Local null controllability of coupled degenerate systems with nonlocal terms and one control force

We deal with a class of one-dimensional nonlinear parabolic systems with nonlocal and weakly degenerate diffusion coefficients. Using Liusternk's Inverse mapping Theorem, we proved a local null controllability result with only one internal control.

The main purpose of this work is to prove the local null controllability of (1.1) by means of one control. Precisely, we will obtain h ∈ L 2 ((0, T ) × (0, 1)) such that the associated state (u, v) = (u(t, x), v(t, x)) of (1.1) satisfies a is sufficiently small, where H 1 a is an appropriate weighted Hilbert space which will be defined later in Section 2 .
The main difficult comes from the fact that the diffusion coefficients degenerate at x = 0 and have nonlocal terms, namely It is important to remark that semilinear nondegenerate problems have been extensively studied over the last decades, see [13,18,19,21,26] for example.
However, it seems to us that there is also a large interest in degenerate operators when the degeneracy occurs at the boundary of the space domain. For instance, in [28], it was developed a study about the Prandtl system for stationary flows, in which the related boundary layer system was reduced to a quasilinear degenerate parabolic equation. Degenerate operators also appear in probabilistic models, see [14,15], and in climate science, see [20].
In the context of degenerated systems, controllability was studied in the case of two coupled equations in [5,22,23]. Recently, Benhassi et al.,in [4], generalize the Kalman rank condition for the null controllability to n-coupled linear degenerate parabolic systems with m-controls.
On the other hand, as it was pointed out in [18], nonlocal terms type can be found in several natural phenomena, such as in the reaction-diffusion systems, see [7], and in nonlinear vibration theory, see [27] for example.
In [1], it was obtained the null controllability for the semilinear equation Based on this work, in [11], we have considered (1.3), replacing the second-order term (au x ) x by a specific degenerate nonlocal operator. In that new context, we have achieved a local null controllability result. For systems of parabolic equations, the main issue is often to reduce the number of control functions acting on the system (see [4,6,9,12], for example), besides that, as it was pointed out in [3], the problem of controlling coupled parabolic equations has a very different behavior with respect to the scalar case, for instance, boundary controllability is not equivalent to distributed controllability, approximate controllability is not equivalent to null controllability, and "the list of open problems is long and there is a lot of work to be done in order to fully understand this challenging subject". In this direction, the current work may be seen as a natural continuation of [11] and a first step in order to understand parabolic system with nonlocal and degenerate diffusion coefficients of the type µ ·, Our main result is the following: Under the assumptions on µ 1 , µ 2 , f 1 and f 2 , the nonlinear system (1.1) is locally null-controllable at any time T > 0, i.e., there exists r > 0 such that, whenever u 0 , v 0 ∈ H 1 a and The proof of Theorem 1.1 will follow standard arguments (see for instance [18], [8], [24], [16] and [17]), based on Lyusternik's Inverse Mapping Theorem, which can be found in [21] and [25]. To be more specific, we will see that the desired result is equivalent to find a solution to the equation where H : E −→ F is a C 1 mapping between two appropriate Hilbert spaces. We emphasize that the resolution of (1.4) is related to a global null controllability result to the linearization of (1.1) (see the system (3.1) below). This approach relies on a suitable Carleman estimate for the solutions of the adjoint problem associated to (3.1) (see Proposition 3.2). This paper is organized as follows: Section 2 contains notations we use and a preliminary result. In Section 3, we prove a Carleman type inequality to solutions of (3.2), which also allows us to obtain an Observality inequality. Section 4 is concerned with the global null controllability of (3.1) as well as two crucial additional estimates. Section 5 is devoted to the proof of Theorem 1.1. In Section 6, we present some comments and remarks.

Carleman and Observability inequalities
In order to prove Theorem 1.1, we must obtain a global null controllability result to the linearization of (1.1), given by where g 1 , g 2 and h belong to appropriate L 2 -weighted spaces which we will specify later on. To this purpose it is crucial to obtain an appropriate Carleman estimate for solutions to x ∈ (0, 1), which is the adjoint problem of (3.1). The well-posedness of problem (3.1) can be found in [12].
Proposition 3.1. Let us assume that there exists ω 1 ⊂⊂ ω such that Then, there exist positive constants C, λ 0 and s 0 such that, for any s ≥ s 0 , λ ≥ λ 0 and any y T , z T ∈ L 2 ((0, T ) × (0, 1)), the corresponding solution (y, z) to (3.2) satisfies Proof. Initially, applying Proposition 2.1 to each equation in (3.2), we obtain, for s and λ sufficiently large, the inequality Thus, it is sufficient to show that there exists a small ε > 0 such that Let us take χ ∈ C ∞ 0 (ω) satisfying 0 ≤ χ ≤ 1 and χ ≡ 1 in ω 1 . Since inf b 21 > 0, we can easily see that Now, multiplying the first equation in (3.2) by e 2sϕ s 3 λ 3 σ 3 χz and integrating over (0, T ) × (0, 1), we get Next, we need to estimate I 1 , I 2 , I 3 and I 4 . Firstly, from Young's inequality, we have In the same way, since b 11 is bounded, it is immediate that Using integration by parts, we will split up I 2 and I 3 in several integrals. In fact, and, recalling that e 2sϕ vanishes at 0 and T and using the second equation of (3.2), we have Thus, Now, it remains estimates these four integrals. It is immediate that and, from Young's inequality, that In order to estimate J 1 , we will analyze each term between brackets. Firstly, we observe that all the terms are multiplied by χ, which vanishes outside of ω. Clearly, Since |σ x | ≤ Cλσ and a ∈ C 1 (ω), after distributing the derivatives with respect to x, we can see that Likewise, the relations |ϕ t | ≤ Cσ 2 and |σ t | ≤ Cσ 2 yield As a conclusion, The last step is to deal with J 2 . To do this, we notice that Hence, we only need to estimate the last integral. Multiplying the first equation in (3.2) by χ 2 e 2sϕ s 5 λ 5 σ 5 y, integrate over (0, T ) × (0, 1) and integrating by parts we get that We can see that all the integrals here are of the same type of those in (3.7). Following the same arguments developed there, we have the result. Now we need to prove a Carleman inequality for solutions of problem (3.2) with weights which do not vanish at t = 0.
Proof. Let us multiply the first equation in (3.1) byρ 2 u and the second one byρ 2 v, and let us integrate over [0,1]. In this case, we obtain Clearly, the terms in the right hand side of (4.4) can be estimated as follows: Now, let us deal with the left hand side of (4.4). Notice that Summing up, we have just checked that Next, we will estimate I. Firstly, we put is a bounded function on [0, 1]. Secondly, we observe that and, for any t ∈ [0, T ], we have From this, we obtain Now, in order to deal with J , we consider the estimate and we getJ ≤ C 1 0 ρ 2 0 a(|u| 2 + |v| 2 ). Recalling inequality (4.5), we conclude that and, integrating in time, we obtain the desired result.
Proof. Multiplying the first equation in (3.1) by ρ 2 * u t and the second one by ρ 2 * v t , , we take Notice that, Proceeding as in the proof of Proposition 4.2, Using Young's inequality with ε, we have Since |ζ t | ≤ Cζ 2 and ζ and Thus, from (4.8), applying Proposition 4.2 and using ρ * ≤ Cρ 0 , we get In order to conclude the proof, it remains to estimate . In fact, it is enough to multiply the first equation in (3.1) by −ρ 2 * (au x ) x and the second one by −ρ 2 * (av x ) x , and proceed as in the first part of this proof.

Main Result
In this section, our goal is to prove Theorem 1.1. Let us define the functions spaces We also consider the Hilbertian norms a . Now, we set the mapping H : E −→ F , given by Applying Lyusternik's Inverse Mapping Theorem, see [2], we will prove that H has a right inverse mapping defined in a small ball contained in F . Due to the choice of the spaces E and F , the existence of that inverse mapping will imply the local null controllability of (1.1). Before doing it, we will establish some results which will guarantee that H satisfies the hypotheses of Lyusternik's Theorem.  Taking k > 0, we quickly get e −k for any t ∈ [0, T ]. From this point, we may argue as in [11] (see Lemma 4.4, on page 533), in order to check that q 1 , q 2 ∈ H 1 (0, T ) and Therefore, the desired result is a consequence of the continuous embedding H 1 (0, T ) → C(0, T ).
As a consequence of Lemma 5.1, we deduce the useful result below: Proof. Take Analagously, a similar estimate also holds to (a) H is well defined; x, u, v), with i = 3, 4. Then, the linear mapping T : E −→ G and S : E −→ G, given by 2ū +f 4 2v , are the Gateaux derivative of H 1 and H 2 at (u, v, h) ∈ E, respectively. Proof.

Some Additional Comments
As a first comment, we note that, in assumption A.1, we have taken a weak type of degeneracy and so that Dirichlet boundary conditions are required in (1.1). However, if we had chosen strong type degeneracy, see [1], (1.1) it would be treated with Neumann conditions. In this context, we believe that analogous results can be obtained.
Another interesting question is concerned with global null controlability to (1.1), which does not seem to be simple. Perhaps, this kind of result relies on a global inverse mapping theorem, see [10], but much more refined estimates are necessary.
Under some changes in the Lema 5.1 and following the arguments presented here, Theorem 1.1 can also be obtained if we consider (1.1) with the diffusion coefficients Other important topics arrise from our current research: • It would be very nice to obtain Theorem 1.1 without imposing µ 1 and µ 2 have separated variables. Nevertheless, it is still an open problem. • In the the system (1.1), we can replace each nonlinearity f i (t, x, u, v) by f i (t, x, u, v, u x , v x ), with i ∈ {1, 2}, in order to analyse whether it is possible to prove results about null controllability. • Previously, in [11], we have obtained a local null controlability result for degenarate parabolic equations with nonlocal tems, which implies, throughout standard arguments, a local null boundary controllability result. However, the same fact can not be directly deduced for systems with a reduced number of controls, see [3]. In other words, the boundary controllability of x, u, v) = 0, (t, x) ∈ (0, T ) × (0, 1), u(t, 1) = v(t, 0) = v(t, 1) = 0, u(t, 0) = h(t) t ∈ (0, T ), u(0, x) = u 0 (x) and v(0, x) = v 0 (x), x ∈ (0, 1).
is a very interesting unknown issue.