Simultaneous identification of damping coefficient and initial value for ...

INTERNATIONAL JOURNAL OF CONTROL,  https://doi.org/./..

Simultaneous identification of damping coefficient and initial value for PDEs from boundary measurement Zhi-Xue Zhaoa,b , Mapundi K. Bandab and Bao-Zhu Guo

c,d

a

School of Mathematical Sciences, Tianjin Normal University, Tianjin, China; b Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa; c The Key Laboratory of System and Control, Academy of Mathematics and Systems Science, Academia Sinica, Beijing, China; d School of Computer Science and Applied Mathematics,University of the Witwatersrand, Johannesburg, South Africa

ABSTRACT

ARTICLE HISTORY

In this paper, simultaneous identification of damping or anti-damping coefficient and initial value for some Riesz spectral systems is considered. An identification algorithm is proposed based on the fact that the output of the system happens to be decomposed into a product of an exponential function and a periodic function. The former contains information of the damping coefficient, while the latter does not. The convergence and error analysis are also developed. Three examples, namely an anti-stable wave equation with boundary anti-damping, the Schrödinger equation with internal antidamping and two connected strings with middle joint anti-damping, are investigated and demonstrated by numerical simulations to show the effectiveness of the proposed algorithm.

Received  August  Accepted  April 

1. Introduction Let H be a Hilbert space with the inner product  ·, · and inner product induced norm ·, and let Y = R (or C). Consider the dynamic system in H:  ˙ ) = A(q)x(t ), x(0) = x0 , x(t (1.1) y(t ) = Cx(t ) + d(t ), where A(q) : D(A(q)) ⊂ H → H is the system operator with compact resolvent depending on the coefficient q, which is assumed to be a generator of C0 -semigroup Tq = (Tq (t ))t∈R+ on H, C : H → Y is the admissible observation operator for Tq (Weiss, 1989), x0  H is the initial value and d(t) is the external disturbance. Various partial differential equation (PDE) control systems with damping mechanism can be formulated into system (1.1), where q is the damping coefficient. For a physical system, if the damping is produced by the material itself that dissipates the energy stored in system, then the system keeps stable. The identification of damping coefficient has been well considered for distributed parameter systems like Kelvin–Voigt viscoelastic damping coefficient in Euler–Bernoulli beam investigated in Banks and Rosen (1987), and a more general theoretical framework for various classes of parameter estimation problems presented in Banks and Ito (1988). In these works, the inverse problems are formulated as least square problems and are solved by finite dimensionalisation. For more relevant works, we can CONTACT Zhi-Xue Zhao

[email protected]

©  Informa UK Limited, trading as Taylor & Francis Group

KEYWORDS

Identification; damping coefficient; anti-stable PDEs; anti-damping coefficient

refer to the monograph (Banks & Kunisch, 1989). Sometimes, however, the source of instability may arise from the negative damping. One example is the thermoacoustic instability in duct combustion dynamics and the other is the stick-slip instability phenomenon in deep oil drilling, see for instance, Bresch-Pietri and Krstic (2014) and the references therein. In such cases, the negative damping will result in all the eigenvalues being located in the righthalf complex plane, and the open-loop plant is hence ‘anti-stable’ (exponentially stable in negative time) and the coefficient q in such kind of system is said to be the anti-damping coefficient. A widely investigated problem in recent years is stabilisation for anti-stable systems by imposing feedback controls. By the back-stepping method, a boundary state feedback control was designed in Smyshlyaev and Krstic (2009) to stabilise an anti-stable wave equation, and a generalisation of Smyshlyaev and Krstic (2009) was made in Guo and Jin (2010) to two connected anti-stable strings with joint anti-damping. Very recently, Guo and Jin (2013, 2015) investigated stabilisation for anti-stable wave equation subject to external disturbance coming through the boundary input, where the sliding mode control and active disturbance rejection control are employed. It is worth pointing out that in all the aforementioned works, the anti-damping coefficients are always supposed to be known. On the other hand, a few stabilisation results for anti-stable systems with unknown anti-damping coefficients are also available. In Krstic (2010), a full state

2

Z.-X. ZHAO

feedback adaptive control was designed for an anti-stable wave equation. By converting the wave equation into a cascade of two delay elements, an adaptive output feedback control and parameter estimator were designed in Bresch-Pietri and Krstic (2014). Unfortunately, no convergence of the parameter update law is provided in these works. It can be seen in Bresch-Pietri and Krstic (2014) and Krstic (2010) that it is the uncertainty of the anti-damping coefficient that leads to complicated design for adaptive control and parameter update law. This comes naturally with the identification of unknown anti-damping coefficient. To the best of our knowledge, there are few studies in this regard. Our focus in the present paper is on simultaneous identification for both anti-damping (or damping) coefficient and initial value for system (1.1), where the coefficient q is assumed to be in a prior parameter set Q and the initial value is supposed to be nonzero. We proceed as follows. In Section 2, we propose an algorithm to identify simultaneously the coefficient and initial value through the measured observation. The system may or may not suffer from a general bounded disturbance. In Section 3, a wave equation with anti-damping term in the boundary is discussed. A Schrödinger equation with internal anti-damping term is investigated in Section 4. Section 5 is devoted to coupled strings with middle joint anti-damping. In all these sections, numerical simulations are presented to verify the performance of the proposed algorithms. Some concluding remarks are presented in Section 6.

To begin with, we suppose that there is no external disturbance in system (1.1), that is, 

˙ ) = A(q)x(t ), x(0) = x0 , x(t y(t ) = Cx(t ).

Theorem 2.2 indicates that identification of the coefficient q and initial value x0 can be achieved exactly simultaneously without error for A(q) with some structure. Theorem 2.2: Suppose x0  0. Let A(q) in system (2.3) generate a C0 -semigroup Tq = (Tq (t ))t∈R+ and suppose that A(q) and the boundary observation operator C satisfy the following conditions: (1) A(q) has a compact resolvent and all its eigenvalues {λn }n∈N (or {λn }n∈Z ) admit the following expansion: λn = f (q) + iμn ,

where f : Q → R is invertible, μn is independent of q, and there exists an L > 0 such that μn L ∈ Z for all n ∈ N. 2π

for all

k ∈ Z,

(2.5)

(2) The corresponding eigenvectors {φn }n∈N form a Riesz basis for H. (3) There exist two positive numbers κ and K such that κ  |κ n |  K for all n ∈ N, where κn := Cφn ,

Theorem 2.1: Assume that the strictly increasing sequence {ωk }k∈Z of real numbers satisfies the gap condition ωk+1 − ωk ≥ γ

· · · < μn < μn+1 < · · · , (2.4)

2. Identification algorithm Before giving the main results, we introduce the following well-known Ingham’s theorem (Ingham, 1936; Komornik & Loreti, 2005; Young, 1980) as Theorem 2.1.

(2.3)

n ∈ N.

(2.6)

Then both coefficient q and initial value x0 can be uniquely determined by the output y(t), t  [0, T], where T > 2L. Precisely, for any L < T1 < T2 − L, q= f

−1



(2.1)

yL2 (T1 ,T2 ) 1 ln L yL2 (T1 −L,T2 −L)

 ,

(2.7)

and for some γ > 0. Then, for all T > 2π/γ , there exist two positive constants C1 and C2 , depending only on γ and T, such that C1





T

|ak |2 ≤

k∈Z

0

 2      ak eiωkt  dt ≤ C2 |ak |2 , (2.2)    k∈Z

k∈Z

for every complex sequence (ak )k∈Z ∈ 2 , where 2T C1 = π

    4π 2 4π 2 8T 1 − 2 2 , C2 = 1+ 2 2 . T γ π T γ

1 1 x0 = L κn n∈N



L

y(t )e

−λn t

 dt φn . (2.8)

0

Proof: Since {φn }n∈N forms a Riesz basis for H, there exists a sequence {ψn }n∈N of eigenvectors of A(q)∗ , which is biorthogonal to {φn }n∈N , that is, φ n , ψ m  = δ nm . In this way, we can express the initial value x0  H as x0 =  n∈N x0 , ψn φn , and the solution of system (2.3) as x(t ) = Tq (t )x0 =

 n∈N

eλnt x0 , ψn φn .

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By Equation (2.5), there exists an increasing sequence {Kn } ⊂ Z such that μn =

2πKn , n ∈ N, L

We next need to show that CA(q)−1 is bounded. For any f  H, we have  2  1     f , ψn Cφn  CA(q)−1 f 2 =    λn n∈N     Cφn 2    ≤ | f , ψn |2  λ  · n n∈N n∈N

2  K 1 ≤  f 2 , M1 |λn |2

(2.9)

which implies that {μn } satisfies the following gap condition μn+1 − μn =

3

2π 2π (Kn+1 − Kn ) ≥  γ . (2.10) L L

n∈N

So generally, we have y(t ) = Cx(t ) = =e





f (q)t

eλnt x0 , ψn Cφn

n∈N

eiμn t x0 , ψn Cφn  e f (q)t PL (t ), (2.11)

n∈N

 where PL (t ) = n∈N eiμn t x0 , ψn Cφn . To show that the boundary observation y(t) in Equation (2.11) is well defined, we need to prove that C is admissible for Tq (Weiss, 1989). We first need to show that for some τ > 0, there exists a constant Kτ  0 such that 

τ

0

y(t )2 dt ≤ Kτ2 x0 2 ,

∀ x0 ∈ D(A(q)).

Actually, by Theorem 2.1 and Equations (2.10) and (2.11), it follows that, for τ > L, 2  τ   τ     y(t )2 dt = eiμn t x0 , ψn Cφn  dt e f (q)t  0 0  n∈N  |x0 , ψn |2 , ≤ Cτ K 2

 where the convergence of n∈N |λ1n |2 is from Equations (2.4) and (2.5). Therefore, C is admissible for Tq (Weiss, 1989), see also Proposition 2 of Guo and Luo (2002a) or Theorem 2 of Guo and Luo (2002b). It follows from Equation (2.9) that PL (t) is a Y-valued function of period L, and so for any T2 − L > T1 > L, 

T2

|x0 , ψn |2 ≤ x0 2 ≤ M2

n∈N



|x0 , ψn |2 . (2.12)



T2 −L T1 −L

  y(t )2 dt,

yL2 (T1 ,T2 ) = e f (q)L yL2 (T1 −L,T2 −L) .

(2.13)

To obtain Equation (2.7), we need to show that yL2 (T1 ,T2 ) = 0 for T2 − T1 > L. Actually, it follows from Equation (2.11) that  y2L2 (T1 ,T2 )

T2

=

 f (q)t 2 e PL (t ) dt

T1

≥ C3



2 f (q)L

 f (q)(t+L) 2 e PL (t ) dt

that is,

where

Since {φn }n∈N forms a Riesz basis for H and so does {ψn }n∈N for H, there are two positive numbers M1 and M2 such that

T2 −L T1 −L

=e



 

2 L2 8τ 1 + 2 · max{1, eτ f (q) } . Cτ = π τ



T1

n∈N

M1

  y(t )2 dt =

T2 T1

 2     eiμnt x0 , ψn Cφn  dt, (2.14)    n∈N

  where C3 = min e2T1 f (q) , e2T2 f (q) > 0. By Theorem 2.1 and the gap condition (2.10), it follows that for T2 − T1 > 2π = L, γ 

T2

T1

 2      iμn t 2 |x0 , ψn |2 , e x , ψ Cφ  0 n n  dt ≥ C1 κ   n∈N

n∈N

(2.15)

n∈N

where

Hence  0

τ

Cτ K 2 y(t ) dt ≤ x0 2 . M1 2

  L2 2(T2 −T1 ) C1 = 1− > 0 for T2 − T1 > L. π (T2 −T1 )2

4

Z.-X. ZHAO

The inequality (2.14) together with (2.15) gives y2L2 (T1 ,T2 ) ≥ C1C3 κ 2



|x0 , ψn |2 .

(2.16)

n∈N

Combining Equation (2.16) with Equation (2.12) yields yL2 (T1 ,T2 ) ≥ Cx0  > 0,

(2.17)

 where C = κ CM1C23 > 0. The identity (2.7) then follows from (2.13). The inequality (2.17) means that system (2.3) is exactly observable for T2 − T1 > L. So the initial value x0 can be uniquely determined by the output y(t), t  [T1 , T2 ]. We show next how to reconstruct the initial value from the output. Actually, it follows from Equation (2.9) that 1 L



L

ei(μm −μn )t dt = δnm .

(2.18)

0

Hence, 

L

y(t )e

0

−λn t



L

dt = 0





e

i(μm −μn )t

x0 , ψm Cφm dt

m∈N

= κn L · x0 , ψn .

(2.19)

Therefore, the initial value x0 can be reconstructed by x0 =

 n∈N

x0 , ψn φn =

1 1 L n∈N κn



L

 y(t )e−λnt dt φn .

0

(2.20)  This completes the proof of the theorem. Remark 2.1: Clearly, Equations (2.7) and (2.8) provide an algorithm to reconstruct q and x0 from the output. It seems that the condition (2.5) is restrictive but it is satisfied by some physical systems discussed in Sections 3–5. Condition (2.5) is only needed for identification of q. For identification of initial value only, this condition can be removed. From numerical standpoint, the function PL (t) in Equation (2.11) can be approximated by finite truncation. Hence, condition (2.5) can be relaxed in numerical algorithm as follows: C . There exists an L such that: every μ2πn L is equal to (or approximately equal to) some integer for n  {1, 2, , N}, for some sufficiently large N. Obviously, the relaxed condition C can still ensure that PL (t) is approximately equal to a function of period L. In this case, some points μn may be very close to each other and the corresponding Riesz basis property of the family of divided differences of exponentials eiμn t developed in Avdonin and Ivanov (1995) (Section II.4),

Avdonin and Moran (2001), and Avdonin and Ivanov (2002) can be used. For the third condition, |Cφn | ≤ K implies that C is admissible for Tq which ensures that the output belongs to L2loc (0, ∞; Y ), and |Cφn | ≥ κ implies that system (2.3) is exactly observable which ensures the unique determination of the initial value. It is easily seen from Equation (2.13) that the coefficient q can always be identified as long as yL2 (T1 ,T2 ) = 0 for some time interval [T1 , T2 ], which shows that the identifiability of coefficient q does not rely on the exact observability. Remark 2.2: The condition T2 − T1 > L in Theorem 2.2 is only used in application of Ingham’s inequality in Equation (2.15) to ensure that yL2 (T1 ,T2 ) = 0. In practical applications, however, this condition is not always necessary. Actually, any L < T1 < T2 is applicable in Equation (2.7) as long as yL2 (T1 ,T2 ) = 0. Similar remark also applies for Theorem 2.3. Remark 2.3: It should be noted that for identification of damping coefficient in Banks and Rosen (1987), Banks and Ito (1988), and Banks and Kunisch (1989), the distributed observations are always required. In Theorem 2.2, however, we use only boundary measurement. In addition, our identification algorithm uses specifically the damping mechanism, i.e. the damping coefficient q can make the measurement have an exponential factor ef(q)t in Equation (2.11). On the other hand, we should also point out that in most of the existing literature where identification of the damping coefficient is dealt with, the initial value is always supposed to be known (Banks & Ito, 1988; Banks & Rosen, 1987). In Theorem 2.2, we not only remove this restrictive condition but also develop an algorithm to reconstruct the unknown initial value. Actually, after q being estimated, there are various methods for initial value reconstruction, see e.g. Ramdani et al. (2010) and Xu (2014) and the references therein. The idea of the algorithm for reconstruction of the initial value here is borrowed from the Riesz basis approach proposed in Xu (2014). Now we turn to system (1.1) with external disturbance, i.e. system (1.1) is corrupted by an unknown general bounded disturbance d(t) in observation. It should be noted that system (1.1) is supposed to be anti-stable in Theorem 2.3, whereas in Theorem 2.2, there is no constraint on the stability of system. Theorem 2.3: Suppose that system (1.1) is anti-stable and all the conditions in Theorem 2.2 are satisfied. If the inverse of f(q) is continuous and the disturbance d(t) is bounded, i.e. |d(t)|  M for some M > 0 and all t  0, then for T1 > L and L < T2 − T1 < +, ˆ 1 , T2 ) = q, lim q(T

T1 →+∞

lim xˆ0T1 − x0  = 0, (2.21)

T1 →+∞

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ye L2 (T1 −L,T2 −L) > dL2 (T1 −L,T2 −L) , we have

where ˆ 1 , T2 ) = f −1 q(T



yL2 (T1 ,T2 ) 1 ln L yL2 (T1 −L,T2 −L)

 , (2.22)

and xˆ0T1

1 1 = L n∈N κn



T1

 dt φn .

(2.23)

Moreover, for sufficiently large T1 , the ˆ 1 , T2 )) − f (q)| and xˆ0T1 − x0  satisfy | f (q(T

errors

T1 −L

y(t )e

−λn t

5

√ 4ML−1 T2 − T1 ˆ 1 , T2 )) − f (q)| < | f (q(T , √ yL2 (T1 −L,T2 −L) − M T2 − T1

(2.24)

yL2 (T1 ,T2 ) ye + dL2 (T1 ,T2 ) = yL2 (T1 −L,T2 −L) ye + dL2 (T1 −L,T2 −L) ye L2 (T1 ,T2 ) + dL2 (T1 ,T2 ) ≤ ye L2 (T1 −L,T2 −L) − dL2 (T1 −L,T2 −L) ≤

eL f (q) + ε(T1 , T2 ) , 1 − ε(T1 , T2 )

(2.28)

where √ M T2 − T1 ε(T1 , T2 ) = . ye L2 (T1 −L,T2 −L)

(2.29)

Similarly, for sufficiently large T1 such that ye L2 (T1 ,T2 ) > dL2 (T1 −L,T2 −L) ,

and xˆ0T1

yL2 (T1 ,T2 ) eL f (q) − ε(T1 , T2 ) ≥ . yL2 (T1 −L,T2 −L) 1 + ε(T1 , T2 )

CM − x0  ≤ √ e− f (q)T1 for some C > 0. (2.25) κ L

Proof: Introduce ye (t ) = CTq (t )x0 = y(t ) − d(t ) = e f (q)t PL (t ), (2.26)

It is clear from Equations (2.27) and (2.29) that, for L < T2 − T1 < +, we have limT1 →+∞ ε(T1 , T2 ) = 0. This together with Equations (2.28) and (2.30) gives yL2 (T1 ,T2 ) = eL f (q) . T1 →+∞ yL2 (T1 −L,T2 −L)

where PL (t) is defined in Equation (2.11). We first show that for T2 − T1 > L, lim ye L2 (T1 ,T2 ) = +∞.

T1 →+∞

(2.27)

Since system (1.1) is anti-stable, the real part of the eigenvalues f(q) > 0. It then follows from Equation (2.26) that  ye 2L2 (T1 ,T2 ) = ≥e

T2

 f (q)t 2 e PL (t ) dt

T1 2 f (q)T1



T2 T1

 2     iμn t e x0 , ψn Cφn  dt.    n∈N

Using the same arguments as Equations (2.14)–(2.17) in the proof of Theorem 2.2, we have ye L2 (T1 ,T2 ) ≥ Ce f (q)T1 x0 ,  C1 > 0. Since f(q) > 0, x0  0, Equation where C = κ M 2 (2.27) holds. On the other hand, since |d(t)|  M, for any finite time interval I,  dL2 (I) =

|d(t )|2 dt

 12

 ≤ M |I|,

I

where |I| represents the length of the time interval I. Therefore, for sufficiently large T1 such that

(2.30)

lim

Since f−1 (q) is continuous, −1

ˆ 1 , T2 ) = f lim q(T



T1 →+∞

= q.

yL2 (T1 ,T2 ) 1 ln lim L T1 →+∞ yL2 (T1 −L,T2 −L)



We next show convergence of the initial value. Similarly with the arguments (2.18)–(2.20) in the proof of Theorem 2.2, we have 1 1 x0 = L n∈N κn



T1 T1 −L

ye (t )e

−λn t

 dt φn , ∀ T1 ≥ L.

It then follows from Equation (2.23) that for arbitrary T1  L, xˆ0T1

1 1 − x0 = L κn n∈N



T1 T1 −L

d(t )e

−λn t

 dt φn .

In view of the Riesz basis property of {φ n }, it follows that xˆ0T1

   2  1  1  T1   2 −λn t − x0  =  d(t )e dt φn   L κn T1 −L n∈N

M2 ≤ 2 2 e−2 f (q)(T1 −L) Lκ

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Z.-X. ZHAO

    ×  n∈N

L



d(t + T1 − L)e

− f (q)t



e

−iμn t

0

2  dt  ,

(2.31) where M2 > 0 is introduced in Equation (2.12). To estimate the last series in Equation (2.31), we need the Riesz property of the exponential system :=  basis (sequence) fn = eiμn t n∈N . There are two cases according to the relation between the sets {Kn }n∈N introduced in Equation (2.9) and integers Z: 2nπ Case 1: {Kn }n∈N = Z, that is, = {ei L t }n∈Z . In this case, since {eint }n∈Z forms a Riesz basis for L2 [ − π, π], forms a Riesz basis for L2 [− L2 , L2 ]. Case 2: {Kn }n∈N   Z. In  this case, it is noted that the exponential system eiμnt n∈N forms a Riesz sequence for L2 [− L2 , L2 ]. In each case above, by properties of Riesz basis and Riesz sequence (see e.g. Young, 1980, pp.32–35, p.154), there exists a positive constant C4 > 0 such that 

|(g, fn )|2 ≤ C4 g2L2 [− L , L ] , 2 2

n∈N

(2.32)

for all g ∈ L2 [− L2 , L2 ], where ( ·, ·) denotes the inner product in L2 [− L2 , L2 ]. We return to the estimation of xˆ0T1 − x0 . By variable substitution of t = L2 − s in Equation (2.31), together with Equation (2.32), we have xˆ0T1 − x0 2 M2 ≤ 2 2 e−2 f (q)(T1 −L) Lκ  2      L2   L f (q)(s− L2 ) −iμn L2 iμn s  d T1 − − s e e ds × e   − L2  2 n∈N     L  2 2  M2C4 d T1 − L −s e f (q)(s− L2 )  ds ≤ 2 2 e−2 f (q)(T1 −L)   Lκ 2 − L2 ≤

M 2 M2C4 −2 f (q)(T1 −L) e . Lκ 2

Therefore,  xˆ0T1 − x0  ≤ e

f (q)L

M2C4 M − f (q)T1 e , L κ

which implies that xˆ0T1 − x0  will tend to zero as T1 → + for f(q) > 0. The √ inequality (2.25) with the positive number C = e f (q)L M2C4 is also concluded. ˆ 1 , T2 )) − f (q)|. Since Finally, we estimate | f (q(T limT1 →+∞ ε(T1 , T2 ) = 0 for L < T2 − T1 < +, setting T1 large enough such that ε(T1 , T2 ) < 1, then it follows

from Equations (2.22) and (2.28) that yL2 (T1 ,T2 ) eL f (q) + ε(T1 , T2 ) ≤ ln yL2 (T1 −L,T 1 − ε(T1 , T2 )   2 −L) (1 + e−L f (q) )ε(T1 , T2 ) = L f (q) + ln 1 + 1 − ε(T1 , T2 ) 2ε(T1 , T2 ) , < L f (q) + 1 − ε(T1 , T2 )

ˆ 1 , T2 )) = ln L f (q(T

(2.33) where the last inequality comes from the fact ln (1 + x) < x over x > 0. Similarly, for T1 large enough and L < T2 − T1 < + such that ε(T1 , T2 ) ≤ 14 , it follows from Equations (2.22) and (2.30) that yL2 (T1 ,T2 ) eL f (q) − ε(T1 , T2 ) ≥ ln yL2 (T1 −L,T 1 + ε(T1 , T2 )   2 −L) (1 + e−L f (q) )ε(T1 , T2 ) = L f (q) + ln 1 − 1 + ε(T1 , T2 ) 4ε(T1 , T2 ) > L f (q) − , 1 + ε(T1 , T2 )

ˆ 1 , T2 )) = ln L f (q(T

(2.34) where the last inequality comes from the fact ln (1 − x)  −2x over x ∈ [0, 12 ]. Combining Equations (2.33) and (2.34), and setting T1 large enough and L < T2 − T1 < + such that ε(T1 , T2 ) ≤ 14 , we have ˆ 1 , T2 )) − f (q)| < | f (q(T

4ε(T1 , T2 ) . L

The error estimation (2.24) comes from the fact √ M T2 − T1 ε(T1 , T2 ) ≤ . √ yL2 (T1 −L,T2 −L) − M T2 − T1  We thus complete the proof of the theorem. Remark 2.4: Theorem 2.3 shows that when system (1.1) ˆ 1 , T2 ) defined in Equation (2.22) is anti-stable, then q(T can be regarded as an approximation of the coefficient q when T1 is sufficiently large. Roughly speaking, the ε(T1 , T2 ) defined in Equation (2.29) reflects the ratio of the energy, in L2 norm, of the disturbance d(t) which is an unwanted signal, with the energy of the real output signal ye (t). We may regard 1/ε(T1 , T2 ) as signal-tonoise ratio (SNR) which is well known in signal analyˆ 1 , T2 ) defined in Equasis. Theorem 2.3 indicates that q(T tion (2.22) is an approximation of the coefficient q when SNR is large enough. However, if system (1.1) is stable, i.e. f(q) < 0, similar analysis shows that the output will be an exponentially decaying oscillation, which implies that the unknown disturbance will account for a large proportion in observation and the SNR cannot be too large. In this case, it is difficult to extract enough useful information from the corrupted observation as that with large SNR. Actually, the anti-stability assumption in

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Theorem 2.3 is necessary since otherwise, we may have the case of y(t ) = Cx(t ) + d(t ) ≡ 0 for which we cannot obtain anything for identification. Remark 2.5: It is well known that the inverse problems are usually ill-posed in the sense of Hadamard, that is, an arbitrarily small error in the measurement data may lead to a large error in solution. Theorem 2.3 shows that if system (1.1) is anti-stable, our algorithm is robust against bounded unknown disturbance in measurement data. Actually, similar to the analysis in Theorem 2.3, it can be shown that when system (1.1) is not anti-stable, the algorithm in Theorem 2.2 is also numerically stable in the presence of small perturbations in the measurement data, as long as the perturbation is relatively small in comparison to the output. Some numerical simulations validate this also in Example 3.1 in Section 3.

7

It is indicated in Xu (2014) that the operator A generates a C0 -group on H. Lemma 3.1 (Xu, 2014 ): Let A be defined by Equation (3.2) and let q  1. Then the spectrum of A consists of all isolated eigenvalues given by ⎧ 1 1+q ⎪ + inπ, q > 1, ⎨ λn = ln 2 q−1 (3.3) 2n + 1 1 1+q ⎪ ⎩ λn = ln +i π, 0 < q < 1, 2 1−q 2 for n ∈ Z, and the corresponding eigenfunctions n (x) are given by   sinh λn x , sinh λn x , ∀ n ∈ Z. (3.4) n (x) = λn Moreover, {n (x)}n∈Z forms a Riesz basis for H.

In this section, we apply the algorithm proposed in previous section to the identification of the anti-damping coefficient and initial values for a one-dimensional vibrating string equation described by Bresch-Pietri and Krstic (2014) and Krstic (2010)

Lemma 3.2 (Xu, 2014): Let A be defined by Equation (3.2) and let q  1. Then the adjoint operator A∗ of A is given by ⎧ ∗  ⎨ A (v, h) =  −(h, v ), 2 ∗ D(A ) = (v, h) ∈ H (0,1) × H 1 (0, 1) | v (0) (3.5) ⎩ = 0, v  (1) + qh(1) = 0 ,

⎧ utt = uxx , 0 < x < 1, t > 0, ⎪ ⎪ ⎨ u(0, t ) = 0, ux (1, t ) = qut (1, t ), t ≥ 0, t ≥ 0, ⎪ y(t ) = ux (0, t ) + d(t ), ⎪ ⎩ u(x, 0) = u0 (x), ut (x, 0) = u1 (x), 0 ≤ x ≤ 1, (3.1)

and σ (A∗ ) = σ (A). The eigenvector  n (x) of A∗ corresponding to λn is given by

sinh λn x n (x) = , − sinh λn x , ∀ n ∈ Z. (3.6) λn

where x denotes the position, t the time, 0 < q  1 the unknown anti-damping coefficient, u0 (x) and u1 (x) the unknown initial displacement and initial velocity, respectively, and y(t) is the boundary measured output corrupted by the disturbance d(t). Let H = HE1 (0, 1) × L2 (0, 1), where HE1 (0, 1) = { f ∈ 1 H (0, 1)| f (0) = 0}, equipped with the inner product ·,· and the inner product induced norm

It is easy to verify that for any n, m ∈ Z, n ,  m  = δ nm . System (3.1) can be written as the following evolutionary equation in H:

3. Application to wave equation



1

( f , g)2 =



| f  (x)|2 + |g(x)|2 dx.

dX (t ) = AX (t ), t > 0, dt

where X(t) = (u( ·, t), ut ( ·, t)), and the solution of Equation (3.7) is given by  X (t ) = eλnt X (0), n n . n∈Z

0

Define the system operator A : D(A)(⊂ H) → H as ⎧  ⎨ A( f , g) =  (g, f ), 2 D(A) = ( f , g) ∈ H (0, 1) × HE1 (0, 1) | f (0) (3.2) ⎩ = 0, f  (1) = qg(1) , and the observation operator C from H to C as C( f , g) = f  (0),

( f , g) ∈ D(A).

X (0) = (u0 , u1 ), (3.7)

Thus, y(t ) =



eλn t X (0), n  + d(t ).

(3.8)

n∈Z

It can be seen from Lemma 3.1 that when q = 1, the real part of the eigenvalues is +, while for 0 < q  1, the real part is finite positive. Hence, we suppose 1Q as usual (see e.g. Bresch-Pietri & Krstic, 2014; Krstic, 2010), where Q is the prior parameter set.

8

Z.-X. ZHAO

We take q  Q = (1, +) as an example to illustrate how to apply the algorithms proposed in previous section to simultaneous identification for the anti-damping coefficient and initial values. The following Corollaries 3.1 and 3.2 are the direct consequences of Theorem 2.2 and Theorem 2.3, respectively, by noticing that for system (3.1), the relevant function and parameters now are f (q) =

1 q+1 ln , μn = nπ, L = 2, κn = 1. 2 q−1

Corollary 3.1: Suppose that d(t) = 0 in system (3.1). Then both the coefficient q and initial values u0 (x) and u1 (x) can be uniquely determined by the output y(t), t  [0, T], where T > 4. Specifically, q can be recovered exactly from q=

yL2 (T1 ,T2 ) + yL2 (T1 −2,T2 −2) , 2 ≤ T1 < T2 − 2, yL2 (T1 ,T2 ) − yL2 (T1 −2,T2 −2) (3.9)

and the initial values u0 (x) and u1 (x) can be reconstructed from u0 (x) =

1 2



n∈Z

u1 (x) =

 1 2

n∈Z

2

y(t )e−λnt dt

0 2

y(t )e

−λn t

 

sinh λn x , λn

dt sinh λn x. (3.10)

0

Note that in Equation (3.10), the observation interval [0, 2] is the minimal time interval for observation to identify the initial values for any identification algorithm. Corollary 3.2: Suppose that q  Q = (1, +) in system (3.1) and the disturbance is bounded, i.e. |d(t)|  M for some M > 0 and all t  0. Then for T1 > 2 and 2 < T2 − T1 < +, ˆ 1 , T2 ) = q, lim q(T

T1 →+∞

lim (uˆ0T1 , uˆ1T1 ) − (u0 , u1 ) = 0,

T1 →+∞

(3.11)

To end this section, we present some numerical simulations for system (3.1) to illustrate the performance of the algorithm. Example 3.1: The observation with random noises when system (1.1) is stable. A simple spectral analysis together with Theorem 2.2 shows that Corollary 3.1 is also valid for q  Q = (−, −1). In this example, the damping coefficient q and initial values u0 (x), u1 (x) are chosen as q = −3,

u0 (x) = −3 sin πx,

u1 (x) = π cos πx.

In this case, the output can be obtained from Equation (3.8) (with d(t) = 0), where the infinite series is approximated by finite truncation, that is, {n ∈ Z} is replaced by {n ∈ Z | − 5000 ≤ n ≤ 5000}. Some random noises are added to the measurement data and we use these data to test the algorithm proposed in Corollary 3.1. Let T1 = 2, T2 = 2.5. Then the damping coefficient q can be recovered from Equation (3.9), and the initial values u0 (x) and u1 (x) can be reconstructed from Equation (3.10). Table 1 lists the numerical results for the damping coefficients (the second column in Table 1) and Figure 1(a)-1(c) for the initial values in various cases of noise levels. In Table 1, the absolute errors of the real damping coefficient and the recovered ones, and the L2 -norm of the differences between the exact initial values and the reconstructed ones are also shown. It is worth pointing out that in reconstruction of the initial values from Equation (3.10), the infinite series is approximated by finite truncation once again, that is, {n ∈ Z} is replaced by {n ∈ Z | |n| ≤ 1000}, which accounts for the zero value of the reconstructed initial velocity at the left end. This is also the reason that the errors of the initial velocity (the last column in Table 1) are relatively large even if there is no random noise in the measured data. Example 3.2: The observation with general bounded disturbance when system (1.1) is anti-stable. The anti-damping coefficient and initial values are chosen as

where yL2 (T1 ,T2 ) + yL2 (T1 −2,T2 −2) ˆ 1 , T2 ) = , (3.12) q(T yL2 (T1 ,T2 ) − yL2 (T1 −2,T2 −2) and  1



sinh λn x , 2 n∈Z T1 −2 λn   T1 (3.13) 1 y(t )e−λnt dt sinh λn x. uˆ1T1 (x) = 2 T1 −2

uˆ0T1 (x) =

n∈Z

T1

y(t )e−λnt dt

q = 3,

u0 (x) = 3 sin πx,

u1 (x) = π cos πx.

and the observation is corrupted by the bounded disturbance: d(t ) = 2 sin

1 + 3 cos 10t. 1+t

The relevant parameters in Corollary 3.2 are chosen to be T2 = T1 + 3, and let T1 be different values increasing from

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9

Table . Absolute errors with different noise levels. Noise level  % %

Recovered q

Errors for q

Errors for u (x)

Errors for u (x)

−. −. −.

.E− .E− .E−

.E− .E− .E−

.E− .E− .E−

2 to 10. The corresponding anti-damping coefficients ˆ 1 , T2 ) recovered from Equation (3.12) are depicted in q(T ˆ 1 , T2 ) converges to the real Figure 2. It is seen that q(T value q = 3 as T1 increases. Setting T1 = 2, 5, 9 in Equation (3.13) and reconstructing the initial values produce results in Figure 2 from which we can see that the reconstructed initial values become closer to the real ones as T1 increases.

4. Application to Schrödinger equation In this section, we consider a quantum system described by the following Schrödinger equation: ⎧ 0 < x < 1, t > 0, ut = −iuxx + qu, ⎪ ⎪ ⎨ ux (0, t ) = 0, u(1, t ) = 0, t ≥ 0, (4.1) y(t ) = u(0, t ) + d(t ), t ≥ 0, ⎪ ⎪ ⎩ u(x, 0) = u0 (x), 0 ≤ x ≤ 1.

0

0

real u (x) 0

−1

0

recovered

−2

−2 −3 0

real u (x)

−1

recovered

−3 0.2

0.4

0.6

0.8

−4 0

1

0.2

0.4

x

0.6

4

1

4

real u (x) 1

2

real u1(x)

2

recovered 0

0

−2

−2

−4 0

0.8

x

0.2

0.4

0.6

0.8

recovered

−4 0

1

0.2

0.4

0.6

x

x

(a)

(b)

0.8

1

0

real u (x) 0

−1

recovered

−2 −3 −4 0

0.2

0.4

0.6

0.8

1

x 4

real u (x) 1

2

recovered

0 −2 −4 0

0.2

0.4

0.6

0.8

1

x

(c)

Figure . The initial values: initial displacement (upper) and initial velocity (lower). (a) Without random noise, (b) with % random error and (c) with % random error.

10

Z.-X. ZHAO

System (4.1) can be rewritten as the following evolutionary equation in H:

3.2 3.1

dX (t ) = AX (t ), t > 0, dt

3 2

4

6

8

10

T

X (0) = u0 ,

(4.5)

and the solution of Equation (4.5) is given by

1



X (t ) =

4

eλnt X (0), φn φn .

n∈N∗

2

real u (x)

T =5

T =2

T =9

0

0 0

0.2

Thus,

1

1

1

0.4

0.6

0.8

y(t ) =

1

√  λt 2 e n u0 , φn  + d(t ).

x 5

0

−5 0

real u1(x)

T1=5

T1=2

T1=9

0.4

0.6

0.8

1

x

Figure . Anti-damping coefficient q and initial values u , u .

where u(x, t) is the complex-valued state, i is the imaginary unit and the potential q > 0 and u0 (x) are the unknown anti-damping coefficient and initial value, respectively. Let H = L2 (0, 1) be equipped with the usual inner product ·,· and the inner product induced norm ·. Introduce the operator A defined by 

The relevant function and parameters in Theorems 2.2 and 2.3 for system (4.1) are 

0.2

 Aφ = −iφ   + qφ, D(A) = φ ∈ H 2 (0, 1) | φ  (0) = φ(1) = 0 .

λn = q + i n −

1 2

π 2,

n ∈ N∗ ,

n ∈ N∗ .

π 2, L =

√ 8 , κn = 2. π

Parallel to Section 3, we have two corollaries corresponding to the exact observation and observation with general bounded disturbance, respectively, for system (4.1). Here, we only list the latter one and the former is omitted. Corollary 4.1: Suppose that q  Q = (0, +) in system (4.1) and the disturbance is bounded, i.e. |d(t)|  M for some M > 0 and all t  0. Then for T1 > π8 and π8 < T2 − T1 < +∞, ˆ 1 , T2 ) = q, lim q(T

lim uˆ0T1 − u0  = 0, (4.7)

T1 →+∞

ˆ 1 , T2 ) = q(T

(4.3)

and the corresponding eigenfunctions φ n (x) are given by   √ 1 φn (x) = 2 cos n − πx, 2

2

T1 →+∞

where

Lemma 4.1 (Krstic, Guo, & Smyshlyaev, 2011): Let A be defined by Equation (4.2). Then the spectrum of A consists of all isolated eigenvalues given by 2

1 f (q) = q, μn = n − 2

(4.2)

A straightforward verification shows that such defined A generates a C0 -semigroup on H.



(4.6)

n∈N∗

(4.4)

yL2 (T1 ,T2 ) π ln , 8 yL2 (T1 − π8 ,T2 − π8 )

and

   T1 π  1 −λn t πx. y(t )e dt cos n − uˆ0T1 (x) = 8 n∈N∗ T1 − π8 2 (4.9) We also give a numerical simulation to test the algorithm proposed in Corollary 4.1 for system (4.1), where the anti-damping coefficient q and initial value u0 (x) are chosen as q = 0.7,

u0 (x) = sin πx + i cos πx,

and the observation is corrupted by the disturbance In addition, {φn (x)}n∈N∗ forms an orthonormal basis for H.

(4.8)

d(t ) = 2 sin

t + 3i cos 20t. 10 + t

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joint vertical force anti-damping (see Guo & Jin, 2010; Guo & Zhu, 1997; Xu & Guo, 2003 for more details). Let H = HE1 (0, 1) × L2 (0, 1) be equipped with the inner product  ·, · and its induced norm

0.7 0.65



0.6 3

4

5

6

7

8

9

10

0.2

Re(u )

T =5.55

T1=2.55

T1=9.55

1

0.4

0.6

0.8

d X (t ) = AX (t ), dt

1

x Im(u )

T1=5.55

T =2.55

T1=9.55

0

1

0

0.2

0.4

0.6

0.8

(5.2)

where X (t ) = (u(·, t ), ut (·, t )) ∈ H and A is defined by

2

−2 0

|u (x)|2 + |v (x)|2 dx,

  where HE1 (0, 1) = u| u ∈ H 1 (0, 1), u(0) = 0 . Then system (5.1) can be rewritten as an evolutionary equation in H as follows:

1 0



0

2

0

1

(u, v )2 =

T1

−1 0

11

A(u, v ) = (v (x), u (x)),

1

x

Figure . Coefficient (upper), real part (middle) and imaginary part (lower) of initial value u (x).

The observation can be obtained from Equation (4.6) by a finite series approximation, that is, {n ∈ N} is replaced by {n ∈ N | n ≤ 5000}. The relevant parameters in Corollary 4.1 are chosen to be T2 = T1 + 1, and T1 increasing from 2.55 to 10. The corresponding antiˆ 1 , T2 ) recovered from Equation damping coefficients q(T ˆ 1 , T2 ) (4.8) are shown in Figure 3. It is obvious that q(T is convergent to the real value q = 0.7 as T1 increases. Setting T1 = 2.55, 5.55, 9.55 in Equation (4.9), the reconstructed initial values are shown in Figure 3 from which it is seen that the errors between the reconstructed initial values and the real ones become smaller as T1 increases.

with the domain  D(A) = (u, v ) ∈ H 1 (0, 1) × HE1 (0, 1)    u(0) = u (1) = 0, u| 1 ∈ H 2 (0, 1 ), [0, 2 ]  2 ,  − +  u|[ 1 ,1] ∈ H 2 ( 12 , 1), u ( 12 ) − u ( 12 ) = qv ( 12 ) 2 (5.4) where u|[a, b] denotes the function u(x) confined to [a, b]. We assume without loss of generality that the prior parameter set for q is Q = (2, +) since the case for Q = (0, 2) is very similar. Lemma 5.1 (Xu & Guo, 2003): Let A be defined by Equations (5.3) and (5.4) and q  Q = (2, +). Then A−1 is compact on H and the eigenvalues of A are algebraically simple and separated, given by λn =

5. Application to coupled strings equation In this section, we consider the following two connected anti-stable strings with joint anti-damping described by ⎧ ⎪ utt(x, t ) = uxx  (x, t ), x ∈ (0, 12 ) ∪ ( 12 , 1), t > 0, ⎪ ⎪ ⎪ − + ⎪ ⎪ t ≥ 0, u 12 , t = u 12 , t , ⎪ ⎪     ⎪ ⎨ 1− 1+ ux 2 , t − ux 2 , t = qut ( 12 , t ), t ≥ 0, ⎪ ⎪ t ≥ 0, u(0, t ) = ux (1, t ) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ (x), u (x, 0) = u u(x, 0) = u 0 t 1 (x), 0 ≤ x ≤ 1, ⎪ ⎪ ⎩ t ≥ 0, y(t ) = ux (0, t ) + d(t ),

(5.3)

1 q+2 ln + inπ, n ∈ Z. 2 q−2

(5.5)

The corresponding eigenfunctions n (x) are given by n (x) = (φn (x), λn φn (x)), ∀ n ∈ Z,

(5.6)

where ⎧√ λn 1 2 ⎪ ⎪ ⎪ 0 0, q  2 is the unknown anti-damping constant. System (5.1) models two connected strings with

and {n (x)}n∈Z forms a Riesz basis for H. In addition, A generates a C0 -semigroup on H.

12

Z.-X. ZHAO

Lemma 5.2 (Xu & Guo, 2003): Let A be defined by Equations (5.3) and (5.4) and q  Q = (2, +). Then the adjoint operator A∗ of A is given by A∗ (u, v ) = −(v, u ),

(5.7)

some M > 0 and all t  0. Then for T1 > 2 and 2 < T2 − T1 < +, ˆ 1 , T2 ) = q, lim q(T

T1 →+∞

lim (uˆ0T1 , uˆ1T1 ) − (u0 , u1 ) = 0,

T1 →+∞

where

with the domain 

  2 yL2 (T1 ,T2 ) + yL2 (T1 −2,T2 −2) ˆ 1 , T2 ) = q(T , (5.10) yL2 (T1 ,T2 ) − yL2 (T1 −2,T2 −2)

D(A∗ ) = (u, v ) ∈ H 1 (0, 1) × HE1 (0, 1)    u(0) = u (1) = 0, u| 1 ∈ H 2 (0, 1 ), [0, 2 ]  2 ,  − +  u|[ 1 ,1] ∈ H 2 ( 12 , 1), u ( 12 ) − u ( 12 ) = −qv ( 12 ) 2 ∗

∗

and σ (A ) = σ (A). The eigenfunctions  n (x) of A corresponding to λn are given by   n (x) = φ n (x), −λn φ n (x) ,

∀ n ∈ Z.

(5.8)

A direct calculation shows that { n (x)} is biorthogonal to {n (x)}. Hence, the solution of Equation (5.2) can be expressed as X (t ) =



eλnt X (0), n n .

and ⎧ ⎪ ⎪1 ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎨ 1 uˆ0T1 (x) = ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎧ 1 ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎨ 1 uˆ1T1 (x) = ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

 

T1

T1 −2 n∈Z   T1 n∈Z

T1 −2

n∈Z

T1 −2

 y(t )e−λn t dt  y(t )e

−λn t

sinh λn x 1 ,0