LOCAL NULL CONTROLLABILITY OF A QUASI-LINEAR SYSTEM AND RELATED NUMERICAL EXPERIMENTS

. This paper concerns the null control of quasi-linear parabolic systems where the diﬀusion coeﬃcient depends on the gradient of the state variable. In our main theoretical result, with some assumptions on the regularity and growth of the diﬀusion coeﬃcient and regular initial data, we prove that local null controllability holds. To this purpose, we consider the null controllability problem for the linearized system, we deduce new estimates on the control and the state and, then, we apply a Local Inversion Theorem. We also formulate an iterative algorithm of the quasi-Newton kind for the computation of a null control and an associated state. We apply this method to some numerical approximations of the problem and illustrate the results with several experiments


Introduction and main results
Let d be an integer with 1 ≤ d ≤ 3 and let Ω ⊂ R d be a non-empty open bounded connected set, with smooth boundary ∂Ω. For a finite time horizon T > 0, we introduce Q :=]0, T [×Ω and Σ :=]0, T [×∂Ω. We will deal with systems where the control acts on a set of the form ]0, T [×ω, where ω ⊂⊂ Ω is another non-empty open set. More precisely, the following nonlinear parabolic system will be considered: y t − ∇ · [a(∇y)∇y] = χ ω v in Q, y = 0 on Σ, y| t=0 = y 0 in Ω.
(1.1) y 0 ∈ H 5 (Ω) ∩ H 1 0 (Ω) (1.3) and the compatibility conditions ∆y 0 , ∆ 2 y 0 ∈ H 1 0 (Ω) (1.4) and is sufficiently small in the sense that To the best of our knowledge, [15] and the present paper are the first works dealing with the null controllability of quasi-linear parabolic PDE's where the diffusion coefficient can be, among other possibilities, a general power law.
Our main result is the following: such that (1.6) holds. Furthermore, given a non-empty open set γ ⊂ ∂Ω, it is possible to select h such that For the proof of Theorem 1.3, we will rewrite the null controllability problem in the form H(y, v) = (0, y 0 ), (y, v) ∈ Y, (1.8) where the Hilbert space Y of state-control pairs (y, v) will be chosen appropriately (in particular, to any couple (y, v) ∈ Y we will require to satisfy y(T , ·) = 0) and H : Y → Z will be an adequate nonlinear mapping (Z is another Hilbert space); actually, we will take H(y, v) = (y t − ∇ · [a(∇y)∇y] − χ ω v, y| t=0 ) for all (y, v). Among other things, we prove that H is well-defined and strictly differentiable at (0, 0), with a surjective derivative H (0, 0) ∈ L(Y ; Z). Accordingly, it will be possible to apply a local inversion theorem to (1.8) and deduce that, for any "small" y 0 , we can steer (1.1) to zero.
Our original contribution in this context is to provide good definitions of the spaces Y and Z, as well as of the mapping H. We must harmonize conflicting relations when specifying Y and Z: indeed, they must be sufficiently small to guarantee the correct definition and needed regularity of H and, also, sufficiently large to ensure that the solutions to the linearized problem belong to Y . Furthermore, we can only establish the well-posedness and differentiability of H after several new weighted energy estimates for the solutions to the linearized system.
Note that it seems difficult to assert the global null controllability of (1.1). We could try to apply a global inversion argument, like for instance in [35], based on the existence and boundedness of H (ȳ,v) for all (ȳ,v). However, this requires in practice the resolution of null controllability problems for systems of the form      y t − ∇ · [a(∇ȳ)∇y + (a (∇ȳ) · ∇y)∇ȳ] = χ ω v in Q, y = 0 on Σ, y| t=0 = y 0 in Ω,

5
with good estimates of v and y, uniform with respecto toȳ. And this is an open and probably difficult problem.
We could also try to apply a fixed-point argument, that is, search for a fixed-point of the mappingỹ → y, where y is now, together with some v, a solution to the null controllability problem for      y t − ∇ · [a(∇ỹ)∇y] = χ ω v in Q, y = 0 on Σ, y| t=0 = y 0 in Ω, assuming that this mapping can be well defined. But, unfortunately, it is difficult to find (and not clear at all) a space where this can be done.
In the second part of the paper, we will be concerned with the computation of null controls for (1.1). In the literature on the subject, the numerical controllability of linear and nonlinear PDE's has been considered in many papers; for instance, see [6-8, 20, 34, 43] and the references therein. Here, we will argue as in [13,26], taking advantage of the surjectivity of H (0, 0).
end return (y n+1 , v n+1 ) In these iterates, we use H (0, 0) −1 , which is by definition an inverse to the left of H (0, 0). The precise definition will be given in Section 3.2.
We remark that ALG 1 is an elementary quasi-Newton method; below, it will be shown to converge in the space Y . We will analyze the convergence and the computational cost of ALG 1 in Section 4 and we will present several numerical experiments to illustrate our results.
The plan of the paper is the following: Study of the linearized system. In Section 2, we carry out a detailed study of the linearization of (1.1) around zero. We recall Fursikov-Imanuvilov's method and solve the corresponding null controllability problem. We emply techniques that make it possible to find controls with high regularity, as long as the initial state is sufficiently smooth. Application of a Local Inversion Theorem. In Section 3, with the help of an adequate functionalanalytic setting, the controllability problem is written in the form (1.8). At this point, we need higher order estimates for the solutions to non-homogeneous linearized problems; as already mentioned, this is one of the main contributions of the paper. Then, we find that the relevant mapping is well-defined and differentiable (in an appropriate sense), whence we can apply a local inversion result, taken from [2]. This yields the desired result.
Computation of null controls. We present in Section 5 some numerical methods for the computation of a solution to the null controllability problem for (1.1). The results of some numerical experiments in spatial dimensions d = 1 and d = 2 are also given.
Comments and concluding remarks. We summarize the achievements in Section 5. There, we have also included several additional comments and pertinent open questions.
In principle, throughout this paper, we regard all derivatives in the distributional sense. For any d-dimensional multi-index β, the symbol D β stands for the corresponding spatial differential operator; explicitly, We will use · to denote the standard norm in L 2 (Ω). In general, for any other Banach space E, the corresponding norm will be denoted by · E . The symbol C will stand for a generic positive constant, depending on Ω, ω, T and the other data appearing in hypotheses H1 and H2. For any given nonnegative integer k and any function g, whenever it makes sense, we set Finally, the integration elements dt, dx, etc. will be usually omitted.

Study of the linearized problem
The linear system considered in this section is the following: where σ is a positive constant. We must also consider the corresponding adjoint system, given by It is well-known that the null controllability of (2.1), together with an estimate of the control, is equivalent to the observability of (2.2).

LOCAL NULL CONTROLLABILITY OF A QUASI-LINEAR SYSTEM AND RELATED NUMERICAL EXPERIMENTS
and let us set where R > α 0 L ∞ + log(2) and λ > 0. We will need in the sequel a global Carleman inequality, where we estimate the quantity The result is the following: There exist positive constants λ 0 , s 0 and C 0 , depending only on Ω, ω 0 , σ and T , such that for any s ≥ s 0 , λ ≥ λ 0 , F ∈ L 2 (Q) and ϕ T ∈ L 2 (Ω), the solution to (2.2) corresponding to the data F and ϕ T satisfies The proof is given in [13]. From now on, we fix s = s 0 and λ = λ 0 .
We will assume that R is sufficiently large to have 2α 1 ≥ α 2 . We note that this also implies (r + 1)α 1 ≥ α 2 , since r ≥ 1 (cf. H2 in Sect. 1). Thus, we have: The following weights will be needed: Theorem 2.3. Let us assume that ρ 3 f ∈ L 2 (Q) and y 0 ∈ L 2 (Ω). There exists a control v ∈ L 2 (]0, T [×ω) which, together with the solution to (2.1) corresponding to v, f and y 0 , satisfies: In particular, v is a control that drives the solution to (2.1) from y 0 exactly to zero at time T . Furthermore, we can choose v satisfying together with the estimates This result is well-known. The proof of the first part is due to Fursikov and Imanuvilov [28]. For the proof of (2.7), see for instance [22].
The proof of this result is given in Appendix A (Appendix A).
3. The local null controllability of (1.1) In this section, we will rewrite the local null controllability of (1.1) as a nonlinear equation in a Hilbert space. Then, we will apply an "abstract" local inversion result. This will lead to the controllability result in Theorem 1.3.

A general local inversion theorem
In this section, Y and Z will stand for generic Hilbert spaces. We will denote the closed ball centered at 0 of radius r in Y (resp. Z) by B Y (0; r) (resp. B Z (0; r)).
We begin by recalling the definition of strict differentiability: If this is the case, then Λ is unique and is usually denoted by H (0).
We can find a proof of this result in [2]. The particular statement we give here is similar to the one in [11]. Obviously, we must view the mapping W as an inverse to the right of H.

Proof of the local null controllability
Let us introduce the weighted L 2 -spaces H k := L 2 (Q, µ k ). They will be of help in the definition of the particular spaces Y and Z needed in the proof of Theorem 1.3. Here, µ k is the absolutely continuous measure in Q whose Radon-Nikodým derivative is ρ 2 k , that is, with integration element dµ k = ρ 2 k d(t, x). Thus, a (class of) function(s) v : Q → R belongs to H k if and only if it is measurable and The space H k is endowed with the natural norm We will set This is a Hilbert space for the norm where, as before, we have set f := y t − a(0)∆y − χ ω v and we take the norms of v and its derivatives over ]0, T [×ω.
In order to define the space Z, let us introduce and the Hilbertian norm We will set with the corresponding Hilbert product structure and we will consider the nonlinear mapping H : In order to apply Theorem 3.2 in this setting, we must first establish some auxiliary results whose proofs are respectively given in Appendices B, C and D: is well-defined and continuous.
Lemma 3.4. The linear mapping Λ : Y → Z, given by is well-defined, continuous and onto. Furthermore, there exists a constant M > 0 such that Lemma 3.5. The mapping H is strictly differentiable at (0, 0) ∈ Y and H (0, 0) = Λ.

Proof of Theorem 1.3
In view of Lemmas 3.3 and 3.5, it is possible to apply Theorem 3.2 (for instance, with = 1/(2M )) and conclude that there exist δ > 0 and W : It is evident that (3.9) and the fact that (y, v) ∈ Y imply that (y, v) solves (1.1) and (1.6), since the weights ρ k blow up to infinity exponentially as t → T .

Proof of Corollary 1.4
Let γ be given and letΩ be a bounded connected open set with a smooth boundary such that Ω ⊂Ω, ∂Ω ∩ ∂Ω = γ andω :=Ω\Ω is a non-empty open set. Let us set There exists a linear continuous extension mapping E : Then, we can apply Theorem 1.3 to the system whence we deduce that there exists η > 0 such that, if ŷ 0 H 5 (Ω) ≤ η, (3.10) admits a control that drives the state exactly to zero at t = T . Consequently, if y 0 satisfies y 0 H 5 ≤ δ := η/Ĉ and y and h are respectively given by the restriction to Q and the trace on ]0, T [×γ of the correspondingŷ, we find that (y, h) is a state-control pair that solves the considered boundary control problem.

The convergence of ALG 1, approximations and experiments
As we already explained in Section 1, arguing as in [13,26], we can introduce an elementary quasi-Newton algorithm for the computation of a solution to the null control problem. To this purpose, we previously have to specify the inverse to the right of H (0, 0). We can do this with the help of the Fursikov-Imanuvilov method [28].
The argument is the following. For any (f, y 0 ) ∈ Z, we can obtain a solution to (2.1) in Y by solving the following extremal problem: Recall that the definition of the weights ρ k is given in (2.1).

13
It is well-known that (4.1) possesses exactly one solution, given by where p is the unique solution to the variational equality (recall the notation introduced in the proof of Thm. 1.3). The existence and uniqueness of a solution to (4.3) is an immediate consequence of Lax-Migram Theorem; see the details in [28]. Accordingly, we set H (0, 0) −1 (f, y 0 ) = (y, v), with y and v respectively given by (4.2)-(4.3). Note that, in ALG 1, for each n the task reduces to solve a problem of the kind (4.3), with The local convergence of ALG 1 is guaranteed by the following result: where ε is the small quantity furnished by Theorem 1.3 ensuring the null controllability of (1.1). There exists then we get linear rate convergence to (y, v) in Y for the sequence {(y n , v n )} n .
The proof of this result is very similar to the proof of Theorem 1.2 in [22]. For brevity, it is omitted.
In order to solve numerically the problems (4.3), a first idea is to construct adequate finite dimensional spaces P h ⊂ P and search for functions p h ∈ P h satisfying the equalities in (4.3) for all w ∈ P h . This is relatively easy when d = 1. However, if d ≥ 2, this is not so simple. The reason is that the functions in P must satisfy L * p ∈ L 2 loc (Q) and, consequently, an approximation based on a standard mesh of Q requires spaces P h of functions that must be globally C 0 in all the variables and globally C 1 in space. Accordingly, for d ≥ 2, it seems convenient to use a mixed formulation, as in [25,26].
For completeness, we have performed numerical experiments for d = 1 with approximations of both kinds (based on subspaces P h and also based on mixed formulations). For d = 2, our tests only concern mixed finite elements.
The details are given in [24,25]; see also [23]. Let N x and N t be two (large) positive integrals and set ∆x = L/N x , ∆t = T /N t and h = (∆x, ∆t). Then, we replace in (4.3) the space P by the finite diensional subspace P h formed by the functions p h ∈ C 0 (Q) such that p h,x ∈ C 0 (Q), the restriction to each rectangle is polynomial of degree ≤ 3 in x and degree ≤ 1 in t and vanish on Σ.

A direct approximation of
This approach was introduced in [24]. A complete analysis of the convergence of the resulting approximations was given there. In order to get a formulation where the finite dimensional spaces are more tractable, we can follow a different process: 1. First, we introduce new variables (z, m) and we rewrite (4.3) accordingly. Indeed, since the weights ρ and ρ 3 blow up as t → T , in practice it is convenient to work with the new variables z = ρ −1 L * p and m = ρ −1 3 p. We make use of the equality z = ρ −1 L * (ρ 3 m) to introduce a multiplier and get a mixed formulation. 2. Then, we integrate by parts and rewrite the formulation in such a way that only first order partial derivatives remain. 3. Finally, we replace the resulting mixed formulation by a finite dimensional approximation using standard Lagrange (C 0 in time and space) finite elements.
This mixed formulation was also introduced in [24] and then reconsidered and adapted to several situations on other papers, see for instance [25].
In the following sections, we will provide the results of some experiments.

Test 1.1
We applied the quasi-Newton method to the solution to the null controllability problem for (1.1) with the following data: At each step of ALG 1, we wrote the problem in the form (4.3). This first test corresponds to experiments performed by direct approximation (Sect. 4.1) with different values of h = (∆x, ∆t). The computations and visualizations have been carried out with the help of the MatLab Library [40].
In this and the other tests, we fix the stopping criterion where y n and y n+1 are the computed states respectively at the n-th and (n + 1)-th steps and κ = 10 −5 . The evolution of the relative errors of the computed solutions for various h is given in Table 1. There, we have assigned the role of "exact" to the solution computed for ∆x = 0.0167 and ∆t = 0.083, which corresponds to a mesh with 3600 points.
The computed state and control for ∆x = 0.02 and ∆t = 0.01 are depicted in Figure 1. The evolution in time of the associated L 2 norms is given in Figure 2.  The computed state and control for ∆x = 0.02 and ∆t = 0.01 are depicted in Fig. 1. The evolution in time of the associated L 2 norms is given in Fig. 2.

Test 1.2
We take the same data except for the initial state and the diffusion coefficient. This time, • y 0 (x) ≡ 0.15 sin(πx).
Again, we approximate (4.3) as in Section 4.1 with the same values of h = (∆x, ∆t). The computed state and control and the evolution in time of the corresponding norms is now given in Fig. 3-4.
A comparison of these results with those in Test 1.1 shows that, although the initial state is larger, the norm of the computed control is smaller. This seems to be the effect of a monotone diffusion coefficient. However, this should be investigated more in detail.

Test 1.2
We take the same data except for the initial state and the diffusion coefficient. This time, A comparison of these results with those in Test 1.1 shows that, although the initial state is larger, the norm of the computed control is smaller. This seems to be the effect of a monotone diffusion coefficient. However, this should be investigated more in detail.
Again, we approximate (4.3) as in Section 4.1 with the same values of h = (∆x, ∆t). The computed state and control and the evolution in time of the corresponding norms is now given in Fig. 3-4.
A comparison of these results with those in Test 1.1 shows that, although the initial state is larger, the norm of the computed control is smaller. This seems to be the effect of a monotone diffusion coefficient. However, this should be investigated more in detail.

Test 1.3
In this test, we take again the data in Test 1.1. At each step, we compute a numerical approximation of the corresponding problem (4.3) using piecewise linear and quadratic mixed finite elements (Section 4.2).
We have performed the computations with the FreeFem++ package, with a mesh adaptation technique. This means that, at each quasi-Newton iterate, the mesh is not regular, but adapted to the behavior of the last computed solution. More precisely, the points are distributed in the space-time domain in accordance with a metric that corresponds to the values of the computed (approximate) Hessian of p n ; related explanations are given in [30]. This technique was already used, for instance, in reference [27] for the solution of another null controllability problem; see Figure 2 there.
For a detailed description of FreeFem++, see [31,47]. The stopping criterion is (4.5), where the notation is self-explanatory. We have started from a regular mesh with 970 vertices and 1844 triangles. The final mesh is displayed in Fig. 5. The computed state and control are displayed in Fig. 6.
They have been compared to the most accurate results in Test 1.1 in Table 2, with excellent agreement. Also, in order to check that the computed control does the work, we have solved numerically the nonlinear problem (1.1) with this control in the right hand side. To this end, we have used the MatLab tool pdepe, that uses second order finite difference approximation in space and a variable-step, fifth-order time discretization scheme based on numerical differentiation, see [48]. The state found this way is depicted in Fig. 7 and is in practice identical to the solution to the test. This shows that the method described in Section 4.2 certainly

Test 1.3
In this test, we take again the data in Test 1.1. At each step, we compute a numerical approximation of the corresponding problem (4.3) using piecewise linear and quadratic mixed finite elements (Sect. 4.2).
We have performed the computations with the FreeFem++ package, with a mesh adaptation technique. This means that, at each quasi-Newton iterate, the mesh is not regular, but adapted to the behavior of the last computed solution. More precisely, the points are distributed in the space-time domain in accordance with a metric that corresponds to the values of the computed (approximate) Hessian of p n ; related explanations are given in [29]. This technique was already used, for instance, in reference [27] for the solution of another null controllability problem; see Figure 2 there.
Again, the solution has been computed for several mesh sizes h = (∆x 1 , ∆x 2 , ∆t). We show the cylinder Q and a mesh in Fig. 8. Several views of the computed state and control are displayed in Fig. 9-11. Also, the evolution in time of the L 2 norms of the state and the control is shown in Fig. 12.
Starting again from (y 0 , v 0 ) = H (0, 0) −1 (0, y 0 ), we have used ALG 1 to compute the solution to the null control problem. The evolution of the relative errors of the computed solutions for several mesh sizes is given in Table 5; again, we have assumed that the "exact" solution corresponds to the finest mesh.
The number of iterates needed to get convergence is indicated in Table 6 for several mesh sizes. Finally, we give in Table 7 this number for y 0 ≡ y 00 sin(πx 1 ) sin(2πx 2 ) and several choices of y 00 when the finest mesh is used.
Again, the solution has been computed for several mesh sizes h = (∆x 1 , ∆x 2 , ∆t). We show the cylinder Q and a mesh in Fig. 8. Several views of the computed state and control are displayed in Fig. 9-11. Also, the evolution in time of the L 2 norms of the state and the control is shown in Fig. 12.
Starting again from (y 0 , v 0 ) = H (0, 0) −1 (0, y 0 ), we have used ALG 1 to compute the solution to the null control problem. The evolution of the relative errors of the computed solutions for several mesh sizes is given in Table 5; again, we have assumed that the "exact" solution corresponds to the finest mesh.
The number of iterates needed to get convergence is indicated in Table 6 for several mesh sizes. Finally, we give in Table 7 this number for y 0 ≡ y 00 sin(πx 1 ) sin(2πx 2 ) and several choices of y 00 when the finest mesh is used.  The stopping criterion is (4.5), where the notation is self-explanatory. We have started from a regular mesh with 970 vertices and 1844 triangles. The final mesh is displayed in Figure 5. The computed state and control are displayed in Figure 6.
They have been compared to the most accurate results in Test 1.1 in Table 2, with excellent agreement. Also, in order to check that the computed control does the work, we have solved numerically the nonlinear problem (1.1) with this control in the right hand side. To this end, we have used the MatLab tool pdepe, that uses second order finite difference approximation in space and a variable-step, fifth-order time discretization scheme based on numerical differentiation, see [48]. The state found this way is depicted in Figure 7 and is in practice identical to the solution to the test. This shows that the method described in Section 4.2 certainly furnishes a null control.   Table 3: Tests 1.3 -Iterates for convergence vs. initial data size. The first and third (resp. second and fourth) columns indicate the numbers of iterates needed to satisfy the stopping test for κ = 10 −5 (resp. κ = 10 −9 ). Convergence fails for y 00 ≥ 2.06.

Test 2.2
In this last Test, we take the same data as in Test 2.1, except for the diffusion coefficient. Now, • a(s) ≡ 1 + 0.7|s| 2 .
We can obtain similar numerical results by applying ALG 1 in combination with a mixed finite element approximation. The computations lead to Tables 8 and 9. In the framework of this test, we have studied how large the initial data can be to get convergence for ALG 1. Specifically, we have taken y 0 (x) ≡ y 00 sin(πx), with several y 00 .
The amount of iterates needed to ensure (4.5) can be found in Table 3. Convergence fails for y 00 ≥ 2.06; it is reasonable to expect that this corresponds to initial data too far from zero to support ALG 1 and, maybe, a continuation algorithm is in order. 1 We have also compared ALG 1 and the usual Newton method, for which the iterates are given by In practice, this means that, at each step, we must solve a problem like (4.3) with a different π(· , ·) and f is as in (4.4). The results of this comparison are given in Table 4. It is readily seen that ALG 1 is clearly preferable both in terms of CPU execution time and similar in terms of convergence rate.

Test 2.1
The data in this Test are the following: Table 3. Tests 1.3 -Iterates for convergence vs. initial data size. The first and third (resp. second and fourth) columns indicate the numbers of iterates needed to satisfy the stopping test for κ = 10 −5 (resp. κ = 10 −9 ). Convergence fails for y 00 ≥ 2.06.
Again, the solution has been computed for several mesh sizes h = (∆x 1 , ∆x 2 , ∆t). We show the cylinder Q and a mesh in Figure 8. Several views of the computed state and control are displayed in Figures 9-11. Also, the evolution in time of the L 2 norms of the state and the control is shown in Figure 12.
It is observed that ALG 1 and Newton algorithm are similar in convergence rate but ALG 1 is much less expensive. An explanation is that the domain of convergence of ALG 1 is larger and increasing initial data slow down Newton iterates before. The CPU time is given in seconds.     Starting again from (y 0 , v 0 ) = H (0, 0) −1 (0, y 0 ), we have used ALG 1 to compute the solution to the null control problem. The evolution of the relative errors of the computed solutions for several mesh sizes is given in Table 5; again, we have assumed that the "exact" solution corresponds to the finest mesh.
The number of iterates needed to get convergence is indicated in Table 6 for several mesh sizes. Finally, we give in Table 7 this number for y 0 ≡ y 00 sin(πx 1 ) sin(2πx 2 ) and several choices of y 00 when the finest mesh is used.

Test 2.2
In this last Test, we take the same data as in Test 2.1, except for the diffusion coefficient. Now, a(s) ≡ 1 + 0.7|s| 2 .

Conclusions, further remarks and open questions
We have established the local null controllability property of the quasi-linear parabolic system (1.1), at any given positive time we prescribe, both with distributed and boundary controls. We found that a general method, which is quite standard nowadays -see [22,15,38,23], to name a few -is also successful in the present context.   We can obtain similar numerical results by applying ALG 1 in combination with a mixed finite element approximation. The computations lead to Tables 8 and 9.

Conclusions, further remarks and open questions
We have established the local null controllability property of the quasi-linear parabolic system (1.1), at any given positive time we prescribe, both with distributed and boundary controls. We found that a general method, which is quite standard nowadays -see [13,21,22,37], to name a few -is also successful in the present context.
In what concerns theoretical control, the main novelty in this paper is the proof of new nontrivial regularity results of the control (and the state), as well as the boot-strapping and verification arguments -the aspects that make it possible to use these techniques.
Indeed, the local inversion argument needed to prove Theorem 1.3 only works after introducing the spaces Y and Z in (3.1)-(3.6) and checking that the mappring H in (3.7) is well-defined and C 1 and H (0, 0) is onto.   By inspection, we see that the proof in Section 3 can be adapted to deal with the system where A : R × R d → R d×d and f : R × R d → R satisfy the following assumptions: H1 A is of class C 4 and f is of class C 3 . H2 There exist C, a 0 > 0 and r, s, r , s ≥ 1 such that for all (u, p, ξ) ∈ R × R d × R d , all multi-indices β, σ subject to the constraints |β| ≤ 4 and |σ| ≤ 3, and all integers α, γ satisfying |α| ≤ 4 and |γ| ≤ 3. Above, ∂ 1 denotes the partial derivative with respect to the one dimensional variable u, while D 2 refers to derivatives with respect to the coordinates of p ∈ R d .
Note in particular that the state norms displayed in Fig. 16 evolve in a clearly different way in the linear and quasi-linear cases.   • y 0 (x) ≡ sin(πx).
Note in particular that the state norms displayed in Fig. 16 evolve in a clearly different way in the linear and quasi-linear cases.   where the assumptions on f remain the same but less regularity on A is required. More precisely, it suffices to assume that A is of class C 3 and satisfies power law limitations, like in H2 or H2 , but now up to derivatives of order three.
Another natural question is whether Theorem 1.3 still holds for similar systems with PDE's of the form y t − ∇ · (a(t, x; ∇y)∇y) = v1 ω , (5.1) that is, with a nonlinear diffusion coefficient non-homogeneous in space or time. The answer is affirmative as long as we suppose enough regularity for a : Q × R d → R. It is also worthwhile to note that the local exact controllability to the (smooth) trajectories is an interesting additional topic. After a change of variable, this can be reduced to the null controllability of a system where the PDE is similar to (5.1) (see some related comments in Sect. 5). For brevity, we will not delve into the technical aspects of this extension. It is possible to demonstrate local null controllability properties for multi-phase quasi-linear parabolic systems, assuming uniform ellipticity of the diffusion coefficients and that all phases are to be controlled. For two-phase  Finally, recall that there are many open problems concerning the null control of semilinear and nonlinear systems. In particular, as already mentioned at the beginning of the paper, it is unknown whether the global controllability property holds for (1.1). With the notations introduced after the statement of Theorem 2.3, we set

Appendix
where k ≥ 1 is an integer that will be fixed below. Therefore, we have L * (ρ kvt ) = L * A 1 + L * A 2 + L * A 3 . It is clear that 24 Figure 15. Additional Test -The computed states corresponding the linear case (left) and the quasi-linear case (right).  Finally, recall that there are many open problems concerning the null control of semilinear and nonlinear systems. In particular, as already mentioned at the beginning of the paper, it is unknown whether the global controllability property holds for (1.1). With the notations introduced after the statement of Theorem 2.3, we set ρ kvt = 2sαe −sα/ −5+k/2 l p − 3e −sα/ −4+k/ ϕ + e −sα/ −3+k/2 p t =:

Appendix
where k ≥ 1 is an integer that will be fixed below. Therefore, we have L * (ρ kvt ) = L * A 1 + L * A 2 + L * A 3 . It is clear that 24 Figure 16. Additional Test -Evolution in time of the L 2 norms of the computed states corresponding the linear case (left) and the quasi-linear case (right).
systems, it is also straightforward to treat the case of zero-order coupling with one distributed internal control force. For more details and some related results, see for instance [3,33]. The techniques developed here may serve to address more general models of chemotaxis, as in [31], possibly leading to results similar to those in [11,12]. In [15], the authors showed that the present method also works in addressing the local null controllability of a generalization of the Navier-Stokes equations.
Regarding the limitations of the present method, the drawbacks are the dimensional constraint d ≤ 3 and the higher regularity demanded to the initial data. In both cases, the effect of nonlinearity is observed. To make this assertion more convincing, we have depicted in Figures 13-16  Note in particular that the state norms displayed in Figure 16 evolve in a clearly different way in the linear and quasi-linear cases.
Finally, recall that there are many open problems concerning the null control of semilinear and nonlinear systems. In particular, as already mentioned at the beginning of the paper, it is unknown whether the global controllability property holds for (1. With the notations introduced after the statement of Theorem 2.3, we set where k ≥ 1 is an integer that will be fixed below. Therefore, we have L * (ρ kvt ) = L * A 1 + L * A 2 + L * A 3 . It is clear that For k ≥ 11, it follows that L * A 1 + L * A 2 ∈ L 2 (Q), with On the other hand, L * p t = (e 2sα/ y) t = e 2sα/ y t − 2sαe 2sα/ −2 y and, consequently, From the estimate (2.9), we can conclude that ρ −1 −7 L * p t ∈ L 2 (Q). Moreover, from the Carleman inequality (2.3), we have ρ −1 −1 p t ∈ L 2 (Q). Finally, we observe that ρ −1 −7 L * p t ∈ L 2 (Q) and Hence, for k ≥ 15, it is true that L * A 3 ∈ L 2 (Q) and Recalling that ρ 15 v = 0 on Σ and (ρ 15v )(T , ·) = 0 in Ω, we conclude that v = χv| ]0,T [×ω satisfies (ρ 15 v t ) t ∈ L 2 (]0, T [×ω) and ρ 15 v t ∈ L 2 (0, T ; H 2 (ω)) and the corresponding estimates are fulfilled.
Let us establish the estimates (d). Henceforth, we fix n ≥ 1, and we let v n , f n and y 0n denote the projections of v, f and y 0 on the space spanned by the first n eigenfunctions of the Dirichlet Laplacian in Ω. Then, we let y n denote the corresponding solution to (2.1).
For the sake of brevity, we drop in the next computations the indices n in v n , f n , y 0n and y n . The constants C arising in the estimates below are independent of n.
We divide the proof into four parts.
This ends the proof.

Appendix B. Proof of Lemma 3.3
We must prove that H takes values in Z. Obviously, it suffices to prove that the first component of H(y, v) belongs to F .
From the definitions of Y and the norm of F , it is clear that R(y, v) ∈ F . Recall the notation H k = L 2 (Q, µ k ), where dµ k = ρ 2 k d(t, x).
Remark B.1. The main reason why we have to impose d 3 is that, in these estimates, we need the following results from Sobolev space theory: (a) H 1 (Ω) → L 6 (Ω), with a dense and continuous embedding. (b) H 2 (Ω) → L ∞ (Ω), with a dense and compact embedding.
For d ≤ 3, they are well known and can be easily deduced from the general results presented in [1].
Using hypothesis H2 and then the estimates in Corollary 2.4, we can deal with all the terms E 1 , ..., E 8 . Thus, let us show what can be done with E 5 ; the other E j can be bounded likewise.
For example, and we can obtain very similar inequalities for the other E 5,i . As a final consequence, we get (D.5) and the proof is done.