Exact controllability to eigensolutions for evolution equations of parabolic type via bilinear control

In a separable Hilbert space X, we study the controlled evolution equation u′(t)+Au(t)+p(t)Bu(t)=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u'(t)+Au(t)+p(t)Bu(t)=0, \end{aligned}$$\end{document}where 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} (σ≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma \ge 0$$\end{document}) is a self-adjoint linear operator, B is a bounded linear operator on X, and p∈Lloc2(0,+∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in L^2_{loc}(0,+\infty )$$\end{document} is a bilinear control. We give sufficient conditions in order for the above nonlinear control system to be locally controllable to the jth eigensolution for any j≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j\ge 1$$\end{document}. We also derive semi-global controllability results in large time and discuss applications to parabolic equations in low space dimension. Our method is constructive and all the constants involved in the main results can be explicitly computed.


Introduction
In a separable Hilbert space X consider the nonlinear control system    u ′ (t) + Au(t) + p(t)Bu(t) = 0, t > 0 u(0) = u 0 . (1.1) where A : D(A) ⊂ X → X is a linear self-adjoint operator on X such that A ≥ −σI, with σ ≥ 0, B belongs to L(X), the space of all bounded linear operators on X, and p(t) is a scalar function representing a bilinear control.We suppose that the spectrum of A consists of a sequence of real numbers {λ k } k∈N * which can be ordered, whithout loss of generality, as −σ ≤ λ k ≤ λ k+1 → ∞ as k → ∞.We denote by {ϕ k } k∈N * the corresponding eigenfunctions, In the recent paper [1], we studied the stabilizability of (1.1) to the j-th eigensolution of the free equation (p ≡ 0), ψ j (t) = e −λ j t ϕ j , for every j ∈ N * .For this purpose, we introduced the notion of j-null controllability in time T > 0 for the pair {A, B}: denoting by y(•; v 0 , p) the solution of the linear system    y ′ (t) + Ay(t) + p(t)Bϕ j = 0, t ∈ [0, T ] y(0) = y 0 , we say that {A, B} is j-null controllable in time T > 0 if for any initial condition y 0 ∈ X there exists a control p ∈ L 2 (0, T ) such that y(T ; v 0 , p) = 0 and p L 2 (0,T ) ≤ N T y 0 , where N T is a positive constant depending only on T .Then, the control cost is given by inf p L 2 (0,T ) : y(T ; y 0 , p) = 0 .
In [1,Theorem 3.7] we have shown that, if {A, B} is j-null controllable, then (1.1) is locally superexponentially stabilizable to ψ j : for all u 0 in some neighborhood of ϕ j there exists a control p ∈ L 2 loc ([0, +∞)) such that the corresponding solution u of (1.1) satisfies for suitable constants ω, M > 0 independent of u 0 .Notice that such a result holds only under the condition of j-null controllability for the pair {A, B}.In particular, no assumptions are required on the behavior of the control cost.Moreover, in [1,Theorem 3.8] we gave sufficient conditions to ensure the j-null controllability of {A, B}: a gap condition for the eigenvalues of A and a rank condition on B.
In this paper, we address the related, more delicate, issue of the exact controllability of (1.1) to the eigensolutions ψ j via bilinear controls.The main differences between the results of this paper and [1,Theorem 3.7] can be summarized as follows: • in addition to assuming the pair {A, B} to be j-null controllable, we further require that the control cost N(•) satisfies N(τ ) ≤ e ν/τ for any 0 < τ ≤ T 0 , with ν, T 0 > 0, • under the above stronger assumptions, not only we prove local exact controllability in any time, but also global exact controllablity in large time for a wide set of initial data.
The following result ensures local exact controllability for problem (1.1) assuming a precise behavior of the control cost for small time.In the last section of this paper, we show that such a behavior of the control cost is typical of parabolic problems in one space dimension.
Then, for any T > 0, there exists a constant R T > 0 such that, for any u 0 ∈ B R T (ϕ j ), there exists a control p ∈ L 2 (0, T ) such that the solution u of (1.1) satisfies u(T ) = ψ j (T ).Moreover, the following estimate holds p L 2 (0,T ) ≤ e −π 2 Γ 0 /T e 2π 2 Γ 0 /(3T ) − 1 , where Γ 0 and R T can be computed as follows R T := e −6Γ 0 /T 1 , ( with D := 2 B e 2σ+(3 B )/2+1/2 max {1, B } . (1.9) The main idea of the proof consists of applying the stability estimates of [1] on a suitable sequence of time intervals of decreasing length T j , such that ∞ j=1 T j < ∞.Such a sequence, which can be constructed only thanks to (1.4), has to be carefully chosen in order to fit the error estimates that we take from [1].We point out that our method is fully constructive, being based on an algorithm that allows to compute all relevant constants.In particular, we make no use of inverse mapping theorems.
In [1], we gave sufficient conditions for j-null controllability.However, the hypotheses of [1,Theorem 3.8] do not guarantee the validity of condition (1.4) for the control cost.In the result that follows, we provide sufficient conditions for N(T ) to satisfy (1.4).It would be interesting to understand if (1.4) is also necessary for the local exact controllability of (1.1).Theorem 1.2.Let A : D(A) ⊂ X → X be such that (1.3) holds and suppose that there exists a constant α > 0 for which the eigenvalues of A fulfill the gap condition (1.10) Let B : X → X be a bounded linear operator such that there exist b, q > 0 for which Then, the pair {A, B} is j-null controllable in any time T > 0, and the control cost N(T ) satisfies (1.4) with and ν = Γ j , where Here Γ(•) is the Gamma function and C is a positive constant independent of T and α.
Observe that assumption (1.11) is stronger that [1, hypothesis (16)].Nevertheless, it is satisfied by all the examples of parabolic problems that we presented in [1].
From Theorems 1.1 and 1.2 we deduce the following Corollary.
Corollary 1.3.Let A : D(A) ⊂ X → X be such that (1.3) holds and suppose that there exists a constant α > 0 for which (1.10) is satisfied.Let B : X → X be a bounded linear operator that verifies (1.11) for some b, q > 0.Then, problem (1.1) is locally controllable to the jth eigensolution ψ j in any time T > 0.
Furthermore, from Theorem 1.1 we deduce two semi-global controllability results in the case of an accretive operator A. In the first one, Theorem 1.4 below, we prove that all initial states lying in a suitable strip can be steered in finite time to the first eigensolution ψ 1 (see Figure 1).Moreover, we give a uniform estimate for the controllability time depending on the size of the projection of the initial datum u 0 on ϕ ⊥ 1 .Theorem 1.4.Let A : D(A) ⊂ X → X be a densely defined linear operator such that (1.3) holds with σ = 0 and let B : X → X be a bounded linear operator.Let {A,B} be a 1-null controllable pair which satisfies (1.4).Then, there exists a constant r 1 > 0 such that for any R > 0 there exists T R > 0 such that for all u 0 ∈ X with ) Figure 1: the colored region represents the set of initial conditions that can be steered to the first eigensolution in time T R .
Our second semi-global result, Theorem 1.5 below, ensures the exact controllability of all initial states u 0 ∈ X \ ϕ ⊥ 1 to the evolution of their orthogonal projection along the first eigensolution.Such a function is defined by where ψ 1 is the first eigensolution.
Theorem 1.5.Let A : D(A) ⊂ X → X be a densely defined linear operator such that (1.3) holds with σ = 0 and let B : X → X be a bounded linear operator.Let {A,B} be a 1-null controllable pair which satisfies (1.4).Then, for any R > 0 there exists T R > 0 such that for all u 0 ∈ X with system (1.1) is exactly controllable to φ 1 , defined in (1.17), in time T R .
Notice that, denoting by θ the angle between the half-lines R + ϕ 1 and which defines a closed cone, say Q R , with vertex at 0 and axis equal to Rϕ 1 (see Figure 2).Therefore, Theorem 1.5 ensures a uniform controllability time for all initial conditions lying in Q R .We observe that, since R is any arbitrary positive constant, all initial conditions u 0 ∈ X \ ϕ ⊥ 1 can be steered to the corresponding projection to the first eigensolution.Indeed, for any u 0 ∈ X \ ϕ ⊥ 1 , we define Then, for any R ≥ R 0 condition (1.18) is fulfilled: Finally, we would like to recall part of the huge literature on bilinear control of evolution equations, referring the reader to the references in [1] for more details.A seminal paper in this field is certainly the one by Ball, Marsden, Slemrod [3], which establishes that system (1.1) is not controllable.More precisely, denoting by u(t; u 0 , p) the unique solution of (1.1), the attainable set from u 0 defined by is shown in [3] to have a dense complement.
As for positive results, we would like to mention Beauchard [5], on bilinear control of the wave equation, and Beauchard, Laurent [7] on bilinear control of the Schrödinger equation (see also [4] for a first result on this topic).The results obtained in these papers rely on linearization around the ground state, the use of the inverse mapping theorem, and a regularizing effect which takes place in both problems.Local controllability is proved for any positive time for the Schrödinger equation and for a sufficiently (optimal) large time for the wave equation.Both papers require the condition to be satisfied, together with a suitable asymptotic behavior with respect to the eigenvalues.Notice that the structure of the second order operator and the fact that the space dimension equals one allow the authors of [5] and [7] to apply Ingham's theory ( [19]) which requires a gap condition on the eigenvalues.We further observe that even if the genericity of assumption (1.19) is proved in both papers [5,7], only few explicit examples of operators B of multiplication type are available in the literature.We refer to [2] where a general constructive algorithm for building potentials which satisfy the infinite non-vanishing conditions (1.19), and further the asymptotic condition (1.11), is established.
If (1.19) is violated then it has been first shown by Coron [17], for a model describing a particle in a moving box, that there exists a minimal time for local exact controllability to hold.This model couples the Schrödinger equation with two ordinary differential equations modeling the speed and acceleration of the box (see also Beauchard,Coron [6] for local exact controllability for large time).A further paper by Beauchard and Morancey [9] for the Schrödinger equation extends [7] to cases for which the above condition is violated, that is, when there exist integers k such that Bϕ 1 , ϕ k = 0.
An example of controllability to trajectories for nonlinear parabolic systems is studied in [18], where, however, additive controls are considered.In such an example, one can obtain controllability to free trajectories by Carleman estimates and inverse mapping arguments.Such a strategy seems hard to adapt to the current setting.
The paper which has the strongest connection with our work is the one by Beauchard and Marbach in [8], where the authors study small-time null controllability for a scalar-input heat equation in one space dimension, with nonlinear lower order terms.Among the results of such paper, we mention null-controllability to the first eigenstate of a heat equation with bilinear control.From this result it would be possible to deduce local controllability only to the first eigenstate of the heat equation subject to Neumann boundary conditions.It is worth noting that [8] addresses a specific parabolic equation.Moreover, the methods developed therein, relying on the so-called source term procedure, are totally different from ours.
We observe that the bilinear controls we use in this paper are just scalar functions of time.This fact explains why applications mainly concern problems in low space dimension, like the results in [4,5,6,7,8,9,17].A stronger control action could be obtained by letting controls depend on time and space.We refer the reader to [12,13] for more on this subject.This paper is organized as follows.In section 2, we have collected some preliminaries as well as results from [1] that we need in order to prove Theorem 1.1.Section 3 contains such a proof, while section 4 is devoted to demonstrate Theorem 1.2.In section 5, we give the proof of our semi-global results (Theorems 1.4 and 1.5).Finally, applications of Theorem 1.1 to parabolic problems are analyzed is section 6.

Preliminaries
In this section, we recall a well-known result for the well-posedness of our control problem and the regularity of the solution as well as some results from [1] that are necessary for the proof of Theorem 1.1.Moreover, we will remind the fundamental definition of j-null controllable pair.
We recall our general functional frame.Let (X, •, • , • ) be a separable Hilbert space, let A : D(A) ⊂ X → X be a densely defined linear operator with the following properties We denote by {λ k } k∈N * the eigenvalues of A, which can be ordered, whithout loss of generality, as −σ ≤ λ k ≤ λ k+1 → ∞ as k → ∞, and by {ϕ k } k∈N * the corresponding eigenfunctions, Let B : X → X be a bounded linear operator.Fixed T > 0, consider the following bilinear control problem We introduce the following notation: The well-posedness of (2.2) is ensured by the following proposition (see [3] for a proof).
Remark 2.2.Under the hypotheses of Theorem 2.1 it is possible to prove that the solution is more regular.Indeed, for every ε and the following identity is satisfied Furthermore, if u 0 = 0 then u ∈ H 1 (0, T ; X)∩L 2 (0, T ; D(A)) (it can be deduced by applying, for instance, [10, Proposition 3.1, p.130]).
Let us now consider the following nonlinear control problem where ϕ j is the jth eigenfunction of A. We denote by v(•; v 0 , p) the solution of (2.4) associated with initial condition v 0 and control p.
The following result establishes a bound for the solution of (2.4) in terms of the initial condition.We give its proof in Appendix A for the sake of clarity and completeness.This proof follows that of [1,Proposition 4.3], with a different presentation, in particular with respect to the assumptions in the statement.
Proposition 2.3.Let T > 0. Let A : D(A) ⊂ X → X be a densely defined linear operator that satisfies (2.1) and let B : X → X be a bounded linear operator.Let v 0 ∈ X and let p ∈ L 2 (0, T ) be such that ) where For any 0 ≤ s 0 ≤ s 1 , we now introduce the linear problem and we denote by y(•; y 0 , s 0 , p) the solution associated with initial condition y 0 at time s 0 and control p.Let us recall that for any fixed T > 0 and j ∈ N * , we say that the pair {A, B} is j-null controllable in time T if there exists a constant N T such that for every y 0 ∈ X there exists a control p ∈ L 2 (0, T ) with for which the solution of (2.7) with s 0 = 0 and s 1 = T satisfies y(T ; y 0 , 0, p) = 0.In this case, we define the control cost as With an approximation argument one realizes that (2.8) holds with N T = N(T ), that is, for every y 0 ∈ X there exists p ∈ L 2 (0, T ) with p L 2 (0,T ) ≤ N(T ) y 0 such that y(T ; y 0 , 0, p) = 0.

Now, consider the following control problem
with v the solution of (2.4).We denote by w(•; 0, p) the solution of (2.10) associated with control p.
In the following proposition we give a quadratic estimate of the solution of (2.10) in terms of the initial condition of the Cauchy problem solved by v.We give its proof in Appendix A for the sake of clarity and completeness.This proof follows that of [ Proposition 2.4.Let T > 0, A : D(A) ⊂ X → X be a densely defined linear operator that satisfies (2.1) and B : X → X be a bounded linear operator.Let p ∈ L 2 (0, T ) verify (2.5) with N T = N(T ) and v 0 ∈ X be such that (2.11) where (2.13)

Proof of Theorem 1.1
Fixed any j ∈ N * and any T > 0, our aim is to prove local exact controllability in time T for the following problem to the jth eigensolution ψ j = e −λ j t ϕ j of A, that is the solution of (3.1) when p = 0 and u 0 = ϕ j .Hereafter, we will denote by u(•; u 0 , p) the solution of (3.1) associated with initial condition u 0 and control p.We recall that A : D(A) ⊂ X → X is a densely defined linear operator that satisfies and we denote by {λ k } k∈N * and {ϕ k } k∈N * the eigenvalues and the eigenfunctions of A, respectively.B : X → X is a bounded linear operator.The pair {A, B} is assumed to be j-null controllable in any time, with control cost that satisfies for some constants ν, T 0 > 0. The proof of Theorem 1.1 is divided into two main parts: the case λ j = 0, that we build by a series of steps, and the case λ j = 0.

Case λ j = 0
If λ j = 0 our reference trajectory will be the stationary function ψ j ≡ ϕ j .Given T > 0, we define T f as where T 0 is the constant in (3.3).We will actually build a control p ∈ L 2 (0, T f ) such that u(T f ; u 0 , p) = ψ j , and then, by taking p(t) ≡ 0 for t > T f , the solution u of (3.1) will remain forever on the target trajectory ψ j .Now, we define and we observe that 0 < T 1 ≤ 1.Then, we introduce the sequence {T j } j∈N * as and the time steps with the convention that 0 j=1 T j = 0. Notice that ∞ j=1 T j = π 2 6 T 1 = T f .Remark 3.1.Note that the sequence of times (T j ) j∈N N * is strictly decaying towards 0, whereas the sequence of times (τ j ) j∈N N * is strictly increasing and converges to T f .Set v := u − ϕ j .We will consider the equation satisfied by v on suitable intervals of time [s 0 , s 1 ] and suitable initial data v 0 at the initial time s 0 , as follows.Given any 0 ≤ s 0 ≤ s 1 ≤ T , and any v 0 in X, v is the solution of the following Cauchy problem We denote by v(•; v 0 , s 0 , p) the solution of (3.8) associated with initial condition v 0 at time s 0 and control p. Observe that proving the controllability of u to The strategy of the proof consists first of building a control p 1 ∈ L 2 (0, T 1 ) such that at time T 1 the solution of (3.8) can be estimated by the square of the initial condition.We then iterate the procedure on consecutive time intervals of the form [τ n−1 , τ n ]: each time we construct a control p n ∈ L 2 (τ n−1 , τ n ) such that the solution of (3.8) on [τ n−1 , τ n ] at time τ n is estimated by the square of the initial condition on such interval.Hence, combining all those estimates and letting n go to infinity, we finally deduce that there exists a control p ∈ L 2 loc (0, +∞) such that v(T f ; v 0 , 0, p) = 0 and so u(T f ; u 0 , p) = ϕ j .In practice, we shall build, by induction, controls p n ∈ L 2 (τ n−1 , τ n ) for n ≥ 1 such that, setting Observe that, by construction,
Note that a suitable choice of constant Γ 0 such that (3.17) holds is (1.6).
We now define the radius of the neighborhood of ϕ j where we take the initial condition u 0 as in (1.7).Let u 0 ∈ B R T (ϕ j ), or equivalently v 0 = u 0 − ϕ j ∈ B R T (0), be chosen arbitrarily.With this choice we have that and (3.14) is satisfied.Therefore, we get that which proves 3. and 4. of (3.10) for n = 1.

Iterative step
Now, suppose that we have built controls p j ∈ L 2 (τ j−1 , τ j ) such that (3.10) holds for each j = 1, . . ., n − 1.In particular, for j = n − 1, there exists We shall now prove that there exists p n ∈ L 2 (τ n−1 , τ n ) such that every item of (3.10) is fulfilled.We defined q n−1 and v n−1 as in (3.9) and we consider the following problem where the control p has still to be suitably chosen.By the change of variables s = t − τ n−1 and the definition (3.7), we shift the problem from [τ n−1 , τ n ] into the interval [0, T n ].We introduce the functions ṽ(s) = v(s + τ n−1 ) and p(s) = p (s + τ n−1 ) and we rewrite (3.20) as    ṽ′ (s) + Aṽ(s) + p(s)Bṽ(s) + p(s)Bϕ j = 0, s ∈ [0, T n ] ṽ(0) = v n−1 . (3.21) Recalling that {A, B} is j-null controllable in any time, there exists a control pn ∈ L 2 (0, T n ) such that pn L 2 (0,Tn) ≤ N(T n ) v n−1 and ỹ(T n , v n−1 , 0, pn ) = 0, where ỹ(•; v n−1 , 0, pn ) is the solution of the linear problem (2.7) on [0, T n ].Furthermore, since where we have used that the constant of the control cost ν is less than or equal to Γ 0 (see Remark 3.2), and the identity which can be easily checked by induction.We now choose the control p = pn in (3.21) and still denote by ṽ the corresponding solution.We set w = ṽ − ỹ.Then, w solves (2.10) with T = T n and p = pn .So, we can apply Proposition 2.4 with T = T n to problem (3.21) and since w(T n ; 0, pn ) = ṽ(T n ; v n−1 , 0, pn ), we obtain that ṽ(T n ; We shift back the problem into the original interval [τ n−1 , τ n ], we define p n (t) := pn (t − τ n−1 ), and we get So, we have proved the first two items of (3.10).Moreover, thanks to 3. of (3.19), we deduce that ) that is the third item of (3.10).Finally, using again (3.24) and 4. of (3.19) we obtain that This concludes the induction argument and the proof of (3.10).
We are now ready to complete the proof of Theorem 1.1 for the case λ j = 0. We observe that for all n where we have used that ∞ j=1 j 2 /2 j = 6.Notice that (3.26) is equivalent to where q n (t) = n j=1 p j (t)χ [τ j−1 ,τ j ] (t).We now take the limit as n → ∞ in (3.27) and we get because v 0 < e −6Γ 0 /T 1 .This means that, we have built a control p ∈ L 2 loc ([0, ∞)), defined by for which the solution u of (3.1) reaches the jth eigensolution ψ j = ϕ j in time T f , less than or equal to T , and stays on it forever.Observe that, thanks to the first item of (3.10) and to (3.22), we are able to yield a bound for the L 2 -norm of such a control: (3.30) Notice that since (3.4) holds, (3.30) implies (1.5).

3.2.
Case λ j = 0 Now, we face the case λ j = 0. We define the operator We proved in [1,Lemma 4.7] that if {A, B} is j-null controllable, then the same holds for the pair {A j , B}.Furthermore, it is easy to check that also condition (3.3) is verified by the control cost associated with {A j , B}, if the same property holds for the control cost associated with the pair {A, B}.
It is possible to check that A j satisfies (3.2) and moreover it has the same eigenfuctions, {ϕ k } k∈N * , of A, while the eigenvalues are given by In particular, µ j = 0. We define the function z(t) = e λ j t u(t), where u is the solution of (3.1).Then, z solves the following problem z(0) = u 0 . (3.31) We define T f as in (3.4) (where T 0 is now the constant associated with the control cost relative to the pair {A j , B}) and R T as in (1.7).We deduce from the previous analysis that, if u 0 ∈ B R T (ϕ j ), then there exists a control p ∈ L 2 ([0, +∞)) that steers the solution z to the eigenstate ϕ j in time T f ≤ T .This implies the exact controllability of u to the eigensolution ψ j (t) = e −λ j t ϕ j : indeed, Remark 3.3.We observe that, from (3.30), it follows that p L 2 (0,T f ) → 0 as T f → 0. This fact is not surprising since as T f approaches 0, also the size of the neighborhood where the initial condition can be chosen goes to zero.

Proof of Theorem 1.2
Before showing the proof of Theorem 1.2, we define formally the following function where M is a positive constant, ω k := λ k − λ 1 , for all k ∈ N * , {λ k } k∈N * are the eigenvalues of A. In Lemma 4.1 below, we investigate the behavior of G M (T ) for small values of T .Such a result will be crucial for the analysis of the control cost N(T ) in Theorem 1.2.
Lemma 4.1.Let A : D(A) ⊂ X → X be such that (1.3) and (1.10) hold and B : X → X be such that (1.11) holds.Then, for any M, T > 0 the series in (4.1) is convergent and there exists a positive constant Γ j , such that Moreover, a suitable choice of Γ j = Γ j (M, b, q, α) is (1.13).
Proof.Thanks to assumption (1.11), we have that For any ω ≥ 0 we set f (ω) = e −ωT +M √ ω .The maximum value of f is attained at So, we can bound G M (T ) as follows Now, for any ω ≥ 0 we define the function g(ω) = ω 2q e −ωT .Its derivative is given by and therefore we deduce that and g has a maximum at ω = (2q)/T .We define the following index: Note that k 1 (T ) goes to ∞ as T converges to 0. We can rewrite the sum in (4.4) as follows For any k ≤ k 1 − 1, we have and for any k So, by using estimates (4.6) and (4.7), (4.5) becomes Furthermore, recalling that g has a maximum for ω = 2q/T , it holds that Finally, the integral term of (4.8) can be rewritten as where by Γ(•) we indicate the Euler integral of the second kind.Therefore, we conclude from (4.9) and (4.10) that there exist two constants C q , C q,α > 0 such that We use this last bound to prove that there exists Γ j > 0 such that ) as claimed.Now we proceed with the proof of Theorem 1.2.
Our aim is to find a control p ∈ L 2 (0, T ) for which y(T ; y 0 , 0, p) = 0, that is equivalent to the following identity Since, by hypothesis, the eigenfunctions of A form an orthonormal basis of X, the above formula reads as By defining q(s) := e λ 1 s p(s) and ω k := λ k − λ 1 ≥ 0, the family of equations (4.14) can be rewritten as Thanks to hypothesis (1.10), we can apply [15,Theorem 2.4] that ensures the existence of a biorthogonal family {σ k } k∈N * to the family of exponentials We claim that the series is convergent in L 2 (0, T ).Indeed, thanks to the following estimate, from [15, Theorem 2.4], for the biorthogonal family , with C > 0 independent of T and α, and Observe that, by Lemma 4.1, the right-hand side of the above estimate is finite for any T > 0. Therefore, we define the control q as q(s) := and we deduce that q ∈ L 2 (0, T ) satisfies (4.15) and furthermore . Finally, returning to p, we obtain that By taking we deduce that {A, B} is j-null controllable in any time T > 0 with associated control cost (4.18).What remains to prove is estimate (1.4) for the control cost N(T ) defined in (4.18), for T small.Let us define T 0 as in (1.12).Then for any 0 < T < T 0 , it holds that We can assume without loss of generality that the constant C ≥ 1, since we can replace it by max {1, C}.We assume for all the sequel that C ≥ 1.
Since 0 < T < T 0 ≤ 1, we claim that there exists M > 0 such that Indeed, we have We set We note that since C ≥ 1, we have 2C α 2 ≤ M .Hence from the two above estimates, we deduce (4.19).Moreover, we easily prove that max 1, e −λ 1 T ≤ e |λ 1 | ∀ T ∈ (0, T 0 ).Therefore, the control cost N(T ) given by (4.18) can be bounded from above as follows where M is defined as in (1.14) and the function G M (•) is defined in (4.1).Finally, thanks to Lemma 4.1, we deduce that N(T ) fulfills property (1.4) with ν = Γ j .

Proof of Theorems 1.4 and 1.5
Before proving Theorem 1.4, let us show a preliminary result that demonstrates the statement in the case of a strictly accretive operator.Lemma 5.1.Let A : D(A) ⊂ X → X be a densely defined linear operator such that (1.3) holds with σ = 0 and let B : X → X be a bounded linear operator.Let {A,B} be a 1-null controllable pair which satisfies (1.4).Furthermore, we assume λ 1 = 0.Then, there exists a constant r 1 > 0 such that for any R > 0 there exists T R > 0 such that for all v 0 ∈ X that satisfy problem (2.4) is null controllable in time T R .
Second step.Let v 0 ∈ X be the initial condition of (2.4).We decompose v 0 as follows and we can directly apply the first step of the proof with T R = 1.Otherwise, we define t R as and in the time interval [0, t R ] we take the control p ≡ 0.Then, for all t ∈ [0, t R ], we have that In particular, for t = t R , it holds that v(t R ) 2 < 2r 2 1 .Now, we define T R := t R + 1 and set v 1 (0) = v(t R ).Thanks to the first step of the proof, there exists a control p 1 ∈ L 2 (0, 1), such that v 1 (1) = 0, where v 1 is the solution of (2.4) on [0, 1] with p replaced by p 1 .
Then v(t) = v 1 (t − t R ) solves (2.4) in the time interval (t R , T R ] with the control p 1 (t − t R ) that steers the solution v to 0 at T R .
Proof (of Theorem 1.4).We start with the case λ 1 = 0. Let u 0 ∈ X satisfy (1.16).Set v(t) := u(t) − ϕ 1 , then v satisfies (2.4) and moreover v 0 := v(0) = u 0 − ϕ 1 fulfills (5.1).Thus, by Lemma 5.1, problem (1.1) is exactly controllable to the first eigensolution ψ 1 ≡ ϕ 1 in time T R .Now, we consider the case λ 1 > 0. As in the proof of Theorem 1.1, we introduce the variable z(t) = e λ 1 t u(t) that solves problem (3.31).For such a system, since the first eigenvalue of A 1 is equal 0, we have the exact controllability to ϕ 1 in time T R .Namely z(T R ) = ϕ 1 , that is equivalent to the exact controllability of u to ψ 1 : 3) The proof is thus complete.
The proof of Theorem 1.5 easily follows from Theorem (1.4).
Note that if u 0 ∈ X satisfies both u 0 ∈ ϕ ⊥ 1 and (1.18), then we have trivially that u 0 ≡ 0. We then choose p ≡ 0, so that the solution of (1.1) remains constantly equal to φ 1 ≡ 0.

Applications
In this section we present some examples of parabolic equations for which Theorem 1.1 can be applied.The hypotheses (1.3),(1.10)and (1.11) have been verified in [1] and [16], to which we refer for more details.We observe that, thanks to [1, Remark 6.1], since the second order operators considered in the examples are accretive ( Ax, x ≥ 0, for all x ∈ D(A)), it suffices to prove the following gap condition which implies (1.10).Moreover, in the case of an accretive operator it suffices to show that there exist b, q > 0 such that Bϕ j , ϕ j = 0 and to have (1.11).Furthermore, we note that the global results Theorem 1.4 and Theorem 1.5 can be applied to any example below.Note also, that the given examples below, are non-exhaustive.

Diffusion equation with Dirichlet boundary conditions
Let I = (0, 1) and X = L 2 (0, 1).Consider the following problem We denote by A the operator defined by and it can be checked that A satisfies (1.3).We indicate by {λ k } k∈N * and {ϕ k } k∈N * the families of eigenvalues and eigenfunctions of A, respectively: It is easy to see that (6.1) holds true (and so (1.10)): Then, there exists b > 0 such that For instance, a suitable function that satisfies (6.4) is µ(x) = x 2 : indeed, in this case Therefore, problem (6.3) is controllable to the j-th eigensolution ψ j in any time T > 0 as long as u 0 ∈ B R T (ϕ j ), with R T > 0 a suitable constant, where ψ j (t, x) = √ 2 sin(jπx)e −j 2 π 2 t .

Diffusion equation with Neumann boundary conditions
Let I = (0, 1), X = L 2 (I) and consider the Cauchy problem x ∈ I. (6.5) The operator A, defined by 3) and has the following eigenvalues and eigenfunctions Thus, the gap condition (6.1) is fulfilled with α = π.Fixed j ∈ N, the j-th eigensolution is the function ψ j (x) = e −λ j t ϕ j (x).We define B : X → X as the multiplication operator by a function µ ∈ H 2 (I), Bϕ = µϕ, such that µ ′ (1) ± µ ′ (0) = 0 and It can be proved that, there exists b > 0 such that For example, µ(x) = x 2 satisfies (6.7).Indeed, it can be shown that and for j = 0 Therefore, problem (6.5) is controllable to the j-th eigensolution ψ j in any time T > 0 as long as u 0 ∈ B R T (ϕ j ), with R T > 0 a suitable constant.

Variable coefficient parabolic equation
Let I = (0, 1), X = L 2 (I) and consider the problem We denote by A : D(A) ⊂ X → X the following operator It can be checked that A satisfies (1.3) and that the eigenvalues and eigenfunctions have the following expression with b a positive constant.For instance, µ(x) = x 2 verifies (6.12) and (6.13): Therefore, by applying Theorem 1.1, we conclude that for any T > 0, the exists a suitable constant R T > 0 such that, if u 0 ∈ B R T (ϕ j ), problem (6.10) is exactly controllable to the j-th eigensolution ψ j in time T .

Degenerate parabolic equation
In this last section we want to address an example of a control problem for a degenerate evolution equation of the form                    u t − (x γ u x ) x + p(t)x 2−γ u = 0, (t, x) ∈ (0, +∞) × (0, 1)  It is possible to prove that A satisfies (1.3) (see, for instance [11]) and furthermore, if we denote by {λ k } k∈N * the eigenvalues and by {ϕ k } k∈N * the corresponding eigenfunctions, it turns out that the gap condition (6.1) is fulfilled with α = 7 16 π (see [19], page 135).If γ ∈ [1, 2), problem (6.14) is called strong degenerate and the corresponding weighted Sobolev space are described as follows: given I = (0, 1) and X = L 2 (I), we define In this case the operator A : D(A) ⊂ X → X is defined by              ∀u ∈ D(A), Au := −(x γ u x ) x , D(A) := u ∈ H 1 γ,0 (I) : x γ u x ∈ H 1 (I) = {u ∈ X : u is absolutely continuous in (0,1] , x γ u ∈ H 1 0 (I), x γ u x ∈ H 1 (I) and (x γ u x )(0) = 0} and it has been proved that (1.3) holds true (see, for instance [14]) and that (6.1) is satisfied for α = π 2 (see [19]).We fix j = 1 and for all γ ∈ [0, 3/2), we define the linear operator B : X → X by Bu(t, x) = x 2−γ u(t, x) and in [16] we have proved that there exists a constant b > 0 such that Finally, by applying Theorem 1.1, we ensure the exact controllability of problem (6.14) to the first eigensolution, for both weakly and strongly degenerate problems.
We now integrate the last inequality from ε to t to obtain Finally, recalling estimate (2.5) for p, we get (2.6).

Figure 2 :
Figure 2: fixed any R > 0, the set of initial conditions exactly controllable in time T R to their projection along the first eigensolution is indicated by the colored cone Q R .

1 ,
Proposition 4.4], with a different presentation and a different hypothesis (2.11) compared to the corresponding ones in the statement of [1, Proposition 4.4].