On the Uniform Controllability of the Inviscid and Viscous Burgers-$\alpha$ Systems

This work is devoted to prove the global uniform exact controllability of the inviscid and viscous Burgers-$\alpha$ systems. The state $y$ is the solution to a regularized Burgers equation, where the transport velocity $z$ consists of a filtered version of $y$ - specifically $z=(Id-\alpha^2\partial^2_ {xx})^{-1}y$ with $\alpha>0$ being a small parameter - in place of $y$. First, a global uniform exact controllability result for the nonviscous Burgers-$\alpha$ system with three scalar controls is obtained, using the return method. Then, global exact controllability to constant states of the viscous system is deduced from a local exact controllability result and a global approximate controllability result for smooth initial and target states.


Introduction
Let L > 0 and T > 0 be given.Let us present the notations used along this work.The symbols C, C and C i , i = 0, 1, . . .stand for positive constants (usually depending on L and T and independent of α).For any r ∈ [1, +∞] and any given Banach space X, • L r (X) will stand for the usual norm in Lebesgue-Bochner space L r (0, T ; X).In particular, the norms in L r (0, L), L r (0, T ) and L r ((0, T ) × (0, L)) will be denoted by • r .In this paper, we will consider the following two families of controlled systems: where I l = {t ∈ [0, T ] : v l (t) > 0} and I r = {t ∈ [0, T ] : v r (t) < 0}, and y t − µy xx + zy x = p(t) in (0, T ) × (0, L), z − α 2 z xx = y in (0, T ) × (0, L), z(•, 0) = y(•, 0) = v l in (0, T ), z(•, L) = y(•, L) = v r in (0, T ), y(0 , •) = y 0 in (0, L). (1.2) These are the so called non-viscous and viscous convectively filtered Burgers equations (also known in the literature as the Burgers-α system or Leray-Burgers equation).The couples (y, z) and the triplets (p, v l , v r ) stand for the corresponding states and controls.The parameter µ > 0 is the fluid viscosity and α > 0 is the characteristic wavelength of the considered Helmholtz filter.For simplicity, throughout this paper we will take µ = 1.All the results can be extended without difficulty to the case where µ is an arbitrary positive number.
Obviously, (1.1) and (1.2) can be regarded as nonlinear regularizations of the non-viscous and viscous Burgers equations.These systems and some related variants have already been studied.More precisely, (1.2) is the b = 0 case of the b-family: or, equivalently, It has been studied in [37] as a model for the nonlinear 1D wave dynamics in a fluid including the effects of convection and stretching; the dimensionless parameter b measures the relative strength of these effects and z can be viewed as the fluid velocity in the x direction (or equivalently the height of the free surface of the fluid above a flat bottom).When b = 2, (1.3) is the so-called viscous Camassa-Holm equation; it describes the unidirectional surface waves at a free surface of shallow water under the influence of gravity, see [10].When b = 3, we are dealing with the viscous Degasperis-Procesi equation, which plays a similar role in water wave theory, see [19].
It is interesting to highlight that this regularization idea was first employed by Leray in [39] to prove the existence of a solution to the Navier-Stokes equations.It has been used to capture shocks in the Burgers equation in [5,7,8,42,43].It has also been employed in other contexts, such as the analysis of compressible Euler equations, scalar conservations laws and aggregation equations, see [4,6,9,18,46].Finally, systems like (1.1) and (1.2) can also be viewed as simplified 1D versions of the so called Leray-α system introduced some time ago to describe turbulent flows, as an alternative to the classical averaged Reynolds models; see [12,29,30].
Our main results deal with the global uniform (with respect to α) exact controllability of (1.1) and (1.2).More precisely, one has: Theorem 1.1.Let α > 0 be given.The non-viscous Burgers-α system (1.1) is globally exactly controllable in C 1 .That is, for any given y 0 , y T ∈ C 1 ([0, L]), there exist a space-independent control p α ∈ C 0 ([0, T ]), a couple of boundary controls ) and an associated state pair (1.4) Moreover, there exists a positive constant C > 0 (depending on L, T , y 0 and y T but independent of α) such that Theorem 1.2.Let α > 0 be given.The viscous Burgers-α system (1.2) is globally exactly controllable in L ∞ to constant trajectories.That is, for any given y 0 ∈ L ∞ (0, L) and N ∈ R, there exist a space-independent control p α ∈ C 0 ([0, T ]), a couple of boundary controls (v α l , v α r ) ∈ H 3/4 (0, T ; R 2 ) and an associated state pair and the following estimate where C is a positive constant (depending on L, T , y 0 and N but independent of α).Moreover, if y 0 ∈ H 1 0 (0, L) then the same conclusion holds with In this paper, we are going to deal with situations that lead to new difficulties compared to previous works on nonlinear parabolic equations.Let us explain these differences: -Nonlocal nonlinearities.In (1.1) and (1.2), the usual convective term is replaced and a filtered (averaged) velocity appears.As a consequence, the arguments in [11] must be modified, as shown below.-Uniform controllability.Performing careful estimates of the controls, global uniform controllability results can be obtained.This way, we will be able to generalize previous control results for nonlinear parabolic equations with nonlocal nonlinearities, see [11,41].
For completeness, let us mention some previous works on the well-posedness and control of the previous systems and other similar models.Global and local well-posedness are respectively established in [23] (when the boundary conditions are homogeneous) and [13,45].Moreover, there are many important works dealing with the controllability properties of parabolic equations and systems (see [22,25,28,31]) and non-viscous and viscous Burgers equations (see [11,17,20,26,31,33,34,38,41,44]).In the case of the Burgers-α system, the local uniform null controllability of the viscous system (1.2) with distributed and boundary controls was studied in [1]; later, the results have been extended to any equation of the b-family in [27].In higher dimensions, local uniform (distributed and boundary) null controlability results for the Leray-α system have been obtained in [2].
The rest of this paper is organized as follows.In Section 2, we prove some results concerning the existence, uniqueness and regularity of the solutions.Sections 3 and 4 deal with the proofs of Theorems 1.1 and 1.2.Finally, in Section 5, we present several additional comments and questions.

Notations and classical results
Let us denote by C 0 b (R) (resp.C 0,1 b (R)) the Banach space of bounded continuous functions on R (resp.bounded Lipschitz-continuous functions on R).Let C 0,1 x ([0, T ] × R) be the space of functions f : [0, T ] × R → R that are continuous in x and t and globally Lipschitz-continuous in space, with a Lipschitz constant independent of t.
In the sequel, for any given f ∈ C 0 ([0, T ] × R), the associated flux function Φ = Φ(s; t, x) is defined as follows: (2.1) The mapping Φ contains all the information on the trajectories of the particles transported by the velocity f .Furthermore, we have the following existence, uniqueness and regularity result: x ([0, T ] × R) and f x belongs to the space C 0 ([0, T ] × R).Then, there exists a unique flux function associated to f , that is, a unique function See [35,36] for other similar results.Under the assumptions in Proposition 2.1, it is well known that for any s, t ∈ [0, T ] the mapping Φ(s; t , Let us now recall a result from Bardos and Frisch [3].To this purpose, let us first note that, for any given Banach space X with norm • X and any function u ∈ C 1 ([0, T ]; X), the following inequality holds: where d/dt + represents the time derivative to the right.
Proposition 2.2 ([3], Lem. 1).Let the velocity v and the source g be given respectively in the spaces Then, any solution in the space C 0 ([0, T ]; to the equation satisfies the following inequality in (0, T ).
Proof.Let Φ be the flow associated to v.For any (s, t, x) Using this identity and the fact that Φ(s; t , •) is a diffeomorphism, we get: Now, the result follows easily from this and from (2.2).
The last result of this section is an immediate consequence of Banach's Fixed-Point Theorem: Theorem 2.3.Let (E, • E ) and (F, • F ) be Banach spaces, with F continuously embedded in E. Let B be a subset of F and let G : B → B be a mapping such that Let us denote by B the closure of B for the norm • E .Then, G can be uniquely extended to a continuous mapping G : B → B that possesses a unique fixed-point in B.
The following result concerns the global existence and uniqueness of a solution to the viscous Burgers-α system: 0, T ) be given.Assume that the following compatibility relations hold: v l (0) = y 0 (0) and v r (0) = y 0 (L).
Then there exists a unique solution (y α , z α ) to the Burgers-α system: (2.4) with Then, the following estimates holds: where the constants C do not depend of α.
Proof.The proof of existence can be reduced to find a fixed-point of an appropriate mapping Λ α .Thus, note first that there exists ξ ∈ L 2 (0, T ; Accordingly, for each ȳ ∈ L ∞ (0, T ; L ∞ (0, L)) there exists exactly one z ∈ L ∞ (0, T ; and, moreover, using a lifting argument together with standard energy estimates for elliptic equations, we get a positive constant C (independent of α) such that: With this z, by applying (for instance) the Faedo-Galerkin method, we can easily prove the existence of a solution y to the linear parabolic equation that satisfies and for some C > 0 that is independent of α.
Arguing as in the proof of Lemma 1 in [1], we can deduce that the solution to (2.7) belongs to the space C 0 ([0, T ]; H 1 (0, L)) and, in particular, y L ∞ (L ∞ ) ≤ M T .Accordingly, we can introduce the bounded closed convex set and the mapping Λ α : K → K, with Λ α (ȳ) = y.Obviously, Λ α is well-defined and continuous and, moreover, we can see from the estimates in (2.8) that From classical results of the Aubin-Lions kind (see [47]), we deduce that G is relatively compact in L ∞ (0, T ; L ∞ (0, L)).Therefore, by Schauder's Fixed Point Theorem, Λ α has a fixed point in K, which obviously implies the existence of a solution to (2.4).
We prove now that the solution is unique.Let (y α , z α ) and ( y α , z α ) be two solutions to (2.4) and let us introduce u := y α − y α and v := z α − z α .Then, (2.9) Using the fact that ) and multiplying the first equation of the system above by u, we get: Therefore, Since u(0 , •) = 0, Gronwall's Lemma implies u ≡ 0 and, consequently, v ≡ 0. Finally, let us check that z α satisfies the regularity properties in (2.5).To get this, let us introduce the function given by Then, we obtain from (2.4) that z α = w α + h, where w α solves and the estimates are uniform, with respect to α, in the space L 2 (0, T ; Now, let us present a result concerning global existence and uniqueness of a (weak) solution with initial conditions in L ∞ (0, L): 0, T ) be given.Then, there exists a unique solution (y α , z α ) to the Burgers-α system: with (2.11) Then, the following estimates holds: where the constant C > 0 is independent of α.
Now, for any ȳ ∈ L 2 (0, T ; L ∞ (0, L)), there exists a unique solution to the elliptic problem and, again, using a lifting argument together with standard energy estimates for elliptic equtions, we get a positive constant C (independent of α) such that: With this z, we can solve (for instance via Faedo-Galerkin method) the linear problem and we find a weak solution y that satisfies Again, as in the proof of Lemma 1 in [1], we can deduce that the solution to (2.13) satisfies Let us introduce the set and the mapping Λ α : K → K with Λ α (ȳ) = y.Then, arguing as in the proof of Proposition 2.4, it is not difficult to prove that Λ α possesses a fixed-point in K.

Controllability of the non-viscous Burgers-α system
In this section we present a proof of the global exact controllability property of the non-viscous Burgers-α system.We split the proof in two parts: (i) a local null controllability result; (ii) an argument based on a time scale-invariance and reversibility in time that leads to the desired global result.

Local null controllability
We have the following result: Proposition 3.1.Let α > 0 be given.Then, there exist δ > 0 and C > 0 (both independent of α) such that the following property holds: for each and The proof is obtained by applying the return method, see [11,15,16,32].It relies on a linearization process in combination with a fixed-point argument: (i) first, we need to find a "good" trajectory (a particular solution for the nonlinear system) steering 0 to 0 such that the linearization around it is controllable; (ii) then, we must recover (for instance by a fixed-point argument) the exact controllability result, at least locally, for the nonlinear system.
In our case, it is not difficult to verify that the linearization around zero is not controllable.Accordingly, we build an appropriate nontrivial trajectory connecting (0, 0) to (0, 0).
To this purpose, let us introduce the set Let us consider the couple ( y(x, t), z(x, t)) := (λ(t), λ(t)) and the triplet ( p(t), v l (t), v r (t)) := (λ (t), λ(t), λ(t)), with λ ∈ Λ L,T,1 and supp λ ⊂ (0, T ).Note that ( y, z) is a particular solution to (1.1), associated to the control ( p, v l , v r ).We have the following general controllability result: Proposition 3.2.Assume that λ ∈ Λ L,T,0 and let α > 0 be given.Then, for any Thanks to Proposition 3.2, one may expect that the null controllability for the nonlinear system (1.1) holds.Indeed, we have the following result from which Proposition 3.1 is an immediate consequence: Proposition 3.3.Assume that λ ∈ Λ L,T,0 and let α > 0 be given.Then, there exist δ > 0 and C > 0 (both independent of α) such that: for any and Proof.We will reformulate the null controllability problem as a fixed-point equation.To do this, we will first introduce some auxiliar functions and establish some helpful results.Thus, to any given It follows from the maximum principle for elliptic equations that e. in [0, T ] and, by applying again the maximum principle for elliptic equations, we get: Then, by combining (3.4) and (3.5), we easily get that Note that this inequality does not lead to contradiction when alpha goes to 0, since the space C 0 ([0, T ]; is not reflexive and, the limit, z can "escape".Since λ ∈ Λ L,T,0 , we can find η ∈ (0, L/2) such that Now, we consider the following extension of z to the closed interval [−η, L + η]: ) and there exists C 1 > 0 (independent of α) such that Then, let χ be given, with χ ∈ C ∞ 0 (−η/2, L + η/2), χ = 1 in [0, L] and 0 ≤ χ ≤ 1.This way, we can introduce and, using (3.8) and (3.9), we see that for some C 2 > 0, again independent of α.
Let us set R := η C2T and let us assume from now on that  This and the fact that φ(s; t, x) is well-defined for all (s, t, x) ∈ [0, T ] × [0, L] × R lead to the desired conclusion.
Let y 0 ∈ C 1 ([0, L]) be given and let us introduce and Then, it is easy to see that y * 0 is a compactly supported extension of y 0 to the whole real line and for some C 3 > 0 independent of α.
Let us set y ∈ C 1 ([0, T ] × R), with y(t, x) := y * 0 (φ * (t; 0, x)) ∀(t, x) ∈ [0, T ] × R. (3.16) Then, we have the following : Claim 3.5.The function y satisfies: Moreover, it is not difficult to see that, for any 0 < η < L/2 such that (3.7) holds, the flux associated to λ satisfies φ(T ; 0, L) < −2η and we obtain from (3.14) that φ * (T ; 0, L) < −η.Since φ * (s; t , •) is increasing for any s, t ∈ [0, T ], we see that This inequality and (3.16) together with the fact that An immediate consequence of (3.12), (3.13), (3.15) and (3.16) is that for a positive constant C 4 depending on R but independent of α.By taking we get easily the function y, defined in (3.16), satisfies the inequality Let us consider now the closed ball B R centered at the origin and radius R in the Banach space . Using the inequality (3.18), we can define the mapping , where y is given by (3.16).
Thanks to (3.16), we have that ).Furthermore, the following holds: Claim 3.6.There exists a positive constant C 5 depending on y 0 C 1 ([0,L]) , L, R and T , but independent of α, such that, for any m ≥ 1 and any h 1 , h 2 ∈ B R , one has where F m denotes F iterated m times.
Proof.The proof relies on an induction argument.Let h i ∈ B R be given for i = 1, 2. Then, let us consider the functions z i, * and y i , respectively given by (3.9) and (3.16) and set y := y 1 − y 2 and z * := z 1, * − z 2, * .By Claim 3.5, we have Therefore, integrating from 0 to t and using that y 2 x ∈ C 0 ([0, T ]; C 0 b (R)) and the maximum principle for elliptic PDE's, we find a positive constant C 5 depending on y 0 C 1 ([0,L]) , L, R and T , but independent of α, such that It follows that and the result is true for m = 1.Then, by making similar computations and using the induction hypothesis, we get the result of the claim.
Let B R be the closure of B R with the norm of C 0 ([0, T ]; C 0 ([0, L])) and let F be the unique continuous extension of F to B R .Note that, since F is uniformly continuous, F is well-defined.
Claim 3.7.The continuous extension F satisfies the following properties:

Proof. Let us begin by proving the item a). For a given
. Therefore, the corresponding elliptic solutions to (3.3) and the associated flows (given in (3.13)) satisfy the convergences Moreover, since the Φ * n , Φ * ∈ C 1 ([0, T ] × [0, T ] × R), the corresponding functions, defined in (3.16), belong to C 1 ([0, T ] × [0, L]), verify the transport equation (3.17) and, furthermore, Thus, it follows easily from this and from the definition of F and F that F(h) = y.
Let us now prove the item b).In fact, notice that, by definition of F, for any h ∈ B R there exists a sequence and

It is not difficult to prove that
On the other hand, since and, consequently, letting n → +∞ and then maximizing in x and x , the following is found: Therefore, using the fact that It follows from Claim 3.6 that F m is a contraction for m large enough.Then, from the Theorem 2.3, F possesses a unique fixed-point y ∈ B R .Finally, taking into account Claim 3.7, the proof of Proposition 3.3 is achieved.
Remark 3.8.Assume that λ ∈ Λ L,T,1 and consider the Banach space ) is small enough, then the fixed-point mapping F can be defined in a closed ball of X centered at zero of radius R > 0.Then, one applies the Theorem 2.3 to the closure of this ball with the norm of Performing similar computations as those in Proposition 3.3, one can deduce that there exists δ > 0 (independent of α) such that, for any for a constant C > 0 that is independent of α.
Remark 3.10.From the proof of the previous result, one sees that z( This is important at least for two reasons: (i) to guarantee that in the limit, as α goes to 0, z and y converge to the same limit; (ii) to build suitable controls for the uniform approximate controllability problem presented in Section 4.2.
Remark 3.11.Notice that the distributed control p α is independent of α, that is, its definition only depends on T , L, the initial condition and the target state.

Smoothing effect
The goal of this section is to prove that, starting from a H 1 0 initial state, there exists a small time where the solution begins to be smooth.More precisely, we have the following result: Proposition 4.1.Let y 0 ∈ H 1 0 (0, L) be given and let (y α , z α ) be the solution to Then, there exist T * ∈ (0, T /2) such that the solution y α belongs to C 0 ([T * , T ]; H 3 (0, L)) and satisfies where Λ : R + → R + is a continuous function satisfying Λ(s) → 0 as s → 0 + .
Proof.We will divide the proof in several steps.Throughout the proof, all the constants are independent of α.
Here, C is a positive constant that depends on L, T, M and τ , but it is independent of α.Moreover, there exists a constant K that depends on L, T and M (independent of α and τ ), such that Proof.The proof is standard.It can be easily obtained, for instance, via a Faedo-Galerkin technique in combination with well known energy estimates.