Exponential stabilization of a Rayleigh beam - actuator and feedback design

Transcription

1 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 Abstract: Our plant is a hinged elastic beam described by the Rayleigh beam equation on the interval [0, π. We assume the presence of two sensors: one measures the angular velocity of the beam at a point ξ (0, π) and the other measures the bending (curvature) at the same point. These two measurements are advantageous because they make the open-loop system exactly observable, regardless of the point ξ. We design the actuators and the feedback law in order to exponentially stabilize this system. Using the theory of colocated feedback developed in a recent paper by Ruth Curtain and the author, we choose colocated actuators, meaning that B = C, where B is the control operator and C is the observation operator. It turns out that the actuators cause a discontinuity of the bending exactly at ξ. This obliges us to use an extension of C to define the output signal in terms of the left and right limit of the bending at ξ. It turns out that for all positive feedback gains in a suitable finite range, the closed-loop system is well-posed and exponentially stable, as follows from the general theory in the paper mentioned earlier. Keywords: Rayleigh beam, well-posed linear system, output feedback, exact controllability and observability, colocated sensors and actuators. 1. Problem statement The physical system that we are modelling consists of a hinged elastic beam with two sensors: one measures the angular velocity of the beam at a point ξ and the other measures the bending (curvature) of the beam at the same point. These two measurements are advantageous because they make the open-loop system exactly observable, as we shall see. Our aim is to design actuators and a feedback law such that the closed-loop system is exponentially stable in a suitable state space. This research was supported by EPSRC (from the UK) under the portfolio partnership grant GR/S6156/01.

2 We model the open-loop system as a homogenous Rayleigh beam situated along the interval [0, π, with the two sensors located at ξ (0, π). This is an extension of an example discussed in Ammari, Liu and Tucsnak [1 and Weiss, Staffans and Tucsnak [3, and these papers contain relevant further references to Rayleigh beam models. As far as we know, [1 was the first paper to consider the stabilization of a beam using two sensors and two actuators which are colocated (not the same sensors as considered here). They considered also the stabilization of Euler-Bernoulli beams by a similar technique. The equations describing the open-loop system are q(x, t) α q x (x, t) + 4 q (x, t) = U(x, t), (1.1) x4 q q(0, t) = q(π, t) = 0, x (0, t) = q (π, t) = 0, t 0, (1.) x y 0 (t) = q (ξ, t), x y 1(t) = q (ξ, t). (1.3) x In these equations q(x, t) represents the transverse displacement of the beam at the position x [0, π and the time t 0, δ ξ is the Dirac mass at the point ξ and α > 0 is a constant, proportional to the moment of inertia of the cross section of the beam (α is proportional to the square of the thickness of the beam and it is often neglected). In (1.1), U denotes the control terms which are to be designed. The scalar signals y 0 and y 1 are the two output signals (measurements). According to our design, the actuators for this system should be such that d U = u 0 dx δ ξ u 1 [αδ ξ + b, (1.4) where u 0 and u 1 are the two (scalar) input signals and b(x) = { ax ξ for x ξ, aπ ax π ξ for x > ξ, where 1 a = 1 ξ + 1 π ξ. (1.5) The graph of b consists of two straight lines, with a peak at x = ξ. The actuators will cause a jump discontinuity of q at ξ, so that the second part of (1.3) becomes x ambiguous. The output signal y 1 must be re-defined in the form y 1 (t) = γ q x (ξ, t) + (1 γ) q (ξ+, t), (1.6) x where γ C, but it seems more reasonable to assume that γ [0, 1. Then the system described above is well-posed, with the state space The two components of the state x(t) are X = H (0, π) H 1 0(0, π) H 1 0(0, π). (1.7) q(, t) and q(, t) + b( )u 1 (t). We call this (open-loop) system the beam system. The strongly continuous semigroup of operators on X which describes the evolution of the state x(t) when u 0 = 0 and u 1 = 0 is unitary. Our main result is the following:

3 Theorem 1.1. There exists a number κ 0 > 0 such that for every κ (0, κ 0 ), the closed-loop system corresponding to the beam system with the feedback u j = κy j + v j, j = 1,, where v 1 and v are the new input signals, is well-posed and exponentially stable. Note that the exponential stability is independent of the position ξ of the two sensors, which is not always the case for stabilizing feedback laws in the literature. To treat the above problem, we need a recently developed theory concerning the exponential stabilization of systems with colocated sensors and actuators, which can be found in Curtain and Weiss [6. The main result of this theory will be recalled in Section. In Section 3 we develop an abstract framework which helps to apply the theory from Section to a specific class of systems described by differential equations of order two in Hilbert spaces. We do this because the beam system fits into this specific class of systems. Finally, in Section 5 we solve the stabilization problem for the beam system, obtaining (also) Theorem A stabilization result from a recent paper In this section, we recall the main results of Curtain and Weiss [6 concerning the stabilization of a special class of well-posed linear systems by colocated feedback. To specify our terminology and notation, we recall that for any well-posed linear system Σ with input space U, state space X and output space Y, all Hilbert spaces, the state trajectories z C([0, ), X) are described by the differential equation ż(t) = Az(t) + Bu(t), (.1) where u L loc([0, ), U) is the input function. The operator A : D(A) X is the generator of a strongly continuous semigroup of operators T on X and the (possibly unbounded) operator B is an admissible control operator for T. In general, the output function y is in L loc([0, ), Y ). If u = 0 and z(0) D(A), then y is given by y(t) = Cz(t) t 0, where C : D(A) Y is an admissible observation operator for T. B is called the control operator of Σ and C is called the observation operator of Σ. If z(0) = 0, then the input and output functions u and y are related by the formula ŷ(s) = G(s)û(s), (.) where a hat denotes the Laplace transform and G is the transfer function of Σ. The formula (.) holds for all s C with Re s sufficiently large. Assumption ESAD. The operator A is essentially skew-adjoint and dissipative, which means that D(A) = D(A ) and there exists a Q L(X) with Q 0 such that Ax + A x = Qx x D(A). 3

4 This implies that T is a contraction semigroup. Note that A is a bounded perturbation of the skew-adjoint operator A + 1 Q. Such a model is often used to describe the dynamics of oscillating systems, such as waves or flexible structures. Assumption COL. Y = U and C = B. Our aim is to show that for certain numbers κ > 0, the static output feedback law u = κy + v stabilizes the system, where v is the new input function. The closed-loop system Σ κ is shown as a block diagram in Figure 1. v u y + Σ κ Figure 1: The open-loop system Σ with negative output feedback via κ. If Σ satisfies ESAD and COL and the number κ > 0 is sufficiently small, then this is a new wellposed linear system Σ κ, called the closed-loop system. If Σ is exactly controllable and exactly observable, then Σ κ is exponentially stable. The main result of [6 is the following: Theorem.1. Suppose that Σ is a well-posed linear system which is exactly controllable, exactly observable and satisfies assumptions ESAD and COL. Then there exists a κ 0 > 0 (possibly κ 0 = ) such that for all κ (0, κ 0 ), the feedback law u = κy + v (where u and y are the input and the output of Σ) leads to a closedloop system Σ κ which is well-posed and exponentially stable. The paper [6 also gives various formulas for κ 0. For this paper, the important formulas are the following: First, we introduce a self-adjoint operator E L(U) by E = 1 [G(0) + G(0) + B (A ) 1 QA 1 B. (.3) Recall that a transfer function G, analytic on the open right half-plane C 0, is called positive if G(s) + G(s) 0 s C 0. The crucial property of the operator E introduced earlier is that G + E is a positive transfer function (see [6 for details). Now let E + be the positive part of E. This means that E + = EP + where P + is the spectral projector corresponding to all the positive spectrum of E (hence E + E but E + E ). Then κ 0 = 1 E + (for E + = 0, take κ 0 = + ). (.4) 4

5 Many models of controlled flexible structures satisfy assumptions ESAD and COL. The feedback u = κy + v is very simple to implement and it is often used in the stabilization of these structures. An extensive list of relevant references is given in [6. Here we mention only some of the papers which refer to beams and plates and which obtain exponential stability by static output feedback through colocated control and observation: Chen [3, 4, Rebarber [10, Triggiani [15, Tucsnak and Weiss [16, Luo, Guo and Morgul [9, Guo and Luo [7, Ammari and Tucsnak [. In many of these examples the approach is a classical Lyapunov one, with the key step being the appropriate PDE formulation so that the energy of the system can play the role of a Lyapunov functional. If one examines these examples carefully, one can recognize that they fit into our framework (the assumptions ESAD and COL are satisfied) and they use the feedback u = κy + v for stabilization. However, it must be pointed out that not all the examples of this type in the literature satisfy our assumptions, since sometimes the open-loop system is not well-posed. In all the published examples that we are aware of, the feedback u = κy + v is stabilizing for all κ > 0, at least in the input-output sense, and often strongly or exponentially. However, in [6, Section 5 there is an example of a simple open-loop system Σ which fits into our framework, moreover it is regular with feedthrough operator zero, but for which the feedback u = κy + v is only exponentially stabilizing for sufficiently small κ > 0. For too large a κ, the closed-loop semigroup has a positive growth rate. 3. A class of undamped second order systems In this section we introduce a class of undamped second order systems satisfying ESAD and COL. For these systems we derive an explicit expression for κ 0 from Theorem.1. The beam system introduced in Section 1 fits into the class introduced here, as we shall see in Section 4. Let U 0, U 1 and H be Hilbert spaces and let A 0 : D(A 0 ) H be positive and boundedly invertible on H. Consider the system described by the following second order differential equation and two output equations: { q + A0 q = C0u 0 + A0 1 C1 u 1, (3.1) y 0 = C 0 q, y 1 = C 1 q, for sufficiently smooth signals u 0, u 1 and compatible initial conditions q(0) and q(0). The input signals u 0, u 1 and the output signals y 0, y 1 are such that u 0 (t) U 0, u 1 (t) U 1, y 0 (t) U 0, y 1 (t) U 1. To formulate these equations in the form { ż = Az + Bu, y = Cz + Du, (3.) which describe a well-posed linear system for sufficiently smooth u and compatible z(0), we have to introduce various spaces and operators, and we have to make some assumptions. For every µ > 0, we put H µ = D(A µ 0), with the norm φ µ = A µ 0φ H. 5

6 We define H µ = H µ (duality with respect to the pivot space H). We put H 0 = H and φ 0 = φ H. We assume that C 0 L(H 1, U 0 ) C 1 L(H 1, U 1 ). We identify U 0 and U 1 with their duals, so that C0 L(U 0, H 1 ), C1 L(U 1, H 1 ). We assume that C 0 and C 1 have extensions C 0 and C 1 such that the operators D 0 = C 0 A 1 0 C 1 L(U 1, U 0 ), D 1 = C 1 A 1 0 C 0 L(U 0, U 1 ) exist. Introduce the input (and output) space U and the state space X: U = U 0 U 1, X = H 1 H. (3.3) Now the equations of the system Σ from (3.1) can be rewritten in the form (3.), where q u0 y0 z = X, u = U, y = U, (3.4) w u 1 y 1 0 I A : D(A) X, A =, (3.5) A 0 0 B = X 1 = D(A) = H 1 H 1 [ 0 A 1 0 C 1 C 0 0 C =, X 1 = H H 1, A = A, 0 L(U, X 1 ), C = B C0 = L(X C 1 0 1, U), (3.6) 0 C0 0 D0, D =. (3.7) C We shall now prove that for sufficiently smooth u, (3.1) is equivalent to (3.). We also show how the condition of compatible initial conditions, Az(0) + Bu(0) X, can be expressed in terms of the functions and operators appearing in (3.1). The significance of this condition for well-posed systems is that if u H 1 loc(0, ; U), then this condition is preserved (i.e., Az(t) + Bu(t) X remains valid for all t 0) and the output function is well defined by the second part of (3.). Proposition 3.1. For u H 1 loc(0, ; U), the equations (3.1) (for a specific t 0) are equivalent to (3.) (for the same t) with z, u, y as in (3.4) and with w(t) = q(t) A 1 0 C 1u 1 (t). The first part of (3.1) is regarded as an equation in H 1, and the first part of (3.) is regarded as an equation in X 1. Moreover, for any t 0, the conditions A 0 q(t) C 0u 0 (t) H, q(t) H 1 (3.8) are equivalent to Az(t) + Bu(t) X. 6

7 Proof. We rewrite the first part of (3.): [ d q(t) 0 I q(t) = dt w(t) A 0 0 w(t) or, equivalently, + [ 0 A 1 0 C 1 C 0 0 [ u0 (t) u 1 (t) q(t) = w(t) + A 1 0 C 1u 1 (t), ẇ(t) = A 0 q(t) + C 0u 0 (t). From these we obtain, by differentiating the first formula, the first equation in (3.1). To derive the last two equations in (3.1), we compute, starting from (3.), [ y0 (t) 0 C0 q(t) 0 C0 A 1 0 C = + 1 u0 (t) y 1 (t) C 1 0 w(t) 0 0 u 1 (t) [ C0 ( q(t) = A 1 0 C1u 1 (t)) + C 0 A 1 0 C1u 1 (t) C0 q(t) =. C 1 q(t) C 1 q(t) To obtain (3.) from (3.1), we just reverse the order of the computations. It is easy to verify that the conditions (3.8) are equivalent to Az(t) + Bu(t) X. Clearly, Σ satisfies ESAD and COL, but it need not be well-posed. The optimal control of systems of this type, but with C 1 = 0, has been studied in [ without assuming well-posedness. Here, we do assume that our system Σ is well-posed. The transfer function of Σ is given by G(s) = C(sI A) 1 B + D, and this is easy to compute in terms of A 0, C 0 and C 1 : for s ρ(a 0 ), C0 s G(s) = (s I + A 0 ) [ 1 C0 A 1 0 C1s. (3.9) C 1 Note the curious fact that G does not depend on the extended operator C 0. This extended operator appears in the representation (3.) of Σ, but (3.9) shows that the system Σ is in fact independent of the choice of the extension C 0. In particular, we see from (3.9) that G(0) = [ 0 0 D 1 0., According to (.3) (with Q = 0) we have E = 1 [ 0 D 1 D 1 0. (3.10) Such systems, but with C 1 = 0 (hence E = 0), have been considered in [, [7, [. The main result of this section is the following: Proposition 3.. Suppose that the well-posed system Σ from (3.) is exactly controllable (or equivalently, exactly observable). Let E L(U) be the operator from (3.10), let E + be the positive part of E and put κ 0 = 1. Then for every κ (0, κ E + 0), K = κi is an admissible feedback operator for Σ and the resulting closed-loop system Σ κ is exponentially stable. 7

8 Proof. The equivalence of exact controllability and exact observability follows by duality, since A = A. Thus, we can apply Theorem.1 and formula (.4) to obtain this proposition. For the example treated in the next section, it will be useful to note the following facts. We have for all s ρ(a), using the expression (3.6) for B, (si A) 1 B = (s I + A 0 ) 1 [ C 0 sa 1 0 C 1 sc 0 C 1 Since the growth bound of the semigroup generated by A is zero, according to a result in [19 there exists a δ > 0 such that (si A) 1 B L(U,X) δ Re s holds for all s C with Re s 1. This together with the above expression for (si A) 1 B implies that (s I + A 0 ) 1 C0 L(U0,H 1 ) (s I + A 0 ) 1 C1 L(U1,H) δ Re s for Re s 1, (3.11) δ Re s for Re s 1, (3.1). s(s I + A 0 ) 1 A 1 0 C1 L(U1,H 1 ) δ Re s for Re s 1. (3.13) 4. Stabilization of a Rayleigh beam Our main task in this section is to show that the beam system from Section 1 fits into the framework of Section 4, in particular, it is well-posed. We also show that it is exactly controllable and exactly observable, so that we can apply Proposition 3.. Here, C 0 will be the operator corresponding to the measurement of the angular velocity at ξ, and C 1 will be the operator corresponding to the measurement of the bending at ξ. We shall see that in order to have the control terms as in (3.1) (which means that they are colocated), we must choose them to be as in (1.4). In order to fit the system described by (1.1) (1.7) into the framework of (3.1), (3.3), let us denote H = H0(0, 1 π), V = H (0, π) H0(0, 1 π). On H we define the inner product such that φ, ψ H = (I α d ) φ, ψ φ, ψ V. dx L We introduce the operator R : L (0, π) V by R = (I α d dx ) 1. 8

9 It is easy to see that R > 0 when regarded as an operator from L (0, π) to L (0, π). We also define the linear operator A 0 : D(A 0 ) H by { D(A 0 ) = φ H 3 (0, π) φ(0) = φ(π) = 0, d φ dx (0) = d } φ dx (π) = 0, The set {φ k k N} with φ k (x) = A 0 φ = d 4 is an orthonormal basis in H, and we have A 0 φ k = dx 4 (Rφ) φ D(A 0). 1 sin kx x [0, π (4.1) 1 + αk π k αk φ k, Rφ k = αk φ k, so that, in particular, A 0 is self-adjoint, strictly positive and it commutes with R. Defining H µ and φ µ for µ R by fractional powers of A 0 and duality, as explained after (3.1), we have H 0 = H = H 1 0(0, π) and H 1 = D(A 0 ), H 1 = V, H 1 = L (0, π), H 1 = H 1 (0, π), H 1.5 = V, the norm 1 being equivalent to the L -norm. Here, V denotes the dual of V with respect to the pivot space L (0, π). For every µ R, A 0 and R have extensions (or restrictions) to H µ and we have A 0 L(H µ, H µ 1 ), R L(H µ, H µ+1 ). The spaces U 0 and U 1 from (3.3) are U 0 = U 1 = C, so that U = C. Corresponding to the two measurements in (1.3), we define the operators C 0 L(H 1, C) and C 1 L(H 1, C) by C 0 φ = dφ dx (ξ), C 1φ = d φ dx (ξ). Then C0 H 1 and C1 H 1 (the adjoints of C 0 and C 1 with respect to the pivot space H) are C0 = R d dx δ ξ, C1 = R d dx δ ξ. Assuming that the terms of (1.1) are in H 1.5 = V and applying R to these terms, we obtain q + A 0 q = RU in H 1 = L (0, π). (4.) Now the first equation in (3.1) shows that the control terms, represented above by RU, should be RU = C0u 0 + A 1 0 C1 u 1 = u 0 R d dx δ ξ u 1 b, (4.3) where b = A 1 0 C 1 H is given by (1.5). If we apply R 1 to the terms of (4.), with RU as in (4.3), we obtain that in H 1.5, q α q x + 4 q x = u d 4 0 dx δ ξ u 1 [αδ ξ + b. (4.4) This equation must also hold in the sense of distributions on (0, π). Together with (1.) and (1.3), it defines our open-loop system with the designed actuators. 9

10 Proposition 4.1. For u H 1 loc(0, ; C ), the equations (4.4) (in H 1.5 ) and (1.3) are equivalent to ż = Az + Bu ( in X 1 ), y = Cz + Du, with z, u, y as in (3.4) and with w = q A 1 0 C1u 1 = q + bu 1. Here, A is as in (3.5), B is as in (3.6) and C, D are as in (3.7). These equations determine a well-posed linear system Σ with input and output space U = C and state space X = H 1 H. This system is exactly controllable and exactly observable. Proof. Assume that (4.4) and (1.3) hold. We have seen that (4.4) (in H 1.5 ) is equivalent to (4.) and (4.3) (in H 1 ). According to Proposition 3.1, these equations are equivalent to ż = Az + Bu (in X 1 ), with z, u as in (3.4) and with w = q A 1 0 C1u 1 (recall that b = A 1 0 C1). According to the same proposition, (1.3) is equivalent to y = Cz + Du, with y as in (3.4). The eigenvalues and the corresponding normalized eigenvectors of A are ±iω k and e ±k, for k N, where k ω k =, e 1 + αk ±k = 1 1 ±iω k φ k φ k and the set {e k, e k k N} is an orthonormal basis in X. (Here, φ k are the eigenvectors of A 0 defined in (4.1).) We introduce the vectors c ±k R by c ±k = Ce ±k = 1 [ C 0 φ k 1 ±iω k C 1 φ k = 1 k 1+αk π 1 k ±iω k 1+αk π cos kξ sin kξ Since ω k k/ α, we see from the last formula that there exist constants C min > 0 and C max > 0 such that C min c ±k C max k N. (4.5) Since the eigenvalues of A have a uniform gap, the second estimate above implies, via the Carleson measure criterion (see, for example, Hansen and Weiss [8, Theorem 1.3) that C is an admissible observation operator for the (unitary) semigroup generated by A. Since A = A and B = C, it follows by duality that B is an admissible control operator for the same semigroup. To check the well-posedness of the system Σ, according to the criterion in [5, it remains to show that its transfer function G from (3.9) is bounded on some right half-plane. This would be difficult to show using eigenfunction expansions, as the discussion (in particular, the counterexamples) in [7 show. Instead, we will estimate G(s) by using an approximation of A 0, as in [1, [3. We introduce A 1 : D(A 0 ) H by A 1 φ = 1 α d φ dx. This operator has the same eigenvectors as A 0, so that it commutes with A 0. A 1 has an extension to all the spaces H µ, and we have A 1 L(H µ, H µ 1 ) for all µ R. The. 10

11 eigenvalues of A 1 are k α, so that the eigenvalues of A 1 A 0 are bounded. Hence, A 1 A 0 is a bounded (positive) operator on any of the spaces H µ. Note that (s I + A 1 ) 1 = (s I + A 0 ) 1 (s I + A 0 ) 1 (A 1 A 0 )(s I + A 1 ) 1. (4.6) Applying C 1 to all the terms and using the dual version of (3.1), we get that for some η > 0 C1 (s I + A 1 ) 1 η for Re s 1. (4.7) L(H,C) Re s For each s ρ(a), G(s) is a matrix and its component G 11 (s) has been estimated on a right half-plane in [1 (in the proof of their Proposition 3.). We give the details for how to estimate G 1 (s), in order to prove its boundedness on the half-plane where Re s 1. First we rewrite (4.6) in the form (s I + A 0 ) 1 = (s I + A 1 ) 1 + (s I + A 1 ) 1 (A 1 A 0 )(s I + A 0 ) 1. (4.8) Recall that G 1 (s) = C 1 (s I + A 0 ) 1 C 0. Using the above formula, we obtain G 1 (s) C 1 (s I + A 1 ) 1 C 0 + C 1 (s I + A 1 ) 1 L(H,C) (A 1 A 0 ) L(H) (s I + A 0 ) 1 C 0 H. Using (3.11), the continuous embedding H 1 H and (4.7), we see that the second term on the right-hand side above is uniformly bounded for Re s 1. Thus, we only have to prove the boundedness (for Re s 1) of the function g(s) = C 1 (s I + A 1 ) 1 C 0. Lemma 4.. The function g defined above is bounded for Re s 1. The proof of this lemma involves some tedious computations, and it will be given in the journal version of this paper. As mentioned before the lemma, this lemma implies that G 1 (s) is bounded on the half-plane where Re s 1. The boundedness of G 1 (s) on the same half-plane can be inferred from the boundedness of G 1 (s), by the following argument: G 1 (s) = C 0 s (s I + A 0 ) 1 A 1 0 C 1 = C 0 [ A 1 0 (s I + A 0 ) 1 C 1 = { C 1 [ A 1 0 (s I + A 0 ) 1 C 0 } = {G1 (0) G 1 (s)}. The last component of G to be estimated is G (s) = sc 1 (s I + A 0 ) 1 b, where we have used (3.9) and the definition of b (after (4.3)). We introduce the function h by replacing A 0 with A 1 in the above formula: h(s) = sc 1 (s I +A 1 ) 1 b. We claim that G (s) h(s) is bounded for Re s 1. Indeed, using (4.8) we have G (s) h(s) = sc 1 (s I + A 1 ) 1 (A 1 A 0 )(s I + A 0 ) 1 b, 11

12 G (s) h(s) C 1 (s I + A 1 ) 1 L(H 1,C) A 1 A 0 L(H 1 ) s(s I + A 0 ) 1 b 1. The factor s(s I + A 0 ) 1 b 1 is bounded for Re s 1, according to (3.13). The operator A 1 A 0 is bounded on any of the spaces H µ, as explained before (4.6). The factor C 1 (s I + A 1 ) 1 L(H 1,C) is bounded for Re s 1, according to the dual version of (3.1), with A 1 in place of A 0, and using the continuous embedding H 1 H. This shows that G (s) h(s) is indeed bounded for Re s 1. Thus, to prove the boundedness of G (s) for Re s 1, it is enough to prove the boundedness of h(s) on this half-plane. This can be done by explicit computations, as in Lemma 4., and we omit the gory details. To prove that Σ is exactly observable, we use the first estimate in (4.5) together with [11, Theorem 5.9, which is based on the vector-valued version of Ingham s theorem. By duality, it follows that Σ is also exactly controllable. Proposition 4.3. Let Σ be the well-posed system from Proposition 4.1 (the beam equation with two sensors and two actuators). Denote D 1 = C 1 A 1 0 C0, κ 0 =. D 1 Then for every κ (0, κ 0 ), K = κi is an admissible feedback operator for Σ and the resulting closed-loop system Σ κ is exponentially stable. Proof. We know from Proposition 4.1 that the system Σ is well-posed, exactly controllable and exactly observable. Let E L(C ) be the matrix from (3.10). Its eigenvalues are ± 1 D 1, so that E + (the positive part of E) is a matrix of rank one with the eigenvalues 1 D 1 and 0. Hence, E + = 1 D 1, κ 0 = D 1. According to Proposition 3., for every κ (0, κ 0 ), K = κi is an admissible feedback operator for Σ and Σ κ is exponentially stable. References [1 K. Ammari, Z. Liu and M. Tucsnak, Decay rates for a beam with pointwise force and moment feedback, Mathematics of Control, Signals and Systems 15 (00), pp [ K. Ammari and M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks, ESAIM COCV 6 (001), pp [3 G. Chen, A note on boundary stabilization of the wave equation, SIAM J. Control and Optim. 19 (1981), pp [4 G. Chen, M.C. Delfour, A.M. Krall and G. Payre, Modelling, stabilization and control of serially connected beams, SIAM J. Control and Optim. 5 (1987), pp [5 R.F. Curtain and G. Weiss, Well-posedness of triples of operators (in the sense of linear systems theory), Control and Estimation of Distributed Parameter Systems (F. Kappel, K. Kunisch, W. Schappacher, eds.), pp , Birkhäuser-Verlag, Basel, [6 R.F. Curtain and G. Weiss, Exponential stabilization of essentially skew-adjoint systems by colocated feedback, submitted in

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