( σ v ) y + σ v div ( y ) . Applying Theorem 2.6 to v with source term ˜ g , for all s ≥ s and λ ≥ λ we get s / λ − / Z O ( ξ ∗ ) − /m | v (0 , · ) | e − sϕ ∗ (0) + sλ Z Z O T ξ | v | e − sϕ + s / λ − / Z T ( ξ ∗ ) − /m e − sϕ ∗ k v k H ( O ) + Z Z O T ξ | rot v | e − sϕ + s − Z Z O T ξ |∇ v | e − sϕ ≤ C (cid:18) s / λ Z Z ω T b ξ | v | e sϕ ∗ − s b ϕ + s − λ − Z Z O T ξ | ˜ g | e − sϕ + s / λ − / Z T ( ξ ∗ ) − /m e − sϕ ∗ k ˜ g k H − ( O ) + s − / λ − / Z Z O T ( ξ ∗ ) − /m | ˜ g | e − sϕ ∗ (cid:19) (2.55)and we are thus reduced to estimate the last terms of the inequality.But we have s − λ − Z Z O T ξ | ˜ g | e − sϕ ≤ C (cid:0) s − λ − Z Z O T ξ | g | e − sϕ + s − λ − Z Z O T ξ | v | e − sϕ + s − λ − Z Z O T ξ |∇ v | e − sϕ (cid:1) ,s − / λ − / Z Z O T ( ξ ∗ ) − /m | ˜ g | e − sϕ ∗ ≤ C (cid:0) s − / λ − / Z Z O T ( ξ ∗ ) − /m | g | e − sϕ ∗ + s − / λ − / Z Z O T ( ξ ∗ ) − /m | v | e − sϕ ∗ + s − / λ − / Z Z O T ( ξ ∗ ) − /m |∇ v | e − sϕ ∗ (cid:1) , in which all the terms in v , ∇ v can be absorbed by the left-hand side of (2.55) for s and λ large enough.We also have, for all t ∈ (0 , T ), k ˜ g ( t ) k H − ( O ) ≤ C k g ( t, · ) k L ( O ) + C k v ( t, · ) k L ( O ) . Hence s / λ − / Z T ( ξ ∗ ) − /m e − sϕ ∗ k ˜ g k H − ( O ) ≤ Cs / λ − / Z Z O T ( ξ ∗ ) − /m e − sϕ ∗ | g | + Cs / λ − / Z Z O T ( ξ ∗ ) − /m e − sϕ ∗ | v | . (2.56)Plugging these last estimates in (2.55), we obtain (2.16) for s and λ large enough. We use the following simplified form of (2.16): for all s ≥ s and λ ≥ λ and all smoothsolutions v of (2.6) with source term g : Z O ( ξ ∗ ) − /m | v (0 , · ) | e − sϕ ∗ (0) + s / λ / Z Z O T ξ | v | e − sϕ ≤ C (cid:18) s λ / Z Z ω T b ξ | v | e sϕ ∗ − s b ϕ + Z Z O T ξ − /m | g | e − sϕ (cid:19) . (2.57)Easy density arguments then show that this result extends to all solutions v of (2.6) withsource term g ∈ L ( O T ) and final data v ( T ) = v T ∈ V (Ω). e then follow the proof of Theorem 2.5 and introduce the functional J St defined by J St ( v T , g ) : def = 12 Z Z O T ξ − /m | g | e − sϕ + s λ / Z Z ω T b ξ | v | e sϕ ∗ − s b ϕ − Z Z O T f · v − Z O u ( · ) · v (0 , · ) , (2.58)defined for data ( v T , g ) ∈ V (Ω) × L ( O T ), where v solves (2.6) with v ( T ) = v T .We then need to define the functional J St on the set X St,obs : def = X St,obs k·k
St,obs , where X St,obs : def = (cid:8) ( v T , g ) ∈ V (Ω) × L ( O T ) } (2.59)and the norm k ( v T , g ) k St,obs is defined by k ( v T , g ) k St,obs : def = Z Z O T ξ − /m | g | e − sϕ + s λ / Z Z ω T b ξ | v | e sϕ ∗ − s b ϕ , where v is the corresponding solution to (2.6).According to (2.57), the functional J St can be extended by continuity on X St,obs if f satisfies (2.21). The functional J St then has a unique minimizer on X St,obs , that wedenote ( V T , G ) and corresponds to a solution V of (2.6). We get, for all smooth solution v of (2.6) corresponding to a source term g ,0 = Z Z O T ξ − /m G · g e − sϕ + s λ / Z Z ω T b ξ V · v e sϕ ∗ − s b ϕ − Z Z O T f · v − Z O u ( · ) · v (0 , · ) . (2.60)In particular, setting u = ξ − /m G e − sϕ , h = − s λ / b ξ V e sϕ ∗ − s b ϕ ω T , (2.61)we obtain a solution in the sense of transposition of the control problem (2.4)–(2.5) witha control term acting only on ω T .Besides, using again the Carleman estimate (2.57) and the fact that J St ( V T , G ) ≤ J St (0 ,
0) = 0, one immediately derives that k ( V T , G ) k obs ≤ Cs / λ / Z Z O T ξ − | f | e sϕ + C Z O ( ξ ∗ ) /m − | u | e sϕ ∗ (0) . (2.62)Hence, using (2.61), the controlled trajectory ( u , h ) satisfies Z Z O T ξ /m − | u | e sϕ + 1 s λ / Z Z ω T b ξ − | h | e s b ϕ − sϕ ∗ ≤ Cs / λ / Z Z O T ξ − | f | e sϕ + Z O ( ξ ∗ ) /m − | u | e sϕ ∗ (0) . (2.63)Finally, we can then derive H ( L ) ∩ L ( H ) estimates on u by applying regularityresults for Stokes equations to the system satisfied by e sϕ ∗ u . The computations are leftto the reader. This section is devoted to explain how to solve the control problem (1.18). As we saidin the introduction, the main difficulty is that we need to provide a controlled trajectorythat can be estimated with the use of the weight functions introduced in Section 2. .1 Basic properties of the flow Let y be the extension of y on [0 , T ] × R and X the corresponding flow, defined in(1.6). As y ∈ C ([0 , T ] × R ), the flow X is continuous with respect to the variables( t, τ, x ) ∈ [0 , T ] × R .We first discuss the stability of property (1.8): Lemma 3.1.
Assume that y ∈ C ([0 , T ] × R ) , and that the flow X defined by (1.6) satisfies (1.8) .There exist ε > , T ∗ > and T ∗ > such that for all T ∈ (0 , T ∗ ) , for all T ∈ (0 , T ∗ ) and for all x ∈ Ω , there exists t ∈ [ T , T − T ] such that d ( X ( t, T , x ) , Ω) ≥ ε .Proof. The proof is done by contradiction. Assume it is false. Then for all ε >
0, thereexist T ε > T ε such that T ε , T ε converge to 0 as ε →
0, and an x ε in Ω such that ∀ t ∈ [ T ε , T − T ε ] , d ( X ( t, T ε , x ε ) , Ω) < ε. (3.1)But x ε is bounded in Ω. Hence, up to a subsequence, it converges to some x in Ω. Asthe flow X is continuous in [0 , T ] × R and the distance function is continuous, for each t ∈ (0 , T ), one could then pass to the limit in (3.1): ∀ t ∈ (0 , T ) , d ( X ( t, , x ) , Ω) = 0 . This is of course in contradiction with (1.8).For b u ∈ L (0 , T ; H ( R )) we denote by b X the flow defined by ∂ t b X ( t, τ, x ) = ( y + b u )( t, b X ( t, τ, x )) , b X ( τ, τ, x ) = x. (3.2)We then show that, provided b u is small enough, the property (1.8) also holds for b X : Lemma 3.2.
Under the setting of Lemma 3.1, there exists ς > such that for all b u ∈ L (0 , T ; H ( R )) , satisfying k b u k L (0 ,T ; L ∞ ( R )) ≤ ς, (3.3) the flow b X defined by (3.2) satisfies the following property: for all T ∈ (0 , T ∗ ) , for all T ∈ (0 , T ∗ ) and for all x ∈ Ω , there exists t ∈ [ T , T − T ] such that d ( b X ( t, T , x ) , Ω) ≥ ε .Proof. Set L = k∇ y k L ∞ (0 ,T ; L ∞ (Ω)) . For τ, t ∈ [0 , T ] with t ≥ τ and x ∈ R , we have: | b X ( t, τ, x ) − X ( t, τ, x ) | = | b X ( t, τ, x ) − b X ( τ, τ, x ) + X ( τ, τ, x ) − X ( t, τ, x ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z tτ (cid:16) ∂ t b X ( t ′ , τ, x ) − ∂ t X ( t ′ , τ, x ) (cid:17) d t ′ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z tτ b u ( t ′ , b X ( t ′ , τ, x )) + y ( t ′ , b X ( t ′ , τ, x )) − y ( t ′ , X ( t ′ , τ, x ))d t ′ (cid:12)(cid:12)(cid:12)(cid:12) ≤ | t − τ | / k b u k L ( τ,t ; L ∞ ( R )) + L Z tτ | b X ( t ′ , τ, x ) − X ( t ′ , τ, x ) | d t ′ . Then Gronwall’s Lemma yields for all t ∈ [0 , T ] and x ∈ R : | b X ( t, τ, x ) − X ( t, τ, x ) | ≤ T / e LT k b u k L ( τ,t ; L ∞ ( R )) . (3.4)According to Lemma 3.1, Lemma 3.2 thus holds by setting ς = T − / e − LT ε/ .2 Construction of the controlled density In this section, we assume that b u ∈ L (0 , T ; H ( R )) and k b u k L (0 ,T ; L ∞ ( R )) ≤ ς, (3.5)where ς is given by Lemma 3.2. We then choose T ∈ (0 , T ∗ ) and T ∈ (0 , T ∗ ), where T ∗ , T ∗ are given by Lemma 3.2.The construction of the controlled density ρ solution of (1.18) is then done as in [9]:we construct a forward solution ρ f and a backward solution ρ b of the transport equationin (1.18) and we glue these two solutions according to the characteristics of the flow.Indeed, we define ρ f as the solution of ∂ t ρ f + ( y + b u ) · ∇ ρ f = − b u · ∇ σ in Ω T ,ρ f ( t, x ) = 0 for t ∈ (0 , T ) , x ∈ ∂ Ω , with ( y ( t, x ) + b u ( t, x )) · n ( x ) < ,ρ f (0) = ρ in Ω , (3.6)and ρ b as the solution of ∂ t ρ b + ( y + b u ) · ∇ ρ b = − b u · ∇ σ in Ω T ,ρ b = 0 for t ∈ (0 , T ) , x ∈ ∂ Ω , with ( y ( t, x ) + b u ( t, x )) · n ( x ) > ,ρ b ( T ) = 0 in Ω . (3.7)We also introduce χ the solution of ∂ t χ + ( y + b u ) · ∇ χ = 0 in Ω T ,χ = 1 t ∈ (0 ,T ) ( t ) for t ∈ (0 , T ) , x ∈ ∂ Ω , with ( y ( t, x ) + b u ( t, x )) · n ( x ) < ,χ (0) = 1 in Ω . (3.8)We finally define ρ ( t, x ) as follows, ρ ( t, x ) : def = (1 − χ ( t, x )) ρ b ( t, x ) + χ ( t, x ) ρ f ( t, x ) . (3.9)It is easy to check that this function ρ satisfies the transport equation (1.18) (1) and therequired initial condition (1.18) (2) . The final condition ρ ( T ) = 0 in (1.18) (3) is satisfieddue to the properties of the flow proved in Lemma 3.2, which guarantees that χ ( T ) = 0.In the next subsections, we describe how to get estimates on the function ρ constructedin (3.9) in the weighted spaces adapted to the Carleman estimates derived in Section 2. To begin with, let us remark that the function χ is explicitly given by: χ ( t, x ) = t < T t ≥ T and X ( τ, t, x ) ∈ Ω for all τ ∈ [ T , t ],0 else, (3.10)so that from Lemma 3.2 we have in particular χ ( t, x ) = 0 and ρ ( t, x ) = ρ b ( t, x ) for ( t, x ) ∈ [ T − T , T ] × Ω . (3.11)We also give explicit expressions for ρ f and ρ b . In order to do that, for t ∈ [0 , T ], weintroduce Ω [0] ( t ) : def = { x ∈ Ω | b X ( τ, t, x ) ∈ Ω for all τ ∈ [0 , t ] } Ω [ T ] ( t ) : def = { x ∈ Ω | b X ( τ, t, x ) ∈ Ω for all τ ∈ [ t, T ] } (3.12) nd for all ( t, x ) ∈ [0 , T ] × Ω: t in ( t, x ) : def = sup { τ ∈ [0 , t ) | b X ( τ, t, x ) ∈ ∂ Ω } ,t out ( t, x ) : def = inf { τ ∈ ( t, T ] | b X ( τ, t, x ) ∈ ∂ Ω } . (3.13)In the above definitions, we use the convention sup ∅ = 0 and inf ∅ = T . This way, t in ( t, x ) = 0 iff x ∈ Ω [0] ( t ) and t out ( t, x ) = T iff x ∈ Ω [ T ] ( t ).Using these notations, ρ f and ρ b are explicitly given by ρ f ( t, x ) = ρ ( b X (0 , t, x )) − Z t ( b u · ∇ σ )( τ, b X ( τ, t, x ))d τ if x ∈ Ω [0] ( t ) , − Z tt in ( t,x ) ( b u · ∇ σ )( τ, b X ( τ, t, x ))d τ else, (3.14) ρ b ( t, x ) = Z tt out ( t,x ) ( b u · ∇ σ )( τ, b X ( τ, t, x ))d τ for x ∈ Ω . (3.15)We are now in position to derive weighted estimates on ρ . In order to derive weighted estimates on ρ based on the Carleman weights ψ , θ , ϕ , ξ described in (2.7)–(2.9)–(2.10)–(2.11), we will need some further assumptions. Assumptions on the weights.
We assume that T and T in the definition of θ in(2.10) satisfy T ∈ (0 , T ∗ ) , T ∈ (0 , T ∗ ) , (3.16)where T ∗ and T ∗ are given by Lemma 3.1.We also assume that the function ψ in (2.7) satisfies the transport equation ∂ t ψ + y · ∇ ψ = 0 in Ω T . (3.17) Assumptions on b u. In order to derive estimates on ρ , we shall assume that b u is ina weighted Sobolev space. According to Theorem 2.2, it is natural to assume ξ − b u e sϕ ∈ L (0 , T ; L (Ω)) , (3.18) b u e sϕ ∗ / ∈ L (0 , T ; H (Ω)) with (cid:13)(cid:13)(cid:13)b u e sϕ ∗ / (cid:13)(cid:13)(cid:13) L (0 ,T ; H (Ω)) ≤ ς. (3.19) Extension of b u. To fit into the setting of Section 3.2, we extend b u on [0 , T ] × R thatwe still denote the same: b u = E ( b u ), where E denotes an extension from H (Ω) to H ( R )such that k E ( v ) k H ( R ) ≤ k v k H (Ω) for all v ∈ H (Ω). This allows us to define the flow b X by (3.2) for ( t, τ, x ) ∈ [0 , T ] × R .Note that, for s large enough, this last assumption is stronger than (3.5) and is thusperfectly compatible with the construction of Section 3.2, as it implies in particular that k θ b u k L (0 ,T ; L ∞ ( R )) ≤ cςe − c sλ , (3.20)where c > s and λ . For the following we suppose that s ≥ s and λ ≥ s large enough such that (3.5) and (3.20) are satisfied. On the flows b X and X . We first establish a lemma on the closeness of b X to X . Lemma 3.3.
There exists c > independent of s and λ such that for all ( τ, t ) ∈ [0 , T ] and x ∈ R : | b X ( τ, t, x ) − X ( τ, t, x ) | ≤ cςe − c sλ . (3.21) Moreover, if T ≤ t ≤ τ ≤ T , we also have θ ( t ) | b X ( τ, t, x ) − X ( τ, t, x ) | ≤ cςe − c sλ . (3.22) roof. Estimate (3.21) is an immediate consequence of (3.4) and (3.20). From (3.4), wealso have θ ( t ) | b X ( τ, t, x ) − X ( τ, t, x ) | ≤ T / e LT θ ( t ) k b u k L ( t,τ ; L ∞ ( R )) , where L = k∇ y k L ∞ (0 ,T ; L ∞ ( R )) . Using the fact that θ is increasing on [ T , T ], θ ( t ) | b X ( τ, t, x ) − X ( τ, t, x ) | ≤ T / e LT k θ b u k L ( t,τ ; L ∞ ( R )) , for all T ≤ t ≤ τ ≤ T , which concludes the proof of Lemma 3.3 by (3.20). On the weight functions.
Here, we shall deeply use the fact that ψ is assumed tosolve the transport equation (3.17), thus implying in particular that ∀ ( t, τ, x ) ∈ [0 , T ] × R , ψ ( t, X ( t, τ, x )) = ψ ( τ, x ) . (3.23)We then show the following lemma: Lemma 3.4.
There exist c > , c > and c > independent of s and λ , and s > such that for all s ≥ s , λ ≥ , the following inequalities hold:1. For all t ∈ [0 , T − T ] , τ ∈ [0 , t ] and x ∈ R , ϕ ( t, x ) − ϕ ( τ, b X ( τ, t, x )) ≤ c ςe − c sλ , (3.24) ξ ( τ, b X ( τ, t, x )) ξ ( t, x ) ≤ e c ςe − c sλ . (3.25)
2. For all t ∈ [ T , T ] , τ ∈ [ t, T ] and x ∈ R , ϕ ( t, x ) − ϕ ( τ, b X ( τ, t, x )) ≤ c ςe − c sλ − c ( θ ( τ ) − θ ( t )) , (3.26) ξ ( τ, b X ( τ, t, x )) ξ ( t, x ) ≤ θ ( τ ) θ ( t ) e c ςe − c sλ . (3.27) Proof.
We focus on the proof of item , the first one being similar and easier because θ takes value in [1 ,
2] close to t = 0. Estimate (3.26) follows from the following computations:for T ≤ t ≤ τ ≤ T , ϕ ( t, x ) − ϕ ( τ, b X ( τ, t, x ))= θ ( t ) (cid:16) λe λ ( m +1) − e λψ ( t,x ) (cid:17) − θ ( τ ) (cid:16) λe λ ( m +1) − e λψ ( τ, b X ( τ,t,x )) (cid:17) = θ ( t ) (cid:16) e λψ ( τ, b X ( τ,t,x )) − e λψ ( t,x ) (cid:17) + ( θ ( t ) − θ ( τ )) (cid:16) λe λ ( m +1) − e λψ ( τ, b X ( τ,t,x )) (cid:17) ≤ θ ( t ) (cid:16) e λψ ( τ, b X ( τ,t,x )) − e λψ ( t,x ) (cid:17) − c ( θ ( τ ) − θ ( t )) , for some c >
0, where we used in the last estimate that θ is increasing on [ T , T ]. Wethen use (3.23) and (3.22): | θ ( t ) (cid:16) e λψ ( τ, b X ( τ,t,x )) − e λψ ( t,x ) (cid:17) | = θ ( t ) (cid:12)(cid:12)(cid:12) e λψ ( τ, b X ( τ,t,x )) − e λψ ( τ,X ( τ,t,x )) (cid:12)(cid:12)(cid:12) ≤ cθ ( t ) λ k∇ ψ k ∞ e λ (6 m +1) | b X ( τ, t, x ) − X ( τ, t, x ) | ≤ c ςe − c sλ , for s large enough, as announced in (3.26). Next, by construction we have ξ ( τ, b X ( τ, t, x )) ξ ( t, x ) = θ ( τ ) θ ( t ) e λ ( ψ ( τ, b X ( τ,t,x )) − ψ ( τ,X ( τ,t,x ))) ≤ θ ( τ ) θ ( t ) e λ k∇ ψ k ∞ | b X ( τ,t,x ) − X ( τ,t,x ) | , (3.28)which immediately yields (3.27) by (3.22).We immediately deduce from Lemma 3.4 the following: roposition 3.5. Introducing the weight function ℵ ( t, x ) : def = ( ξ ( t, x )) − e sϕ ( t,x ) , (3.29) there exist s ≥ and c > independent of s and λ such that for all λ ≥ , s ≥ s , forall ( τ, t, x ) ∈ [0 , T ] × [0 , T ] × Ω satisfying τ ≤ t ≤ T − T or T ≤ t ≤ τ , ℵ ( t, x ) ≤ c ℵ ( τ, b X ( τ, t, x )) . (3.30) Proof. If τ ≤ t ≤ T − T then (3.30) follows immediately from (3.24) and (3.25).If T ≤ t ≤ τ then (3.30) follows from (3.26) and (3.27): ℵ ( t, x ) ≤ (cid:18) θ ( τ ) e − c sθ ( τ ) θ ( t ) e − c sθ ( t ) (cid:19) e c ς ( s +2) e − sλc ℵ ( τ, b X ( τ, t, x )) . But, for s ≥ /c , the function x x e − c sx is decreasing on [1 , + ∞ ) and then, since θ is increasing on [ T , T ], θ ( τ ) e − c sθ ( τ ) ≤ θ ( t ) e − c sθ ( t ) . On the controlled trajectory ρ . We now derive estimates on the controlled trajec-tory ρ given by Section 3.2: Theorem 3.6.
Let ψ , θ , ϕ , ξ are defined in (2.7) – (2.9) – (2.10) – (2.11) and assume (3.16) , (3.17) . Further assume that b u satisfies (3.18) and (3.19) with s ≥ s , λ ≥ and s largeenough such that (3.5) and (3.20) are satisfied.There exists c > independent of s , λ and b u such that the solution ρ given by Section3.2 satisfies kℵ ρ k L (Ω T ) ≤ C (cid:16) kℵ b u k L (0 ,T ; L (Ω)) + e sϕ ∗ (0) k ρ k L (Ω) (cid:17) , (3.31) where ℵ is given by (3.29) , and (cid:13)(cid:13)(cid:13) e sλe λ ( m +1) θ ( t ) / ρ (cid:13)(cid:13)(cid:13) L ∞ (Ω T ) ≤ C (cid:18)(cid:13)(cid:13)(cid:13) e sλe λ ( m +1) θ ( t ) / b u (cid:13)(cid:13)(cid:13) L (0 ,T ; L ∞ (Ω)) + e sλe λ ( m +1) k ρ k L ∞ (Ω) (cid:17) . (3.32) Proof.
The proof of Theorem 3.6 follows from the precise description of ρ f and ρ b givenin (3.14)–(3.15).Let us begin with the proof of estimate (3.32). On one hand, as t sλe λ ( m +1) θ ( t ) isnon-increasing on (0 , T − T ), from (3.14) we get, for all ( t, x ) ∈ (0 , T − T ) × Ω e sλe λ ( m +1) θ ( t ) | ρ f ( t, x ) | ≤ e sλe λ ( m +1) θ ( t ) k ρ k L ∞ (Ω) + 2 k∇ σ k L ∞ (Ω T ) Z t e sλe λ ( m +1) θ ( τ ) k b u ( τ, · ) k L ∞ (Ω) d τ. On the other hand, using that t sλe λ ( m +1) θ ( t ) is non-decreasing on ( T , T ), from(3.15), similarly, we have, for all ( t, x ) ∈ ( T , T ) × Ω, e sλe λ ( m +1) θ ( t ) | ρ b ( t, x ) | ≤ k∇ σ k L ∞ (Ω T ) Z Tt e sλe λ ( m +1) θ ( τ ) k b u ( τ, · ) k L ∞ (Ω) d τ. Together with the fact that the solution χ of (3.8) takes value in [0 ,
1] on Ω T and theproperties (3.10), these two estimates easily yield (3.32).We then focus on the proof of (3.31), that mainly relies on the two following estimates:for all time t ∈ (0 , T − T ), we get Z Ω | ρ f ( t ) | ℵ ( t )d x ≤ C (cid:18) e sϕ ∗ (0) Z Ω | ρ | d x + Z Z Ω T | b u | ℵ d x d τ (cid:19) , (3.33) nd for all time t ∈ ( T , T ), Z Ω | ρ b ( t ) | ℵ ( t )d x ≤ c Z Z Ω T | b u | ℵ d x d τ. (3.34)Indeed, once estimates (3.33)–(3.34) are proved, we can bound the L (Ω T )-norm of ℵ ρ bythe sum of the L ∞ ((0 , T − T ); L (Ω))-norm of ρ f and of the L ∞ (( T , T ); L (Ω))-normof ρ b , and estimate (3.31) immediately follows.Let us first present the proof of (3.33). We fix t ∈ [0 , T − T ]. From (3.14) and (3.30)we deduce that, for x ∈ Ω [0] ( t ), | ρ f ( t, x ) | ℵ ( t, x ) ≤ C (cid:18) | ρ ( b X (0 , t, x )) | ℵ (0 , b X (0 , t, x )) + Z t | b u ( τ, b X ( τ, t, x )) | ℵ ( τ, b X ( τ, t, x ))d τ (cid:19) , whereas for x ∈ Ω \ Ω [0] ( t ), | ρ f ( t, x ) | ℵ ( t, x ) ≤ C Z tt in ( t,x ) | b u ( τ, b X ( τ, t, x )) | ℵ ( τ, b X ( τ, t, x ))d τ. Combining these two estimates, for all t ∈ (0 , T − T ) we get: Z Ω | ρ f ( t, x ) | ℵ ( t, x )d x ≤ C Z Ω [0] ( t ) | ρ ( b X (0 , t, x )) | ℵ (0 , b X (0 , t, x ))d x + C Z t Z Ω [ t in ( t,x ) ,t ] ( τ ) | b u ( τ, b X ( τ, t, x )) | ℵ ( τ, b X ( τ, t, x ))d x d τ. (3.35)Since y + b u is divergence free in Ω T , the Jacobian of x b X ( t, τ, x ) equals 1 identically.Therefore, Z Ω [0] ( t ) | ρ ( b X (0 , t, x )) | ℵ (0 , b X (0 , t, x ))d x = Z b X (0 ,t, Ω [0] ( t )) | ρ ( x ) | ℵ (0 , x )d x ≤ Z Ω | ρ ( x ) | ℵ (0 , x )d x. Similarly, we get Z t Z Ω [ t in ( t,x ) ,t ] ( τ ) | b u ( τ, b X ( τ, t, x )) | ℵ ( τ, b X ( τ, t, x ))d x d τ ≤ Z t Z Ω | b u ( τ, x ) | ℵ ( τ, x )d τ d x Estimate (3.33) then follows from (3.35).The proof of (3.34) is based on (3.15) and follows the same lines. It is therefore leftto the reader.
We are now in position to prove Theorem 1.1. The idea is to construct suitable convexsets which are invariant by the mapping F = F ( ρ , u ) in (1.20) and relatively compactfor a topology making F continuous. In all this section, we assume the assumptions ofTheorem 1.1. In the introduction, we introduced formally a mapping F . We are now in position todefine it precisely.In order to do this, the first step in the proof of Theorem 1.1 is to construct a weightfunction ˜ ψ which is suitable for both Section 2 and Section 3, i.e. suitable in the sametime for controlling the velocity equation and the density equation. We claim the followingresult, proved in Section 4.2: emma 4.1. Let Ω be a smooth bounded domain. Further assume the regularity condition (1.10) on ( σ, y ) , the geometric condition (1.8) and condition (1.9) .Then one can find a smooth ( C ) bounded domain O satisfying (2.1) such that thereexists a C ( O T ) -function ˜ ψ satisfying the transport equation (3.17) and satisfying assump-tions (2.7) to (2.8) for ω T = [0 , T ] × ω and ˜ ω T = [0 , T ] × ˜ ω where ω , ˜ ω are two subdomainsof O\ Ω such that ˜ ω ⋐ ω . Next, we take T ∗ , T ∗ and ς > T ∈ (0 , T ∗ ) and T ∈ (0 , T ∗ ). We then use the function ψ , θ , ϕ and ξ given by (2.9), (2.10), (2.11) for m ≥ s ≥ s , λ ≥ λ , and the notations given in (2.14)–(2.15). Moreover, we supposethat s , λ are large enough given by Theorem 2.2 and Theorem 3.6. Now, we define thespaces X s,λ and Y s,λ depending on positive parameters s ≥ s and λ ≥ λ as follows: X s,λ : def = { u ∈ L (Ω T ) , with div ( u ) = 0 in Ω T , (4.1) s / ξ /m − e sϕ u ∈ L (Ω T ) ,e sϕ ∗ / u ∈ L (0 , T ; H (Ω)) ∩ H (0 , T ; L (Ω)) } , endowed with the norm k u k X s,λ : def = k e sϕ ∗ / u k L ( H ) ∩ H ( L ) + s / (cid:13)(cid:13)(cid:13) ξ /m − e sϕ u (cid:13)(cid:13)(cid:13) L (Ω T ) , and Y s,λ : def = { ρ ∈ L ∞ (Ω T ) , with ξ − e sϕ ρ ∈ L (Ω T ) and e sλe λ ( m +1) θ/ ρ ∈ L ∞ (Ω T ) } , endowed with the norm k ρ k Y s,λ : def = k ξ − e sϕ ρ k L (Ω T ) + (cid:13)(cid:13)(cid:13) e sλe λ ( m +1) θ/ ρ (cid:13)(cid:13)(cid:13) L ∞ (Ω T ) . We also introduce the space F s,λ defined by F s,λ : def = { f ∈ L (0 , T ; L (Ω)) , with ξ − f e sϕ ∈ L (0 , T ; L (Ω)) } endowed with the norm k f k F s,λ : def = (cid:13)(cid:13) ξ − f e sϕ (cid:13)(cid:13) L ( L ) . Note that, in the above definitions as well as in the following results, we keep the depen-dence in both parameters λ and s to be consistent with notations of Section 2. However,only the dependence in s will be needed in this section.We then derive the following results. Theorem 4.2 (On the mapping F ) . Fix ρ ∈ L ∞ (Ω) . For all b u ∈ X s,λ with k b u k X s,λ ≤ ς , the construction in Section 3.2 yields ρ = F ( b u , ρ ) solution of the control problem (1.18) . Besides, ρ ∈ Y s,λ and for some constant C independent of s ≥ s and λ ≥ λ , k ρ k Y s,λ ≤ C (cid:18) s / k b u k X s,λ + e sϕ ∗ (0) k ρ k L ∞ (Ω) (cid:19) . (4.2) Furthermore, the application F satisfies the following compactness property: If b u n is asequence of functions in X s,λ with k b u n k X s,λ ≤ ς which weakly converges to some b u in X s,λ , the corresponding sequence ρ n = F ( b u n , ρ ) strongly converges to F ( b u , ρ ) in all L q (Ω T ) for q ∈ [1 , ∞ ) . The proof of Theorem 4.2 is done in Section 4.3. Let us point out that the compactnessproperty stated in Theorem 4.2 is of primary importance for our result and follows from[3, Theorem 4].We then focus on the study of the mapping F : Theorem 4.3 (On the mapping F ) . We can define a bounded linear mapping F : F s,λ × V (Ω) → X s,λ such that for all u ∈ V (Ω) and f ∈ F s,λ , u = F ( f , u ) solvesthe control problem (1.19) and satisfies, for some constant C > independent of s ≥ s and λ ≥ λ , k u k X s,λ ≤ C (cid:16) k f k F s,λ + e sϕ ∗ (0) k u k H (Ω) (cid:17) . (4.3) heorem 4.3 is a direct consequence of Theorem 2.2: the mapping F is obtained byrestricting the controlled trajectory given by Theorem 2.2 to (0 , T ) × Ω. Of course, thisdepends on the extension O of Ω, but this choice is done once for all. Estimate (4.3) isthen a rewriting of Theorem 2.2 by taking into account that f and u are extended byzero outside Ω.We are then able to derive the following properties on the mapping F in (1.20), whoseproof is postponed to Section 4.4: Theorem 4.4.
Let ρ ∈ L ∞ (Ω) and u ∈ V (Ω) .Then for all s ≥ s and λ ≥ λ the mapping F in (1.20) is well-defined for all b u ∈ X s,λ with k b u k X s,λ ≤ ς . Besides, for all b u ∈ X s,λ with k b u k X s,λ ≤ ς , u = F ( b u ) belongs to X s,λ ,and satisfies, for some constant C independent of s and λ , k u k X s,λ ≤ C (cid:18) s / k b u k X s,λ + k b u k X s,λ + e sϕ ∗ (0) k ρ k L ∞ (Ω) + e sϕ ∗ (0) k ρ k L ∞ (Ω) + e sϕ ∗ (0) k u k H (Ω) (cid:17) . (4.4) Moreover, if b u n is a sequence of functions in X s,λ with k b u n k X s,λ ≤ ς which weakly con-verges to some b u in X s,λ , the corresponding sequence u n = F ( b u n ) strongly converges to u = F ( b u ) in L (0 , T ; L (Ω)) . We may then conclude the proof of Theorem 1.1. For R ∈ (0 , ς ), we introduce theclosed convex set X Rs,λ = { u ∈ X s,λ with k u k X s,λ ≤ R } . We then choose R small enough such that C R ≤ /
4, where C is the constant in (4.4), λ = λ and s ≥ s large enough to guarantee C ≤ s / /
4. We then get from (4.4) thatfor all b u ∈ X Rs,λ , u = F ( b u ) satisfies k u k X s,λ ≤ R C (cid:16) e sϕ ∗ (0) k ρ k L ∞ (Ω) + e sϕ ∗ (0) k ρ k L ∞ (Ω) + e sϕ ∗ (0) k u k H (Ω) (cid:17) . Thus, choosing ε > F maps X Rs,λ to itself.We then check that the set X Rs,λ is compact in L (0 , T ; L (Ω)) as H (0 , T ; L (Ω)) ∩ L (0 , T ; H (Ω)) is compactly embedded in L (0 , T ; L (Ω)) due to Rellich’s compactnesstheorem and Aubin-Lions’ theorem.Besides, the mapping F is continuous on X Rs,λ endowed with the L (0 , T ; L (Ω))-topology from Theorem 4.4. Indeed, if b u n is a sequence of functions in X Rs,λ whichstrongly converges to b u in L (0 , T ; L (Ω)), it necessarily weakly converges in X Rs,λ . Thus,from the last item of Theorem 4.4, u n = F ( b u n ) strongly converges to u in L (0 , T ; L (Ω)).Schauder’s fixed point theorem then implies the existence of a fixed point to the map-ping F , and concludes the proof of Theorem 1.1. We do it in several steps.
Construction of O . In a neighborhood of Γ c , according to Assumption (1.9), thereexists a C extension O of Ω such that • Ω ⊂ O ; • Γ ⊂ ∂ Ω ∩ ∂ O and for all t ∈ (0 , T ) and x ∈ ∂ Ω ∩ ∂ O , y ( t, x ) · n ≥ γ/ • ∂ O ∩ ∂ Ω and
O \
Ω have a finite number of connected components.Let ω , ˜ ω be two subdomains of O\ Ω such that ˜ ω ⋐ ω and fix d = dist(˜ ω, Ω).
Construction of an extension y e of y in O . We then construct an extension y e ∈ C ([0 , T ] × R ) of y outside Ω T (i.e y e ≡ y in Ω T ) satisfying k y e k C ([0 ,T ] ×O ) < ∞ , inf [0 ,T ] × ∂ O y e · n > , (4.5)and y e ≡ , T ) × ˜ ω. (4.6) efore going into the detailed construction of y e , let us remark that y e cannot be diver-gence free as it would not be compatible with the condition inf [0 ,T ] × ∂ O y e · n > y e , we proceed as follows. First, we consider anyextension of y in C ([0 , T ] × R ). By continuity, there exists d > t, x ) ∈ (0 , T ) × ∂ O with d ( x, Ω) < d , y ( t, x ) · n ≥ γ/
3. We also introduce a function m in C ([0 , T ] × R ) such that m · n = 1 on the whole boundary ∂ O and m ≡ ω , anda smooth non-negative cut-off function η = η ( x ) taking value 1 in Ω and 0 for all x ∈ O with d ( x, Ω) > min { d , d } , and we then consider y e ( t, x ) = η ( x ) y ( t, x ) + (1 − η ( x )) m ( x ) . This function indeed belongs to C ([0 , T ] × R ). Besides,inf [0 ,T ] × ∂ O y e · n ≥ min n γ , o , and (4.6) is trivially satisfied as m ≡ η ≡ ω . Construction of ˜ ψ . We then construct a function b ψ T = b ψ T ( x ) such that • b ψ T is a non-negative C ( O ) function; • The critical points of b ψ T all belong to ˜ ω ; • b ψ T satisfies the following conditions on the boundary ∂ O : b ψ T ( x ) = 0 on ∂ O , y e ( T, x ) · ∇ b ψ T ( x ) = − ∂ O ,∂ t y e ( T, x ) · ∇ b ψ T ( x ) − ( y e ( T, x ) · ∇ ) b ψ T ( x ) = 0 on ∂ O . (4.7) • inf O b ψ T = ( b ψ T ) | ∂ O = 0.Note that such function exists according to the construction of Fursikov and Imanuvilovin [14] suitably modified to handle the conditions on the first and second order derivativeson the boundary of O . This can be done easily following the lines of [22, Appendix III].We then consider the solution b ψ of ∂ t b ψ + y e · ∇ b ψ = 0 in O T , b ψ ( t, x ) = t − T on Γ T , b ψ ( T ) = b ψ T in O . (4.8)Note that this problem is well-posed as, by construction, y e ( t, x ) · n > t, x ) ∈ (0 , T ) × ∂ O . We then want to check that • ∂ n b ψ ( t, x ) ≤ t, x ) ∈ (0 , T ) × ∂ O ; • b ψ belongs to C ([0 , T ] × O ); • For all t ∈ [0 , T ], the critical points of b ψ ( t, · ) belong to ˜ ω ; • For all t ∈ [0 , T ], inf O b ψ ( t, · ) = b ψ ( t ) | ∂ O ;Indeed, providing these properties are true, one can choose a > b ∈ R such thatthe function ˜ ψ = a b ψ + b is suitable for Lemma 4.1.Using the equation (4.8) and the fact that tangential derivatives of b ψ vanish due tothe boundary conditions, we get, for all ( t, x ) ∈ (0 , T ) × ∂ O , y e ( t, x ) · n ∂ n b ψ ( t, x ) = − ∂ t b ψ ( t, x ) = − . Using (4.5), we thus deduce that ∀ ( t, x ) ∈ (0 , T ) × ∂ O , ∂ n b ψ ( t, x ) ≤ − [0 ,T ] × ∂ O y e ( t, x ) · n < . (4.9) o describe more precisely the function b ψ , we will introduce the flow X e correspondingto y e , i.e. the solution of ∀ ( t, τ, x ) ∈ [0 , T ] × R , ∂ t X e ( t, τ, x ) = y e ( t, X e ( t, τ, x )) , X e ( τ, τ, x ) = x. (4.10)The fact that b ψ ∈ C ([0 , T ] × O ) follows from the following lemma, whose proof ispostponed to Appendix B: Lemma 4.5.
Under the above assumptions, b ψ ∈ C ([0 , T ] × O ) . We then have to check that the critical points of b ψ ( t, · ) all belong to ˜ ω .We first remark that (4.9) implies that there is no critical point on the boundary ∂ O .We then remark that ∇ b ψ solves the equation ∂ t ∇ b ψ + ( y e · ∇ ) ∇ b ψ + D y e ∇ b ψ = 0 in O T . (4.11)From the equation (4.11), if the point x c is a critical point for b ψ ( t c , · ), then for all t in a neighborhood around t c , X e ( t, t c , x c ) is a critical point for b ψ ( t, · ). Note that thisneighborhood actually correspond to the set I c of time t ∈ [0 , T ] such that the trajectory τ X e ( τ, t c , x c ) stays in O for τ between t and t c .Since there is no critical point on the boundary ∂ O and thanks to conditions (4.5),for all time t c ∈ [0 , T ], the critical points x c of b ψ ( t c , · ) are linked by a trajectory τ X e ( τ, t c , x c ) to a critical point x c,T of b ψ T , that is x c = X e ( t c , T, x c,T ). By construc-tion of b ψ T , x c,T necessarily belongs to ˜ ω . But, according to condition (4.6), as long as X e ( t, T, x c,T ) ∈ ˜ ω , ∂ t X e ( t, T, x c,T ) = 0 , so that X e ( t, T, x c,T ) = x c,T for all t ∈ [0 , T ]. This implies that the set of critical pointsof b ψ ( t, · ) is invariant through the flow X e and is then included in ˜ ω .We finally check the condition inf O b ψ ( t, · ) = b ψ ( t ) | ∂ O for all t ∈ [0 , T ] by contradiction.If this were wrong, there would exist t ∈ [0 , T ] and x t ∈ O such that x t ∈ Argmin b ψ ( t, · ).Thus, x t would be a critical point, and as above, X e ( T, t, x t ) would belong to O and bea critical point of b ψ T . Following, b ψ ( t, x t ) = b ψ T ( X e ( T, t, x t )) would be larger than 0 dueto the assumption on b ψ T . But from the boundary conditions, it follows that inf O b ψ ( t )cannot be strictly smaller than b ψ ( t ) | ∂ O , which is negative for all time t ∈ [0 , T ). According to Section 3, the construction in Section 3.2 yields ρ = F ( b u , ρ ) solution ofthe control problem (1.18) for b u satisfying (3.5). This condition is indeed satisfied for b u ∈ X s,λ with k b u k X s,λ ≤ ς , see (3.18)–(3.19)–(3.20).Theorem 3.6 immediately provides estimate (4.2), as λe λ ( m +1) θ/ ≤ ϕ ∗ /
4, see (2.13).We then focus on the proof of the compactness property. According to the constructionin Section 3.2, we introduce ρ f,n the solution of ∂ t ρ f,n + ( y + b u n ) · ∇ ρ f,n = − b u n · ∇ σ in Ω T ,ρ f,n ( t, x ) = 0 for t ∈ (0 , T ) , x ∈ ∂ Ω , with ( y ( t, x ) + b u n ( t, x )) · n ( x ) < ,ρ f,n (0) = ρ in Ω , (4.12) ρ b,n the solution of ∂ t ρ b,n + ( y + b u n ) · ∇ ρ b,n = − b u n · ∇ σ in Ω T ,ρ b,n = 0 for t ∈ (0 , T ) , x ∈ ∂ Ω , with ( y ( t, x ) + b u n ( t, x )) · n ( x ) > ,ρ b,n ( T ) = 0 in Ω , (4.13)and χ n the solution of ∂ t χ n + ( y + b u n ) · ∇ χ n = 0 in Ω T ,χ n = 1 t ∈ (0 ,T ) ( t ) for t ∈ (0 , T ) , x ∈ ∂ Ω , with ( y ( t, x ) + b u n ( t, x )) · n ( x ) < ,χ n (0) = 1 in Ω . (4.14) ince b u n is a bounded sequence of H (0 , T ; L (Ω)) ∩ L (0 , T ; H (Ω)), which is compact in L (0 , T ; L (Ω)), up to a subsequence still denoted the same for simplicity, b u n converge to b u weakly in H (0 , T ; L (Ω)) ∩ L (0 , T ; H (Ω)) and strongly in L (0 , T ; L (Ω)). Then [3,Theorem 4] applies and for all q ∈ [1 , + ∞ ) the sequence χ n strongly converges towards χ in L q (Ω T ) solution of (3.8).Next, to pass to the limit in (4.12), we notice that σ f,n : def = σ + ρ f,n solves ∂ t σ f,n + ( y + b u n ) · ∇ σ f,n = 0 in Ω T ,σ f,n ( t, x ) = 0 for t ∈ (0 , T ) , x ∈ ∂ Ω , with ( y ( t, x ) + b u n ( t, x )) · n ( x ) < ,σ f,n (0) = ρ in Ω . (4.15)Thus, by applying again [3, Theorem 4] we deduce that, for all q ∈ [1 , + ∞ ), the sequence σ f,n is strongly convergent in L q (Ω T ) to the solution σ f of ∂ t σ f + ( y + b u ) · ∇ σ f = 0 in Ω T ,σ f ( t, x ) = 0 for t ∈ (0 , T ) , x ∈ ∂ Ω , with ( y ( t, x ) + b u ( t, x )) · n ( x ) < ,σ f (0) = ρ in Ω . (4.16)It follows that ρ f,n strongly converges in all L q (Ω T ) for q ∈ [1 , ∞ ) to ρ f = σ f − σ , whichsolves (3.6) by construction.Of course, the same can be done to show that ρ b,n strongly converges in all L q (Ω T )for q ∈ [1 , ∞ ) to the solution ρ b of (3.7). Consequently, the sequence ρ n = F ( b u n , ρ )converges to ρ = F ( b u , ρ ) in L q (Ω T ) for all q ∈ [1 , ∞ ). u n and u are uniformly bounded in X s,λ , so the convergence of u n to u actually isweak in X s,λ . Let ρ ∈ L ∞ (Ω), u ∈ V (Ω) and b u ∈ X s,λ with k b u k X s,λ ≤ ς .According to Theorem 4.2, ρ = F ( b u , ρ ) belongs to Y s,λ and is bounded in that spaceby (4.2). Thus, according to Theorem 4.3, for F to be well-defined, we have to check that f ( ρ, b u ) given in (1.13) belongs to F s,λ , and we will get estimates on u = F ( b u ) from anestimate of f ( ρ, b u ) in F s,λ according to (4.3). We thus estimate f ( ρ, b u ) in F s,λ term byterm from estimates on ρ ∈ Y s,λ and b u ∈ X s,λ .We easily check (cid:13)(cid:13) ξ − e sϕ ρ ( ∂ t b u + ( y + b u ) · ∇ b u + b u · ∇ y ) (cid:13)(cid:13) L ( L ) ≤ (cid:13)(cid:13)(cid:13) e sλe λ ( m +1) θ/ ρ (cid:13)(cid:13)(cid:13) L ∞ (cid:13)(cid:13)(cid:13) ξ − e sϕ − sλe λ ( m +1) θ/ ( ∂ t b u + (( y + b u ) · ∇ ) b u + b u · ∇ y ) (cid:13)(cid:13)(cid:13) L ( L ) ≤ C k ρ k Y s,λ (cid:13)(cid:13)(cid:13) e sϕ ∗ / b u (cid:13)(cid:13)(cid:13) L ( H ) ∩ H ( L ) (cid:13)(cid:13)(cid:13) ξ − e sϕ − sλe λ ( m +1) θ/ − sϕ ∗ / (cid:13)(cid:13)(cid:13) L ∞ , where we used that y + b u is bounded in L ∞ (0 , T ; L (Ω)) due to Sobolev’s embedding as b u belongs to L (0 , T ; H (Ω)) ∩ H (0 , T ; L (Ω)) and is of norm bounded by ς , and that (cid:13)(cid:13)(cid:13) e sϕ ∗ / ∇ b u (cid:13)(cid:13)(cid:13) L ( L ) ≤ C (cid:13)(cid:13)(cid:13) e sϕ ∗ / b u (cid:13)(cid:13)(cid:13) L ( H ) ∩ H ( L ) . According to (2.13), sϕ − sλe λ ( m +1) θ/ − sϕ ∗ / ≤ − sϕ/
4, and thus there exists someconstant C independent of s and λ such that (cid:13)(cid:13)(cid:13) ξ − e sϕ − sλe λ ( m +1) θ/ − sϕ ∗ / (cid:13)(cid:13)(cid:13) L ∞ ≤ C. Following, (cid:13)(cid:13) ξ − e sϕ ρ ( ∂ t b u + ( y + b u ) · ∇ b u ) (cid:13)(cid:13) L ( L ) ≤ C k ρ k Y s,λ k b u k X s,λ . (4.17) ext, we estimate σ ( b u · ∇ ) b u . Similarly as above, we write (cid:13)(cid:13) ξ − e sϕ σ b u · ∇ b u (cid:13)(cid:13) L ( L ) ≤ C (cid:13)(cid:13)(cid:13) e sϕ ∗ / b u (cid:13)(cid:13)(cid:13) L ∞ ( L ) (cid:13)(cid:13)(cid:13) e sϕ ∗ / ∇ b u (cid:13)(cid:13)(cid:13) L ( L ) (cid:13)(cid:13)(cid:13) ξ − e sϕ − sϕ ∗ / (cid:13)(cid:13)(cid:13) L ∞ ≤ C k b u k X s,λ . (4.18)Last, we estimate ρ ( ∂ t y + ( y · ∇ ) y ): (cid:13)(cid:13) ξ − e sϕ ρ ( ∂ t y + ( y · ∇ ) y ) (cid:13)(cid:13) L ( L ) ≤ C (cid:13)(cid:13) ξ − e sϕ ρ (cid:13)(cid:13) L ≤ C k ρ k Y s,λ . (4.19)Putting estimates (4.17)–(4.19) together, we obtain: k f ( ρ, b u ) k F s,λ = (cid:13)(cid:13) ξ − e sϕ f ( ρ, b u ) (cid:13)(cid:13) L ( L ) ≤ C ( k ρ k Y s,λ + k ρ k Y s,λ + k b u k X s,λ ) . (4.20)Combined with estimates (4.2) and (4.3), this yields the well-posedness of the mapping F for b u ∈ X s,λ with k b u k X s,λ ≤ ς and the estimate (4.4).We now focus on the last part of Theorem 4.4. Let b u n is a sequence of X s,λ with k b u n k X s,λ ≤ ς which weakly converges to b u . Note that this weak convergence implies that k b u k X s,λ ≤ ς , so that F ( b u ) is well-defined.Besides that, according to Theorem 4.2, the sequence ρ n = F ( b u n , ρ ) strongly con-verges in all L q (Ω T ) with q < ∞ to ρ = F ( b u , ρ ) and the sequence ρ n is uniformlybounded in Y s,λ .We then have to check that f ( ρ n , b u n ) weakly converges in F s,λ to f ( ρ, b u ). But (4.20)shows that the sequence f ( ρ n , b u n ) is bounded in F s,λ , and thus we only need to provethat the sequence f ( ρ n , b u n ) weakly converges in D ′ (Ω T ) to f ( ρ, b u ). To obtain this con-vergence result in D ′ (Ω T ), as ρ n strongly converges to ρ in all L q (Ω T ) with q < ∞ and b u n weakly converges to b u in H (0 , T ; L (Ω)) ∩ L (0 , T ; H (Ω)), we only have to fo-cus on the convergence of the term ( σ + ρ n ) b u n · ∇ b u n . But, using the compactness of H (0 , T ; L (Ω)) ∩ L (0 , T ; H (Ω)) in L (0 , T ; L (Ω)), we have the convergences σ + ρ n −→ n →∞ σ + ρ strongly in L q (Ω T ) , q ∈ [1 , ∞ ) , b u n −→ n →∞ b u strongly in L (0 , T ; L (Ω)) , ∇ b u n −→ n →∞ ∇ b u weakly in L (0 , T ; L (Ω)) , so that, choosing q = 4 for instance, we obtain the weak convergence of ( σ + ρ n ) b u n · ∇ b u n to ( σ + ρ ) b u · ∇ b u .Following, f ( ρ n , b u n ) weakly converges in F s,λ to f ( ρ, b u ) and, since F : F s,λ × V (Ω) → X s,λ is a linear bounded operator, we obtain that u n = F ( b u n ) = F ( f ( ρ n , b u n ) , u )weakly converges to F ( f ( ρ, b u ) , u ) = F ( b u ) = u in X s,λ . Finally, as X s,λ is compact in L (0 , T ; L (Ω)), u n strongly converges to u in L (0 , T ; L (Ω)). A Proofs of Theorems 2.4 and 2.5
For simplicity, we make the proof of Theorems 2.4 and 2.5 for ν of equal to 1.This can bedone without loss of generality by replacing σ and f by σ/ν and f/ν if needed. A.1 Proof of Theorem 2.4
Let z be a smooth function on [0 , T ] × O satisfying z = 0 on (0 , T ) × ∂ O and set f : def = − σ∂ t z − ∆ z, ( t, x ) ∈ (0 , T ) × O , (A.1)Set then w = e − sϕ z. (A.2)According to the definition of θ in (2.10), w satisfies w ( T, x ) = 0 , ∇ w ( T, x ) = 0 , x ∈ O , (A.3) n addition to the conditions w ( t, x ) = 0 on (0 , T ) × ∂ O .Besides, with f as in (A.1), w satisfies e − sϕ f = e − sϕ ( − σ∂ t z − ∆ z ) = e − sϕ ( − σ∂ t ( e sϕ w ) − ∆( e sϕ w )) = P ϕ w, where the operator P ϕ is given by P ϕ w = − σ∂ t w − sσ∂ t ϕw − ∆ w − s ∇ ϕ · ∇ w − s |∇ ϕ | w − s ∆ ϕw. (A.4)We now set P , P and R the operators: P w = − σ∂ t w − s ∇ ϕ · ∇ w + 2 sλ |∇ ψ | ξw, (A.5) P w = − ∆ w − sσ∂ t ϕw − s |∇ ϕ | w, (A.6) Rw = sλ ∆ ψξw − sλ |∇ ψ | ξw, (A.7)so that P ϕ = P + P + R. We then use that P w + P w = fe − sϕ − Rw and then Z Z O T | P w | + Z Z O T | P w | + 2 Z Z O T P wP w = Z Z O T | fe − sϕ − Rw | ≤ Z Z O T | f | e − sϕ + 2 Z Z O T | Rw | . (A.8)The main part of the proof then consists in computing the scalar product of P w with P w and estimate it from below. Computations.
We write
Z Z O T P wP w = X i,j =1 I ij , where I i,j is the scalar product of the i -th term of P w with the j -th term of P w . Computation of I . I = Z Z O T σ∂ t w ∆ w = − Z Z O T σ∂ t (cid:18) |∇ w | (cid:19) − Z Z O T ∂ t w ∇ σ · ∇ w = 12 Z O σ (0) |∇ w (0) | + 12 Z Z O T ∂ t σ |∇ w | − Z Z O T ∂ t w ∇ σ · ∇ w. (A.9) Computation of I . I = s Z Z O T σ ∂ t w∂ t ϕw = − s Z O σ (0) ∂ t ϕ (0) | w (0) | − s Z Z O T σ ∂ tt ϕ | w | − s Z Z O T σ∂ t σ∂ t ϕ | w | . (A.10) Computation of I . I = s Z Z O T σ∂ t w |∇ ϕ | w = − s Z O σ (0) |∇ ϕ (0) | | w (0) | − s Z Z O T σ∂ t (cid:0) |∇ ϕ | (cid:1) | w | (A.11) − s Z Z O T ∂ t σ |∇ ϕ | | w | . omputation of I . I = 2 s Z Z O T ∇ ϕ · ∇ w ∆ w = 2 s Z Γ T ∂ n ϕ | ∂ n w | − s Z Z O T ∇ ( ∇ ϕ · ∇ w ) · ∇ w = 2 s Z Γ T ∂ n ϕ | ∂ n w | − s Z Z O T D ϕ ( ∇ w, ∇ w ) − s Z Z O T ∇ ϕ · ∇ (cid:0) |∇ w | (cid:1) = s Z Γ T ∂ n ϕ | ∂ n w | − s Z Z O T D ϕ ( ∇ w, ∇ w ) + s Z O ∆ ϕ |∇ w | . (A.12) Computation of I . I = 2 s Z Z O T σ ∇ ϕ · ∇ w∂ t ϕw = − s Z Z O T div ( σ∂ t ϕ ∇ ϕ ) | w | = − s Z Z O T σ div ( ∂ t ϕ ∇ ϕ ) | w | − s Z Z O T ∇ σ · ∇ ϕ∂ t ϕ | w | . (A.13) Computation of I . I = 2 s Z Z O T ∇ ϕ · ∇ w |∇ ϕ | w = − s Z Z O T div (cid:0) |∇ ϕ | ∇ ϕ (cid:1) | w | . (A.14) Computation of I . I = − sλ Z Z O T |∇ ψ | ξw ∆ w = 2 sλ Z Z O T |∇ ψ | ξ |∇ w | + 2 sλ Z Z O T ∇ ( |∇ ψ | ξ ) w · ∇ w. (A.15) Computation of I . I = − s λ Z Z O T σ |∇ ψ | ξ∂ t ϕ | w | . (A.16) Computation of I . I = − s λ Z Z O T |∇ ψ | ξ |∇ ϕ | | w | . (A.17)Combining the above computations (A.9)–(A.17), we obtain the following: Z Z O T P wP w = 12 Z O σ (0) |∇ w (0) | + 12 Z O | w (0) | σ (0) (cid:0) − s |∇ ϕ (0) | − sσ (0) ∂ t ϕ (0) (cid:1) (A.18) − s Z Z O T D ϕ ( ∇ w, ∇ w ) + s Z Z O T (∆ ϕ + 2 λ |∇ ψ | ξ ) |∇ w | (A.19)+ Z Z O T | w | s (cid:0) − div ( |∇ ϕ | ∇ ϕ ) − λ |∇ ψ | ξ |∇ ϕ | (cid:1) (A.20)+ s σ (cid:0) − ∂ t (cid:0) |∇ ϕ | (cid:1) − (∆ ϕ + 2 λ |∇ ψ | ξ ) ∂ t ϕ (cid:1) (A.21)+ sσ (cid:18) − ∂ tt ϕ (cid:19) ! (A.22)+ s Z T Z ∂ O ∂ n ϕ | ∂ n w | + I R . (A.23) here I R = 12 Z Z O T ∂ t σ |∇ w | + 2 sλ Z Z O T ∇ ( |∇ ψ | ξ ) w · ∇ w − s Z Z O T σ∂ t σ∂ t ϕ | w | − s Z Z O T ∂ t σ |∇ ϕ | | w | − s Z Z O T ∇ σ ∇ ϕ∂ t ϕ | w | − Z Z O T ∂ t w ∇ σ · ∇ w. (A.24) Positivity.
Our main goal now is to check that the coefficients in the above integralsare positive, except perhaps on the observation set ω T . At this step, we will strongly relyupon the choice of the weight function ϕ in (2.11), and on the formula ∂ t ϕ = ∂ t θθ ϕ − λ∂ t ψξ, ∂ t ξ = ∂ t θθ ξ + λ∂ t ψξ. (A.25)In the following, to simplify notations, we will denote by C generic positive large con-stants that do not depend on s or λ and by c generic positive small constants independentof s and λ . The constants may change from line to line. Positivity of the terms (A.18) at t = 0 . Explicit computations yield − ∂ t ϕ (0) = µT ( λe λ ( m +1) − e λψ (0) ) + 2 λ∂ t ψ (0) e λψ (0) ≥ csλ e λ (12 m +2) whereas |∇ ϕ (0) | ≤ Cλ | ξ (0) | ≤ Cλ e λ (6 m +1) . Thus, with (2.2), for some λ >
0, taking λ ≥ λ ≥ O (cid:8) − s |∇ ϕ (0) | − sσ (0) ∂ t ϕ (0) (cid:9) ≥ cs λ e λ (6 m +1) , (A.26)and, following,12 Z O σ (0) | w (0) | (cid:0) − s |∇ ϕ (0) | − sσ (0) ∂ t ϕ (0) (cid:1) ≥ cs λ e λ (6 m +1) Z O | w (0) | . (A.27) Positivity of the terms (A.19) involving the gradient.
For η ∈ R N , we have − sD ϕ ( η, η ) + s (∆ ϕ + 2 λ |∇ ψ | ξ ) | η | = 2 sλ ξ |∇ ψ · η | + sλ ξ |∇ ψ | | η | + 2 sλξD ψ ( η, η ) − sλξ ∆ ψ | η | . (A.28)Using (2.8), we get the existence of λ = λ ( α, k D ψ k ∞ ) ≥ λ such that for all λ ≥ λ and η ∈ R N , ∀ ( t, x ) ∈ O T \ ˜ ω T , − sD ϕ ( η, η ) + s (∆ ϕ + 2 λ |∇ ψ | ξ ) | η | ≥ csλ | η | ξ, (A.29)whereas there exists a positive constant C = C ( α, k D ψ k ∞ ) such that ∀ η ∈ R N , ∀ ( t, x ) ∈ ˜ ω T , − sD ϕ ( η, η ) + s (∆ ϕ + 2 λ |∇ ψ | ξ ) | η | ≥ csλ ξ | η | − Csλ ξ | η | . Hence we obtain, for all λ ≥ λ , − s Z Z O T D ϕ ( ∇ w, ∇ w ) + s Z Z O T (∆ ϕ + 2 λ |∇ ψ | ξ ) |∇ w | ≥ csλ Z Z O T ξ |∇ w | − Csλ Z Z ˜ ω T ξ |∇ w | . (A.30) Positivity of the terms (A.20) involving w with scale s . Using ∇ ϕ = − λ ∇ ψξ , we have − div ( |∇ ϕ | ∇ ϕ ) = 3 λ |∇ ψ | ξ + λ ξ div ( |∇ ψ | ∇ ψ ) ,λ |∇ ψ | ξ |∇ ϕ | = λ |∇ ψ | ξ . Hence − div ( |∇ ϕ | ∇ ϕ ) − λ |∇ ψ | ξ |∇ ϕ | = λ |∇ ψ | ξ + λ ξ div ( |∇ ψ | ∇ ψ ) . (A.31) sing (2.8), we thus get the existence of λ = λ ( α, k D ψ k ∞ ) ≥ λ such that for λ ≥ λ , ∀ ( t, x ) ∈ O T \ ω T , − div ( |∇ ϕ | ∇ ϕ ) − λ |∇ ψ | ξ |∇ ϕ | ≥ cλ ξ . (A.32)whereas there exists a positive constant C = C ( α, k D ψ k ∞ ) such that ∀ ( t, x ) ∈ ˜ ω T , − div ( |∇ ϕ | ∇ ϕ ) − λ |∇ ψ | ξ |∇ ϕ | ≥ cλ ξ − Cλ ξ . (A.33)We thus obtain, for all λ ≥ λ , s Z Z O T | w | (cid:0) − div ( |∇ ϕ | ∇ ϕ ) − λ |∇ ψ | ξ |∇ ϕ | (cid:1) ≥ cs λ Z Z O T ξ | w | − Cs λ Z Z ˜ ω T ξ | w | . (A.34) Terms (A.21) involving w in the scale s . We have to estimate − ∂ t (cid:0) |∇ ϕ | (cid:1) − (∆ ϕ + 2 λ |∇ ψ | ξ ) ∂ t ϕ. Explicit computations yield: − ∂ t (cid:0) |∇ ϕ | (cid:1) − (∆ ϕ + 2 λ |∇ ψ | ξ ) ∂ t ϕ = − λ ξ ∂ t ψ |∇ ψ | − λ ξ ∇ ψ · ∇ ∂ t ψ − λ ξ ∂ t ψ ∆ ψ (A.35)+ ∂ t θθ (cid:0) − λ ξϕ |∇ ψ | + λξ ∆ ψϕ − λ ξ |∇ ψ | (cid:1) . (A.36)Before going further, let us remark that, using ξ ≥
1, there exists a positive constant C ,only depending on the C -norm of ψ such that for all λ ≥
1, for all ( t, x ) ∈ (0 , T ) × O , (cid:12)(cid:12) − λ ξ ∂ t ψ |∇ ψ | − λ ξ ∇ ψ · ∇ ∂ t ψ − λ ξ ∂ t ψ ∆ ψ − λ ξ |∇ ψ | (cid:12)(cid:12) ≤ Cλ ξ . This estimate is sufficient to handle the terms in (A.35).We will then focus on the terms in (A.36). First remark that on ( T , T − T ), ∂ t θ ≡ T − T , T ), we use the fact that there exists a constant C > ∀ t ∈ ( T − T , T ) , | ∂ t θ | ≤ Cθ . Hence there exists C = C ( k∇ ψ k ∞ , k ∆ ψ k ∞ ) such that for all ( t, x ) ∈ ( T − T , T ) × O , (cid:12)(cid:12)(cid:12)(cid:12) ∂ t θθ (cid:0) − λ ξϕ |∇ ψ | + λξ ∆ ψϕ − λ ξ |∇ ψ | (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cλ θξϕ ≤ Cλ ξ , (A.37)where for the last inequality we have used | θϕ | ≤ λξ , which is a consequence of (2.13).On (0 , T ), we are going to use that ∂ t θ ≤ θ ∈ [1 ,
2] and thus the term in (A.36)has the good sign outside ˜ ω T . Indeed, using (2.8), we can find λ = λ ( α, k ∆ ψ k ∞ ) ≥ λ such that for all λ ≥ λ , for all ( t, x ) ∈ (0 , T ) × O such that ( t, x ) / ∈ ˜ ω T , − (cid:0) − λ ξϕ |∇ ψ | + λξ ∆ ψϕ − λ ξ |∇ ψ | (cid:1) ≥ cλ ξϕ, whereas it is bounded by Cλ ξϕ everywhere in O T . We thus derive, for all λ ≥ λ , s Z Z O T | w | σ (cid:0) − ∂ t (cid:0) |∇ ϕ | (cid:1) − (∆ ϕ + 2 λ |∇ ψ | ξ ) ∂ t ϕ (cid:1) ≥ cs λ Z T Z O | ∂ t θ | ξϕ | w | − Cs λ Z Z O T ξ | w | − Cs λ Z Z ˜ ω T ∩{ t ∈ (0 ,T ) } | ∂ t θ | ξϕ | w | . (A.38) Term (A.22) involving w in the scale s . We have to estimate − ∂ tt ϕ . ∂ tt ϕ = ∂ tt θθ ϕ − λ ∂ t θθ ∂ t ψξ − λ∂ tt ψξ − λ ( ∂ t ψ ) ξ (A.39) et us first remark that we immediately have (cid:12)(cid:12) − λ∂ tt ψξ − λ ( ∂ t ψ ) ξ (cid:12)(cid:12) ≤ Cλ ξ . For t ∈ (0 , T ), we further have ∀ t ∈ (0 , T ) , | ∂ tt θ | ≤ Cs λ e λ (12 m − , | ∂ t θ | ≤ Csλ e λ (6 m − , so that, on (0 , T ) | ∂ tt ϕ | ≤ Cs λ e λ (12 m − e λ ( m +1) + Csλ e λ (6 m − ξ + Cλ ξ ≤ Cs λ ξ . For t ∈ ( T − T , T ), we have ∀ t ∈ ( T − T , T ) , | ∂ tt θ | ≤ Cθ and | ∂ t θ | ≤ Cθ . Hence, using (2.13) and θϕ ≤ λξ , for some positive constant C = C ( k ∂ t ψ k ∞ ), ∀ ( t, x ) ∈ ( T − T , T ) × O , | ∂ tt ϕ | ≤ Cθ ϕ + Cλθξ + Cλ ξ ≤ Cλ ξ . Combining all these estimates, we get s Z Z O T σ | w | (cid:18) − ∂ tt ϕ (cid:19) ≥ − Cs λ Z Z O T ξ | w | . (A.40) Positivity of the terms (A.20) – (A.21) – (A.22) involving w . Here we combine the esti-mates in (A.34), (A.38), (A.40) in order to derive suitable estimates for the sum of theterms in (A.20)–(A.21)–(A.22). To simplify notations, let us set I w the sum of the termsin (A.20)–(A.21)–(A.22): I w : def = Z Z O T | w | s (cid:0) − div ( |∇ ϕ | ∇ ϕ ) − λ |∇ ψ | ξ |∇ ϕ | (cid:1) + s σ (cid:0) − ∂ t (cid:0) |∇ ϕ | (cid:1) − (∆ ϕ + 2 λ |∇ ψ | ξ ) ∂ t ϕ (cid:1) + sσ (cid:18) − ∂ tt ϕ (cid:19) ! . (A.41)Putting together (A.34), (A.38), (A.40), we deduce that there exist s ≥ λ ≥ λ such that for s ≥ s and λ ≥ λ , I w ≥ cs λ Z Z O T ξ | w | + cs λ Z T Z O | ∂ t θ | ξϕ | w | (A.42) − Cs λ Z Z ˜ ω T ξ | w | − Cs λ Z Z ˜ ω T ∩{ t ∈ (0 ,T ) } | ∂ t θ | ξϕ | w | . (A.43) Positivity of the boundary terms (A.23) . Here, we only have to remark that ∂ n ϕ ≥ ∂ n ψ ≤ A bound on I R in (A.24) We also provide an upper bound on I R .First, we shall of course use the immediate estimate12 Z Z O T ∂ t σ |∇ w | ≤ C Z Z O T |∇ w | . Using ∇ ( |∇ ψ | ξ ) ≤ Cλξ , one easily checks that (cid:12)(cid:12)(cid:12)(cid:12) sλ Z Z O T ∇ ( |∇ ψ | ξ ) w · ∇ w (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cs λ Z Z O T ξ | w | + Cλ Z Z O T ξ |∇ w | . (A.44)Using (A.25), we have | ∂ t ϕ | ≤ sλ e λ (6 m − λe λ ( m +1) + Cλξ on (0 , T ) ,Cλξ on ( T , T − T ) ,θλe λ ( m +1) + Cλξ on ( T − T , T ) , o that | ∂ t ϕ | ≤ Csλξ everywhere. Hence (cid:12)(cid:12)(cid:12)(cid:12) s Z Z O T σ∂ t σ∂ t ϕ | w | (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cs λ Z Z O T ξ | w | . (A.45)Moreover, using |∇ ϕ | ≤ Cλξ , (A.25) and θϕ ≤ λξ we also obtain (cid:12)(cid:12)(cid:12)(cid:12) s Z Z O T ∂ t σ |∇ ϕ | | w | (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cs λ Z Z O T ξ | w | , (cid:12)(cid:12)(cid:12)(cid:12) s Z Z O T ∇ σ ∇ ϕ∂ t ϕ | w | (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cs λ Z T Z O ξϕ | ∂ t θ || w | + Cs λ Z Z O T ξ | w | . Finally, we also have (cid:12)(cid:12)(cid:12)(cid:12)Z Z O T ∂ t w ∇ σ · ∇ w (cid:12)(cid:12)(cid:12)(cid:12) ≤ C sλ Z Z O T ξ | ∂ t w | + Csλ
Z Z O T ξ |∇ w | , (A.46)and combining all the above estimates, | I R | ≤ Csλ
Z Z O T ξ | ∂ t w | + Csλ
Z Z O T ξ |∇ w | + Cs λ Z T Z O ξ | ∂ t θ | ϕ | w | + Cs λ Z Z O T ξ | w | . (A.47) A lower bound for the cross-product RR P wP w . This step simply consists in puttingtogether all the above estimates: for all s ≥ s and λ ≥ λ ,2 Z Z O T P wP w ≥ Z O |∇ w (0) | + cs λ e λm +2 Z O | w (0) | + csλ Z Z O T ξ |∇ w | − Csλ Z Z ˜ ω T ξ |∇ w | + cs λ Z Z O T ξ | w | + cs λ Z T Z O | ∂ t θ | ξϕ | w | − Cs λ Z Z ˜ ω T ξ | w | − Cs λ Z Z ˜ ω T ∩{ t ∈ (0 ,T ) } | ∂ t θ | ξϕ | w | − | I R | . Thus, using (A.47), for some s ≥ s and λ ≥ λ , for all s ≥ s and λ ≥ λ Z Z O T P wP w ≥ Z O |∇ w (0) | + cs λ e λm +2 Z O | w (0) | + csλ Z Z O T ξ |∇ w | − Csλ Z Z ˜ ω T ξ |∇ w | + cs λ Z Z O T ξ | w | + cs λ Z T Z O | ∂ t θ | ξϕ | w | − Cs λ Z Z ˜ ω T ξ | w | − Cs λ Z Z ˜ ω T ∩{ t ∈ (0 ,T ) } | ∂ t θ | ξϕ | w | − Csλ Z T Z O ξ | ∂ t w | . (A.48) Conclusion.
We first derive a Carleman estimate on w with gradient observations,and then explains how to remove this term using a suitable multiplier. A Carleman estimate on w with gradient observations. According to estimates (A.8) nd (A.48), for all s ≥ s and λ ≥ λ , Z Z O T (cid:0) | P w | + | P w | (cid:1) + c Z O |∇ w (0) | + cs λ e λm +2 Z O | w (0) | + csλ Z Z O T ξ |∇ w | + cs λ Z Z O T ξ | w | + cs λ Z T Z O | ∂ t θ | ϕξ | w | ≤ C Z Z O T | f | e − sϕ + C Z Z O T | Rw | + Csλ Z Z ˜ ω T ξ |∇ w | + Cs λ Z Z ˜ ω T ξ | w | + Cs λ Z Z ˜ ω T ∩{ t ∈ (0 ,T ) } | ∂ t θ | ξϕ | w | + Csλ
Z Z O T ξ | ∂ t w | . To handle the term k Rw k L , we recall that Rw is given by (A.7), hence Z Z O T | Rw | ≤ Cs λ Z Z O T ξ | w | . where C = C ( k∇ ψ k ∞ , k ∆ ψ k ∞ ) is a positive constant.Also note that1 sλ Z Z O T ξ | ∂ t w | ≤ Csλ
Z Z O T | P w | + Csλ
Z Z O T ξ |∇ w | + Csλ Z Z O T ξ | w | . In particular, for some s ≥ s , for all s ≥ s and λ ≥ λ , Z Z O T (cid:0) | P w | + | P w | (cid:1) + c Z O |∇ w (0) | + cs λ e λm +2 Z O | w (0) | + csλ Z Z O T ξ |∇ w | + cs λ Z Z O T ξ | w | + cs λ Z T Z O | ∂ t θ | ξϕ | w | ≤ C Z Z O T | f | e − sϕ + Csλ Z Z ˜ ω T ξ |∇ w | + Cs λ Z Z ˜ ω T ξ | w | + Cs λ Z Z ˜ ω T ∩{ t ∈ (0 ,T ) } | ∂ t θ | ξϕ | w | . (A.49)In (A.49), the observation is done on ˜ ω T and concerns both w and ∇ w . Below, we shallexplain that this observation can be done only on w provided we take an observation setslightly larger. A Carleman estimate on w without gradient observations. Recall that ˜ ω T ⋐ c ω T , thenthere exists a nonnegative smooth function η = η ( t, x ) taking value in [0 ,
1] such that η = 1 on ˜ ω T , and η = 0 in (0 , T ) × O \ c ω T . We then compute the scalar product of P w and ηsλ ξw : Z Z O T P w ( ηsλ ξw ) = sλ Z Z O T ηξ |∇ w | − sλ Z Z O T ∆( ηξ ) | w | − s λ Z Z O T ησ∂ t ϕξ | w | − s λ Z Z O T η |∇ ϕ | ξ | w | . In particular, using (A.25) and (2.13), sλ Z Z O T ηξ |∇ w | + cs λ Z T Z O ση | ∂ t θ | ξϕ | w | ≤ Z Z O T P w ( ηsλ ξw ) + sλ Z O T | ∆( ηξ ) || w | + s λ e λ ( m +1) Z TT − T Z O ησ | ∂ t θ | ξ | w | + s λ Z Z O T ησ | ∂ t ψ | ξ | w | + s λ Z Z O T η |∇ ψ | ξ | w | . f course, this implies that sλ Z Z ˜ ω T ξ |∇ w | + s λ Z Z ˜ ω T ∩{ t ∈ (0 ,T ) } ση | ∂ t θ | ξϕ | w | ≤ √ s Z Z O T | P w | + Cs / λ Z Z O T η ξ | w | + 2 Csλ Z Z O T | ∆( ηξ ) || w | + 2 Cs λ e λ ( m +1) Z TT − T Z O η | ∂ t θ | ξ | w | + 2 Cs λ Z Z O T η | ∂ t ψ | ξ | w | + 2 Cs λ Z Z d ω T |∇ ψ | ξ | w | . But there exists a constant C = C ( k η k L ∞ ( C ) , k∇ ψ k ∞ , k ∆ ψ k ∞ , k ∂ t ψ k ∞ ) such that | ∆( ηξ ) | ≤ Cλ ξ , sup [ T − T ,T ) (cid:26) | ∂ t θ | θ (cid:27) ≤ C, hence, using the fact that η is supported on c ω T , s / λ Z Z O T η ξ | w | + sλ Z Z O T | ∆( ηξ ) || w | + s λ Z Z O T η | ∂ t ψ | ξ | w | ≤ Cs λ Z Z b ω T ξ | w | , whereas s λ e λ ( m +1) Z TT − T Z O η | ∂ t θ | ξ | w | ≤ Cs λ e λ ( m +1) Z TT − T Z O ηθ ξ | w | ≤ Cs λ Z Z b ω T ξ | w | . Hence, by combining above estimates with (A.49), for some s ≥ s and λ ≥ λ , thereexists a constant C such that for all s ≥ s and λ ≥ λ , Z O |∇ w (0) | + s λ e λ (12 m +2) Z O | w (0) | + sλ Z Z O T ξ |∇ w | + s λ Z Z O T ξ | w | + s λ Z T Z O | ∂ t θ | ξϕ | w | ≤ C Z Z O T | f | e − sϕ + Cs λ Z Z b ω T ξ | w | . (A.50) Back to the function z . We now go back to the function z = we sϕ . For that, let us firstremark that there exists a constant C = C ( k∇ ψ k ∞ ) such that for all ( t, x ) ∈ (0 , T ) × O , | z | e − sϕ = | w | , |∇ z | e − sϕ ≤ |∇ w | + 2 s |∇ ϕ | | w | ≤ |∇ w | + 2 Cs λ ξ | w | . We immediately deduce from (A.50) that for all s ≥ s and λ ≥ λ , for some positiveconstant C , Z O |∇ z (0) | e − sϕ (0) + s λ e λ (12 m +2) Z O | z (0) | e − sϕ (0) + sλ Z Z O T ξ |∇ z | e − sϕ + s λ Z Z O T ξ | z | e − sϕ + s λ Z T Z O | ∂ t θ | ξϕ | z | e − sϕ ≤ C Z Z O T | f | e − sϕ + Cs λ Z Z d ω T ξ | z | e − sϕ . (A.51)We conclude the proof of Theorem 2.4 by setting s = s and λ = λ . .2 Proof of Theorem 2.5 We divide the proof in several steps.
A duality approach.
To solve the control problem (2.26)–(2.27), we first rewrite thecontrol problem under a weak form. Multiplying y solution of (2.26) by smooth functions z on [0 , T ] × O such that z = 0 on [0 , T ] × ∂ O , we get: Z O σ ( T ) y ( T ) z ( T ) + Z Z O T y ( − σ∂ t z − ∆ z ) = Z Z O T fz + Z Z b ω T hz. (A.52)In particular, since σ ( T ) >
0, the null-controllability requirement (2.27) is satisfied ifand only if for all smooth functions z on [0 , T ] × O such that z = 0 on [0 , T ] × ∂ O Z Z O T y ( − σ∂ t z − ∆ z ) = Z Z O T fz + Z Z b ω T hz. (A.53)The trick now is to introduce a functional J whose Euler Lagrange equation coincidewith (A.53): For smooth functions z on [0 , T ] × O such that z = 0 on [0 , T ] × ∂ O , wedefine J ( z ) = 12 Z Z O T | ( − σ∂ t − ∆) z | e − sϕ + s λ Z Z b ω T ξ | z | e − sϕ − Z Z O T fz. (A.54)But the set of smooth functions z on [0 , T ] × O such that z = 0 on [0 , T ] × ∂ O is nota Banach space. We thus introduce X obs = { z ∈ C ∞ ([0 , T ] × O ) such that z = 0 on [0 , T ] × ∂ O} k·k obs (A.55)where k·k obs is the Hilbert norm defined by k z k obs = Z Z O T | ( − σ∂ t − ∆) z | e − sϕ + s λ Z Z b ω T ξ | z | e − sϕ . (A.56)The set X obs is then endowed with the Hilbert structure given by k·k obs . Note that herewe use the fact that k·k obs is a norm, which is a consequence of the Carleman estimate(2.25). Also note that X obs and k·k obs strongly depends on s and λ and we shall followthese dependences carefully in the sequel.The functional J can be extended as a continuous functional on X obs provided (2.28).Indeed, due to (2.25), we easily have, for some constant C > s and λ , (cid:12)(cid:12)(cid:12)(cid:12)Z Z O T fz (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k z k obs (cid:18) s λ Z Z O T ξ − | f | e sϕ (cid:19) / . (A.57)It follows that, if condition (2.28) is satisfied, the functional J can be uniquely extended asa continuous functional (still denoted the same) on X obs . Besides, (A.57) also implies thecoercivity of J on X obs . Since it is also strictly convex on X obs since k·k obs is an Hilbertnorm, J admits a unique minimizer Z on X obs .Setting Y = ( − σ∂ t − ∆) Ze − sϕ and H = − s λ ξ Ze − sϕ b ω T , (A.58)writing the Euler Lagrange equation of J at Z , for all smooth functions z on [0 , T ] × O such that z = 0 on [0 , T ] × ∂ O ,0 = Z Z O T Y ( − σ∂ t z − ∆ z ) − Z Z b ω T Hz − Z Z O T fz, (A.59)which coincides with (A.53).In particular, (A.59) holds for all smooth functions z on [0 , T ] × O such that z = 0 on[0 , T ] × ∂ O with z ( T ) ≡
0, which implies that Y solves the equation (2.26) with h = H in the sense of transposition. By uniqueness of solutions in the sense of transposition, his is the solution of (2.26) in the classical sense. In particular, since H ∈ L ( O T ), Y is C ([0 , T ]; L ( O )). Then, using again (A.59), we remark that it coincides with (A.53),hence Y solves the control requirement (2.27).Besides, using (A.57) and the fact that J ( Z ) ≤ J (0) = 0, s λ Z Z O T | Y | e sϕ + Z Z b ω T ξ − | H | e sϕ ≤ C Z Z O T ξ − | f | e sϕ . (A.60) Estimates on ∇ Y . In the previous step, we found (
Y, H ) satisfying the equations ∂ t ( σY ) − ∆ Y = f + H b ω T , in O T ,Y = 0 , in Γ T ,Y (0 , · ) = 0 , in O ,Y ( T, · ) = 0 , in O . (A.61)and the estimates (A.60).Our goal now is to obtain an estimate on ∇ Y . In order to do this, for ε >
0, weintroduce ϕ ε ( t, x ) : def = θ ε ( t ) (cid:16) λe λ ( m +1) − e ψ ( t,x ) (cid:17) , ξ ε ( t ) : def = θ ε ( t ) e ψ ( t,x ) and θ ε is given by: θ ε : def = θ ε ( t ) such that ∀ t ∈ [0 , T ] , θ ε ( t ) = 1 + (cid:18) − tT (cid:19) µ , ∀ t ∈ [ T , T − T + ε ] , θ ε ( t ) = 1 , ∀ t ∈ [ T − T + ε, T ) , θ ε ( t ) = θ ( t − ε ) ,µ as in (2.12) . We then multiply the equation (A.61) by ξ − ε Y e sϕ ε : − Z Z O T | Y | ∂ t (cid:0) σξ − ε e sϕ ε (cid:1) + Z Z O T | Y | ∂ t σξ − ε e sϕ ε + Z Z O T ξ − ε |∇ Y | e sϕ ε − Z Z O T | Y | ∆ (cid:0) ξ − ε e sϕ ε (cid:1) = Z Z O T fξ − ε Y e sϕ ε + Z Z b ω T Hξ − ε Y e sϕ ε . Following, multiplying by sλ , sλ Z Z O T ξ − ε |∇ Y | e sϕ ε − sλ Z T Z O σ | Y | ∂ t (cid:0) ξ − ε e sϕ ε (cid:1) = sλ Z TT Z O σ | Y | ∂ t (cid:0) ξ − ε e sϕ ε (cid:1) + sλ Z Z O T | Y | ∆ (cid:0) ξ − ε e sϕ ε (cid:1) + sλ Z Z O T fξ − ε e sϕ ε Y + sλ Z Z b ω T Hξ − ε Y e sϕ ε − sλ Z Z O T | Y | ∂ t σξ − ε e sϕ ε . (A.62)We then compute explicitly: − e − sϕ ε ∂ t (cid:0) ξ − ε e sϕ ε (cid:1) = 2 sλξ − ε ∂ t ψ − sξ − ε ∂ t θ ε θ ε ϕ ε + 2 ∂ t θ ε θ ε ξ − ε + 2 λ∂ t ψξ − ε . (A.63)On (0 , T ), we remove the dependence in ε > θ ε = θ on (0 , T ). Using (2.13), ∂ t θ ≤ θ ∈ [1 ,
2] in [0 , T ] we have, for all s ≥ s and t ∈ (0 , T ), − sξ − ∂ t θθ ϕ + 2 ∂ t θθ ξ − ≥ cs | ∂ t θ | ξ − ϕ, hereas (cid:12)(cid:12) sλξ − ∂ t ψ + 2 λ∂ t ψξ − (cid:12)(cid:12) ≤ Csλξ − . Hence − sλ Z T Z O σ | Y | ∂ t (cid:0) ξ − ε e sϕ ε (cid:1) ≥ cs λ Z T Z O σ | ∂ t θ | ξ − ϕ | Y | e sϕ − Cs λ Z T Z O | Y | e sϕ . (A.64)On ( T , T ), from the identity (A.63), using | ∂ t θ ε | ≤ Cθ ε , we derive (cid:12)(cid:12)(cid:12)(cid:12) sλξ − ε ∂ t ψ − sξ − ε ∂ t θ ε θ ε ϕ + 2 ∂ t θ ε θ ε ξ − ε + 2 λ∂ t ψξ − ε (cid:12)(cid:12)(cid:12)(cid:12) ≤ Csλ
We thus obtain (cid:12)(cid:12)(cid:12)(cid:12) sλ Z TT Z O σ | Y | ∂ t (cid:0) ξ − ε e sϕ ε (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cs λ Z TT Z O | Y | e sϕ ε . (A.65)Straightforward computations yield (cid:12)(cid:12) ∆ (cid:0) ξ − ε e sϕ ε (cid:1)(cid:12)(cid:12) ≤ Cs λ e sϕ ε , from which we get (cid:12)(cid:12)(cid:12)(cid:12) sλ Z Z O T | Y | ∆ (cid:0) ξ − ε e sϕ ε (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cs λ Z Z O T | Y | e sϕ ε . (A.66)Using Cauchy-Schwarz estimates, (cid:12)(cid:12)(cid:12)(cid:12) sλ Z Z O T (cid:0) ( f + H b ω T ) ξ − ε e sϕ ε (cid:1) Y (cid:12)(cid:12)(cid:12)(cid:12) (A.67) ≤ s λ Z Z O T | Y | e sϕ ε + Cs Z Z O T ξ − ε | f | e sϕ ε + Cs Z Z b ω T ξ − ε | H | e sϕ ε . Since we obviously have (cid:12)(cid:12)(cid:12)(cid:12) sλ Z Z O T | Y | ∂ t σξ − ε e sϕ ε (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cs λ Z Z O T | Y | e sϕ ε , (A.68)combining estimates (A.64)–(A.65)–(A.66)–(A.67)–(A.68) and plugging (A.62), we obtain sλ Z Z O T ξ − ε |∇ Y | e sϕ ε + s λ Z T Z O σ | ∂ t θ | ξ − | Y | e sϕ ≤ Cs λ Z Z O T | Y | e sϕ ε + C Z Z b ω T ξ − ε | H | e sϕ ε + C Z Z O T ξ − ε | f | e sϕ ε . Since the constant C is independent of ε >
0, we can pass to the limit ε →
0, and using(A.60) and the fact that σ is bounded from below away from 0, we get: sλ Z Z O T ξ − |∇ Y | e sϕ + s λ Z T Z O | ∂ t θ | ξ − ϕ | Y | e sϕ ≤ C Z Z O T ξ − | f | e sϕ . (A.69) Estimates on ∆ Y , ∂ t Y . Multiplying the equation (A.61) by − ξ − ε ∆ Y e sϕ ε /s , − s Z Z O T ∂ t ( σξ − ε e sϕ ε ) |∇ Y | + 1 s Z Z O T ∂ t Y ∇ Y · ∇ ( σξ − ε e sϕ ε )+ 1 s Z Z O T ξ − ε | ∆ Y | e sϕ ε = − s Z Z O T ( f + V b ω T − ∂ t σY ) ξ − ε ∆ Y e sϕ ε . (A.70) s in (A.63), we compute explicitly − ∂ t ( ξ − ε e sϕ ε ). Arguing as in (A.64), we get − s Z T Z O σ∂ t ( ξ − ε e sϕ ε ) |∇ Y | ≥ c Z T Z O σ | ∂ t θ | ξ − ϕ |∇ Y | e sϕ − Cλ Z T Z O ξ − |∇ Y | e sϕ . (A.71)Besides, arguing as in (A.65), we get (cid:12)(cid:12)(cid:12)(cid:12) − s Z TT Z O σ∂ t ( ξ − ε e sϕ ε ) |∇ Y | (cid:12)(cid:12)(cid:12)(cid:12) ≤ Csλ Z Z O T ξ − ε |∇ Y | e sϕ ε . (A.72)One can also easily check that (cid:12)(cid:12)(cid:12)(cid:12) − s Z Z O T ∂ t σ ( ξ − ε e sϕ ε ) |∇ Y | (cid:12)(cid:12)(cid:12)(cid:12) ≤ Csλ Z Z O T ξ − ε |∇ Y | e sϕ ε . (A.73)We then estimate the cross-term of (A.70): (cid:12)(cid:12)(cid:12)(cid:12) s Z Z O T ∂ t Y ∇ Y · ∇ ( σξ − ε e sϕ ε ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ σ s Z Z O T ξ − ε | ∂ t Y | e sϕ ε + Csλ Z Z O T ξ − ε |∇ Y | e sϕ ε , (A.74)where σ min : def = min O T σ . From the equation (A.61), ∂ t Y = 1 σ (∆ Y + f + H ω T − ∂ t σY ) , (A.75)and thus we deduce σ s Z Z O T ξ − ε | ∂ t Y | e sϕ ε ≤ s Z Z O T ξ − ε | ∆ Y | e sϕ ε + Cs Z Z O T | Y | e sϕ ε + Cs Z Z O T ξ − ε | f | e sϕ ε + Cs Z Z d ω T ξ − ε | H | e sϕ ε . (A.76)And of course, (cid:12)(cid:12)(cid:12)(cid:12) − s Z Z O T ( f + H ω T − ∂ t σY ) ξ − ε ∆ Y e sϕ ε (cid:12)(cid:12)(cid:12)(cid:12) ≤ s Z Z O T ξ − ε | ∆ Y | e sϕ ε + Cs Z Z O T ξ − ε | f | e sϕ ε + Cs Z Z d ω T ξ − ε | H | e sϕ ε + Cs Z Z O T | Y | e sϕ ε . (A.77)Combining all the above estimates, we get12 s Z Z O T ξ − ε | ∆ Y | e sϕ ε ≤ Csλ Z Z O T ξ − ε |∇ Y | e sϕ ε + Cs λ Z Z O T | Y | e sϕ ε + C Z Z O T ξ − ε | f | e sϕ ε + C Z Z d ω T ξ − ε | H | e sϕ ε . Since the constant C does not depend on ε >
0, we can pass to the limit ε → s Z Z O T ξ − | ∆ Y | e sϕ ≤ Csλ Z Z O T ξ − |∇ Y | e sϕ + Cs λ Z Z O T | Y | e sϕ + C Z Z O T ξ − | f | e sϕ + C Z Z d ω T ξ − | H | e sϕ . sing now estimates (A.60), (A.69) and (A.76), we get1 s Z Z O T ξ − ( | ∂ t Y | + | ∆ Y | ) e sϕ ≤ C Z Z O T ξ − | f | e sϕ . (A.78) Estimates on ∂ n Y in L (Γ T ) . Let η : O 7→ R N such that η ∈ C ( O ; R N ) and η = ~n on ∂ O . Since Y vanishes on Γ T , we have the following identity: for all ε > Z Γ T ξ − ε | ∂ n Y | e sϕ ε = Z Z O T ξ − ε ∆ Y η · ∇
Y e sϕ ε + Z Z O T D (cid:0) ξ − ε ηe sϕ ε (cid:1) ( ∇ Y, ∇ Y ) − Z Z O T div ( ηξ − ε e sϕ ε ) |∇ Y | . Hence λ Z Γ T ξ − ε | ∂ n Y | e sϕ ε ≤ s Z Z O T ξ − ε | ∆ Y | e sϕ ε + Csλ Z Z O T ξ − ε |∇ Y | e sϕ ε . Passing to the limit in ε → λ Z T Z ∂ O ξ − | ∂ n Y | e sϕ ≤ C Z Z O T ξ − | f | e sϕ . (A.79) Conclusion.
Estimates (A.60), (A.69), (A.78) and (A.79) yield (2.29).
B Regularity of the weight function
Proof of Lemma 4.5.
The first remark is that the flow X e is C ([0 , T ] × [0 , T ] × R ) since y e ∈ C ([0 , T ] × R ).In order to study the regularity of b ψ , we will introduce the function t out = t out ( t, x )defined for ( t, x ) ∈ (0 , T ) × O as the supremum of the time τ ∈ ( t, T ] for which ∀ t ′ ∈ ( t, τ ) , X e ( t ′ , t, x ) ∈ O . It is not difficult to check that this time t out can also be charac-terized as the solution of ∂ t t out + y e · ∇ t out = 0 in O T ,t out ( t ) = t on Γ T ,t out ( T ) = T in O . (B.1)For convenience, we also set x out ( t, x ) = X e ( t out ( t, x ) , t, x ) . (B.2)We first prove that t out is continuous in O T . In order to do that, let us remark that X e is C ([0 , T ] × [0 , T ] × R ) and for all ( t, τ ) ∈ [0 , T ] , X e ( t, τ, · ) is a C diffeomorphism of R . In particular, O T can be decomposed into O T = O T, ∪ O T, ∪ Σ T , with O T, = { ( t, x ) ∈ (0 , T ) × O , x ∈ X e ( t, T, O ) } , O T, = { ( t, x ) ∈ (0 , T ) × O , x ∈ X e ( t, T, R \ O ) } , Σ T = { ( t, x ) ∈ (0 , T ) × O , x ∈ X e ( t, T, ∂ O ) } . (B.3)In (B.3), O T, and O T, are open sets whereas Σ T = O T, ∩O T, is closed and of dimension2. For ( t, x ) ∈ O T, ∪ Σ T , t out ( t, x ) = T and t out is thus continuous on O T, . The continuityon O T, is more involved. If ( t, x ) ∈ O T, , then x out ( t, x ) belongs to ∂ O . Due to thecondition (4.5), for any ε >
0, there exists a neighborhood V ε of ( t out ( t, x ) , x out ( t, x )) in[0 , T ] × O such that | t out ( t ′ , x ′ ) − t out ( t, x ) | < ε for all ( t ′ , x ′ ) ∈ V ε . In particular, for some t ε ∈ (0 , T ) close to t out ( t, x ), V ε is a neighborhood of ( t ε , X e ( t ε , t out ( t, x ) , x out ( t, x ))) =( t ε , X e ( t ε , t, x )). Following, { X e ( t − t ε + t ′ , t ′ , x ′ ) , ( t ′ , x ′ ) ∈ V ε } is a neighborhood of( t, X e ( t, t ε , X e ( t ε , t, x ))) = ( t, x ) on which t out is at distance at most ε of t out ( t, x ). hus, t out is continuous in O T . As b ψ solution of (4.8) can be written as b ψ ( t, x ) = (cid:26) b ψ T ( x out ( t, x )) if t out ( t, x ) = T,t out ( t, x ) − T if t out ( t, x ) < T, (B.4)the continuity of b ψ in O T follows from the first compatibility condition in (4.7). Also notethat b ψ is obviously C in O T, .We then focus on the C regularity of b ψ . In order to do this, we remark that ∇ t out solves ∂ t ∇ t out + ( y e · ∇ ) ∇ t out + D y e ∇ t out = 0 in O T , ∇ t out ( t, x ) = − n ( x ) y e ( t, x ) · n ( x ) on Γ T , ∇ t out ( T ) = 0 in O . (B.5)In particular, ∇ t out can be computed for any ( t, x ) ∈ O T, by solving for τ between t and t out ( t, x ) the ODE ddτ (cid:0) ∇ t out ( τ, X e ( τ, t, x )) (cid:1) = − D y e ( τ, X e ( τ, t, x )) ∇ t out ( τ, X e ( τ, t, x )) , τ ∈ ( t, t out ( t, x )) , with ∇ t out ( t out ( t, x ) , x out ( t, x )) = − n ( x out ( t, x )) y e ( t out ( t, x ) , x out ( t, x )) · n ( x out ( t, x )) . One then easily obtains that ∇ t out is C on O T, and from the equation (B.1) we deducethat t out is C in O T, . From there, we derived immediately from (B.4) that b ψ is C on O T, and that it can be extended as a C funtion on O T, as follows: ∇ b ψ can be computedfor any ( t, x ) ∈ Σ T by solving for τ between t and T the ODE: ddτ (cid:16) ∇ b ψ ( τ, X e ( τ, t, x )) (cid:17) = − D y e ( τ, X e ( τ, t, x )) ∇ b ψ ( τ, X e ( τ, t, x )) , τ ∈ ( t, T ) , (B.6)with ∇ b ψ ( T, X e ( T, t, x )) = − n ( X e ( T, t, x )) y e ( T, X e ( T, t, x )) · n ( X e ( T, t, x )) . (B.7)On the other hand, b ψ solves the equation (4.11), and can be extended as a C functionon O T, . For ( t, x ) ∈ Σ T , this yields ∇ b ψ ( t, x ) as the solution of the ODE (B.6) with ∇ b ψ ( T, X e ( T, t, x )) given. But, as b ψ ( T ) is constant on the boundary and satisfies thesecond compatibility condition in (4.5), we get again (B.7) for ( t, x ) ∈ Σ T . Following, ∇ b ψ is continuous across Σ T , hence on O T . Using the equation (4.8), b ψ belongs to C ( O T ).The proof of the C regularity follows the same path and is left to the reader. References [1] P. Albano and D. Tataru. Carleman estimates and boundary observability for acoupled parabolic-hyperbolic system.
Electron. J. Differential Equations , pages No.22, 15 pp. (electronic), 2000.[2] F. Boyer. Trace theorems and spatial continuity properties for the solutions of thetransport equation.
Differential Integral Equations , 18(8):891–934, 2005.[3] F. Boyer and P. Fabrie. Outflow boundary conditions for the incompressible non-homogeneous Navier-Stokes equations.
Discrete Contin. Dyn. Syst. Ser. B , 7(2):219–250 (electronic), 2007.[4] F. Boyer and P. Fabrie.
Mathematical tools for the study of the incompressible Navier-Stokes equations and related models , volume 183 of
Applied Mathematical Sciences .Springer, New York, 2013.[5] F. W. Chaves-Silva, L. Rosier, and E. Zuazua. Null controllability of a system ofviscoelasticity with a moving control. http://arxiv.org/abs/1303.3452.
6] J.-M. Coron. On the controllability of the 2-D incompressible Navier-Stokes equationswith the Navier slip boundary conditions.
ESAIM Control Optim. Calc. Var. , 1:35–75(electronic), 1995/96.[7] J.-M. Coron and A. V. Fursikov. Global exact controllability of the 2D Navier-Stokesequations on a manifold without boundary.
Russian J. Math. Phys. , 4(4):429–448,1996.[8] B. Desjardins. Linear transport equations with initial values in Sobolev spaces andapplication to the Navier-Stokes equations.
Differential Integral Equations , 10(3):577–586, 1997.[9] S. Ervedoza, O. Glass, S. Guerrero, and J.-P. Puel. Local exact controllability for theone-dimensional compressible Navier-Stokes equation.
Arch. Ration. Mech. Anal. ,206(1):189–238, 2012.[10] E. Fern´andez-Cara. Motivation, analysis and control of the variable density Navier-Stokes equations.
Discrete Contin. Dyn. Syst. Ser. S , 5(6):1021–1090, 2012.[11] E. Fern´andez-Cara and S. Guerrero. Global Carleman inequalities for parabolic sys-tems and applications to controllability.
SIAM J. Control Optim. , 45(4):1399–1446(electronic), 2006.[12] E. Fern´andez-Cara, S. Guerrero, O. Y. Imanuvilov, and J.-P. Puel. Local exactcontrollability of the Navier-Stokes system.
J. Math. Pures Appl. (9) , 83(12):1501–1542, 2004.[13] E. Fern´andez-Cara, S. Guerrero, O. Y. Imanuvilov, and J.-P. Puel. Some controlla-bility results for the N -dimensional Navier-Stokes and Boussinesq systems with N − SIAM J. Control Optim. , 45(1):146–173 (electronic), 2006.[14] A. V. Fursikov and O. Y. Imanuvilov.
Controllability of evolution equations , volume 34of
Lecture Notes Series . Seoul National University Research Institute of MathematicsGlobal Analysis Research Center, Seoul, 1996.[15] V. Girault and P.-A. Raviart.
Finite element methods for Navier-Stokes equations.Theory and algorithms . Springer, Berlin, 1986.[16] M. Gonz´alez-Burgos, S. Guerrero, and J.-P. Puel. Local exact controllability tothe trajectories of the Boussinesq system via a fictitious control on the divergenceequation.
Commun. Pure Appl. Anal. , 8(1):311–333, 2009.[17] O. Y. Imanuvilov. Remarks on exact controllability for the Navier-Stokes equations.
ESAIM Control Optim. Calc. Var. , 6:39–72 (electronic), 2001.[18] O. Y. Imanuvilov and J.-P. Puel. Global Carleman estimates for weak solutions ofelliptic nonhomogeneous Dirichlet problems.
Int. Math. Res. Not. , 16:883–913, 2003.[19] O. Y. Imanuvilov, J.-P. Puel, and M. Yamamoto. Carleman estimates for parabolicequations with nonhomogeneous boundary conditions.
Chin. Ann. Math. Ser. B ,30(4):333–378, 2009.[20] O. Y. Imanuvilov and M. Yamamoto. Carleman inequalities for parabolic equationsin Sobolev spaces of negative order and exact controllability for semilinear parabolicequations.
Publ. Res. Inst. Math. Sci. , 39(2):227–274, 2003.[21] P. Martin, L. Rosier, and P. Rouchon. Null controllability of the structurally dampedwave equation with moving control.
SIAM J. Control Optim. , 51(1):660–684, 2013.[22] M. Tucsnak and G. Weiss.
Observation and Control for Operator Semigroups , vol-ume XI of
Birk¨auser Advanced Texts . Springer, 2009.. Springer, 2009.