Dipartimento di Matematica

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1 Dipartimento di Matematica A. FAVINI, L. PANDOLFI ON-LINE RECONSTRUCTION OF INPUT SIGNALS TO DISTRIBUTED PARAMETER SYSTEMS WITH COLOCATED INPUTS AND OUTPUTS Rapporto interno N. 29, dicembre 26 Politecnico di Torino Corso Duca degli Abruzzi, Torino-Italia

2 On-line reconstruction of input signals to distributed parameter systems with colocated inputs and outputs A. Favini L. Pandolfi Abstract. In this paper we consider a distributed system whose evolution is described by a holomorphic semigroup and colocated scalar inputs and outputs. We present an algorithm for on-line reconstruction of the input, based on measures taken on the output. keywords: Input identification, boundary observation, frequency domain techniques AMS subject classification: 49N45 1 Introduction and assumptions Reconstruction of input signals to linear dynamical systems is an important problem. In fact, in the time invariant case, it is a special instance of the deconvolution problem. Several algorithms have been proposed to achieve this goal, which can be classified in two large classes: off-line and on-line reconstruction algorithms. Off-line algorithms accumulate all the available pieces of information and, after the process has come to an end, these pieces of information are elaborated so to obtain an estimate of the input signal. An example of this is Tikonov method. On-line methods instead at each time t produce an estimate v(t) of the This paper fits the research programs of GNAMPA-INDAM University of Bologna, Department of Mathematics, Piazza di Porta San Donato, 5, Bologna, Italy favini@dm.unibo.it Politecnico di Torino, Department of Mathematics, Corso Duca degli Abruzzi, 24, 1129 Torino, Italy, luciano.pandolfi@polito.it 1

3 unknown input u at the time t solely on the basis of observations taken at previous times. Of course, only the algorithms in this class can be used for regulation and control purposes, see applications in [5, 15] for the case of finite dimensional systems. It is obvious that on-line deconvolution methods will require stronger conditions than those used to justify off-line algorithms. When the impulse response of the system is known, on-line reconstruction algorithms have been studied for example in [3, 4, 7, 17], under suitable assumptions. In this paper instead we are concerned with the case that the time invariant system is described in state space form, i.e. ẋ = Ax + bu, y = Cx (1) where x belongs to a Hilbert space X (precise assumptions will be specified below), u is a scalar input and y a scalar output. When the system is finite dimensional, an efficient on-line method for the reconstruction of u has been studied in [14] for finite dimensional systems. The key assumption used there is that the full state is available for measurement, i.e. y = x (in fact, special cases of measures taken on an input and some classes of nonlinear systems have been considered too). Extension of this algorithms to distributed systems, still in the unrealistic case y = x, can be found in [9, 11, 12]. References cited up to now also used the assumption that the input should take values in a known bounded set, an assumption then removed in [1, 13], still in the case that y = x. In the case of finite dimensional systems, the assumption y = x, and the assumption that a bound for u is explicitly known, have been removed in [2]. The algorithm used in [2] is different from the algorithm in the previously cited papers, and it is the algorithm we use here for distributed parameter systems. However, the results in [2] cannot be completely extended to distributed systems since they rely on Morse quasi-canonical form. Morse quasi-canonical form strictly depends on the assumption that the system is finite dimensional and there is no hope for an extension to distributed systems. For this reason we here confine to study the case of colocated inputs and outputs. Input reconstruction is an ill posed problem so that any reconstruction algorithm must introduce a penalization parameter α > and reconstruct a candidate approximant v α of the input u. The point is to prove that, under suitable assumptions, v α converges to u for α +. This is a consistency result for the algorithm we are going to propose 2

4 but, of course, in any practical case we have to fix a (small) value of α and we then choose the corresponding function v α (t) as an approximant of the unknown input u. See [15] for an on line adaptive method for the determination of α in the finite dimensional case. After these preliminaries, we can describe the goal of this paper: we want to extend as much as we can the algorithm for online reconstruction of the input u proposed in [2] to the case of the distributed system (1). In order to give a specific goal, we state here that we would like to identify the input u to a thermal process using boundary observation of the flux of heat. I.e., we would like an on-line algorithm which identifies u when the system is θ t = θ + bu(t) in Ω θ Ω =, (2) and the observation y(t) is an average of the heat flux on the boundary. This goal will be achieved in Section 3.2. In order to study this general case however we proceed by steps: in Section 2 we shall first study the case that b X and C is a bounded operator. The condition b X means for example, in a concrete application, internal distributed heat sources, which is of some interest in certain applications; but C bounded means observations taken inside the body, which of course is not feasible: the arguments in Section 2 concern a problem which is not physically significant but which is here studied for most of clarity. The boundedness assumption on C is then removed in Section 3.2. A different recursive algorithm for an input-output system of parabolic type with bounded operator C is given in [18]. The crucial assumptions in this paper are as follows: the operator A is a selfadjoint generator of a holomorphic semigroup, A = A, u and y are scalar quantities and colocated inputs and output operators. When C is a bounded operator, colocated inputs and outputs means that the output operator C is a (nonzero) multiple of the operator B (B is defined by Bu = bu); i.e. Cx = µ b, x with µ. This definition will be extended to the case of boundary observation in Section 3.2. Of course, the identification of u is equivalent to the identification of µu so that it is not restrictive to assume µ = 1. This will be done below. Furthermore, we assume that x() =. We could also consider the case x() = x, a known initial condition, but in practice the initial condition will be the equilibrium position x =. In any practical application, the semigroup is bounded and often even exponentially stable. We did not assumed this explicitly, but we 3

5 can always reduce ourselves to this case since y 1 (t) = e st y(t) is the output of ξ = ( si + A)ξ + Bu 1 (t), y 1 (t) = Cξ(t) when u 1 (t) = e st u(t). The semigroup generated by ( si + A) is exponentially stable if s is large enough. Hence we can assume without restriction e At < Me at, a >. This implies that the resolvent of A is holomorphic in a halfplane Re λ > s, s >. It follows that we can assume, without restriction, A = A and ρ(a). A noticeable feature of this paper is that the proofs of Section 2 are given in the time domain. This gives a transparent interpretation for the formula we choose to approximate the unknown function u, but the proofs require quite heavy assumptions. Once that we are convinced that this formula has an interest, we can give a frequency domain proof under much weaker assumptions, see Section 3. Finally we note that when B = then v(t) = produces the same output as u(t) and the identification problem is trivial. Hence we shall explicitly assume B 2 Distributed observation: the formula for v α In this section the output operator C is bounded. We recall that B L(R, X) and C X are defined by Bu = bu, y = Cx = B x = b, x. We fix a time interval [, T ] over which the system evolve and we fix a penalization parameter α. For each value of α we construct a function v α (t), t [, T ] which under suitable assumption will converge to the unknown input u for α +. In order to keep the notations to a minimum, the dependence of v on α is not explicitly indicated during the computations. It will be indicated in the statements of the theorems. As already noted, the crucial idea is as follows: we associate a model to system (1), which is a copy of the system. This model 4

6 is forced to track the output of the system and we hope in this way that we can track also the input u. So, the model is ẇ = Aw + Bv, w() =, z = Cw. (3) As in [2], we associate a Liapunov type function to the system and its model. This is now ɛ(t) = Cw(t) Cx(t) 2 + α v 2 (s) ds. Here α is a regularization parameter. We introduce the error e = w x which solves ė = Ae + Bv Bu. The input v does depend on the penalization parameter α. Hence we have also e = e α although we shall use the simpler notation e. Note that if the input u is regular then x(t) is a classical solution of (1) but we cannot say that e is a classical solution since v has not yet been determined. We try to force ɛ to decrease as fast as it can. We compute its derivative and we act on v so to minimize the derivative. We make this computation as if e where a classical solution. ɛ(t) = 2 Ce, CAe + 2 C Ce, B(v u) + αv 2. This computation is performed (just formally!) at each time t. Now we choose v so to keep this derivative as small as we can, i.e. v(t) = v α (t) = 1 α B C Ce(t) = 1 α B BB e(t) = γ α B e(t), γ = B B >. (4) The number γ is positive since we assumed b. Once the expression for v is replaced in the equation of e, we find (e() = and) ė = Ae γ α BB e(t) Bu(t). (5) Remark 1 We see from here that if u is smooth, then e is a classical solution. 5

7 The goal of this paper is the proof that, under suitable assumptions, lim v α = u in L 2 (, T ). α + After that, we are justified in choosing v α with small α as an approximant of u. Remark 2 Note that measures are always corrupted by noise. For most of clarity we present the proofs in the ideal case of noiseless measures. The effect of the noise is then taken into account in Section Time domain analysis We now prove that v = v α is an approximant of the unknown function u using state space techniques. This kind of proof is very transparent but requires heavy assumption, to be removed later on. Namely we prove Theorem 3 In addition to the assumption stated in Section 1 we assume u C 1 (, T ) and For each T > we have in L 2 (, T ). b dom( A) 1/2. lim Bv α = Bu α + The assumption made on B in this theorem has been used already in the literature on identification, see for example [8, Condition (3.2)] and [18, Assumption 3.1]. We recall moreover that we are considering the case of noiseless measures and that e is a classical solution of (5) because u is smooth and that A = A since we can assume, without restriction, that e At is exponentially stable. As a first observation toward the proof of Theorem 3, we apply BB to both the sides of Eq. (5) and we see that [BB e] = { γ } α BB [BB e] + (BB )Ae γbu(t) 6

8 so that Bv(t) = γ α BB e(t) = γ2 α γ α e γ(bb )(t s)/α (BB )Ae(s) ds. e γ(bb )(t s)/α Bu(s) ds We recall that B B = γ where γ is a strictly positive number so that convergence of v to u is equivalent to the convergence of γv = B Bv to γu = B Bu and γv(t) = B Bv(t) = γ2 α We observe that B e γ(bb )(t s)/α Bu(s) ds γ α B e γ(bb )(t s)/α Ae(s) ds B [BB ] n = [B B] n B = γ n B, B e γ(bb )(t s)/α B = γe γ2 (t s)/α. Hence γv(t) = B Bv(t) = γ2 α e γ2 (t s)/α [γu(s)] ds γ2 α e γ2 (t s)/α B Ae(s) ds. We recall that in this section we are assuming smoothness of u. Hence, γ 2 α e γ2 (t s)/α [γu(s)] ds = γu(t) e γ2t/α γu() e γ2 (t s)/α [γu (s)] ds. The right hand side converges to γu(t) in L 2 (, T ), even uniformly on every (ɛ, T ], ɛ >. The mere convergence in L 2 in fact does not need smoothness of u. Consequently, the required convergence of v to u follows if we can prove that the following integral converges to : Now, γ 2 α + e γ2 (t s)/α B Ae(s) ds. (6) γ 2 α e γ2 t/α dt = 1 7

9 and the integral (6) converges to in L 2 (, T ), thanks to Young inequalities, if we can prove that B Ae(s) converges to for α +, in the norm of L 2 (, T ). This is now proved in five steps. Step 1. We prove that e = e(t) remains bounded in L (, T ) for α +. We noted that e is a classical solution of (5). We multiply both the sides with e(t) and we see that ( e() 2 = and) 1 2 d dt e(t) 2 = e(t), Ae(t) γ α B e(t) 2 e(t), Bu(t) e(t), Bu(t). We integrate both the sides and we see that Hence we have e(t) 2 e(t) 2 2 e(s) 2 ds + (7) e(s), Bu(s) ds. (8) Boundedness follows from Gronwall inequality. Bu(s) 2 ds. Step 2. We prove that ( A) 1/2 e(t) remains bounded in L 2 (, T ) for α +. We integrate both the sides of (7) and we see that e(s), Ae(s) ds γ α = 1 2 e(t) 2 + Taking into account the signs we see that In particular, ( A) 1/2 e(s) 2 ds+ γ α T ( A) 1/2 e(s) 2 ds The result now follows from Step 1. B e(s) 2 ds e(s), Bu(s) ds. B e(s) 2 ds = 1 2 e(t) 2 8 T e(s), Bu(s) ds. (9) e(s), Bu(s) ds. (1)

10 Step 3. We prove that sup t [,T ] b, e(t) = O( α). We take the scalar product of both the sides of (5) with the fixed vector b. We find d dt b, e(t) = ( A)1/2 b, ( A) 1/2 e(t) 1 α γ2 [ b, e(t) ] γu(t). We consider this equality as an ordinary differential equation in the unknown b, e(t) so that b, e(t) e γ2 (t s)/α { ( A) 1/2 b, ( A) 1/2 e(s) + γ u(s) } ds (11) Uniform boundedness of u(t) shows that e γ2 (t s)/α γ u(s) ds Mα ; (12) L 2 boundedness of ( A) 1/2 e(s) shows that e γ2 (t s)/α ( A) 1/2 b, ( A) 1/2 e(s) ds M α, as wanted. Remark 4 Note that the condition b dom( A) 1/2 has been used here. Step 4. We prove that T b, e(t) dt Mα. We consider again inequality (11). We integrate both the sides. Using (12) we see that the required estimate holds for the second integral. We improve the estimate of the first integral. We consider that T e γ2 (t s)/α ( A) 1/2 b, ( A) 1/2 e(s) ds dt [ T ] T = e γ2 (t s)/α dt ( A) 1/2 b, ( A) 1/2 e(s) ds Mα. s Step 5. Now, ( A) 1/2 e(t) in L 2 (, T ) for α +, see inequality (1) and Step 4. The property stated in Step 5. implies that, for α +, we have e γ2 (t s)/α B Ae(s) ds in L 2 (, T ) and completes the proof that v α u in L 2 (, T ), thanks to the fact that b dom ( A) 1/2. 9

11 3 Frequency domain analysis As usual in systems theory, problems can be efficiently attached in the frequency domain, i.e. after Laplace transformation. In particular, using frequency domain techniques we can remove the regularity assumptions on b and on u. We need some preliminaries for this. First of all we need to work on [, + ) now. If our data are on [, T ] then we extend u to a square integrable function for t > T. The simplest way is to put u(t) = for t > T. Note that the extended function is not smooth in general, but this assumption is not needed here. This frequency domain approach also removes the smoothness assumption on b which now is any element of X and works also with a larger class of output operators. We shall use the assumption that e At is exponentially stable, an assumption which is not restrictive, as noted above. In order to work with the Laplace transform, we need y(t), x(t) and u(t) on [, + ) and it might seem that we are doing an off-line reconstruction of u now. In fact, we are still doing an on-line reconstruction because v(t), the candidate approximant of u(t), is still given by formula (4). Hence at every time t, v(t) is computed solely on the basis of the available pieces of information. Laplace transformation is only used in the proofs. The crucial property that will be exploited is that a certain transfer function is positive real; i.e. it takes real values for real λ and leaves invariant the closed right half plane Π + = {λ : Re λ }. This property is very important in systems theory. 3.1 A first case: general b and bounded output operators The goal of this subsection is twofold: to convince the reader of the importance of frequency domain techniques in input reconstruction problems and to present a clear sketch of the general arguments to be used in Section 3.2 in a more general case. The proof we present now removes the assumption that u is smooth and, more important, the smoothness assumption on b. The input v is still given by formula (4) and C = B. We give a short sketch of the key ideas. The details of the computations are in Section 3.2, under more general assumptions. 1

12 We recall the standing assumption A = A and the nonrestrictive assumption that e At is exponentially stable so that A = A. The result to be proved in this section is as follows: Theorem 5 Let b X, C = B and u L 2 (, + ; IR). We have lim v α = u α + in L 2 (, + ). We note that the operator (A γ α BB ) which appears in (5) is selfadjoint negative, with the spectrum on the left of s if the spectrum of A is on the left of s ; and s does not depend on α. Consequently for each α this operator generates a holomorphic semigroup which is exponentially stable and (λi A + γ α BB ) 1 is holomorphic in Re λ > s. We extended u to [, + ) so to have a square integrable function. This implies that for each α the functions e(t) and v(t) are square integrable. Even more, if the Laplace transform of u exists in Re λ > ω then also ê(λ) and ˆv(λ) do exist in this half plane. We compute: Hence, [ ê(λ) = λ A + γ ] 1 α BB Bû(λ) ˆv(λ) = γ α B [λ A + γ α BB ] 1 Bû(λ) = γ α [ I + γ α B (λi A) 1 B] 1 B (λi A) 1 Bû(λ). ˆv(λ) û(λ) = α [ αi + γb (λi A) 1 B ] 1 û(λ). We recall that (λi A) 1 is well defined in Re λ > s so that T (λ) = B (λi A) 1 B = b, (λi A) 1 b is a holomorphic function of λ on this halfplane. 11

13 Let Π + = {λ : Re λ }. It is a fact, to be proved in Section 3.2 in more generality, that T (λ) transforms Π + into itself. It follows that Re [α+γb (λi A) 1 B] α so that α α + γb (λi A) 1 B 1 (13) on Π + (including the imaginary axis) when α >, and converges to zero for α +. This shows in particular that ˆv(λ) belongs to H 2 (of the right half plane) so that we can use Parseval identity (on the imaginary axis) in order to prove that v u L 2 converges to zero. In fact, + ˆv(iω) û(iω) 2 dω = + α α + γb (iωi A) 1 B 2 û(iω) 2 dω for α +, thanks to Lebesgue Theorem, using (13) and the fact that u(iω) 2 is integrable. 3.2 Boundary observations Distributed observation is usually impossible and we would like to extend the previous arguments to a class of boundary observations. Let b X be fixed. The computations in the previous sections suggest that similar arguments might be extendable to the case that the observation is given by y = c, x = b, ( A) σ x (14) where σ [, 1) (σ = was the case in subsection 3.1). This we are going to prove now. It is well known that output operators of this kind include certain cases of boundary observation. For example, let A be the Laplace operator with Dirichlet boundary condition, defined on a region Ω with smooth boundary. Let b H 2 (Ω) be a harmonic function. Note that b dom ( A) σ for every σ < 1/4. In this case, for every x dom A, we have y = b, ( A) σ x = ( A) σ 1 b, ( A)x = b, Ax x = b(s) x(s) ds = Ω Ω ν b dγ. 12

14 Hence, y is an average of the flux x ν on the boundary of Ω, as required in the introduction. We sum up: the system we have now is ẋ = Ax + bu, y = Cx = b, ( A) σ x = B ( A) σ x (15) and γ = c, b = b 2 >. We use the same form for v as in (4) (but now C is different!) v(t) = γ C[w x] where ẇ = Aw + Bv (16) α i.e. we consider the new equation for the model system ẇ = Aw b γ α b, ( A)σ (w x). (17) The equation for the error e(t) = w(t) x(t) is ė = Ae b γ α b, ( A)σ e bu(t). (18) We shall prove in the appendix that the operator e Ae b γ α b, ( A)σ e generates a C semigroup which is exponentially stable. This is very important, see Remark 9. We now compute: [ ê(λ) = λ A + γ ] 1 α BB ( A) σ Bû(λ) ˆv(λ) = γ α B ( A) σ [λ A + γ α BB ( A) σ ] 1 Bû(λ) = γ α B ( A) σ [I + γ α (λi A) 1 BB ( A) σ ] 1 (λi A) 1 Bû(λ) = γ α [ I + γ α B ( A) σ (λi A) 1 B] 1 B ( A) σ (λi A) 1 Bû(λ). For each fixed value of α, the computations above make sense in a halfplane Re λ > s 1 with a large enough s 1 (which depends on α). In fact, lim λ + ( A)σ (λi A) 1 = and the inverse in square bracket exists for Re λ > s 1. Even more, we shall see below that B ( A) σ (λi A) 1 B transforms the right half 13

15 plane into itself so that ˆv(λ) has an analytic extension to Re λ and on this halfplane we also have ˆv(λ) = γ [ I + γ ] 1 α α T (λ) T (λ)û(λ), T (λ) = B ( A) σ (λi A) 1 B. Remark 6 The previous argument, based on analytic continuation, has been given here in order not to blur the presentation with details. We shall see in the Appendix a direct proof of the existence of the inverses on the right hand side, and the practical interest of this fact. Consequently, ˆv(λ) û(λ) = α [α + γt (λ)] 1 û(λ). (19) Our goal is the proof that v(t) u(t) 2 L 2 (,+ ) tends to zero for α +. We can use the same argument as in Sect. 3.1 provided that we can prove that λ Π + = T (λ) Π +. (2) The proof of this property relays on the following equality, which is valid for every σ 1: ( A) σ (λ A) 1 b, b = b, ( A) σ ( λ A) 1 b. The computation which shows property (2) is as follows: 2Re T (λ) = b, ( A) σ (λi A) 1 b + b, ( A) σ (λi A) 1 b = b, ( A) σ (λi A) 1 b + ( A) σ (λi A) 1 b, b b, ( A) σ (λi A) 1 b + b, ( A) σ ( λi A) 1 b = b, ( A) σ [ (λi A) 1 + ( λi A) 1] b = (λ + λ) b, ( A) σ ( λi A) 1 (λi A) 1 b 2 b, ( A) σ ( λi A) 1 A(λI A) 1 b = (λ + λ) ( A) σ (λi A) 1 b, (λi A) 1 b +2 b, ( A) σ ( λi A) 1 ( A) 1 σ ( A) σ ( λi A) 1 b = (λ + λ) ( A) σ (λi A) 1 b z, ( A) 1 σ z where z = ( A) σ ( λi A) 1 b. Hence we have the sum of two positive terms. The previous computations show that ˆv(λ) û(λ) belongs to H 2 of the right half plane and that Parseval equality can be used on the imaginary axis in order to prove that v u L 2, precisely as in Subsection

16 3.2.1 Noisy observation Finally, we examine the effect of the noise in the observation. This is quite standard and we sketch the usual idea, in the general case of Section 3.2. In general the observation is corrupted by errors of known tolerance h. Hence the observation is y = Cx + θ = b, ( A) σ x + θ, θ < h t T. The norm of θ is often the L (, T ) norm, sometime the L 2 (, T ) norm. A bound on the L norm implies a bound on the L 2 norm, so that we can consider this last case. Now v depends on α and on θ. In the presence of noise in the observation we cannot expect lim α + v = u. In fact, if α + while the tolerance h of the noise remains fixed, we have lim α + v L 2 (,T ) = +, see [6]. Hence, the consistency result to be proved now is that lim v = u when both α and the tolerance h tends to + while respecting suitable consistency conditions. The function v, the candidate approximant of u is still given by formula (16), but we have to take into account the effect of the noise in the measures of y so that we now have v = γ α [Cw y] = γ α C[w x] + γ α θ. Although this function v is a function of α and θ, it is a common practice to denote it v α,h. We replace this new expression of v in the definition of w, hence of e and we note that all the computations in Section 3.2 can be repeated, with u replaced by u (γ/α)θ. Hence we get the following consistency result: Theorem 7 If α + and h + while respecting the condition then we have lim v α,h = u. h α, Appendix We collect in this appendix important results which, for the sake of clarity, have not been proved before. 15

17 We first prove that the operator A 1 = A γ α BB ( A) σ generates a holomorphic semigroup for every σ [, 1) and every γ/α >. We then prove that this semigroup is exponentially stable. Remark 9 discuss the interest of this fact. The proof that A 1 generates a holomorphic semigroup is based on [1, Theorem ], i.e. on the following result. Let A generates a holomorphic semigroup. Let the linear operator B, defined on dom A, satisfy the following property: for every χ > there exists ν such that Bx χ Ax + ν x for every x dom A. (21) Then the operator A + B, defined on dom A, defines a holomorphyc semigroup. I.e., the requested condition is that the A-bound of B should be zero. In order to apply this theorem, we consider B = γ α BB ( A) σ. We must prove that the A-bound of this operator is zero. This follows from inequality [16, pag. 73]: for every positive ρ we have ( ( A) σ x C ρ σ x + 1 ) Ax x dom A ρ1 σ The constant C does not depend on ρ. Hence, γ α B B( A) σ x γ α C ρ σ BB x + γ α C 1 Ax. ρ1 σ We see from here that, for each fixed α >, the A-bound of B is zero, as wanted. We consider now the spectrum of A 1. We prove that, for each fixed α > we have the inclusion ρ(a 1 ) Π + = {λ, Re λ }. We fix any α > and any λ with non negative real part. We show first that λ cannot be either an eigenvalue or an element of the continuous spectrum. In fact, in both these cases we could find a sequence {x n } in dom A, x n = 1, such that λx n Ax n + γ α BB ( A) σ x n (if λ would be an eigenvalue then (x n ) would be a stationary sequence). We take the scalar product with ( A) σ x n and we see that 16

18 λ ( A) σ x n, x n + γ α B ( A) σ x n 2 + ( A) σ x n, ( A) 1 σ ( A) σ x n. Each of the three terms has non negative real part (in fact the second and third terms are real). So, if this sum tends to zero then in particular the last addendum tends to zero. We show that this is not possible as follows. We recall that A is boundedly invertible so that ( A) σ is boundedly invertible too. Using this observation we see that ( A) (1 σ)/2 ( A) σ x n 2 m 1 ( A) σ x n 2 m 2 x n 2 = m 2 x n 2 = m 2 where m 1 and m 2 are positive. Hence, λ is neither an eigenvalue nor a member of the continuous spectrum. Now we prove that λ does not belong to the residual spectrum. By contradiction, let it be possible to find y which is orthogonal to the image of (λi A 1 ). In this case, for every x dom A we have = y, [λi A+ γ α BB ( A) σ ]x = y, [I+ γ α BB ( A) σ (λi A) 1 ](λi A)x. Here λ is fixed while x is any element in dom A. Hence, ξ = (λi A)x is arbitrary so that y ker[i + γ α BB ( A) σ (λi A) 1 ] = I + γ α ( A)σ (λi A) 1 BB i.e. y = γ α ( A)σ (λi A) 1 BB y. This shows that BB y. We take the inner product of both the sides with BB y and we get BB y, ( A) σ (λi A) 1 BB y = α γ B y 2 < We show that this equality is impossible if Re λ. We proceed as in the proof of (2) in order to compute the real part of the right hand side, which is BB y, ( A) σ ( λi A) 1 [2Re λ 2A] (λi A) 1 BB y. This shows that λ does not belong to the residual spectrum. We now recall now that a holomorphic semigroup is exponentially stable when its spectrum is contained in Re λ <, see [1, p. 347]. Hence 17

19 Theorem 8 For each fixed α > the semigroup exp {(A γ α BB ( A) σ )t} is exponentially stable. Remark 9 This last observation is very important for the applications. In fact in every practical application the values of v and e have to be numerically computed. If the semigroup would not be exponentially stable the resulting errors would be a great obstacle to the practical application of the algorithm. References [1] Arendt, W., Batty, C.J.K., Hieber, M., and F. Neubrander: Vector-valued Laplace transforms and Cauchy problems (Monographs in Mathematics n. 96). Basel: Birkhäuser Verlag 21. [2] Fagnani, F. and L. Pandolfi: A singular perturbation approach to a recursive deconvolution problem., SIAM J. Control Optim. 4 (22), [3] Fagnani, F. and L. Pandolfi: A recursive algorithm for the approximate solutions of Volterra integral equations of first kind of convolution type. Inverse problems, 19 (23), [4] Fagnani, F. and L. Pandolfi: On the solution of a class of Volterra integral equations of the first kind. J. Inverse Ill-Posed Probl. 11 (23), [5] Fagnani, F., Maksimov, V. and L. Pandolfi: A recursive deconvolution approach to disturbance reduction. IEEE Trans. Automat. Control 49 (24), [6] Groetsch, C.W.: The theory of Tikhonov regularization for Fredholm equations of the first kind, Boston: Pitman [7] Lamm P.K.: Future-sequential regularization methods for ill-posed Volterra equations. J. Mathematical Analysis Appl. 195 (1995), [8] Grasselli, M., Kabanikhin, S.I. and A. Lorenzi: An inverse hyperbolic integrodifferential problem arising in geophysics II. Nonlinear Analysis TMA 15 (199),

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