Null controllability for the singular heat equation with a memory term

In this paper we focus on the null controllability problem for the heat equation with the so-called inverse square potential and a memory term. To this aim, we first establish the null controllability for a nonhomogeneous singular heat equation by a new Carleman inequality with weights which do not blow up at t=0. Then the null controllability property is proved for the singular heat equation with memory under a condition on the kernel, by means of Kakutani's fixed-point Theorem.


Introduction
In this paper, we address the null controllability for the following singular heat equation with memory: a(t, s, x)y(s, x) ds + 1 ω u, (t, x) ∈ Q, y(t, 0) = y(t, 1) = 0, t ∈ (0, T ), y(0, x) = y 0 (x), x ∈ (0, 1), (1.1) where y 0 ∈ L 2 (0, 1), T > 0 is fixed, µ is a real parameter, Q := (0, T ) × (0, 1) and 1 ω stands for a characteristic function of a nonempty open subset ω of (0, 1). Here y and u are the state variable and the control variable respectively, a is a given L ∞ function defined on (0, T ) × Q. The analysis of evolution equations involving memory terms is a topic in continuous development. In the last decades, many researchers have started devoting their attention to this branch of mathematics, motivated by many applications in modelling phenomena in which the processes are affected not only by its current state but also by its history. Indeed, there is a large spectrum of situations in which the presence of the memory may render the description of the phenomena more accurate. This is particularly the case for models such as heat conduction in materials with memory, viscoelasticity, theory of population dynamics and nuclear reactors, where there is often a need to reflect the effects of the memory of the system (see for instance [3,7,29,35]).
Controllability problems for evolution equations with memory terms have been extensively studied in the past. Among other contributions, we mention [4,20,23,25,26,28,30,36,38] which, as in our case, deal with parabolic type equations. We also refer to [34] for an overview of the bibliography on control problems for systems with persistent memory. The first results for a degenerate parabolic equation with memory can be found in [1].
In this work, for the first time to our knowledge, we study the null controllability for (1.1). We underline that here we consider not only a memory term but also a singular potential one. In other words, given any y 0 ∈ L 2 (0, 1), we want to show that there exists a control function u ∈ L 2 (Q) such that the corresponding solution y to (1.1) satisfies y(T, x) = 0 for every x ∈ [0, 1]. First results in this direction are obtained in [42] in the absence of a memory term when µ ≤ 1 4 (see also [41] for the wave and Schrödinger equations and [10] for boundary singularity). Indeed, for the equation with associated Dirichlet boundary conditions in a bounded domain Ω ⊂ R N containing the singularity x = 0 in the interior, the value of the parameter µ determines the behavior of the equation: if µ ≤ 1/4 (which is the optimal constant of the Hardy inequality, see [8]) global positive solutions exist, while, if µ > 1/4, instantaneous and complete blowup occurs (for other comments on this argument we refer to [40]). In the case of global positive solutions, hence if µ ≤ 1 4 and using Carleman estimates, in [42] it has been proved that such equations can be controlled (in any time T > 0) by a locally distributed control.
In this case, as shown in [23,45], there exists a set of initial conditions such that the null controllability property for (1.3) fails whenever the control region ω is fixed, independent of time. For some related works in this respect we also refer to [11,26,44]. Nevertheless, since the positive controllability results are important in real world applications, it is natural to analyze whether it is possible that control properties for (1.1) could be obtained. For this reason, under suitable conditions on the singularity parameter µ and on the kernel a, we establish that (1.1) is null controllable.
Our approach is inspired from the techniques presented in the work [38] for the Laplace operator, suitably adapted in order to deal with the additional inverse-square potential. In particular, the technique that we will use is based on appropriate Carleman estimates and on the fixed-point Theorem of Kakutani.
The paper is organized as follows: Section 2 is devoted to the study of null controllability for a nonhomogeneous singular heat equation without memory via new Carleman estimates. In Section 3, the null controllability for the singular heat equation with memory (1.1) is proved.
A final comment on the notation: by C we shall denote universal positive constants, which are allowed to vary from line to line.

Nonhomogeneous singular heat equation
In this section, we prove the null controllability for a nonhomogeneous singular heat equation using a new modified Carleman inequality. This null controllability result is the key tool for the controllability of the heat equation with memory. Thus, as a first step, we consider the following problem: x ∈ (0, 1), where f ∈ L 2 (Q) is a given source term. Prior to null controllability is the well-posedness of (2.1), a question we address in the next subsection.

Functional framework and well-posedness
We analyze here existence and uniqueness of solutions for the heat problem (2.1). To simplify the presentation, we first focus on the well-posedness of the following inhomogeneous singular problem x ∈ (0, 1).

Carleman estimates for a singular problem
In this subsection we prove a new Carleman estimate for the adjoint parabolic equation associated to (2.1), which will provide that the nonhomogeneous singular heat equation (2.1) is null controllable. Hence, in the following, we concentrate on the next adjoint problem x ∈ (0, 1). (2.12) Following [42], for every 0 < γ < 2, let us introduce the weight function c > 0 and d > 1. A more precise restriction on the parameters k, c and d will be needed later. Observe that lim Using the previous weight functions and the following improved Hardy-Poincaré inequality given in [40]: one can prove the following Carleman estimate for the case of a purely singular parabolic equation: is not positive, then the estimate (2.16) is not of great importance. In fact, the Hardy inequality (2.3) only ensures the positivity of of the quantity However, from [40, Remark 3] and similarly as in [24], we will rewrite the result given in Lemma 2.1 in a more practical way.
. Then, there exist C > 0 and s 0 > 0 such that, for all s ≥ s 0 , every solution z of (2.12) satisfies . Here γ is as in (2.14) Proof. Case 1: If µ < 1 4 . Let Z = ze sϕ . In order to prove [40,Theorem 5.1], the author has derived the following estimate By (2.20) and (2.21), we obtain On the other hand, from (2.15), for all η > 0 there exists a constant c 0 = c 0 (η) > 0 such that Using the definition of Z, we have
(2.27) Hence, also in this case the conclusion follows.
We point out that the Carleman estimates stated above are not appropriate to achieve our goal. In fact, all these estimates does not have the observation term in the interior of the domain. However, we use them to obtain the main Carleman estimate stated in Proposition 2.2. More precisely, from the boundary Carleman estimates (2.17), we will deduce a global Carleman estimate for the adjoint problem (2.12) with a distributed observation on a subregion ω ′ := (α ′ , β ′ ) ⊂⊂ ω. (2.28) To do so, we recall the following weight functions associated to nonsingular Carleman estimates which are suited to our purpose: where θ is defined in (2.14) and Ψ(x) = e ρσ − e 2ρ σ ∞ . Here ρ > 0, σ ∈ C 2 ([0, 1]) is such that σ(x) > 0 in (0, 1), σ(0) = σ(1) = 0 and σ x (x) = 0 in [0, 1] \ω, beingω an arbitrary open subset of ω.
In the following, we choose the constant c in (2.14) so that By this choice one can prove that the function ϕ defined in (2.13) satisfies the next estimate Thanks to this property, we can prove the main Carleman estimate of this paper whose proof is based also on the following Caccioppoli's inequality: Proposition 2.1 (Caccioppoli's inequality). Let ω ′ and ω ′′ be two nonempty open subsets of (0, 1) such that ω ′′ ⊂ ω ′ and φ(t, x) = θ(t)̺(x), where ̺ ∈ C 2 (ω ′ , R). Then, there exists a constant C > 0 such that any solution z of (2.12) satisfies
For our purposes in the next section, we concentrate now on a Carleman inequality for solutions of (2.12) obtained via weight functions not exploding at t = 0. To this end, we will apply a classical argument that can be found, for instance, in [21] and recently in [1] for a degenerate parabolic equation with memory. More precisely, let us consider the function: Letz =τ z, thenz satisfies x ∈ (0, 1). (2.43) Thanks to the estimate of sup t∈[0,T ] z(t) 2 L 2 (0,1) (see the energy estimate (2.4)), we have By using the boundedness of θ in To conclude, it suffices to remark that for c > 0, the function x → s 3 e −cs is nonincreasing for s sufficiently large. So, since ν(t) ≤ θ(t) by taking s large enough, one has which, together with (2.46), provides the desired inequality.

Null controllability result
Following the classical method as in [21], with the modified Carleman inequality proved in the previous subsection, we can get a null controllability result for (2.1). However, as explained in [38], this null controllability result cannot help to solve the controllability for integro-differential equations. Indeed, we will need to prove the null controllability of the singular heat equation (2.1), for more regular solutions. For this reason, to formulate our results we introduce the following function space where the controllability will be solved: X s := y ∈ Z : e −sΦ y ∈ L 2 (Q) equipped with the norm y Xs := e −sΦ y L 2 (Q) .
Observe that, sinceΦ < 0, we have that the function e −sΦ tends to +∞ for t → T − . Therefore, y ∈ X s requires that the solution y has more regularity than the one in Lemma 2.1. Moreover, if y ∈ X s then y(T, x) = 0 in (0, 1).
From now on, we denote by s 0 the parameter defined in Lemma 2.3. Our first result, stated as follows, ensures the null controllability for (2.1).

(2.48)
Proof. Following the ideas in [9,38], fixed s ≥ s 0 , let us consider the functional where (y, u) satisfies By means of standard arguments, it is easy to prove (see [31,32]) that J attains its minimizer at a unique point denoted as (ȳ,ū).
We set L µ y := y t − y xx − µ x 2 y in Q.
We will first prove that there exists a dual variablez such that where L ⋆ µ is the (formally) adjoint operator of L µ . Let us start by introducing the following linear space and introduce the bilinear form a: Then, if the functionsȳ andū given by (2.51) satisfy the parabolic problem (2.50), we must have The key idea in this proof is to show that there exists exactly onez satisfying (2.52) in an appropriate class. We will then defineȳ andū using (2.51) and we will check that the couple (ȳ,ū) fulfills the desired properties.
Observe that the modified Carleman inequality (2.41) holds for all z ∈ P 0 . Consequently, In particular, a(·, ·) is a strictly positive and symmetric bilinear form, that is, a(·, ·) is a scalar product in P 0 . Denote by P the Hilbert space which is the completion of P 0 with respect to the norm associated to a(·, ·) (which we denote by · P ). Let us now consider the linear form l, given by By the Cauchy-Schwarz inequality and in view of (2.53), we have that and then l is a linear continuous form on P. Hence, in view of Lax-Milgram's Lemma, there exists one and only onez ∈ P satisfying a(z, z) = l(z), ∀ z ∈ P.
(2.54) Moreover, we have Let us setȳ = e 2sΦ L ⋆ µz andū = −1 ω s 3 ν 3 e 2sΦz . (2.56) With these definitions and by (2.55), it is easy to check thatȳ andū satisfy which implies (2.48). It remains to check thatȳ is the solution of (2.50) corresponding toū. First of all, it is immediate thatȳ ∈ X s andū ∈ L 2 (Q). Denote byỹ the (weak) solution of (2.1) associated to the control function u =ū, thenỹ is also the unique solution of (2.1) defined by transposition. In other words,ỹ is the unique function in L 2 (Q) satisfying where z is the solution to x ∈ (0, 1).
Proposition 3.1. Assume that µ ≤ 1 4 . If y 0 ∈ L 2 (0, 1) and u ∈ L 2 (Q), then there exists a unique solution y of (1.1) such that y ∈ C [0, T ]; L 2 (0, 1) ∩ L 2 0, T ; H 1,µ 0 (0, 1) . Now, we pass to derive our main result, which concerns the null controllability of the singular heat equation with memory (1.1). Hence, in what follows, we assume that the function a satisfies e 4 k scd where c, d, k are the constants defined in (2.14) and s is the same of Theorem 2.2.
Remark 1. It is worth mentioning that, from the results in Guerrero and Imanuvilov [23], it seems that the null controllability property of parabolic equations with memory may fail without any additional conditions on the kernel. On the other hand, observe that the condition (3.1) just restricts the function a very near T , which is due to the fact that the function ν blows up only at t = T .

(3.2)
By Theorem 2.2 we first derive a null controllability result for (3.2); then, as a second step, we will obtain the same controllability result for (1.1) applying Kakutani's fixed point Theorem.
Our main result is thus the following.
Let us now introduce, for every w ∈ X s,R , the multivalued map Λ : X s,R ⊂ X s → 2 Xs with Λ(w) = y ∈ X s : for some u ∈ L 2 (Q) satisfying Observe that if y ∈ Λ(w), then y(T, ·) = 0 in (0, 1) via (2.47). To achieve our goal, it will suffice to show that Λ possesses at least one fixed point. To this purpose, we shall apply Kakutani's fixed point Theorem (see [9,Theorem 2.3]).
It is readily seen that Λ(w) is a nonempty, closed and convex subset of L 2 (Q) for every w ∈ X s,R . Then, we prove that Λ(X s,R ) ⊂ X s,R with sufficiently large R > 0. By (2.48) and condition (3.1), and arguing as before we have Now, choosing the constant c (see (2.14)) in the interval which is not empty for ρ sufficiently large, we have 16 15 k+1 (e 2ρ σ ∞ − e ρ σ ∞ ).
Therefore, taking the parameters d and k defined in (2.14) in such a way that d > 3 and 2 < k < ln(4/3) ln(16/15) − 1, we infer that Hence for s sufficiently large, increasing the parameter s 0 if necessary, we obtain Q y 2 e −2sΦ dx dt + Qω s −3 ν −3 u 2 e −2sΦ dx dt ≤ 1 2 Then, for s and R large enough, we obtain Q y 2 e −2sΦ dx dt ≤ R 2 .
It follows that Λ(X s,R ) ⊂ X s,R . Furthermore, let {w n } be a sequence of X s,R . The regularity assumption on y 0 and Theorem 2.1, imply that the associated solutions {y n } are bounded in H 1 0, T ; L 2 (0, 1) ∩ L 2 0, T ; D(A) . Therefore, Λ(X s,R ) is a relatively compact subset of L 2 (Q) by the Aubin-Lions Theorem [37].
In order to conclude, we have to prove that Λ is upper-semicontinuous under the L 2 topology. First, observe that for any w ∈ X s,R , we have at least u ∈ L 2 (Q) such that the corresponding solution y ∈ X s,R . Hence, taking {w n } a sequence in X s,R , we can find a sequence of controls {u n } such that the corresponding solutions {y n } is in L 2 (Q). Thus, let {w n } be a sequence satisfying w n → w in X s,R and y n ∈ Λ(w n ) such that y n → y in L 2 (Q). We must prove that y ∈ Λ(w). For every n, we have a control u n ∈ L 2 (Q) such that the system        y n,t − y n,xx − µ x 2 y n = t 0 a(t, s, x)w n (s, x) ds + 1 ω u n , (t, x) ∈ Q, y n (t, 0) = y n (t, 1) = 0, t ∈ (0, T ), y n (0, x) = y 0 (x), x ∈ (0, 1) has a least one solution y n ∈ L 2 (Q) that satisfies y n (T, ·) = 0 in (0, 1).
Passing to the limit in (3.5), we obtain a control u ∈ L 2 (Q) such that the corresponding solution y to (3.2) satisfies (3.3). This shows that y ∈ Λ(w) and, therefore, the map Λ is upper-semicontinuous. Hence, the multivalued map Λ possesses at least one fixed point, i.e., there exists y ∈ X s,R such that y ∈ Λ(y). By the definition of Λ, this implies that there exists at least one pair (y, u) satisfying the conditions of Theorem 3.1. The uniqueness of y follows by Proposition 3.1. This ends the proof of Theorem 3.1.
As a consequence of the previous theorem one has the next result. . If the function a satisfies (3.1), then for any y 0 ∈ L 2 (0, 1), there exists a control function u ∈ L 2 (Q) such that the associated solution y ∈ W of (1.1) satisfies y(T, ·) = 0 in (0, 1).