Rapid pointwise stabilization of vibrating strings and beams

This paper is devoted to the study of pointwise observability, controllability and uniform stabilization of vibrating systems. It was pointed out earlier by Haraux and Jaffard [6], [7], [8] that the observability and controllability properties depend heavily on the location of the observation or control point. For the stabilization another difficulty appears because the suitable function spaces, as we will show, are not Sobolev spaces. In order to establish satisfactory stabilization theorems we will introduce functions spaces depending on the arithmetical properties of the stabilization point. Working in this framework, we will be able to adapt a method developed by Komornik [11] to vibrating strings, beams and also to a coupled string-beam system; as a result we will construct pointwise feedbacks leading to arbitrarily large prescribed decay rates. Many works were devoted to the construction of explicit feedback laws and to the proof of exponential decay by different methods; see, e.g., [1], [2], [3], [4], [10], [15], [14], [18]. It is known that this type of feedback does not yield arbitrarily large decay rates.


Introduction
This paper is devoted to the study of pointwise observability, controllability and uniform stabilization of vibrating systems.It was pointed out earlier by Haraux and Jaffard [6], [7], [8] that the observability and controllability properties depend heavily on the location of the observation or control point.For the stabilization another difficulty appears because the suitable function spaces, as we will show, are not Sobolev spaces.In order to establish satisfactory stabilization theorems we will introduce functions spaces depending on the arithmetical properties of the stabilization point.Working in this framework, we will be able to adapt a method developed by Komornik [11] to vibrating strings, beams and also to a coupled string-beam system; as a result we will construct pointwise feedbacks leading to arbitrarily large prescribed decay rates.

Alia BARHOUMI
First we study a vibrating string with pointwise control, modeled by the following system: in R × (0, π), y(t, 0) = y(t, π) = 0 for t ∈ R, y(0, x) = y 0 (x) and y t (0, x) = y 1 (x) for x ∈ (0, π) (1.1) where δ ξ denotes the Dirac mass at some given point ξ ∈ (0, π) and v(t) is a control function in L 2 loc (R).In order to formulate our result, we assume that ξ/π is irrational, so that sin kξ, doesn't vanish for any k = 1, 2, . . ., we denote by Z the linear hull of the functions w k (x) := 2/π sin kx, k = 1, 2, . . ., and we denote by D α ξ and (D α ξ ) ′ for every α ∈ R the Hilbert spaces obtained by completing Z with respect to norms given by the following formulae: If we identify L 2 (0, π) with its dual and take into account that then (D α ξ ) ′ is the dual space of D α ξ .Now we may state our first result: the problem (1.1) is well posed in H ξ and its solutions satisfy the inequality for all (y 0 , y 1 ) ∈ H ξ and t ≥ 0.
Next we consider a beam model: for x ∈ (0, π). (1.3) Using the same notation as above, we will prove the following stabilization theorem: Theorem 1.2 Fix η ∈ (0, π) such that η/π is irrational and introduce the Hilbert space for all (y 0 , y 1 ) ∈ H η and t ≥ 0.
Remark 1.4 It follows from some results of Komornik and Loreti that the system (1.5) can not be exactly controllable for some exceptional choices of the parameters A, B, C, D: see [12] and [13] for explicit counter examples concerning an equivalent observability problem.
Since the Hilbert spaces in the above three theorems were defined in an abstract way, let us mention the following inclusions between these spaces and usual Sobolev spaces: (b) Furthermore, for almost every irrational ξ/π, we have for every ε > 0 the following inclusions: (c) Moreover, if ξ/π is a quadratic irrational number, then The plan of the paper is the following.In the next section we recall some results concerning a method analogous to HUM developed in [11] which useful for the proof of our results.The following three sections contain the proofs of Theorems 1.1, 1.2 and 1.3, respectively.In Section 6 we also state and prove two new observability and controllability theorems.Proposition 1.5 is proved in the last section 7.

Review of some abstract results
We consider the abstract observability problem in a complex Hilbert space H, where A is a (bounded or unbounded) linear operator defined on some linear subspace D(A) of H, with values in H and B is a linear operator, defined on some linear subspace D(B) of H with values in another Hilbert space G.We make the following four assumptions: (H1) The operator A generates a strongly continuous group of automorphisms e tA in H.
(H2) D(A) ⊂ D(B), and there exists a constant c such that BU 0 G ≤ c AU 0 H for all U 0 ∈ D(A).
(H3) There exist a non degenerate bounded interval I and a constant c I such that the solutions of (2.1) satisfy the inequality (H4) There exists a bounded interval I ′ and a positive number c ′ such that the solutions of (2.1) satisfy the inequality and set The following result is a special case of a theorem obtained in [11].
Theorem 2.1 Assume (H1)-(H4) and fix ω > 0 arbitrarily.Then the problem is well-posed in H ′ .Furthermore, there exists a constant M such that the solutions of (2.2) satisfy the estimates for all v 0 ∈ H ′ and for all t ≥ 0.
In other words, this theorem asserts that the feedback law with a decay rate at least equal to ω.
The well-posedness means here that (2.2) has a unique solution v ∈ C(R; H ′ ) for every v 0 ∈ H ′ .

Proof of Theorem 1.1
We consider the following system: If the initial data are given by the formulae with only finitely many nonvanishing coefficients a k and b k , then a simple computation shows that the solution is given by the formula In the sequel we write A ≍ B if there exist two positive constants c 1 , c 2 satisfying c 1 A ≤ B ≤ c 2 A. The constants are assumed to be independent of the particular choice of the initial data or of the parameter k.
If T ≥ 2π, then using Parseval's equality it follows that It can be rewritten in the form We rewrite (3.1) as a first-order system in the usual way, by setting We are going to verify the hypotheses (H1)-(H4) of Theorem 2.1 for the Hilbert spaces Proof.In order to obtain hypothesis (H1), we show that if (u 0 , u 1 ) ∈ H then (u(s), u t (s)) ∈ H for all s ∈ R.This follows from Parseval's equality: and from the observation that the last expression does not depend on s ∈ R.
It follow by a straightforward computation that for all (u, v) then the hypothesis (H2) is satisfied.In deed, using the norms of D 0 ξ and D −1 ξ and writing and therefore, using the Cauchy-Schwarz inequality at the first step, It follow from the equality (3.2) that we have (H3)-(H4).
We may now apply Theorem 2.1 and we have (1.2).In order to write down explicitly the stabilization result, we multiply the equation (1.1) by u and we integrate by parts as follows (we use all conditions in (1.1) and (3.1)): This shows that if we write (3.1) in the form (2.1), then its dual (2.2) corresponds to (1.1).Furthermore, writing the operator we have v(t) = −(P y t + Qy)(t, ξ).
We consider the following system: where ∆u = u xx .If the initial data are given by the formulae with only finitely many nonvanishing coefficients a k and b k , then a simple computation shows that the solution is given by the formula If T > 0, then using a result of Haraux in [7] it follows that It can be rewritten in the form We rewrite (4.1) as a first-order system in the usual way, by setting where ∆ 2 u = u xxxx .Furthermore, we introduce the Hilbert spaces H := D 0 η × D −2 η , G := R and we define the domain of definition of the linear operators A and B by Let us show that hypotheses (H1)-(H4) of Theorem 2.1 are satisfied.
Proof.We show that if (u 0 , u 1 ) ∈ H then (u(s), u t (s)) ∈ H for all s ∈ R.This follows by applying a result of Haraux in [7] as follows: and by observing that the last expression does not depend on s ∈ R. Then hypothesis (H1) is verified.
In order to obtain (H2) it suffices to establish for all (u, v) This follows by a straightforward computation, using the norms of D 0 η and D −2 η : writing k 4 a k sin kx and therefore, using the Cauchy-Schwarz inequality at the first step, It follow from the equality (4.2) that hypotheses (H3)-(H4) are verified .
We may now apply Theorem 2.1 and we obtain (1.4).In order to write down explicitly the stabilization result, we multiply the equation (1.3) by u and we integrate by parts as follows (we use all conditions in (1.3) and (4.1)).This shows that if we write (4.1) in the form (2.1), then its dual (2.2) corresponds to (1.3).Furthermore, writing the operator we have v(t) = −(P y t + Qy)(t, η).

Proof of Theorem 1.3
We consider the following system: for t ∈ R. (5.1) If the initial data are given by the formula with only finitely many non vanishing coefficients a k b k , α k β k , then a simple computation shows that and If T > 2π, then using Parseval's equality and a result of Haraux [7] it follows that

It can be rewritten in the form
We rewrite (5.1) as a first-order system U 0 := (u 10 , u 20 , u 11 , u 21 ), and We introduce the Hilbert spaces and we define the domain of definition of the linear operators A and B by We are going to check the validity of hypotheses (H1)-(H4) of Theorem 2.1.
Proof.We have to show that if for all s ∈ R.This follows by applying Parseval's inequality and Haraux's result as in the preceding cases: and by observing that the last expression does not depend on s ∈ R. Then we have (H1).
In order to verify hypothesis (H2) it suffices to establish the estimate so that the desired estimate is equivalent to the inequalities Fix a real number 1 < p < 3 and choose two numbers ξ, η ∈ (0, π) such that ξ/π, η/π are irrational, and for k → ∞.It is well-known from the theory of Diophantine approximation (see, e.g., [5]) that almost all numbers ξ, η ∈ (0, π) have this property.It follows from this choice that sin 2 kξ sin 2 kη = o k 6 and sin 2 kη sin 2 kξ = o k 6 . (5.4) We are going to prove that there exists a positive integer k 0 such that for all k > k 0 .
It remains to ensure the validity of (5.3) for each k = 1, . . ., k 0 .It suffices to choose the parameters (A, B, C, D) so that none of the determinants vanishes.Indeed, then we have for each k the implication since all norms are equivalent on finite-dimensional vector spaces, this implies that both and therefore for every k = 1, . . ., k 0 with a suitable constant c.The exceptional parameters (A, B, C, D) for which one of these determinants vanishes form a finite number of three-dimensional manifolds in R 4 , hence a set of Lebesgue measure zero.We conclude that for almost all choices of ξ, η ∈ (0, π), hypothesis (H2) holds for almost all choices of (A, B, C, D) ∈ R 4 .
It follows from (5.2) that hypotheses (H3)-(H4) are satisfied.Theorem 1.3 now follows by applying Theorem 2.1.In order to write down explicitly the stabilization result, we multiply the equation (1.5) by u and we integrate by parts as follows (we use all conditions in (1.5) and (5.1)).This shows that if we write (5.1) in the form (2.1), then its dual (2.2) corresponds to (1.5).Furthermore, writing the operator Rapid pointwise stabilization 57 in the matrix form we have (5.7)

Observability and controllability results of the coupled string-beam system
In this section we state and prove two new observability and controlability theorems concerning the coupled string-beam system.Theorem 6.1 Fix ξ, η ∈ (0, π) such that ξ/π and η/π are irrational and introduce the space such that the corresponding solution of (1.5) satisfies the final condition (y 1 , y 1t , y 2 , y 2t )(T ) = (z 10 , z 11 , z 20 , z 21 ).
Remark 6.3 Inequality (6.1) is called observability inequality, which plays a fundamental role in control theory, it is not only sufficient but also necessary condition of exact controllability.
Proof.Theorem 6.4 is an abstract form of the Hilbert Uniqueness Method of Lions [16], [17], see also [18] for a general duality principle between observability and controllability.

Proof of Proposition 1.5
By the definition of the norms the three inclusions in part (a) are equivalent to the inequalities Turning to (b), as we have already mentioned in the preceding section, by a theorem on Diophantine approximation almost every ξ ∈ (0, π) satisfies for all ε > 0 Finally, the inclusions of part (c) are obtained by establish the same proof of (b) which ε = 0.