A spectral approach for the exact observability of infinite-dimensional systems with skew-adjoint generator
|
|
- Susanna Hicks
- 5 years ago
- Views:
Transcription
1 Journal of Functional Analysis 6 ( A spectral approach for the exact observability of infinite-dimensional systems with skew-adjoint generator K. Ramdani, T. Takahashi, G. Tenenbaum, M. Tucsnak Institut Elie Cartan, Université Henri Poincaré Nancy 1, BP 39, Vandoeuvre lès Nancy 5456, France Received 1 August 4; accepted 16 February 5 Communicated by P. Malliavin Available online 7 April 5 Abstract Let A be a possibly unbounded skew-adjoint operator on the Hilbert space X with compact resolvent. Let C be a bounded operator from D(A to another Hilbert space Y. We consider the system governed by the state equation ż(t=az(t with the output y(t=cz(t. We characterize the exact observability of this system only in terms of C and of the spectral elements of the operator A. The starting point in the proof of this result is a Hautus-type test, recently obtained in Burq and Zworski (J. Amer. Soc. 17 ( and Miller (J. Funct. Anal. 18 ( ( We then apply this result to various systems governed by partial differential equations with observation on the boundary of the domain. The Schrödinger equation, the Bernoulli Euler plate equation and the wave equation in a square are considered. For the plate and Schrödinger equations, the main novelty brought in by our results is that we prove the exact boundary observability for an arbitrarily small observed part of the boundary. This is done by combining our spectral observability test to a theorem of Beurling on nonharmonic Fourier series and to a new number theoretic result on shifted squares. 5 Elsevier Inc. All rights reserved. MSC: 93C5; 93B7; 93C; 11N36 Keywords: Boundary exact observability; Boundary exact controllability; Hautus test; Schrödinger equation; Plate equation; Wave equation Corresponding author. Fax: address: marius.tucsnak@iecn.u-nancy.fr (M. Tucsnak. -136/$ - see front matter 5 Elsevier Inc. All rights reserved. doi:1.116/j.jfa.5..9
2 194 K. Ramdani et al. / Journal of Functional Analysis 6 ( Introduction and statement of the main results Let X be a Hilbert space endowed with the norm X, and let A : D(A X be a skew-adjoint operator. Assume that Y is another Hilbert space equipped with the norm Y and let C L(D(A, Y be an observation operator. According to Stone s theorem, A generates a strongly continuous group of isometries in X denoted T = (T t t. This paper is concerned with infinite-dimensional observation systems described by the equations ż(t = Az(t, z( = z, (1.1 y(t = Cz(t. (1. Here, a dot denotes differentiation with respect to the time t. The element z X is called the initial state, z(t is called the state at time t and y is the output function. Such systems are often used as models of vibrating systems (e.g., the wave equation, electromagnetic phenomena (Maxwell s equations or in quantum mechanics (Schrödinger s equation. In several particular cases we will also consider the control system which is the dual of (1.1, (1.. However, in order to avoid technicalities, we do not use the general form of the dual control system and we do not detail the duality arguments (we refer, for instance, to [5] for a brief discussion of these issues. By a solution of (1.1 we mean that z(t = T t z (this is a mild solution. In order to give a sense to (1., we make the assumption that C is an admissible observation operator in the following sense (see [6]: Definition 1.1. The operator C in system (1.1 (1. is an admissible observation operator if for every T > there exists a constant K T such that T y(t Y dt K T z X z D(A. (1.3 If C is bounded, i.e. if it can be extended such that C L(X, Y, then C is clearly an admissible observation operator. Definition 1.. System (1.1 (1. is exactly observable in time T if there exists k T > such that T y(t Y dt k T z X z D(A. (1.4 System (1.1 (1. is exactly observable if it is exactly observable in some time T >. The exact observability property is dual to the exact controllability property, as it has been shown in [9]. By using the above duality, the exact controllability of a system
3 K. Ramdani et al. / Journal of Functional Analysis 6 ( governed by partial differential equations reduces to the observability estimate (1.4 (called inverse inequality in [18]. Most of the literature tackling exact observability and exact controllability for systems governed by partial differential equations is based on a time domain approach. This means that one considers directly solutions of (1.1 (or of a dual equation which are manipulated in various ways: nonharmonic Fourier series ([] and references therein, multipliers method [16,18] or microlocal analysis techniques [6]. Only few papers in the area of controllability and observability of systems governed by partial differential equations have considered a frequency domain approach, related to the classical Hautus test in the theory of finite dimensional systems (see [1]. Roughly speaking, a frequency domain test for the observability of (1.1 (1. is formulated only in terms of the operators A, C and of a parameter (the frequency. This means that the time t does not appear in such a test and that we do not have to solve an evolution equation. In the case of a bounded observation operator C, such frequency domain methods have been proposed in [19,]. In the case of an unbounded observation operator C a Hautus-type test has been recently obtained in [8,1]. The aim of this paper is to use Hautus-type tests in order to characterize the exact observability property only in terms of C and of the spectral elements of the operator A. This will be done provided that the operator A has a compact resolvent and therefore, that the spectrum of A is formed only by eigenvalues. More precisely, since A is skewadjoint, it follows that the spectrum of A is given by σ(a = { iμ n n Λ } with Λ = Z or N and where (μ n n Λ is a sequence of real numbers. The main result of this paper reads as follows: Theorem 1.3. Assume that A is skew-adjoint with compact resolvent and that the operator C is admissible for system (1.1 (1.. Moreover, assume that (Φ n n Λ is an orthonormal sequence of eigenvectors of A associated to the eigenvalues (iμ n n Λ. For ω R and ε >, set J ε (ω = {m Λ such that μ m ω < ε}. (1.5 Then system (1.1, (1. is exactly observable if and only if one of the following equivalent assertions holds: (1 There exists ε > and δ > such that for all ω R and for all z = c m Φ m : m J ε (ω Cz Y δ z X. (1.6 ( There exists ε > and δ > such that for all n Z and for all z = c m Φ m : m J ε (μ n Cz Y δ z X. (1.7
4 196 K. Ramdani et al. / Journal of Functional Analysis 6 ( Remark 1.4. The above theorem can be seen as a generalization of several results in the literature. More precisely, in the particular case of a bounded observation operator C, the result in Theorem 1.3 follows, via a standard argument, from Theorem 3. in []. For unbounded C, but with the additional assumption that the sequence (μ n satisfies the gap condition (i.e., there exists γ > such that μ n μ m > γ for all m, n Λ, m = n, the necessity of condition (1.6 in Theorem 1.3 is a consequence of Theorem 4.4 from Russell and Weiss [3]. An important part of this paper is devoted to the application of the spectral criteria in Theorem 1.3 to systems governed by partial differential equations. The Schrödinger equation, the Bernoulli Euler plate equation and the wave equation in a square are considered. For the plate and Schrödinger equations, the main novelty brought in by our results is that we show that the exact observability property can hold for an arbitrarily small observed part of the boundary. More precisely, in the case of the plate equation, our observability result implies the following exact controllability result. Theorem 1.5. Consider the square Ω = (, π (, π and let Γ be an open subset of Ω. Consider the following control problem: ẅ + Δ w =, x Ω, t >, (1.8 w(x, t =, x Ω, t >, (1.9 Δw(x, t =, x Ω \ Γ, t >, (1.1 Δw(x, t = u, x Γ, t >, (1.11 w(x, = w (x, ẇ(x, = w 1 (x, x Ω, (1.1 where the input is the function u L (, T ; L (Γ. Then the following assertions are equivalent: (1 For all T >, the above system is exactly controllable in H 1 (Ω H 1 (Ω in time T. This means that, for all (w, w 1 H 1 (Ω H 1 (Ω, we can find u L (, T ; L (Γ such that w(x, T =, ẇ(x, T = x Ω. ( The control region Γ contains both a horizontal and a vertical segment of nonzero length. The proof of the above result is based on a consequence of Theorem 1.3 combined to a theorem of Beurling on nonharmonic Fourier series and to a new number theoretic result (a theorem on shifted squares. Let us mention that in the case of a control acting in an arbitrary open subset of the square Ω, an exact controllability result for the plate equation has been given in [13].
5 K. Ramdani et al. / Journal of Functional Analysis 6 ( Moreover, we consider a system governed by the wave equation in a square. We give a very simple proof (it uses only Parseval s theorem of the boundary observability of this system. The paper is organized as follows. Section is devoted to the proof of our main result, namely Theorem 1.3. In Section 3, this result is applied to study the boundary observability of Shrödinger equation in a square. The case of a Dirichlet boundary observation and the Neumann one are successively considered. In Section 4, we derive the counterpart of Theorem 1.3 for second-order systems (see Proposition 4.5. Thanks to this result, we tackle in Section 5 the problem of the boundary observability for the Bernoulli Euler plate equation in a square. A second application of the spectral criteria provided by Proposition 4.5 is detailed in Section 6. This application concerns the boundary observability of the wave equation in a square. Finally, Section 7 is devoted to the proof of Proposition 7.1, which is one of the main ingredients used to establish our observability results in Sections 3 and 5.. Proof of Theorem 1.3 The basic tool in the proof of Theorem 1.3 is a recent Hautus-type test. This result, given in [1], concerns the observability of systems with skew-adjoint generator and with unbounded observation operator. We first recall this result (see [8,1] for the proof. Theorem.1. Assume that A is a skew-adjoint operator in the Hilbert space X and that C : D(A Y is an admissible observation operator. Then system (1.1, (1. is exactly observable if and only if there exists a constant δ > such that (A iωiz X + Cz Y δ z X ω R z D(A. (.1 In order to prove Theorem 1.3, we need the following consequence of the admissibility property. Lemma.. Assume that the operators A and C are as in Theorem.1. Then there exists M > such that C(A I iωi 1 L(X,Y M ω R. (. Proof. Our proof is a slight variation of the proof of Proposition.3 in [3]. Let us fix T > and z X. Then, for any n N we have that nt n 1 e t/ CT t z Y dt (k+1t e kt CT t z Y dt. (.3 k= kt
6 198 K. Ramdani et al. / Journal of Functional Analysis 6 ( By using definition (1.3 of admissibility, combined to the fact that T is a group of isometries we have that (k+1t kt CT t z Y dt K T z X k N. The above inequality combined to (.3 implies that e t/ CT t z K Y dt T 1 e T z X z X. (.4 On the other hand, C(A I iωi 1 z Y = e t CT t z dt which, by applying the Cauchy Schwartz inequality, yields: C(A I iωi 1 z Y ( ( e t dt e t/ CT t z Y dt. Combined to (.4, the above relation clearly implies the desired conclusion (., with K T M =. 1 e T Y, We will also need the following result which can be seen as a generalization of Lemma 4.6 in [3]: Lemma.3. Assume that the operators A and C are as in Lemma.. For each ε > and ω R, we define the subspace V (ω X by V (ω = {Φ m m J ε (ω}, (.5 where J ε (ω is defined in (1.5. We denote by A ω the part of A in V (ω, i.e., and A ω : D(A V (ω V (ω A ω z = Az z D(A V (ω. Then, there exists M > such that C(A ω iωi 1 L(V (ω,y M ω R. (.6
7 K. Ramdani et al. / Journal of Functional Analysis 6 ( Proof. Given ω R, set s = 1 + iω. Then, thanks to the resolvent identity, we have (A ω iωi 1 = (A ω si 1 [ I (A ω iωi 1]. (.7 We first show that Indeed, let f = m/ J ε (ω (A ω iωi 1 L(V (ω 1 ε. (.8 f m Φ m be an element of V (ω. Then (A ω iωi 1 f = m/ J ε (ω f m μ m ω. The above relation and the fact that μ m ω ε for m / J ε (ω clearly imply (.8. On the other hand, we clearly have C(A ω si 1 L(V (ω,y C(A si 1 L(X,Y and thus, by using Lemma., we obtain that there exists a constant M > such that C(A ω si 1 L(V (ω,y M ω R. The above relation, (.7 and (.8 yield then (.6. We are now in a position to prove Theorem 1.3. Proof of Theorem 1.3. We first show that assertions (1 and ( in Theorem 1.3 are equivalent. It is clear that assertion (1 implies assertion ( (take ω = μ n. Conversely, assume that assertion ( holds true for some ε >, and let ω R. Then, either J ε/ (ω is empty, or there exists n J ε/ (ω and in this latter case, one can easily check that J ε/ (ω J ε (μ n. Consequently, in both cases, assertion (1 holds true. It remains to show that the exact observability of system (1.1, (1. is equivalent to assertion (1. To achieve this, we use the characterization of exact observability provided by Theorem.1. Assume that system (1.1, (1. is exactly observable. By Theorem.1, there exists a constant δ > such that for all ω R, and for all z D(A. (A iωiz X + Cz Y δ z X (.9
8 K. Ramdani et al. / Journal of Functional Analysis 6 ( On the other hand, for z = (A iωiz X = By applying (.9 to z = m J ε (ω m J ε (ω m J ε (ω c m Φ m and for ε small enough, we have that i(μ m ωc m ε z X δ z X. (.1 c m Φ m and by using (.1, we obtain that assertion (1 holds. Let us now assume that system (1.1, (1. is not exactly observable. Then, by Theorem.1, condition (.1 is not satisfied, i.e., there exists sequences (ω n n N in R and (z n n N in D(A such that z n X = 1 n N (.11 and satisfying lim n (A iω nz n X =, lim Cz n Y =. (.1 n We introduce then the following orthogonal decomposition of z n = m Λ c n m Φ m: z n = z n + z n (.13 with z n = m J ε (ω n c n m Φ m, z n = m/ J ε (ω n c n m Φ m, where ε > and J ε (ω is defined for all ω R by relation (1.5. Let us prove that the sequences (ω n n N and (z n n N contradict assertion (1. First of all, we note that the orthogonality of (A iω n z n and (A iω n z n implies that (A iω n z n X (A iω n z n X = The above relation and (.1 imply that m/ J ε (ω n (μ m ω n c n m ε z n X. lim n (A iω n z n X = (.14
9 K. Ramdani et al. / Journal of Functional Analysis 6 ( and that lim n z n X =. Thanks to (.11 and (.13, the above relation yields On the other hand, (.13 implies that lim n z n X = 1. (.15 Cz n Y Cz n Y + C z n Y. (.16 Moreover, using the notation of Lemma.3, we have C z n = C(A ωn iω n 1 (A ωn iω n z n. Consequently, Lemma.3 implies that there exists M > such that The above relation and (.14 imply that C z n Y M (A ωn iω n z n X n N. lim n C z n Y =. This fact, together with (.1 and (.16 yield lim n Cz n Y =. The above relation and (.15 show that the sequences (ω n n N and (z n n N contradict assertion (1 in Theorem Boundary observability of the Schrödinger equation in a square 3.1. Dirichlet boundary observation Consider the square Ω = (, π (, π and let Γ be an open subset of Ω. We consider the following initial and boundary value problem: ż + iδz =, x Ω, t, (3.1
10 K. Ramdani et al. / Journal of Functional Analysis 6 ( with the output z ν =, x Ω, t, (3. z(x, = z (x, x Ω (3.3 y = z Γ. (3.4 This system can be described by equations of form (1.1, (1., if we introduce the appropriate spaces and operators. We first define the state space X = L (Ω and the operator A : D(A X by D(A = { φ H (Ω } φ ν =, (3.5 Aφ = iδφ φ D(A. (3.6 We next define the output space Y = L (Γ and the observation operator C L(D(A, Y Cφ = φ Γ φ D(A. (3.7 Proposition 3.1. With the above notation, C is an admissible observation operator. In other words, for all T there exists a constant K T > such that the if z, y satisfy (3.1 (3.4 then T y dγ dt KT z L Γ (Ω z D(A. We skip the proof of the above result since it can be easily obtained from the Fourier series expansion of the solution of (3.1 (3.3. The main result in this subsection is: Proposition 3.. For any nonempty open subset Γ of Ω, the system described by (3.1 (3.4 is exactly observable. In other words there exists T > and a constant k T > such that if z, y satisfy (3.1 (3.4 then T z dγ dt kt z L Γ (Ω z D(A. Proof. We have seen in Proposition 3.1 that C is an admissible observation operator for (3.1 (3.4 in the sense of Definition 1.1. On the other hand, A is clearly skew-adjoint. Moreover, since the imbedding H 1 (Ω L (Ω is compact, A has a compact resolvent.
11 K. Ramdani et al. / Journal of Functional Analysis 6 ( Consequently, according to Theorem 1.3, it suffices to check that the operators A and C defined by (3.5, (3.6 and (3.7 satisfy condition ( in Theorem 1.3. The eigenvalues of A are μ m,n = i(m + n m, n N. A corresponding orthonormal basis of X = L (Ω formed by eigenfunctions of A is Φ m,n (x 1, x = π cos (mx 1 cos (nx m, n N. In order to check that condition ( in Theorem 1.3 holds, we have to show that there exists ε, δ > such that for all (q, r N N and for all z = c m,n Φ m,n, (m,n J ε (μ q,r we have Cz Y = c m,n Φ m,n Γ (m,n J ε (μ q,r dγ δ c m,n, (3.8 (m,n J ε (μ q,r where J ε (μ q,r = {(m, n N N ; (m + n q r < ε}. It is clear that if we choose ε < 1 then J ε (μ q,r = {(m, n N N ; m + n = q + r }. Moreover, without loss of generality, we can assume that there exists α, β (, π with α < β and (α, β {} Γ. Let S denote the set of squares of positive integers. For q, r N we set Λ qr = {m N q + r m S} (3.9 and for m Λ qr we put f (m = q + r m. (3.1
12 4 K. Ramdani et al. / Journal of Functional Analysis 6 ( We have Cz Y 4 β π c m,f (m cos(mx 1 α m Λ qr dx 1. (3.11 By using Proposition 7.1, the above relation implies that there exists a constant δ > such that 4 π β c m,f (m cos(mx 1 α m Λ qr dx 1 δ m +n =q +r c m,n. (3.1 From (3.11 and (3.1, we clearly get the desired estimate (3.8. By a standard duality argument, the above proposition implies that the following exact controllability holds. Corollary 3.3. For any nonempty open subset Γ of Ω, the system ż + iδz =, x Ω, t >, z ν =, x Ω \ Γ, t >, z ν = u L (, T ; L (Γ, x Γ, t >, z(x, = z (x, x Ω. is exactly controllable in some time T in the state space L (Ω. 3.. Neumann boundary observation The example studied in this subsection differs from the case considered in the previous one only by the boundary condition and the choice of the observation operator. We prove that, in order to get exact observability we need a supplementary assumption on the observed part of the boundary. Consider the square Ω = (, π (, π and let Γ be an open nonempty subset of Ω. We consider the following initial and boundary value problem: ż + iδz =, x Ω, t, (3.13 z =, x Ω, t, (3.14 z(x, = z (x, x Ω (3.15
13 K. Ramdani et al. / Journal of Functional Analysis 6 ( with the output y(t = z ν. (3.16 Γ The system can be described by equations of form (1.1, (1., if we introduce the appropriate spaces and operators. Indeed, let us first define the state space X = H 1 (Ω and the operator A : D(A X by D(A = {φ H 3 (Ω H 1 (Ω Δφ = on Ω}, (3.17 Aφ = iδφ φ D(A. (3.18 Next, we define the output space Y = L (Γ and the corresponding observation operator C L(D(A, Y by Cφ = φ ν φ D(A. (3.19 Γ Proposition 3.4. With the above notation, C is an admissible observation operator, i.e. for all T there exists a constant K T > such that if z, y satisfy (3.13 (3.16 then T y dγ dt KT z L Γ (Ω z D(A. The above result is classical (see, for instance, [17], so we skip its proof. The observability properties of system (3.13 (3.16 are different from those encountered in the study of system (3.1 (3.4. More precisely, if we denote by Γ 1 = ([, π] {} ([, π] {π} the horizontal part of Ω and by Γ = ({} [, π] ({π} [, π] the vertical part of Ω, then the following result holds: Proposition 3.5. The system described by (3.13 (3.16 is exactly observable if and only if Γ Γ i =, for i {1, }. In other words the following assertions are equivalent: (1 There exists T > and a constant k T > such that for all z, y satisfying (3.13 (3.16 we have T y dγ dt kt z H Γ 1 (Ω z D(A. ( The control region Γ contains both a horizontal and a vertical segment of nonzero length.
14 6 K. Ramdani et al. / Journal of Functional Analysis 6 ( Proof. We have seen in Proposition 3.1 that, for any open subset Γ of Ω, C is an admissible observation operator for (3.1 (3.4 in the sense of Definition 1.1. Moreover, A is clearly skew-adjoint and it has compact resolvent. Therefore, we can apply condition ( in Theorem 1.3. The eigenvalues of A are μ m,n = i(m + n m, n N. A corresponding family of normalized (in X = H 1 (Ω eigenfunctions are Φ m,n (x 1, x = π m + n sin (mx 1 sin (nx m, n N. We first show the necessity of condition Γ Γ i = for i = 1,. Indeed, if this condition fails then we can assume, without loss of generality, that Γ Γ 1. We notice that Consequently, CΦ n,1 Y Γ 1 Φ n,1 ν dγ = 8 π n π sin (nx 1 dx 1. (3. lim CΦ n,1 Y =, n which contradicts condition ( in Theorem 1.3. We next show that condition Γ Γ i = for i = 1, implies that the operators A and C defined by (3.17, (3.18 and (3.19 satisfy condition ( in Theorem 1.3. In this case, without loss of generality we can assume that Γ ( [α 1, β 1 ] {} ( {} [α, β ] with < α i < β i < π, for i {1, }. For q, r N, we recall the notation J ε (μ q,r = {(m, n N N ; (m + n q r < ε}. It is clear that if we choose ε < 1, then J ε (μ q,r = {(m, n N N ; m + n = q + r }.
15 If z = (m,n J ε (μ q,r K. Ramdani et al. / Journal of Functional Analysis 6 ( c m,n Φ m,n then Cz Y 4 π β1 α 1 β + α m Λ qr n Λ qr f (mc m,f (m m + f (m sin(mx 1 f (nc f (n,n f (n + n sin(nx dx 1 dx, (3.1 where Λ qr and f are defined in (3.9 and in (3.1. On the other hand, by using Proposition 7.1, we obtain that there exists a constant δ > such that β1 α 1 n Λ qr f (mc m,f (m m + f (m sin(mx 1 dx 1 δ = δ m Λ qr f (m f (m + m c m,f (m m +n =q +r n m + n c m,n and β α n Λ qr f (nc f (n,n f (n + n sin(nx dx δ = δ n Λ qr f (n f (n + n c f (n,n m +n =q +r m m + n c m,n. By taking the sum of the two above inequalities we get β1 α 1 δ m Λ qr f (mc m,f (m m + f (m sin(mx 1 β dx 1 + α n Λ qr f (nc f (n,n f (n + n sin(nx m +n =q +r c m,n. (3. By using (3.1 and (3. we obtain that Cz Y δ z X which concludes the proof. By a standard duality argument, the above proposition implies that the following exact controllability holds. dx
16 8 K. Ramdani et al. / Journal of Functional Analysis 6 ( Corollary 3.6. With the notation in Proposition 3.5, the system ż + iδz =, x Ω, t >, z =, x Ω \ Γ, t >, z = u L (, T ; L (Γ, x Γ, t >, z(x, = z (x, x Ω is exactly controllable in some time T > (in the state space X = H 1 (Ω if and only if Γ Γ i =, for i {1, }. Remark 3.7. It can be shown, by using techniques similar to those in [16,17], that the observability and the controllability results in this section hold for any T >. 4. Frequency domain tests for the exact observability of second-order systems In this section we investigate an important particular case fitting in the framework of Theorem 1.3. This case is obtained by considering second-order evolution equations occurring in the study of vibrating systems. More precisely, let H be a Hilbert space equipped with the norm and let A : D(A H be a self-adjoint, positive and boundedly invertible operator, with compact resolvent. Consider the initial value problem: ẅ(t + A w(t =, (4.1 w( = w, ẇ( = w 1, (4. which can be seen as a generic model for the free vibrations of elastic structures such as strings, ( beams, membranes, plates or three-dimensional elastic bodies. Moreover, let C L D(A 1, Y be an observation operator. We first show the equivalence of two conditions which will be used to define a concept of admissibility for observed systems described by second-order differential equations. Proposition 4.1. With the above notation, the following conditions are equivalent: (1 For every T > there exists a constant K T such that the solutions w of (4.1, (4. satisfy T C ẇ(t Y dt K T ( w D(A 1 + w 1 (w, w 1 D(A D(A 1. (4.3
17 K. Ramdani et al. / Journal of Functional Analysis 6 ( ( For every T > there exists a constant K T such that the solutions w of (4.1, (4. satisfy T C w(t Y dt K T ( w + w 1 D(A 1 w D(A 1 w 1 H, (4.4 where D(A 1 stands for the dual space of D(A 1 with respect to the pivot space H. Proof. We first show that assertion (1 implies assertion (. If w D(A 1, w 1 H then the solution w of (4.1, (4. satisfies w C([, T ]; D(A 1 C 1 ([, T ], H. Define Clearly, we have v(t = t w(s ds A 1 w 1. v(t + A v(t =, v( = A 1 w 1 D(A, v( = w D(A 1. Since we supposed that assertion (1 holds true it follows that T T C w(s Y ds = C v(s Y ds K T ( A 1 w 1 D(A 1 + w H for all (w, w 1 D(A D(A 1. Since A is an isometry from D(A 1 onto D(A 1, the above inequality implies that assertion ( holds true. We still have to show that assertion ( implies assertion (1. First, assume that w D(A 3, w 1 D(A. Then, the solution w of system (4.1, (4. satisfies w C 1 ([, T ]; D(A C ([, T ]; D(A 1. If we set v(t = ẇ(t then
18 1 K. Ramdani et al. / Journal of Functional Analysis 6 ( v C([, T ], D(A C 1 ([, T ], D(A 1 satisfies v(t + A v(t =, v( = w 1 D(A, v( = A w D(A 1. Since we supposed that assertion ( holds, we deduce that T T C ẇ(s Y ds = C v(s Y ds K T ( = K T ( w 1 + A w 1 [D(A ] w 1 + w [D(A 1 ] for all w D(A 3, w 1 D(A. A density argument shows that assertion (1 holds. In the remaining part of this paper we consider systems of form (4.1, (4. with one of the two following outputs: y = C w, (4.5 or y = C ẇ, (4.6 We are now in a position to give a definition of the admissibility for second-order problems: Definition 4.. C is an admissible observation operator for (4.1, (4. if it satisfies one of the equivalent conditions in Proposition 4.1. We next state the equivalence of two conditions which will be used in order to define a concept of exact observability for observed systems described by second-order differential equations. Proposition 4.3. With the notation in Proposition 4.1, the following conditions are equivalent:
19 K. Ramdani et al. / Journal of Functional Analysis 6 ( (1 For every T > there exists a constant k T > such that the solutions w of (4.1, (4. satisfy T C ẇ(t Y dt k T ( w D(A 1 + w 1 (w, w 1 D(A D(A 1. (4.7 ( For every T > there exists a constant k T > such that the solutions w of (4.1, (4. satisfy T C w(t Y dt k T ( w + w 1 D(A 1 w D(A 1 w 1 H. (4.8 We skip the proof of the above result since it is completely similar to the proof of Proposition 4.1. Definition 4.4. The system described by (4.1, (4. and (4.5 is exactly observable in time T if it satisfies one of the equivalent conditions in Proposition 4.3. We can now state the main result of this section. Proposition 4.5. Let A : D(A H be a self-adjoint, positive and boundedly invertible operator, with compact resolvent and let C L(D(A 1, Y be an admissible observation operator for (4.1 (4.. Let us denote by (λ n n N the increasing sequence formed by the eigenvalues of A 1 and by ( n n N a corresponding sequence of eigenvectors, forming an orthonormal basis of H. For all ω > and all ε >, let us define the set I ε (ω = {m N such that λ m ω < ε}. (4.9 Then, the following propositions are equivalent: (i System (4.1 (4.5 is exactly observable. (ii There exists a constant δ > such that φ D(A, ω > : (ω A φ + ω C φ Y δ ωφ. (4.1 (iii There exists ε > and δ > such that for all ω > and all φ = c m m : m I ε (ω C φ Y δ φ. (4.11
20 1 K. Ramdani et al. / Journal of Functional Analysis 6 ( (iv There exists ε > and δ > such that for all n N and all φ = c m m : m I ε (λ n C φ Y δ φ. (4.1 Remark 4.6. The fact that condition (ii in the above proposition is equivalent to the exact observability can be seen as a generalization of Theorem 3.4 in [19], where a similar Hautus-type result has been proved in the case of a bounded observation operator C. Proof of Proposition 4.5. It can be easily checked that system (4.1 (4.5 can be written ( in form (1.1 (1. provided that we define the state of the system by z(t = w(t and that we make the following choice of spaces and operators: ẇ(t X = D (A 1 H, A = ( I, C = ( C. (4.13 A The Hilbert space X is endowed here with the norm X defined by ( z X = A 1 φ φ + ψ z = X. ψ (i (ii. By Theorem.1 there exists a constant δ > such that (A iωz X + Cz Y δ z X ω R z D(A. (4.14 Taking in the above relation z = ( φ iωφ, where φ D(A 1, we obtain that Cz Y = ωc φ Y, z X ωφ while (A iωz X = (ω A φ.
21 K. Ramdani et al. / Journal of Functional Analysis 6 ( Therefore, (4.14 implies (4.1, and (ii holds true. (ii (iii. Let ε < λ 1. Then it is easy to see that if ω < ε, then I ε(ω = and (iii holds. If ω ε then for all φ = c m m, we have (ω A φ = m I ε (ω m I ε (ω ω λ m c m ε Consequently, for ε small enough and for φ = By applying condition (ii to φ = m I ε (ω m I ε (ω (ω A φ δ ωφ. m I ε (ω (ω + λ m c m 9ε ωφ. c m m, we have c m m and by using the above equation, we obtain (iii. (iii (iv. This implication obviously holds (take ω = λ n. (iv (i. In order to prove this assertion we use Theorem 1.3. Suppose that (iv holds true. Without loss of generality, we can assume that the constant ε in (iv satisfies ε < λ 1. Let A and C be defined by (4.13, it can be easily checked that the eigenvalues of A are (iμ n n Z where μ n = { λn if n N, λ n if ( n N. If we set n = n, for all n N, then an orthonormal family (in X of eigenvectors (Φ n n Z of A is given by ( Φ n = 1 1 iμ n n n n Z. In order to prove (i it suffices, by Theorem 1.3, to show that there exists ε > and δ > such that for all n Z and for all z = m J ε (μ n c m Φ m = 1 1 c m iμ m m J ε (μ n m (4.15 c m m m J ε (μ n
22 14 K. Ramdani et al. / Journal of Functional Analysis 6 ( we have Cz Y δ z X. Let us consider first the case where μ n > in (4.15. Then, μ n = λ n, and thus we have J ε (μ n = I ε (λ n (since ε < λ 1. Let us denote by φ the second component of z Then, it can be easily checked that φ = 1 m I ε (λ n c m m. Cz = C φ and that z X = φ. Consequently, by applying then (iv to φ we get that Cz Y δ z X. The case μ n < can be treated similarly, and the proof is thus complete. 5. Boundary observability for the Bernoulli Euler plate equation in a square Consider the square Ω = (, π (, π and let Γ be an open subset of Ω. We consider the following initial and boundary value problem: with the output ẅ + Δ w =, x Ω, t, (5.1 w(x, t = Δw(x, t =, x Ω, t, (5. w(x, = w (x, ẇ(x, = w 1 (x, x Ω (5.3 y(t = ẇ ν. (5.4 Γ
23 K. Ramdani et al. / Journal of Functional Analysis 6 ( System (5.1 (5.4 can be written in form (4.1, (4., (4.5. More precisely, we define H = H 1 (Ω, D(A = {φ H 5 (Ω H 1 (Ω Δφ = Δ φ = on Ω}, Y = L (Γ, A φ = Δ φ φ D(A. With the above choice of spaces and operators, one can easily check that A is selfadjoint, positive, boundedly invertible and that D(A 1 = {φ H 3 (Ω H 1 (Ω Δφ = on Ω}. Moreover, the dual space of D(A 1 with respect to the pivot space H is D(A 1 = H 1 (Ω. The output operator corresponding to (5.4 is C φ = φ ν φ D(A 1. Γ Proposition 5.1. With the above notation, C L(D(A 1, Y is an admissible observation operator, i.e. for all T there exists a constant K T > such that if w, y satisfy (5.1 (5.4 then T y dγ dt KT Γ for all (w, w 1 D(A D(A 1. ( w H 3 (Ω + w 1 H 1 (Ω The above result is classical and for its proof we refer, for instance, to [18, p. 87]. In order to state the observability properties of system (5.1 (5.4, let us denote by Γ 1 = ([, π] {} ([, π] {π} the horizontal part of Ω and by Γ = ({} [, π] ({π} [, π] its vertical part. Then, the following result holds: Proposition 5.. The system described by (5.1 (5.4 is exactly observable if and only if Γ Γ i =, for i {1, }. In other words, the following statements are equivalent (1 There exists T > and a constant k T > such that if z, y satisfy (5.1 (5.4 then T y dγ dt kt Γ (w, w 1 D(A D(A 1. ( w H 3 (Ω + w 1 H 1 (Ω
24 16 K. Ramdani et al. / Journal of Functional Analysis 6 ( ( The control region Γ contains both a horizontal and a vertical segment of nonzero length. Proof. By Proposition 5.1, C is an admissible observation operator for the system described by (5.1 (5.4. Moreover, the imbedding D(A 1 H is clearly compact. Consequently, we can apply Proposition 4.5. The eigenvalues of A 1 are λ m,n = m + n m, n N. A corresponding family of normalized (in H = H 1 (Ω eigenfunctions are m,n (x = π m + n sin (mx 1 sin (nx m, n N x = (x 1, x Ω. We first show the necessity of condition Γ Γ i = for i = 1,. If this condition fails then we can assume, without loss of generality, that Γ Γ 1. We notice that C n,1 Y Γ 1 n,1 ν dγ = 8 π n π sin (nx 1 dx 1. (5.5 Consequently, lim C n n,1 Y =, which contradicts condition (iii in Proposition 4.5. In the remaining part of the proof, we show that if Γ Γ i =, for i {1, }, then the operators A and C satisfy condition (iv in Proposition 4.5. More precisely, we prove that for all ε (, 1 there exists δ > such that, for all q, r N and for all φ = a m,n m,n, we have where (m,n I ε (λ q,r C φ Y = Γ (m,n I ε (λ q,r a m,n m,n ν dγ δ (m,n I ε (λ q,r a m,n, (5.6 I ε (λ q,r = {(m, n N N m + n = q + r }.
25 K. Ramdani et al. / Journal of Functional Analysis 6 ( Without loss of generality, we can assume that Γ ( [α 1, β 1 ] {} ( {} [α, β ] with < α i < β i < π, for i {1, }. Then, we have C φ Y 4 π β1 α 1 β + α m Λ qr n Λ qr f (ma m,f (m m + f (m sin(mx 1 f (na f (n,n f (n + n sin(nx dx 1 dx, (5.7 where the set Λ qr and the function f are defined in (3.9 and (3.1. The desired inequality (5.6 follows now directly from (3. which was established in the proof of Proposition 3.5. We conclude this section by remarking that Theorem 1.5 stated in Section 1 follows directly from the results already proved in this section. More precisely, the fact that the system is exactly controllable is some time T > follows from Proposition 5. by a standard duality argument. Showing that T can be chosen arbitrarily small can be achieved by slightly adapting a classical argument (see for instance [16, p. 81] or the appendix written by Zuazua in [18]. 6. Boundary observability of the wave equation in a square In this section, we consider the problem of observability of the wave equation with Neumann boundary observation for the wave equation. This problem has been tackled by a large number of papers by using various methods (see for instance [18] and references therein. However, besides the one-dimensional case, no direct Fourier seriesbased proof seems to exist in the literature. We give such a proof in the case where the space domain is a square. If we except the use of Proposition 4.5, the basic ingredients of the proof are very simple (we only need Parseval s theorem. Consider the square Ω = (, π (, π and let Γ = ([, π] {} ({} [, π]. We consider the following initial and boundary value problem: ẅ Δw =, x Ω, t, (6.1 w =, x Ω, t, (6. w(x, = w (x, ẇ(x, = w 1 (x, x Ω (6.3
26 18 K. Ramdani et al. / Journal of Functional Analysis 6 ( with the output y = w ν. (6.4 Γ System (6.1 (6.4 can be written in form (4.1 (4.5 if we introduce the following notation: H = H 1 (Ω, D(A = {φ H 3 (Ω H 1 (Ω Δφ = on Ω}, Y = L (Γ, A φ = Δφ C φ = φ ν Γ φ D(A, φ D(A. One can easily check that, with the above choice of the spaces and operators, we have that A is self-adjoint, positive and boundedly invertible and D(A 1 = H (Ω H 1 (Ω, D(A 1 = L (Ω. Proposition 6.1. With the above notation, C L(D(A 1, Y is an admissible observation operator, i.e. for all T there exists a constant K T > such that if w, y satisfy (6.1 (6.4 then T y dγ dt KT Γ ( w H 1(Ω + w 1 L (Ω for all (w, w 1 (H (Ω H 1 (Ω H 1 (Ω. The above proposition is classical (see, for instance, [18, p. 44], so we skip the proof. The main result of this section is the following. Theorem 6.. The system described by (6.1 (6.4 is exactly observable. In other words, there exists T > and k T > such that T y dγ dt kt Γ ( w H 1(Ω + w 1 L (Ω for all (w, w 1 (H (Ω H 1 (Ω H 1 (Ω.
27 K. Ramdani et al. / Journal of Functional Analysis 6 ( Proof. By Proposition 6.1, C is an admissible observation operator for the system described by (6.1 (6.4. Moreover, A is clearly self-adjoint, positive, and boundedly invertible, whereas the resolvent of A is clearly compact. Consequently, we can apply Proposition 4.5. The eigenvalues of A 1 are λ m,n = m + n m, n N. A corresponding family of normalized (in H = H 1 (Ω eigenfunctions are m,n (x = π m + n sin (mx 1 sin (nx m, n N x = (x 1, x Ω. (6.5 In the remaining part of the proof, we show that the operators A and C satisfy condition (iii in Proposition 4.5. More precisely, we prove that there exists ε, δ > such that for all ω > and for all φ = a m,n m,n, we have C φ Y = Γ (m,n I ε (ω a m,n m,n ν (m,n I ε (ω dγ δ (m,n I ε (ω a m,n, (6.6 where I ε (ω = {(m, n N N ; λ m,n ω < ε}. Let us introduce some notation. We first set K ε (ω = { m N n N with (m, n I ε (ω }. It is clear that if m K ε (ω then m < ω + ε. For m K ε (ω we introduce the set L(m defined by L(m = { n N (m, n I ε (ω } = {n N } m + n ω ε. (6.7 Then, we have L(m = {n N (ω ε m n } (ω + ε m (6.8
28 K. Ramdani et al. / Journal of Functional Analysis 6 ( if m ω ε and L(m = {n N n } (ω + ε m if ω ε < m < ω + ε. By using (6.5 and the above notation we get that C φ Y = 4 π π + 4 π π m K ε (ω n K ε (ω n L(m m L(n na mn sin(mx 1 m + n ma mn sin(nx m + n dx 1 dx. (6.9 We are going to prove that if ε (, 1 1 and if m K ε(ω satisfies m < (ω + ε/, then the cardinal κ m of the set L(m defined in (6.7 satisfies κ m = 1. Assume that ε (, 1 1 and m K ε(ω satisfies m < (ω + ε/. We first remark that (ω + ε/ ω ε. (6.1 Indeed, if the above inequality is not satisfied, then we get that ω < 9ε, and consequently, ω + ε < 1ε < 1. On the other hand, the fact that m K ε (ω implies that m < ω + ε < 1, which is a contradiction. We have thus shown that (6.1 holds. Consequently, L(m satisfies (6.8, and its cardinal κ m satisfies κ m (ω + ε m (ω ε m + 1 = 4ωε (ω + ε m (6.11 (ω ε m On the other hand, since m (ε + ω/, we have that and therefore, by (6.11, we obtain that (ω + ε m > ω κ m 4 ε + 1. Since ε (, 1 1, the above relation implies that κ m = 1 for all m K ε (ω such that m (ε + ω/. The unique element of L(m is then denoted by l m. This fact,
29 K. Ramdani et al. / Journal of Functional Analysis 6 ( combined to (6.9 and to the orthogonality of the family (sin(mx m 1 in L (, π, yields the existence of a constant δ > such that C φ Y π m Kε(ω m (ω+ε/ l m a mlm m + l m + π n Kε(ω n (ω+ε/ l n a ln n. l n + n The above relation and the fact that there exists C > such that for m (ε + ω/, we have l m m + l m C, implies the existence of δ > such that C φ Y δ m Kε(ω m (ω+ε/ amlm + n Kε(ω n (ω+ε/ aln n. (6.1 Using the fact that for all (m, n I ε (ω, we have either m ε + ω obtain that or n ε + ω, we (m,n I ε (ω a mn m Kε(ω m (ω+ε/ amlm + n Kε(ω n (ω+ε/ aln n. The desired inequality (6.6 follows then from the above relation, together with relation ( An Ingham Beurling-type result and a theorem on shifted squares The following result plays a central rôle in the proof of the observability results in Sections 3 and 5. Proposition 7.1. For q, r N, we set Λ qr = {m N q + r m S},
30 K. Ramdani et al. / Journal of Functional Analysis 6 ( where S denotes the set of squares of positive integers. Then, for any nonempty interval I, there exists a constant δ >, depending only on I, such that the inequality I a n e inx n Λ qr dx δ n Λ qr a n, holds for all sequence (a n l (C. The main ingredients of the proof of the above result are a version of a famous theorem of Beurling [7] on nonharmonic Fourier series, and a number theoretic theorem concerning shifted squares. Let us first state the version of Beurling s result given in Theorem 1.5 in [5]. For the proof, we refer to [5]. Theorem 7.. Let (λ n n Z be a strictly increasing sequence of real numbers such that λ n+1 λ n γ n Z for some γ >. Moreover, assume that exists γ γ and M N such that λ n+m λ n γm n N. Then, for any interval I of length l(i > π γ γ, γ, M and I, such that, there exists δ > depending only on I a n e iλ nx n Z holds for all sequence (a n l (C. dx δ a n n Z Note that, due to a misprint, the condition l(i > π γ written l(i > πγ in [5]. in the above theorem has been Remark 7.3. The result in Theorem 7. can be seen as a generalization of a classical inequality proved by Ingham [1]. For other generalizations and related questions we refer to Avdonin and Moran [1,3], Baiocchi et al. [4], Jaffard et al. [14] and Kahane [15]. Next, we give the second main ingredient of the proof of Proposition 7.1.
31 K. Ramdani et al. / Journal of Functional Analysis 6 ( Theorem 7.4. For positive integers M, N, V, let Z = Z(M, N, V denote the set of those integers n such that M < n M + N and V n is a square. For a suitable positive absolute constant C 1, we have Z C N log(n, where Z denotes the cardinality of the set Z. For the sake of clarity, we postpone the proof of this theorem to the end of this section. A useful consequence of Theorem 7.4 is the following. Corollary 7.5. Let M N and λ < λ 1 < < λ M be M + 1 consecutive elements of Λ qr. Then, we have λ M λ M C log (M, (7.1 where C is the constant appearing in Theorem 7.4. Proof. For N N, denote by U(N the cardinal number of the set By Theorem 7.4 we clearly have Consequently, {j 1 λ j λ + N}. U(N C N log (N N N. M = U(λ M λ C (λ M λ log [(λ M λ ]. Now observe that, since C 1, (7.1 plainly holds if λ M λ > M. Otherwise we have M C (λ M λ log (M, so that (7.1 is still valid. We are now in a position to prove Proposition 7.1. Proof of Proposition 7.1. Take γ > π, where l(i denotes the length of the interval l(i I. Since log(m 3 M for all M 1, Corollary 7.5 implies that if M > 9C 4 γ and
32 4 K. Ramdani et al. / Journal of Functional Analysis 6 ( if λ < λ 1 < < λ M are M + 1 consecutive elements of Λ qr, then λ M λ Mγ. Moreover, the distance between any two distinct elements of Λ qr is at least one. Therefore, we can apply Theorem 7. to get the desired inequality. In order to prove Theorem 7.4, we first introduce some notation. For any prime number p, let (Z/pZ be the (cyclic multiplicative group of invertible residues modulo p and let Q p denote the subset of (Z/pZ comprising all nonzero quadratic residues. Recall that the Legendre symbol is the mapping from Z onto { 1,, 1} defined by the formula ( n := p 1 if n Q p (mod p if p n 1 if n (Z/pZ \ Q p (mod p n Z. A classical result states that, for all odd primes p and all integers n such that p n, we have ( n n (p 1/ (mod p. (7. p This will be used in the proof of the following lemma. The result is known see, for instance, [, Exercise 3.3.] or [11, Theorem 7.8.] but, for convenience of the reader, we provide a short proof. Lemma 7.6. For any odd prime p and all a (Z/pZ, we have ( n + a = 1. p n<p Proof. Denote the sum on the left by S p (a. By (7., we have S p (a (n + a (p 1/ n<p ( (p 1/ n j a (p 1/ j n<p j (p 1/ ( j (p 1/ a (p 1/ j n j (mod p. j (p 1/ j n<p Now observe that the inner sum is zero modulo p unless when j = (p 1/, in which case it is 1. This is a well-known consequence of the fact that (Z/pZ is cyclic
33 K. Ramdani et al. / Journal of Functional Analysis 6 ( and we omit the details. We thus obtain S p (a 1 (mod p. Since S p (a p, this leaves the two possibilities S p (a = p 1 and S p (a = 1. However the former case can only happen if exactly one of the Legendre symbols is while all others have value 1. If this holds, then we have, for some integer h [, p[, ( h + a =. p Since p a, we must have h =. Thus h / p h (mod p and obviously ( (p h + a =, p a contradiction. Hence S p (a = 1, as required. We can now embark on the proof of Theorem 7.4. Proof of Theorem 7.4. Our initial strategy consists in showing that, for all primes p such that p 3 (mod 4, the subset E p of Z/pZ comprising those residue classes which contain at least one element of Z is small in size. We consider two cases, according to whether p V or not. To deal with the first instance, we observe that ( 1 = ( 1 (p 1/ = 1, p so n is not a quadratic residue modulo p if p n. Thus V n can only be a square if it is divisible by p and in fact by p. Therefore, we have E p = 1 if p V. In the second case, we have ( V n n E p = 1 or. p
34 6 K. Ramdani et al. / Journal of Functional Analysis 6 ( Since there are at most two solutions of the equation V n (mod p, we plainly derive E p n<p { 1 + ( V n } p ( 1 ( n V = 1 (p p n<p p = 1 (p + 3, where, in the last stage, we have appealed to Lemma 7.6. We have therefore shown that, for all primes p 3 (mod 4, the set Z is excluded from 1 (p 3 residue classes modulo p. By the large sieve (see e.g. [4, Corollary I.4.6.1] this yields, for all Q 1, Z (N + Q /L (7.3 with L = L(Q := g(q, 1 q Q where μ(q p 3 if p q p 3 (mod 4 p + 3 g(q := p q otherwise ( q 1. Here, as usual in number theory, the letter p denotes a generic prime number and q μ(q denotes the Möbius function. It remains to evaluate L as a function of Q. To this end, we introduce the Dirichlet series associated to g, viz G(s := q 1 g(q q s = p 3 (mod 4 ( 1 + p 3 (p + 3p s, where s is a complex parameter, with initially Res > 1. We need to express this quantity in terms of the Riemann zeta function ζ(s. This can be achieved by introducing the unique nonprincipal character modulo 4, defined by χ(p = ( 1 (p 1/, and the
35 corresponding L-function K. Ramdani et al. / Journal of Functional Analysis 6 ( L(s, χ := (n + 1 s = n p 3 ( 1 n ( 1 χ(p/p s 1. We have, still for Res > 1, G(s = ( {1 χ(p}(p (p + 3p s p 3 = p p 3(1 s 1/ (1 χ(pp s 1/ H (s = ζ(s 1/ L(s, χ 1/ (1 s 1/ H (s with H (s := p 3 = = (1 p s 1/ (1 χ(pp s 1/ ( 1 + p 3 (mod 4 p 3 (mod 4 ( 1 p s 1/ ( 1 + p 3 (p + 3p s 1 + p s ( 1 p s ( 1/ 1 6ps + p 3 (p + 3p s {1 χ(p}(p 3 (p + 3p s. Since L(s, χ 1/ has analytic continuation in the region σ 1 c/ log(3 + τ (s = σ + iτ for a suitable positive absolute constant c (see e.g. [4, notes on Sections II.8. and II.8.3] and since the product H (s converges for σ > 1 and is bounded in any halfplane σ 1 + δ with δ >, we are in a position to apply Selberg Delange type estimates, as given in [4, Theorem III.5.3]. This yields L(Q = g(q = AQ ( 1 } {1 + O q Q log Q log Q Q (7.4 with A := H (1 = πl(1, χ π p 3 (mod 4 ( 1 p ( 1/ 1 7p 3 p. (p + 3
36 8 K. Ramdani et al. / Journal of Functional Analysis 6 ( Inserting (7.4 into (7.3 and selecting Q := N furnishes the bound Z B { ( 1 } N log N 1 + O log N N with B := A = π p 3 (mod 4 ( 1 p ( 1/ 7p (p 1(p p 3 This finishes the proof of Theorem 7.4. References [1] S. Avdonin, W. Moran, Ingham-type inequalities and Riesz bases of divided differences, Internat. J. Appl. Math. Comput. Sci. 11 ( Mathematical methods of optimization and control of large-scale systems (Ekaterinburg,. [] S.-A. Avdonin, S.-A. Ivanov, Families of Exponentials The Method of Moments in Controllability Problems for Distributed Parameter Systems, Cambridge University Press, Cambridge, [3] S.A. Avdonin, W. Moran, Simultaneous control problems for systems of elastic strings and beams, Systems Control Lett. 44 ( [4] C. Baiocchi, V. Komornik, P. Loreti, Ingham type theorem and applications to control theory, Boll. Un. Mat. Ital. B 8 ( [5] C. Baiocchi, V. Komornik, P. Loreti, Ingham-Beurling type theorems with weakened gap conditions, Acta Math. Hungar. 97 ( [6] C. Bardos, G. Lebeau, J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control. Optim. 3 ( [7] A. Beurling, The collected works of Arne Beurling, in: L. Carleson, P. Malliavin, J. Neuberger, J. Wermer (Eds., Harmonic Analysis, Vol., Contemporary Mathematicians, Birkhäuser Boston Inc., Boston, MA, [8] N. Burq, M. Zworski, Geometric control in the presence of a black box, J. Amer. Math. Soc. 17 ( (electronic. [9] S. Dolecki, D.L. Russell, A general theory of observation and control, SIAM J. Control Optim. 15 ( [1] M. Hautus, Controllability and observability conditions of linear autonomous systems, Nederl. Akad. Wetensch. Proc. Ser. A 7 ( [11] L.K. Hua, Introduction to Number Theory, Springer, Berlin, Heidelberg, NY, 198. [1] A. Ingham, Some trigonometrical inequalities with applications to the theory of series, Math. Z. 41 ( [13] S. Jaffard, Contrôle interne exact des vibrations d une plaque rectangulaire, Portugal Math. 47 ( [14] S. Jaffard, M. Tucsnak, E. Zuazua, On a theorem of Ingham, J. Fourier Anal. Appl. 3 ( [15] J.-P. Kahane, Pseudo-périodicité et séries de Fourier lacunaires, Ann. Sci. École Norm. Sup. 79 (3 ( [16] V. Komornik, Exact Controllability and Stabilization The Multiplier Method, Wiley and Masson, Chichester and Paris, [17] G. Lebeau, Contrôle de l équation de Schrödinger, J. Math. Pures Appl. 71 (9 (
37 K. Ramdani et al. / Journal of Functional Analysis 6 ( [18] J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, Recherches en Mathématiques Appliquées, Vol. 8, Masson, Paris, [19] K. Liu, Locally distributed control and damping for the conservative systems, SIAM J. Control Optim. 35 ( [] K. Liu, Z. Liu, B. Rao, Exponential stability of an abstract nondissipative linear system, SIAM J. Control Optim. 4 ( [1] L. Miller, Controllability cost of conservative systems: resolvent condition and transmutation, J. Funct. Anal., 18 ( ( [] I. Niven, H.S. Zuckerman, H.L. Montgomery, An Introduction to the Theory of Numbers, fifth ed., Wiley, New York, [3] D.L. Russell, G. Weiss, A general necessary condition for exact observability, SIAM J. Control Optim. 3 ( [4] G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge Studies in Advanced Mathematics, Vol. 46, Cambridge University Press, Cambridge, [5] M. Tucsnak, G. Weiss, Simultaneous exact controllability and some applications, SIAM J. Control Optim. 38 ( [6] G. Weiss, Admissible observation operators for linear semigroups, Israel J. Math. 65 (
A spectral approach for the exact observability of infinite dimensional systems with skew-adjoint generator.
A spectral approach for the exact observability of infinite dimensional systems with skew-adjoint generator. Karim Ramdani, Takeo Takahashi, Gérald Tenenbaum, Marius Tucsnak To cite this version: Karim
More informationSpectrum and Exact Controllability of a Hybrid System of Elasticity.
Spectrum and Exact Controllability of a Hybrid System of Elasticity. D. Mercier, January 16, 28 Abstract We consider the exact controllability of a hybrid system consisting of an elastic beam, clamped
More informationA remark on the observability of conservative linear systems
A remark on the observability of conservative linear systems Enrique Zuazua Abstract. We consider abstract conservative evolution equations of the form ż = Az, where A is a skew-adjoint operator. We analyze
More informationOn the bang-bang property of time optimal controls for infinite dimensional linear systems
On the bang-bang property of time optimal controls for infinite dimensional linear systems Marius Tucsnak Université de Lorraine Paris, 6 janvier 2012 Notation and problem statement (I) Notation: X (the
More informationExponential stabilization of a Rayleigh beam - actuator and feedback design
Exponential stabilization of a Rayleigh beam - actuator and feedback design George WEISS Department of Electrical and Electronic Engineering Imperial College London London SW7 AZ, UK G.Weiss@imperial.ac.uk
More informationHilbert Uniqueness Method and regularity
Hilbert Uniqueness Method and regularity Sylvain Ervedoza 1 Joint work with Enrique Zuazua 2 1 Institut de Mathématiques de Toulouse & CNRS 2 Basque Center for Applied Mathematics Institut Henri Poincaré
More informationNonlinear stabilization via a linear observability
via a linear observability Kaïs Ammari Department of Mathematics University of Monastir Joint work with Fathia Alabau-Boussouira Collocated feedback stabilization Outline 1 Introduction and main result
More informationControllability of linear PDEs (I): The wave equation
Controllability of linear PDEs (I): The wave equation M. González-Burgos IMUS, Universidad de Sevilla Doc Course, Course 2, Sevilla, 2018 Contents 1 Introduction. Statement of the problem 2 Distributed
More informationConservative Control Systems Described by the Schrödinger Equation
Conservative Control Systems Described by the Schrödinger Equation Salah E. Rebiai Abstract An important subclass of well-posed linear systems is formed by the conservative systems. A conservative system
More informationOn the observability of time-discrete conservative linear systems
On the observability of time-discrete conservative linear systems Sylvain Ervedoza, Chuang Zheng and Enrique Zuazua Abstract. We consider various time discretization schemes of abstract conservative evolution
More informationAnalysis Preliminary Exam Workshop: Hilbert Spaces
Analysis Preliminary Exam Workshop: Hilbert Spaces 1. Hilbert spaces A Hilbert space H is a complete real or complex inner product space. Consider complex Hilbert spaces for definiteness. If (, ) : H H
More informationRecall that any inner product space V has an associated norm defined by
Hilbert Spaces Recall that any inner product space V has an associated norm defined by v = v v. Thus an inner product space can be viewed as a special kind of normed vector space. In particular every inner
More informationFinite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product
Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )
More informationObservation and Control for Operator Semigroups
Birkhäuser Advanced Texts Basler Lehrbücher Observation and Control for Operator Semigroups Bearbeitet von Marius Tucsnak, George Weiss Approx. 496 p. 2009. Buch. xi, 483 S. Hardcover ISBN 978 3 7643 8993
More informationBoundedly complete weak-cauchy basic sequences in Banach spaces with the PCP
Journal of Functional Analysis 253 (2007) 772 781 www.elsevier.com/locate/jfa Note Boundedly complete weak-cauchy basic sequences in Banach spaces with the PCP Haskell Rosenthal Department of Mathematics,
More informationInégalités spectrales pour le contrôle des EDP linéaires : groupe de Schrödinger contre semigroupe de la chaleur.
Inégalités spectrales pour le contrôle des EDP linéaires : groupe de Schrödinger contre semigroupe de la chaleur. Luc Miller Université Paris Ouest Nanterre La Défense, France Pde s, Dispersion, Scattering
More informationAsymptotic Behavior of a Hyperbolic-parabolic Coupled System Arising in Fluid-structure Interaction
International Series of Numerical Mathematics, Vol. 154, 445 455 c 2006 Birkhäuser Verlag Basel/Switzerland Asymptotic Behavior of a Hyperbolic-parabolic Coupled System Arising in Fluid-structure Interaction
More informationA quantitative Fattorini-Hautus test: the minimal null control time problem in the parabolic setting
A quantitative Fattorini-Hautus test: the minimal null control time problem in the parabolic setting Morgan MORANCEY I2M, Aix-Marseille Université August 2017 "Controllability of parabolic equations :
More informationOptimal shape and position of the support for the internal exact control of a string
Optimal shape and position of the support for the internal exact control of a string Francisco Periago Abstract In this paper, we consider the problem of optimizing the shape and position of the support
More informationCompact perturbations of controlled systems
Compact perturbations of controlled systems Michel Duprez, Guillaume Olive To cite this version: Michel Duprez, Guillaume Olive. Compact perturbations of controlled systems. Mathematical Control and Related
More informationSimultaneous boundary control of a Rao-Nakra sandwich beam
Simultaneous boundary control of a Rao-Nakra sandwich beam Scott W. Hansen and Rajeev Rajaram Abstract We consider the problem of boundary control of a system of three coupled partial differential equations
More informationRemarks on the Rademacher-Menshov Theorem
Remarks on the Rademacher-Menshov Theorem Christopher Meaney Abstract We describe Salem s proof of the Rademacher-Menshov Theorem, which shows that one constant works for all orthogonal expansions in all
More informationStability of an abstract wave equation with delay and a Kelvin Voigt damping
Stability of an abstract wave equation with delay and a Kelvin Voigt damping University of Monastir/UPSAY/LMV-UVSQ Joint work with Serge Nicaise and Cristina Pignotti Outline 1 Problem The idea Stability
More informationNew phenomena for the null controllability of parabolic systems: Minim
New phenomena for the null controllability of parabolic systems F.Ammar Khodja, M. González-Burgos & L. de Teresa Aix-Marseille Université, CNRS, Centrale Marseille, l2m, UMR 7373, Marseille, France assia.benabdallah@univ-amu.fr
More information1 Math 241A-B Homework Problem List for F2015 and W2016
1 Math 241A-B Homework Problem List for F2015 W2016 1.1 Homework 1. Due Wednesday, October 7, 2015 Notation 1.1 Let U be any set, g be a positive function on U, Y be a normed space. For any f : U Y let
More informationStabilization and Controllability for the Transmission Wave Equation
1900 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 46, NO. 12, DECEMBER 2001 Stabilization Controllability for the Transmission Wave Equation Weijiu Liu Abstract In this paper, we address the problem of
More informationSPECTRAL PROPERTIES OF THE LAPLACIAN ON BOUNDED DOMAINS
SPECTRAL PROPERTIES OF THE LAPLACIAN ON BOUNDED DOMAINS TSOGTGEREL GANTUMUR Abstract. After establishing discrete spectra for a large class of elliptic operators, we present some fundamental spectral properties
More informationAbstract In this paper, we consider bang-bang property for a kind of timevarying. time optimal control problem of null controllable heat equation.
JOTA manuscript No. (will be inserted by the editor) The Bang-Bang Property of Time-Varying Optimal Time Control for Null Controllable Heat Equation Dong-Hui Yang Bao-Zhu Guo Weihua Gui Chunhua Yang Received:
More informationMath 259: Introduction to Analytic Number Theory Primes in arithmetic progressions: Dirichlet characters and L-functions
Math 259: Introduction to Analytic Number Theory Primes in arithmetic progressions: Dirichlet characters and L-functions Dirichlet extended Euler s analysis from π(x) to π(x, a mod q) := #{p x : p is a
More informationHilbert space methods for quantum mechanics. S. Richard
Hilbert space methods for quantum mechanics S. Richard Spring Semester 2016 2 Contents 1 Hilbert space and bounded linear operators 5 1.1 Hilbert space................................ 5 1.2 Vector-valued
More informationRolle s Theorem for Polynomials of Degree Four in a Hilbert Space 1
Journal of Mathematical Analysis and Applications 265, 322 33 (2002) doi:0.006/jmaa.200.7708, available online at http://www.idealibrary.com on Rolle s Theorem for Polynomials of Degree Four in a Hilbert
More informationProducts of ratios of consecutive integers
Products of ratios of consecutive integers Régis de la Bretèche, Carl Pomerance & Gérald Tenenbaum 27/1/23, 9h26 For Jean-Louis Nicolas, on his sixtieth birthday 1. Introduction Let {ε n } 1 n
More information1 Compact and Precompact Subsets of H
Compact Sets and Compact Operators by Francis J. Narcowich November, 2014 Throughout these notes, H denotes a separable Hilbert space. We will use the notation B(H) to denote the set of bounded linear
More informationThis article was published in an Elsevier journal. The attached copy is furnished to the author for non-commercial research and education use, including for instruction at the author s institution, sharing
More informationCESARO OPERATORS ON THE HARDY SPACES OF THE HALF-PLANE
CESARO OPERATORS ON THE HARDY SPACES OF THE HALF-PLANE ATHANASIOS G. ARVANITIDIS AND ARISTOMENIS G. SISKAKIS Abstract. In this article we study the Cesàro operator C(f)() = d, and its companion operator
More informationThe primitive root theorem
The primitive root theorem Mar Steinberger First recall that if R is a ring, then a R is a unit if there exists b R with ab = ba = 1. The collection of all units in R is denoted R and forms a group under
More informationarxiv: v1 [math.ap] 27 Nov 2018
arxiv:1811.11571v1 [math.ap] 27 Nov 2018 Internal observability of the wave equation in tiled domains Anna Chiara Lai Dipartimento di Scienze di Base e Applicate per l Ingegneria, Sapienza Università di
More informationDecomposition of Riesz frames and wavelets into a finite union of linearly independent sets
Decomposition of Riesz frames and wavelets into a finite union of linearly independent sets Ole Christensen, Alexander M. Lindner Abstract We characterize Riesz frames and prove that every Riesz frame
More informationObservation problems related to string vibrations. Outline of Ph.D. Thesis
Observation problems related to string vibrations Outline of Ph.D. Thesis András Lajos Szijártó Supervisor: Dr. Ferenc Móricz Emeritus Professor Advisor: Dr. Jenő Hegedűs Associate Professor Doctoral School
More informationarxiv: v1 [math.oc] 22 Sep 2016
EUIVALENCE BETWEEN MINIMAL TIME AND MINIMAL NORM CONTROL PROBLEMS FOR THE HEAT EUATION SHULIN IN AND GENGSHENG WANG arxiv:1609.06860v1 [math.oc] 22 Sep 2016 Abstract. This paper presents the equivalence
More informationThe spectral zeta function
The spectral zeta function Bernd Ammann June 4, 215 Abstract In this talk we introduce spectral zeta functions. The spectral zeta function of the Laplace-Beltrami operator was already introduced by Minakshisundaram
More informationStrong Stabilization of the System of Linear Elasticity by a Dirichlet Boundary Feedback
To appear in IMA J. Appl. Math. Strong Stabilization of the System of Linear Elasticity by a Dirichlet Boundary Feedback Wei-Jiu Liu and Miroslav Krstić Department of AMES University of California at San
More informationControl, Stabilization and Numerics for Partial Differential Equations
Paris-Sud, Orsay, December 06 Control, Stabilization and Numerics for Partial Differential Equations Enrique Zuazua Universidad Autónoma 28049 Madrid, Spain enrique.zuazua@uam.es http://www.uam.es/enrique.zuazua
More informationPart II. Number Theory. Year
Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 Paper 3, Section I 1G 70 Explain what is meant by an Euler pseudoprime and a strong pseudoprime. Show that 65 is an Euler
More informationWhere is matrix multiplication locally open?
Linear Algebra and its Applications 517 (2017) 167 176 Contents lists available at ScienceDirect Linear Algebra and its Applications www.elsevier.com/locate/laa Where is matrix multiplication locally open?
More informationOn the Three-Phase-Lag Heat Equation with Spatial Dependent Lags
Nonlinear Analysis and Differential Equations, Vol. 5, 07, no., 53-66 HIKARI Ltd, www.m-hikari.com https://doi.org/0.988/nade.07.694 On the Three-Phase-Lag Heat Equation with Spatial Dependent Lags Yang
More informationOn Riesz-Fischer sequences and lower frame bounds
On Riesz-Fischer sequences and lower frame bounds P. Casazza, O. Christensen, S. Li, A. Lindner Abstract We investigate the consequences of the lower frame condition and the lower Riesz basis condition
More informationc 2009 Society for Industrial and Applied Mathematics
SIAM J. CONTROL OPTIM. Vol. 48, No. 3, pp. 163 1659 c 9 Society for Industrial and Applied Mathematics SOLVING INVERSE SOURCE PROBLEMS USING OBSERVABILITY. APPLICATIONS TO THE EULER BERNOULLI PLATE EQUATION
More informationMATH 117 LECTURE NOTES
MATH 117 LECTURE NOTES XIN ZHOU Abstract. This is the set of lecture notes for Math 117 during Fall quarter of 2017 at UC Santa Barbara. The lectures follow closely the textbook [1]. Contents 1. The set
More informationWaves on 2 and 3 dimensional domains
Chapter 14 Waves on 2 and 3 dimensional domains We now turn to the studying the initial boundary value problem for the wave equation in two and three dimensions. In this chapter we focus on the situation
More informationCzechoslovak Mathematical Journal
Czechoslovak Mathematical Journal Oktay Duman; Cihan Orhan µ-statistically convergent function sequences Czechoslovak Mathematical Journal, Vol. 54 (2004), No. 2, 413 422 Persistent URL: http://dml.cz/dmlcz/127899
More informationDiscreteness of Transmission Eigenvalues via Upper Triangular Compact Operators
Discreteness of Transmission Eigenvalues via Upper Triangular Compact Operators John Sylvester Department of Mathematics University of Washington Seattle, Washington 98195 U.S.A. June 3, 2011 This research
More informationVectors in Function Spaces
Jim Lambers MAT 66 Spring Semester 15-16 Lecture 18 Notes These notes correspond to Section 6.3 in the text. Vectors in Function Spaces We begin with some necessary terminology. A vector space V, also
More informationL p MAXIMAL REGULARITY FOR SECOND ORDER CAUCHY PROBLEMS IS INDEPENDENT OF p
L p MAXIMAL REGULARITY FOR SECOND ORDER CAUCHY PROBLEMS IS INDEPENDENT OF p RALPH CHILL AND SACHI SRIVASTAVA ABSTRACT. If the second order problem ü + B u + Au = f has L p maximal regularity for some p
More informationDirichlet s Theorem. Martin Orr. August 21, The aim of this article is to prove Dirichlet s theorem on primes in arithmetic progressions:
Dirichlet s Theorem Martin Orr August 1, 009 1 Introduction The aim of this article is to prove Dirichlet s theorem on primes in arithmetic progressions: Theorem 1.1. If m, a N are coprime, then there
More informationRIESZ BASES AND UNCONDITIONAL BASES
In this paper we give a brief introduction to adjoint operators on Hilbert spaces and a characterization of the dual space of a Hilbert space. We then introduce the notion of a Riesz basis and give some
More informationSingle input controllability of a simplified fluid-structure interaction model
Single input controllability of a simplified fluid-structure interaction model Yuning Liu Institut Elie Cartan, Nancy Université/CNRS/INRIA BP 739, 5456 Vandoeuvre-lès-Nancy, France liuyuning.math@gmail.com
More informationSwitching, sparse and averaged control
Switching, sparse and averaged control Enrique Zuazua Ikerbasque & BCAM Basque Center for Applied Mathematics Bilbao - Basque Country- Spain zuazua@bcamath.org http://www.bcamath.org/zuazua/ WG-BCAM, February
More informationStabilization of Heat Equation
Stabilization of Heat Equation Mythily Ramaswamy TIFR Centre for Applicable Mathematics, Bangalore, India CIMPA Pre-School, I.I.T Bombay 22 June - July 4, 215 Mythily Ramaswamy Stabilization of Heat Equation
More informationThe Dirichlet-to-Neumann operator
Lecture 8 The Dirichlet-to-Neumann operator The Dirichlet-to-Neumann operator plays an important role in the theory of inverse problems. In fact, from measurements of electrical currents at the surface
More informationUNIFORM DISTRIBUTION MODULO 1 AND THE UNIVERSALITY OF ZETA-FUNCTIONS OF CERTAIN CUSP FORMS. Antanas Laurinčikas
PUBLICATIONS DE L INSTITUT MATHÉMATIQUE Nouvelle série, tome 00(4) (206), 3 40 DOI: 0.2298/PIM643L UNIFORM DISTRIBUTION MODULO AND THE UNIVERSALITY OF ZETA-FUNCTIONS OF CERTAIN CUSP FORMS Antanas Laurinčikas
More informationAveraged controllability of parameter dependent wave equations
Averaged controllability of parameter dependent wave equations Jérôme Lohéac a and Enrique Zuazua b,c a LUNAM Université, ICCyN UM CNS 6597 Institut de echerche en Communications et Cybernétique de Nantes),
More informationFourier Series. 1. Review of Linear Algebra
Fourier Series In this section we give a short introduction to Fourier Analysis. If you are interested in Fourier analysis and would like to know more detail, I highly recommend the following book: Fourier
More informationThe universality of quadratic L-series for prime discriminants
ACTA ARITHMETICA 3. 006) The universality of quadratic L-series for prime discriminants by Hidehiko Mishou Nagoya) and Hirofumi Nagoshi Niigata). Introduction and statement of results. For an odd prime
More informationAffine and Quasi-Affine Frames on Positive Half Line
Journal of Mathematical Extension Vol. 10, No. 3, (2016), 47-61 ISSN: 1735-8299 URL: http://www.ijmex.com Affine and Quasi-Affine Frames on Positive Half Line Abdullah Zakir Husain Delhi College-Delhi
More informationCOMPACT OPERATORS. 1. Definitions
COMPACT OPERATORS. Definitions S:defi An operator M : X Y, X, Y Banach, is compact if M(B X (0, )) is relatively compact, i.e. it has compact closure. We denote { E:kk (.) K(X, Y ) = M L(X, Y ), M compact
More informationTraces and Duality Lemma
Traces and Duality Lemma Recall the duality lemma with H / ( ) := γ 0 (H ()) defined as the trace space of H () endowed with minimal extension norm; i.e., for w H / ( ) L ( ), w H / ( ) = min{ ŵ H () ŵ
More informationarxiv: v3 [math.ac] 29 Aug 2018
ON THE LOCAL K-ELASTICITIES OF PUISEUX MONOIDS MARLY GOTTI arxiv:1712.00837v3 [math.ac] 29 Aug 2018 Abstract. If M is an atomic monoid and x is a nonzero non-unit element of M, then the set of lengths
More informationBoundary Stabilization of Coupled Plate Equations
Palestine Journal of Mathematics Vol. 2(2) (2013), 233 242 Palestine Polytechnic University-PPU 2013 Boundary Stabilization of Coupled Plate Equations Sabeur Mansouri Communicated by Kais Ammari MSC 2010:
More informationHILBERT SPACES AND THE RADON-NIKODYM THEOREM. where the bar in the first equation denotes complex conjugation. In either case, for any x V define
HILBERT SPACES AND THE RADON-NIKODYM THEOREM STEVEN P. LALLEY 1. DEFINITIONS Definition 1. A real inner product space is a real vector space V together with a symmetric, bilinear, positive-definite mapping,
More informationOn the simplest expression of the perturbed Moore Penrose metric generalized inverse
Annals of the University of Bucharest (mathematical series) 4 (LXII) (2013), 433 446 On the simplest expression of the perturbed Moore Penrose metric generalized inverse Jianbing Cao and Yifeng Xue Communicated
More informationEIGENVALUES AND EIGENVECTORS OF SEMIGROUP GENERATORS OBTAINED FROM DIAGONAL GENERATORS BY FEEDBACK
COMMUNICATIONS IN INFORMATION AND SYSTEMS c 211 International Press Vol. 11, No. 1, pp. 71-14, 211 5 EIGENVALUES AND EIGENVECTORS OF SEMIGROUP GENERATORS OBTAINED FROM DIAGONAL GENERATORS BY FEEDBACK CHENG-ZHONG
More informationOn The Weights of Binary Irreducible Cyclic Codes
On The Weights of Binary Irreducible Cyclic Codes Yves Aubry and Philippe Langevin Université du Sud Toulon-Var, Laboratoire GRIM F-83270 La Garde, France, {langevin,yaubry}@univ-tln.fr, WWW home page:
More informationNonarchimedean Cantor set and string
J fixed point theory appl Online First c 2008 Birkhäuser Verlag Basel/Switzerland DOI 101007/s11784-008-0062-9 Journal of Fixed Point Theory and Applications Nonarchimedean Cantor set and string Michel
More informationThis document is downloaded from DR-NTU, Nanyang Technological University Library, Singapore.
This document is downloaded from DR-NTU, Nanyang Technological University Library, Singapore. Title Composition operators on Hilbert spaces of entire functions Author(s) Doan, Minh Luan; Khoi, Le Hai Citation
More informationLong Time Behavior of a Coupled Heat-Wave System Arising in Fluid-Structure Interaction
ARMA manuscript No. (will be inserted by the editor) Long Time Behavior of a Coupled Heat-Wave System Arising in Fluid-Structure Interaction Xu Zhang, Enrique Zuazua Contents 1. Introduction..................................
More informationUniform polynomial stability of C 0 -Semigroups
Uniform polynomial stability of C 0 -Semigroups LMDP - UMMISCO Departement of Mathematics Cadi Ayyad University Faculty of Sciences Semlalia Marrakech 14 February 2012 Outline 1 2 Uniform polynomial stability
More informationMathematics 324 Riemann Zeta Function August 5, 2005
Mathematics 324 Riemann Zeta Function August 5, 25 In this note we give an introduction to the Riemann zeta function, which connects the ideas of real analysis with the arithmetic of the integers. Define
More informationSTOKES PROBLEM WITH SEVERAL TYPES OF BOUNDARY CONDITIONS IN AN EXTERIOR DOMAIN
Electronic Journal of Differential Equations, Vol. 2013 2013, No. 196, pp. 1 28. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu STOKES PROBLEM
More informationYour first day at work MATH 806 (Fall 2015)
Your first day at work MATH 806 (Fall 2015) 1. Let X be a set (with no particular algebraic structure). A function d : X X R is called a metric on X (and then X is called a metric space) when d satisfies
More informationThe effect of Group Velocity in the numerical analysis of control problems for the wave equation
The effect of Group Velocity in the numerical analysis of control problems for the wave equation Fabricio Macià École Normale Supérieure, D.M.A., 45 rue d Ulm, 753 Paris cedex 5, France. Abstract. In this
More informationPERIODIC DAMPING GIVES POLYNOMIAL ENERGY DECAY. 1. Introduction Consider the damped Klein Gordon equation on [0, ) R n :
PERIODIC DAMPING GIVES POLYNOMIAL ENERGY DECAY JARED WUNSCH Abstract. Let u solve the damped Klein Gordon equation 2 t 2 x j + m Id +γx) t ) u = on R n with m > and γ bounded below on a 2πZ n -invariant
More informationMathematica Bohemica
Mathematica Bohemica Cristian Bereanu; Jean Mawhin Existence and multiplicity results for nonlinear second order difference equations with Dirichlet boundary conditions Mathematica Bohemica, Vol. 131 (2006),
More informationEstimates for the resolvent and spectral gaps for non self-adjoint operators. University Bordeaux
for the resolvent and spectral gaps for non self-adjoint operators 1 / 29 Estimates for the resolvent and spectral gaps for non self-adjoint operators Vesselin Petkov University Bordeaux Mathematics Days
More informationSome recent results on controllability of coupled parabolic systems: Towards a Kalman condition
Some recent results on controllability of coupled parabolic systems: Towards a Kalman condition F. Ammar Khodja Clermont-Ferrand, June 2011 GOAL: 1 Show the important differences between scalar and non
More informationFunctional Analysis. Franck Sueur Metric spaces Definitions Completeness Compactness Separability...
Functional Analysis Franck Sueur 2018-2019 Contents 1 Metric spaces 1 1.1 Definitions........................................ 1 1.2 Completeness...................................... 3 1.3 Compactness......................................
More informationEXISTENCE OF SOLUTIONS TO ASYMPTOTICALLY PERIODIC SCHRÖDINGER EQUATIONS
Electronic Journal of Differential Equations, Vol. 017 (017), No. 15, pp. 1 7. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS TO ASYMPTOTICALLY PERIODIC
More informationSubsequences of frames
Subsequences of frames R. Vershynin February 13, 1999 Abstract Every frame in Hilbert space contains a subsequence equivalent to an orthogonal basis. If a frame is n-dimensional then this subsequence has
More informationALMOST PERIODIC SOLUTIONS OF HIGHER ORDER DIFFERENTIAL EQUATIONS ON HILBERT SPACES
Electronic Journal of Differential Equations, Vol. 21(21, No. 72, pp. 1 12. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu ALMOST PERIODIC SOLUTIONS
More informationMath 350 Fall 2011 Notes about inner product spaces. In this notes we state and prove some important properties of inner product spaces.
Math 350 Fall 2011 Notes about inner product spaces In this notes we state and prove some important properties of inner product spaces. First, recall the dot product on R n : if x, y R n, say x = (x 1,...,
More informationOn the digital representation of smooth numbers
On the digital representation of smooth numbers Yann BUGEAUD and Haime KANEKO Abstract. Let b 2 be an integer. Among other results, we establish, in a quantitative form, that any sufficiently large integer
More informationEQUIVALENT CONDITIONS FOR EXPONENTIAL STABILITY FOR A SPECIAL CLASS OF CONSERVATIVE LINEAR SYSTEMS
EQUIVALENT CONDITIONS FOR EXPONENTIAL STABILITY FOR A SPECIAL CLASS OF CONSERVATIVE LINEAR SYSTEMS G. WEISS Λ, M. Tucsnak y Λ Dept. of Electrical and Electronic Engineering Imperial College London Exhibition
More informationSufficient conditions for functions to form Riesz bases in L 2 and applications to nonlinear boundary-value problems
Electronic Journal of Differential Equations, Vol. 200(200), No. 74, pp. 0. ISSN: 072-669. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Sufficient conditions
More informationSPECTRAL THEOREM FOR COMPACT SELF-ADJOINT OPERATORS
SPECTRAL THEOREM FOR COMPACT SELF-ADJOINT OPERATORS G. RAMESH Contents Introduction 1 1. Bounded Operators 1 1.3. Examples 3 2. Compact Operators 5 2.1. Properties 6 3. The Spectral Theorem 9 3.3. Self-adjoint
More informationDensity results for frames of exponentials
Density results for frames of exponentials P. G. Casazza 1, O. Christensen 2, S. Li 3, and A. Lindner 4 1 Department of Mathematics, University of Missouri Columbia, Mo 65211 USA pete@math.missouri.edu
More informationOctober 25, 2013 INNER PRODUCT SPACES
October 25, 2013 INNER PRODUCT SPACES RODICA D. COSTIN Contents 1. Inner product 2 1.1. Inner product 2 1.2. Inner product spaces 4 2. Orthogonal bases 5 2.1. Existence of an orthogonal basis 7 2.2. Orthogonal
More information08a. Operators on Hilbert spaces. 1. Boundedness, continuity, operator norms
(February 24, 2017) 08a. Operators on Hilbert spaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/real/notes 2016-17/08a-ops
More informationANALYSIS QUALIFYING EXAM FALL 2017: SOLUTIONS. 1 cos(nx) lim. n 2 x 2. g n (x) = 1 cos(nx) n 2 x 2. x 2.
ANALYSIS QUALIFYING EXAM FALL 27: SOLUTIONS Problem. Determine, with justification, the it cos(nx) n 2 x 2 dx. Solution. For an integer n >, define g n : (, ) R by Also define g : (, ) R by g(x) = g n
More informationON WEAKLY NONLINEAR BACKWARD PARABOLIC PROBLEM
ON WEAKLY NONLINEAR BACKWARD PARABOLIC PROBLEM OLEG ZUBELEVICH DEPARTMENT OF MATHEMATICS THE BUDGET AND TREASURY ACADEMY OF THE MINISTRY OF FINANCE OF THE RUSSIAN FEDERATION 7, ZLATOUSTINSKY MALIY PER.,
More informationA CHARACTERIZATION OF STRICT LOCAL MINIMIZERS OF ORDER ONE FOR STATIC MINMAX PROBLEMS IN THE PARAMETRIC CONSTRAINT CASE
Journal of Applied Analysis Vol. 6, No. 1 (2000), pp. 139 148 A CHARACTERIZATION OF STRICT LOCAL MINIMIZERS OF ORDER ONE FOR STATIC MINMAX PROBLEMS IN THE PARAMETRIC CONSTRAINT CASE A. W. A. TAHA Received
More information