Superexponential stabilizability of evolution equations of parabolic type via bilinear control

We study the stabilizability of a class of abstract parabolic equations of the form u′(t)+Au(t)+p(t)Bu(t)=0,t≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u'(t)+Au(t)+p(t)Bu(t)=0,\qquad t\ge 0 \end{aligned}$$\end{document}where the control p(·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\cdot )$$\end{document} is a scalar function, A is a self-adjoint operator on a Hilbert space X that satisfies A≥-σI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A\ge -\sigma I$$\end{document}, with σ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma >0$$\end{document}, and B is a bounded linear operator on X. Denoting by {λk}k∈N∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\lambda _k\}_{k\in {\mathbb {N}}^*}$$\end{document} and {φk}j∈N∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\varphi _k\}_{j\in {\mathbb {N}}^*}$$\end{document} the eigenvalues and the eigenfunctions of A, we show that the above system is locally stabilizable to the eigensolutions ψj=e-λjtφj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi _j={\mathrm{e}}^{-\lambda _j t}\varphi _j$$\end{document} with doubly exponential rate of convergence, provided that the associated linearized system is null controllable. Moreover, we give sufficient conditions for the pair {A,B}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{A,B\}$$\end{document} to satisfy such a property, namely a gap condition for A and a rank condition for B in the direction φj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi _j$$\end{document}. We give several applications of our result to different kinds of parabolic equations.


Introduction
In the field of control theory of dynamical systems a huge amount of works is devoted to the study of models in which the control enters as an additive term (boundary or internal locally distributed controls), see, for instance, the books [16], [17] by J.L. Lions.On the other hand, these kinds of control systems are not suitable to describe processes that change their physical characteristics for the presence of the control action.This issue is quite common for the so-called smart materials and in many biomedical, chemical and nuclear chain reactions.Indeed, under the process of catalysis some materials are able to change their principal parameters (see the examples in [14] for more details).
To deal with these situations, an important role in control theory is played by multiplicative controls that appear in the equations as coefficients.
Due to a weaker control action, exact controllability results are not to be expected with multiplicative controls.On the other hand, approximate controllability has been obtained for different types of initial/target conditions.For instance, in [12] the author proved a result of non-negative approximate controllability of the 1D semilinear parabolic equation.In [13], the same author proved approximate and exact null controllability for a bilinear parabolic system with the reaction term satisfying Newton's law.Paper [11] is devoted to the study of global approximate multiplicative controllability for nonlinear degenerate parabolic problems.In [5] and [6], results of approximate controllability of a one dimensional reaction-diffusion equation via multiplicative control and with sign changing data are proved.
An even more specific and weaker class of controls are bilinear controls which enter the equation as scalar functions of time as, for instance, in the following system u (t) + Au(t) + p(t)Bu(t) = 0, t > 0 u(0) = u 0 . ( A fundamental result in control theory for this type of evolution equations is the one due to Ball, Marsden and Slemrod [1] which establishes that the system (1) is not controllable.Indeed, if u(t; p, u 0 ) denotes the unique solution of (1), then the attainable set from u 0 defined by S(u 0 ) = {u(t; p, u 0 ); t ≥ 0, p ∈ L r loc ([0, +∞), R), r > 1} has a dense complement.
On the other hand, when B is unbounded, the possibility of proving a positive controllability result remains open.This idea of exploiting the unboundness of the operator B was developed by Beauchard and Laurent in [3] for the Schrödinger equation iu t (t, x) + u xx (t, x) + p(t)µ(x)u(t, x) = 0, (t, x) ∈ (0, T ) × (0, 1) u(t, 0) = u(t, 1) = 0. ( For such an equation the authors proved the local exact controllability around the ground state in a stronger topology than the natural one of X = H 2 ∩ H 1 0 (0, 1) for which the multiplication operator Bu(t, x) = µ(x)u(t, x) is unbounded.In other terms, the above result could be regarded as a description of the attainable set from an initial submanifold of the original Banach space.
Following the same strategy, Beauchard in [2] studied the wave equation showing that for T > 2 the system is locally controllable in a stronger topology than the natural one for this problem and for which the operator Bu(t, x) = µ(x)u(t, x) is unbounded.In both papers [2] and [3] a key point of the analysis is the application of the inverse mapping theorem which is made possible by the controllability of the linearized problem.This is the reason why, for parabolic problems, the above strategy meets an obstruction: the spaces for which one can prove controllability of the linearized equation are not well-adapted to the use of the inverse map technique.
We recall that some results of approximate controllability of hyperbolic equations with bilinear control have been achieved (see, for instance, [8]).
For other nonlinear equations in fluid dynamics, namely the Navier-Stokes equations with additive controls, the exact controllability to the uncontrolled solution of the equations was shown by Fernández-Cara, Guerrero, Imanuvilov and Puel in [10].
In this paper, we are interested in studying the possibility of steering the solution of (1), with a bilinear control, to a specific uncontrolled trajectory of the equation, namely the ground state solution.
To be more precise, let X be a separable Hilbert space, A : D(A) ⊂ X → X be a self-adjoint accretive operator with compact resolvent (see section 2 for more on notation and assumptions) and let {λ k } k∈N * be the eigenvalues of A, (λ k ≤ λ k+1 , ∀k ∈ N * ), with associated eigenfunctions {ϕ k } k∈N * .Since it is customary to call ϕ 1 the ground state of A, we refer to the function ψ 1 (t) = e −λ 1 t ϕ 1 as the ground state solution.
Our main result (Theorem 3.4 below) ensures that, if {λ k } k∈N * satisfy a suitable gap condition (see condition (13)) and B spreads the ground state in all directions (see condition ( 14)), then system (1) is locally stabilizable to ψ 1 at superexponential rate, that is, one can find a control p ∈ L 2 loc (0, ∞) such that the corresponding solution u(•) of (1) satisfies for suitable constants C, ω > 0.
An important point to underline is that our approach -based on the moment method for the linearized system -is fully constructive.First, we use the gap condition (13) to build a biorthogonal family {σ k (t)} k∈N * to the exponentials e λ k t .Then, we apply such a family to construct a control p(•) that steers the linearized system of (1) exactly to the ground state solution in finite time.Finally, we repeatedly apply such exact controls for the linearized system in order to build a control p(•) for (1) which achieves (3).
We point out that our method applies to both cases λ 1 = 0 and λ 1 > 0, giving an even faster decay rate in the latter case.
The above stabilizability result can be used to study several classes of parabolic problems, for which checking the validity of the assumptions on A and B is usually straightforward.For instance, we can treat the heat equation with a controlled source term of the form with Dirichlet or Neumann boundary conditions, as well as operators with variable coefficients or even 3D problems with radial data symmetry such as Furthermore, we believe that the method we develop in this paper has potentials to be adapted to more general problems, such as a possibly unbounded operator B and a degenerate principal part A.
The outline of the paper is the following.In section 2, we introduce the notation and the preliminary assumptions on the data.Section 3 is devoted to our main result and its proof.Finally, in section 4 we give applications to several examples of parabolic problems.

Preliminaries
Let (X, •, • ) be a separable Hilbert space.We denote by || • || the associated norm on X.Let A : D(A) ⊂ X → X be a densely defined linear operator with the following properties: We recall that under the above assumptions A is a closed operator and D(A) is itself a Hilbert space with the scalar product Moreover, −A is the infinitesimal generator of a strongly continuous semigroup of contractions on X which will be denoted by e −tA .Furthermore, e −tA is analytic.
In view of the above assumptions, there exists an orthonormal basis {ϕ k } k∈N * in X of eigenfunctions of A, that is, ϕ k ∈ D(A) and Aϕ k = λ k ϕ k ∀k ∈ N * , where {λ k } k∈N * ⊂ R denote the corresponding eigenvalues.We recall that λ k ≥ 0, ∀k ∈ N * and we supposewithout loss of generality -that {λ k } k∈N * is ordered so that 0 The associated semigroup has the following representation For any s ≥ 0, we denote by A s : D(A s ) ⊂ X → X the fractional power of A (see [18]).Under our assumptions, such a linear operator is characterized as follows Let T > 0 and consider the problem where u 0 ∈ X and f ∈ L 2 (0, T ; X).We now recall two definitions of solution of problem (7): • the function u ∈ C([0, T ], X) defined by is called the mild solution of (7), • u is a strong solution of (7) in L 2 (0, T ; X) if there exists a sequence {u k } ⊆ H 1 (0, T ; X)∩ L 2 (0, T ; D(A)) such that The well-posedness of the Cauchy problem ( 7) is a classical result (see, for instance, [4]).
Theorem 2.1.Let u 0 ∈ X and f ∈ L 2 (0, T ; X).Under hypothesis (4), problem (7) has a unique strong solution in L 2 (0, T ; X).Moreover u belongs to C([0, T ]; X) and is given by the formula Furthermore, there exists a constant C 0 (T ) > 0 such that and C 0 (T ) is non decreasing with respect to T .
Given T > 0, we consider the bilinear control problem where u is the state variable and p ∈ L 2 (0, T ) is the control function and the bilinear stabilizability problem with p ∈ L 2 loc ([0, +∞)).We recall that, in general, the exact controllability problem for system (10) has a negative answer as shown by Ball, Marsden and Slemrod in [1].
For any j ∈ N * we set ψ j (t) = e −λ j t ϕ j and we call ψ 1 the ground state solution.Observe that ψ j solves (11) with p = 0 and u 0 = ϕ j .We shall study the superexponential stabilizability of (11) to the trajectory ψ 1 .
We observe that if there exists ν > 0 such that Ax, x ≥ ν||x|| 2 , for all x ∈ D(A), then the semigroup generated by −A satisfies If we consider any initial condition u 0 ∈ X, then the evolution of the free dynamics with initial condition u 0 can be represented by the action of the semigroup, u(t) = e −tA u 0 .Therefore, one can prove easily that, when A is strictly accretive, choosing the control p = 0, system (11) is locally exponentially stabilizable to the trajectory ψ 1 .Indeed, and this quantity tends to 0 as t goes to +∞.On the contrary, in the general case of an accretive operator A, we do not have a straightforward choice of p to deduce any stabilizability property of system (11) to the ground state ψ 1 .
The novelty of our work is the construction of a control function p that brings u(t) arbitrary close to ψ 1 (t) in a very short time.Namely, we prove that ( 11) is locally superexponentially stabilizable to the ground state solution.This can be seen as a weak version of the exact controllability to trajectories.
Let B : X → X be a bounded linear operator.From now on we denote by ||Bϕ|| and, without loss of generality, we suppose C B ≥ 1.
We can now state our main result.
Theorem 3.4.Let A : D(A) ⊂ X → X be a densely defined linear operator satisfying hypothesis (4) and suppose that there exists a constant α > 0 such that the eigenvalues of A fulfill the gap condition Let B : X → X be a bounded linear operator and let τ > 0 be such that Then, system (11) is superexponentially stabilizable to ψ 1 .
Moreover, for every ρ > 0 there exists R ρ > 0 such that any u 0 ∈ B Rρ (ϕ 1 ) admits a control p ∈ L 2 loc ([0, +∞)) such that the corresponding solution u(•; u 0 , p) of (11) satisfies where M and ω are positive constants depending only on A and B.
To prove Theorem 3.4 we first start assuming that the first eigenvalue of A is zero, λ 1 = 0, and we prove the local superexponential stabilizability of (11) to the trajectory ϕ 1 .Then, we will recover the general case from this one.
The proof of Theorem 3.4 will be built through a series of propositions.The first result is the well-posedness of the problem We introduce the following notation: , then there exists a unique mild solution of (16), i.e. a function u ∈ C([0, T ]; X) such that the following equality holds in X for every t ∈ [0, T ], Moreover, there exists a constant Hereafter, we denote by C a generic positive constant which may differ from line to line even if the symbol remains the same.Constants which play a specific role will be distinguished by an index i.e., C 0 , C B , . . . .The proof of the existence of the mild solution of ( 16) is given in [1].For what concerns the bound for the solution u of ( 16), it turns out that if C 0 (T )C B ||p|| L 2 (0,T ) ≤ 1/2, then we have inequality (18) with C 1 = C 2 defined by Otherwise, to obtain (18), we proceed subdividing the interval [0, T ] into smaller subintervals for which C 0 (T )C B ||p|| L 2 ≤ 1/2 in all of them, and in this case the constant C 1 of inequality ( 18) is defined by where N is the number of subintervals.
Consider the system and the trajectory ϕ 1 that is a solution of (21) when p = 0, u 0 = ϕ 1 and λ 1 = 0. Set v := u − ϕ 1 , we observe that v is the solution of the following Cauchy problem Remark 3.6.Applying Theorem 2.1, we find that v ∈ C([0, T ]; X) is a mild solution of (22), that is . Therefore, for every ε ∈ (0, T ), v ∈ H 1 (ε, T ; X) and for almost every t ∈ [ε, T ] the following equality holds Showing the stabilizability of the solution u of (21) to the trajectory ϕ 1 is equivalent to proving the stabilizability to 0 of system (22): we have to prove that there exists δ > 0 such that, for every initial condition v 0 that satisfies ||v 0 || ≤ δ, there exists a trajectory-control pair (v, p) such that lim t→+∞ ||v(t)|| = 0.
For this purpose, we consider the following linearized system For this linear system we are able to prove the following null controllability result.
Let us recall the notion of biorthogonal family and a result we will use to show the null controllability of the linearized system (25).Definition 3.8.Let {ζ j } and {σ k } be two sequences in a Hilbert space H.We say that the two families are biorthogonal or that where δ j,k is the Kronecker delta.
The notion of biorthogonal family was used by Fattorini and Russell in [9], where they introduced the moment method.Such a technique was developed later by several authors.We recall below the result proved in [7].Theorem 3.9.Let {ω k } k∈N be an increasing sequence of nonnegative real numbers.Assume that there exists a constant α > 0 such that Then, there exists a family {σ j } j≥0 which is biorthogonal to the family {e Furthermore, there exist two constants C α , C α (T ) > 0 such that Remark 3.10.For all T ∈ R we define the quantity and we observe that if there exists τ > 0 such that (14) holds then, for every T > τ , Λ T < +∞.Furthermore, if λ 1 > 0 then Λ T → 0 as T → +∞.
Thanks to Theorem 3.9 and Remark 3.10 we are able to prove Proposition 3.7: Proof (of Proposition 3.7).For any v 0 ∈ X and p ∈ L 2 (0, T ), it follows from Proposition 3.5 that there exists a unique mild solution v ∈ C 0 ([0, T ], X) of (25) that can be represented by the formula We want to find p ∈ L 2 (0, T ) such that v(T ) = 0, thus the following equality must hold Since {ϕ k } k∈N * is an orthonormal basis of the space X, the equality must hold in every direction and it follows that for every k ∈ N * .Therefore, proving null controllability of the linearized system reduces to finding a function p ∈ L 2 (0, T ) that satisfies for all k ∈ N * .Thanks to assumption (13), there exists α > 0 such that the gap condition λ k+1 − √ λ k ≥ α holds for all k ∈ N * .Then, Theorem 3.9 ensures the existence of a family {σ k } k∈N * that is biorthogonal to {e λ k s } k∈N * .Taking p(s) = k∈N * c k σ k (s) one finds that the coefficients c k are given by c k = v 0 ,ϕ k Bϕ 1 ,ϕ k , ∀k ∈ N * .Thus, in order to show that is a solution of (32), it suffices to prove that the series is convergent in L 2 (0, T ).Indeed, and we appeal to estimate (27) for {σ k } k∈N * , with ω k = λ k for all k ∈ N * , to obtain that that is finite thanks to hypothesis (14) and Remark 3.10.Thus, the following bound for the L 2 -norm of p holds true: In Proposition 3.7 we have found a control p that steers the solution of the linearized system to 0 in time T .We use such a control in the nonlinear system (22) to obtain a uniform estimate for the solution v(t).Proposition 3.11.Let A and B satisfying hypotheses (4), ( 13), ( 14) and furthermore we assume λ 1 = 0. Let p ∈ L 2 (0, T ) be defined by the following formula where {σ k } k∈N * is the biorthogonal family to {e λ k t } k∈N * given by Theorem 3.9.
Then, the solution v of (22) satisfies where C B ≥ 1 is the norm of the operator B, C 3 (T ) := 2 √ T C B C α (T ), and C 4 (T ) := C B C 2 α (T ).Proof.We consider the equation in ( 22).Thanks to Remark 3.6, since (24) is satisfied for almost every t ∈ [ε, T ], we are allowed to take the scalar product with v: Thus, using that B is bounded, we get (37) and therefore, since A is accretive, we have that We integrate the last inequality from ε to t: and by Gronwall's inequality, we obtain and taking the limit ε → 0 we find that Thus, taking the supremum over the interval [0, T ], the last inequality becomes and finally, recalling the estimate (26) for the L 2 -norm of p from Proposition 3.7, we get sup We want now to measure the distance at time T of the solutions of the nonlinear system and the linearized one when using the same control function p built by solving of the moment problem in Proposition 3.7.
Therefore, we introduce the function w(t) := v(t)−v(t) that satisfies the following Cauchy problem We define the constant ). Proposition 3.12.Let A and B satisfy hypotheses (4), ( 13), ( 14), and furthermore we assume λ 1 = 0. Let T > τ , p be defined by (34), and let v 0 ∈ X be such that Then, it holds that ||w(T Proof.Observe that w ∈ C([0, T ]; X) is the mild solution of (40).Moreover w ∈ H 1 (0, T ; X)∩ L 2 (0, T ; D(A)) and thus w satisfies the equality for almost every t ∈ [0, T ].We multiply equation ( 43) by w(t) and we obtain 1 2 Therefore, applying Gronwall's inequality, taking the supremum over [0, T ] and using ( 35) and (26), we get We can suppose, without loss of generality, that C α (T ) ≥ 1.Thus, from (41), we obtain that Λ T ||v 0 || ≤ 1.Therefore, sup Recalling that v(T ) = 0, we deduce from (42) that and, moreover, We observe that we can apply Proposition 3.12 to problem (22) defined in the interval [T, 2T ].Indeed, v T := v(T ) that was computed by solving (22), is the initial condition of the problem We shift this problem to the interval [0, T ] by introducing the variable s := t − T in the above system.If we set ṽ(s) := v(s + T ) and p := p(s + T ), then ṽ solves ṽt (s) + Aṽ(s) + p(s)Bṽ(s) + p(s)Bϕ Here the control p is given by Proposition 3.11, with initial condition v T , that is: where {σ k (s)} k∈N * is the biorthogonal family to {e thanks to the estimate for {σ k (s)} k∈N * given in Theorem 3.9.Therefore, for the control p of the linearized system associated to (49), it holds that Finally, thanks to (48), the hypotheses of Proposition 3.12 for problem (49) are satisfied and we obtain that ||v(2T and we can repeat this argument for the next intervals [2T, 3T ], [3T, 4T ], . . ., [(n−1)T, nT ], . . . .Therefore, we deduce that Now, we want to obtain an estimate as (47) for the solution v of problem (22) defined in time intervals of the form [nT, (n + 1)T ], with n ≥ 1.
Proof of Theorem 3.4.We first consider the case in which the first eigenvalue of A is zero.Let θ ∈ (0, 1) and let ρ > 0 be the value for which θ = e −2ρ .Then, from Proposition 3.14, there exist a constant where M T , ω T > 0 are constants that depend only on T .With the notation of the previous propositions, we have that Now, in order to deal with a general operator A satisfying (4), we introduce the operator We observe that A 1 : D(A 1 ) ⊂ X → X is self-adjoint, accretive and −A 1 generates a strongly continuous analytic semigroup of contraction.Its eigenvalues are given by (in particular, µ 1 = 0) and it has the same eigenfunctions as A, {ϕ k } k∈N * .Moreover, the family {µ k } k∈N * satisfies the same gap condition (13) that is satisfied by the eigenvalues of A. Indeed, it holds that Thus, the operator A 1 satisfies the hypotheses that are required in Theorem 3.4.
We observe that if we introduce the function z(t) = e λ 1 t u(t), where u is the solution of (11), then z solves So, we can apply the previous analysis to this problem and deduce that there exist M T , ω T > 0 such that, for all ρ > 0 there exists R ρ > 0 such that, if We claim that the local superexponetial stabilizability of z to the stationary trajectory ϕ 1 implies the same property of u to the ground state solution ψ 1 .Indeed, it holds that ρe ω T t +λ 1 t) , ∀t ≥ 0 and this concludes the proof also in the case of a strictly accretive operator A. Remark 3.15.Even in the case when A : D(A) ⊆ X → X has a finite number of negative eigenvalues, we can define the operator A 1 := A − λ 1 I.A 1 has nonnegative eigenvalues and we can perform the proof of Theorem 3.4 and deduce the superexponential stabilizability of the solution u of the problem with diffusion operator A to the ground state solution.In this case ψ 1 (t) = e λ 1 t ϕ 1 blows up as t → ∞ since λ 1 < 0, and the same occurs for the controlled solution u.

Applications
In this section we discuss examples of bilinear control systems to which we can apply Theorem 3.4.The first problems we study are 1D parabolic equations of the form in the state space X = L 2 (0, 1), with Dirichlet or Neumann boundary conditions and with B the following multiplication operators: Then, we prove the superexponential stabilizability of the following one dimensional equation with variable coefficients u t (t, x) − ((1 + x) 2 u x (t, x)) x + p(t)Bu(t, x) = 0 with Dirichlet boundary condition.
Finally, we apply Theorem 3.4 to the following parabolic equation for radial data in the 3D unit ball B 3 .
In each example, we will denote by {λ k } k∈N * and {ϕ k } k∈N * , respectively, the eigenvalues and eigenfunctions of the second order operator associated with the problem under investigation.We will take (ū, p) = (ψ 1 , 0) as reference trajectory-control pair, where ψ 1 = e −λ 1 t ϕ 1 is the solution of the uncontrolled problem with initial condition u(0, x) = ϕ 1 .

Dirichlet boundary conditions.
Let Ω = (0, 1), X = L 2 (Ω) and consider the problem where p ∈ L 2 (0, T ) is the control function, u the state variable, and µ is a function in H 3 (Ω).We denote by A the operator defined by A satisfies all the properties in (4): in particular, it is strictly accretive and its eigenvalues and eigenvectors have the following explicit expressions It is straightforward to prove that the eigenvalues fulfill the required gap property.Indeed, So, (13) is satisfied.
In order to apply Theorem 3.4 to system (73) and deduce the superexponential stabilizability to the trajectory ψ 1 , we need to prove that there exists τ > 0 such that: For this purpose, let us compute the scalar product Observe that the last integral term above represents the k th -Fourier coefficient of the integrable function (µ(x)ϕ 1 (x)) and thus, it converges to zero as k goes to infinity.Therefore, if we assume µ (1) ± µ (0) = 0 and then, we deduce that µϕ 1 , ϕ k is of order 1/k 3 as k → ∞.
Remark 4.1.An example of a function which satisfies (75) is µ(x) = x 2 .Indeed, in this case 2 , k = 1 and so x 2 ϕ 1 , ϕ k = 0 for all k ∈ N * and furthermore We conclude that, under assumption (75), and thanks to the polynomial behavior of the bound, the series Therefore, all the hypotheses of Theorem 3.4 are satisfied and system (73) is superexponentially stabilizable to the trajectory ψ 1 .
Remark 4.2.Assumption (76) for problem (73) is not too restrictive.In fact, it is possible to prove that the set of functions in H 3 (Ω) for which (76) holds is dense in H 3 (Ω).For a proof of this fact, see Appendix A in [3].

Dirichlet boundary conditions, variable coefficients
In this example, we analyze the superexponential stabilizability of a parabolic equation in divergence form with nonconstant coefficients in the second order term.
The gap condition holds true because , ∀k ∈ N * .Now, we check the hypotheses on the operator Bϕ = µϕ needed to apply Theorem 3.4.We recall that we want to prove that: • Bϕ 1 , ϕ k = 0, for all k ∈ N * , • there exists τ > 0 such that the series , for all k ∈ N * .Thus, series (80) is finite for all τ > 0. This concludes the verification of the hypotheses of Theorem 3.4, that imply the superexponential stabilizability of (79) to ψ 1 .