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1 SIAM J. CONTROL OPTIM. Vol. 49, No. 4, pp c 11 Society for Industrial and Applied Mathematics BOUNDARY CONTROLLERS AND OBSERVERS FOR THE LINEARIZED SCHRÖDINGER EQUATION MIROSLAV KRSTIC, BAO-ZHU GUO, AND ANDREY SMYSHLYAEV Abstract. We consider a problem of stabilization of the linearized Schrödinger equation using boundary actuation and measurements. We propose two different control designs. First, a simple proportional collocated boundary controller is shown to exponentially stabilize the system. However, the decay rate of the closed-loop system cannot be prescribed. The second, full-state feedback boundary control design, achieves an arbitrary decay rate. We formally view the Schrödinger equation as a heat equation in complex variables and apply the backstepping method recently developed for boundary control of reaction-advection-diffusion equations. The resulting controller is then supplied with the backstepping observer to obtain an output-feedback compensator. The designs are illustrated with simulations. Key words. distributed parameter systems, backstepping, stabilization AMS subject classification. 35Q93 DOI / Introduction. We consider a problem of stabilization of the linearized Schrödinger equation using boundary actuation and measurements. The existing results on the boundary control of the Schrödinger equation usually use the simple passive damper feedback [17, 6] most commonly used for stabilization of the wave equation. For exact controllability and observability results, see [1, 19, 16]. We propose two novel control designs. First, a proportional collocated boundary controller is shown to exponentially stabilize the system. The controller is simple and uses only boundary measurements of the state rather than measurements of the time derivative of the state in case of passive damper controllers). However, the decay rate of the closed-loop system cannot be arbitrarily prescribed. Our second approach relies on the backstepping method recently developed for parabolic PDEs [, 1]. The idea of the design is to formally consider the Schrödinger equation as a complex-valued heat equation and apply the method developed in []. This method uses invertible Volterra integral transformation together with the boundary feedback to convert the unstable plant into a well-damped target system. The kernel of this transformation satisfies a certain PDE, which turns out to be solvable in closed form. The proposed full-state design is explicit and achieves an arbitrary decay rate of the closed-loop system. The interest in achieving arbitrary decay rates in stabilization of distributed parameter systems goes back to at least Theorem.1 in []. Received by the editors October, 7; accepted for publication in revised form) March 31, 11; published electronically July 5, 11. A short version of this paper has appeared in Proceedings of the 46th IEEE Conference on Decision and Control [8]. Department of Mechanical and Aerospace Engineering, University of California at San Diego, 95 Gilman Drive, La Jolla, CA krstic@ucsd.edu, asmyshly@ucsd.edu). Academy of Mathematics and Systems Science, Academia Sinica, Beijing 119, China, School of Mathematical Sciences, Shanxi University, Taiyuan 36, China, and School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 5, Johannesburg, South Africa bzguo@iss.ac.cn). The research of this author was supported by the National Natural Science Foundation of China, the National Basic Research Program of China with grant 11CB88, and the National Research Foundation of South Africa. 1479

2 148 M. KRSTIC, B.-Z. GUO, AND A. SMYSHLYAEV We also develop dual backstepping observers that require only boundary sensing. These observers are combined with the backstepping controllers to obtain a collocated output-feedback compensator.. Collocated boundary controller. Consider the linearized Schrödinger equation 1) ψ t x, t) = jψ xx x, t), <x<1, where ψ is a complex-valued state and j is the imaginary unit. following boundary conditions: We assume the ) 3) ψ x,t)=, ψ1,t)=ut), where u is the control input. With u =, this system displays oscillatory behavior and is not asymptotically stable all the eigenvalues lie on the imaginary axis). The idea for our first control design comes from the consideration of the L energy of the system 4) Et) = 1 ψx, t) dx. Differentiating E with respect to time, we get 5) Ėt) =jut)ψ x 1,t). If we take 6) ut) = j c 1 ψ x 1,t), c 1 >, then 7) Ėt) = 1 c 1 ψ x 1,t). The above inequality certainly does not imply that the controller 6) achieves asymptotic stability; it only gives an indication that the proposed controller is potentially useful. We are going to investigate the properties of the closed-loop system 8) 9) 1) ψ t x, t) = jψ xx x, t), ψ x,t)=, ψ x 1,t)= jc 1 ψ1,t) using the Riesz spectral method. Define the system operator A in H = L, 1) as 11) A ϕx) = jϕ x) ϕ DA ), DA )={ϕ H, 1) ϕ ) =,ϕ 1) = jc 1 ϕ1)}. Then the system 8) 1) can be written as an evolutionary equation in H: 1) d dt ψ,t)=a ψ,t).

3 BOUNDARY CONTROLLERS FOR SCHRÖDINGER EQUATION 1481 Lemma.1. Let the operator A be defined by 11). Then the following hold. i) A 1 is compact on H and hence σa ), the spectrum of A, consists of isolated eigenvalues of finite algebraic multiplicity only. ii) A is dissipative and generates a C -semigroup of contractions on H. Proof. The proof of a generalized version of this lemma for multidimensional systems) can be found in [1]. For completeness, here we give a short proof for the one-dimensional case. A straightforward computation shows that 13) A 1 fx) = c 1x 1+c 1 j sfs)ds+j x 1 x s)fs)ds j sfs)ds f H. Statement i) follows from the Sobolev embedding theorem. Another direct computation shows that 14) Re A ϕ, ϕ = c 1 ϕ1) ; therefore ii) follows from the Lumer Phillips theorem [18, Theorem 4.3, p. 14]. Let us compute the eigenvalues of operator A : 15) A ϕ = λϕ = ρ ϕ, i.e., 16) 17) 18) ϕ x) =jρ ϕx), ϕ ) =, ϕ 1) = jc 1 ϕ1). The solution to 16) 18) is 19) ϕx) =cosh jρx, where ρ satisfies ) e jρ =1 Suppose that jc 1 jρ + jc1. 1) jρ =nπj + O n 1 ), where n is a sufficiently large integer. Substituting 1) into ), we obtain ) On 1 )= c 1 + On ). nπ + c 1 Hence, 3) jρ = nπj k nπ + c 1 + On ), and therefore 4) λ = λ n = c 1 +nπ) j + On 1 ).

4 148 M. KRSTIC, B.-Z. GUO, AND A. SMYSHLYAEV The corresponding eigenfunctions are 5) ϕx) =ϕ n x) =cosnπx + On 1 ). Since {cos nπx, n } forms an orthogonal) Riesz basis for H, we immediately obtain the following result by Theorem 6.3 of [4] see also [3]). Theorem.. Let the operator A be defined by 11). Then, there exists a sequence of generalized eigenfunctions of A, which forms a Riesz basis for H. Moreover, the following statements hold: i) All eigenvalues with sufficiently large modules are algebraically simple. ii) Eigenvalues have the asymptotic expansion 4), where n s are positive integers. iii) The spectrum-determined growth condition holds true for the semigroup e At : SA )=ωa ),wheresa ) is the spectral bound of A and ωa ) is the growth order of e At. iv) The semigroup e At is exponentially stable: e At Me ωt t, where M,ω are positive constants. Therefore, the energy of the system 1) decays exponentially in the following sense: Et) Me ωt E). Proof. The statements i) iii) follow from Theorem 6.3 of [4] and 5). To show iv), it suffices to show that there are no eigenvalues of A on the imaginary axis. Obviously, zero is not the eigenvalue. Let λ = ja, wherea is real and positive; then ρ = ja. Substituting this into ), we get e j a =1 c 1 a + c1. The right-hand side of this equation is real, while the left-hand side is on the unit circle; therefore, it can only be equal to 1 or 1. We get 6) = c 1 a + c1 or = c 1 a + c1. It is clear that neither equation is satisfied when c 1 >, and therefore there are no eigenvalues on the imaginary axis. Remark.3. We should point out that the result iv) of Theorem. has previously been proved in a much more general setting; see [1, 13]. However, Theorem. shows that for this one-dimensional case the Riesz basis approach provides more profound results than just exponential stability. These results are summarized in statements i) iii). The control law 6) is simple and requires only boundary observation. However, we should stress that the asymptotic expansion 4), while clearly indicating that high eigenvalues have a negative real part proportional to the feedback gain c 1,does not give us a clue on how the low-number eigenvalues behave when c 1 increases. Only the numerical analysis can provide us with the clear picture. In Figure 1 the closed-loop eigenvalues are shown for different values of c 1.One can see that by increasing c 1 it is possible to move almost all the eigenvalues arbitrarily

5 BOUNDARY CONTROLLERS FOR SCHRÖDINGER EQUATION Imσ) Reσ) Fig. 1. The closed-loop eigenvalue map of the Schrödinger equation with damping boundary feedback for different values of c 1 : c 1 =.5 ), c 1 =.3 ), c 1 =4+). to the left, except for the eigenvalue with the lowest imaginary part. One can only move the real part of that eigenvalue to 1.4 forc 1.3), and further increasing c 1 just moves this eigenvalue back to the imaginary axis. Since the mode corresponding to that eigenvalue carries the most energy, it is desirable to have the controller that can move it arbitrarily to the left. 3. Backstepping design. Consider again the plant 7) 8) 9) ψ t x, t) = jψ xx x, t), ψ x,t)=, ψ1,t)=ut). We propose considering 7) 9) formally as a heat equation with the imaginary diffusion coefficient and solving the stabilization problem using the backstepping control design for parabolic PDEs []. Consider the transformation 3) vx, t) =ψx, t) x kx, y)ψy, t) dy, where kx, y) is a complex-valued function that satisfies the PDE 31) 3) 33) k xx x, y) k yy x, y) =cjkx, y), k y x, ) =, kx, x) = cj x

6 1484 M. KRSTIC, B.-Z. GUO, AND A. SMYSHLYAEV with c>. It is straightforward to show that this transformation maps 1), ) into the following target system: 34) 35) 36) v t x, t) = jv xx x, t) cvx, t), v x,t)=, v1,t)=. The eigenvalues of this system are 37) σ = c + j π n +1), n =, 1,,...; 4 therefore, the design parameter c allows us to move them arbitrarily to the left in the complex plane. The control law is obtained by setting x = 1 in 3): 38) ψ1,t)= k1,y)ψy, t) dy, where k1,y) is obtained from the solution to the PDE 31) 33), which is [] cjx ) I 1 y ) cjx y )) m kx, y) = cjx = cjx cjx cjx y ) 4 m m!m +1)! 39) = x c x y ) m=1 [ ) )] j 1)ber 1 c x y ) 1 + j)bei 1 c x y ). Here I 1 ) is the modified Bessel function and ber 1 ) and bei 1 ) are the Kelvin functions, which are defined in terms of I 1 as follows [3]: { )} { )} 1+j 1+j 4) ber 1 x) = Im I 1 x, bei 1 x) =Re I 1 x. From the second equality in 39) we see that the kernel k is C in both variables in the triangle region defined by the inequalities y x 1 since the series cjx y )) m 4 m m!m +1)! m=1 and the corresponding series obtained by term-by-term differentiation with respect to x and y is absolutely convergent. Therefore, k1,y) makes sense in the feedback law 38). The precise statement of stability of the closed-loop system 1), ), 38), 39) is given by the following theorem. Theorem 3.1. Consider the system d 41) dt ψ,t)=aψ,t) in H = L, 1), where the operator A is defined as 4) Aϕx) = jϕ x) ϕ DA), { DA) = ϕ H, 1) ϕ ) =,ϕ1) = } k1,y)ϕy)dy

7 BOUNDARY CONTROLLERS FOR SCHRÖDINGER EQUATION 1485 and k is given by 39). Then, there exists a sequence of eigenfunctions of A, which forms a Riesz basis for H. Moreover, the following statements hold: i) The eigenvalues of A are given by 37). ii) The spectrum-determined growth condition holds true for the semigroup e At. iii) The semigroup e At is exponentially stable in the sense e At Me ct t, where M > and c is an arbitrary positive design parameter. Proof. The main idea of the proof is to first establish well-posedness and stability of the target system 34) 36) and then to use the fact that the transformation 3) is invertible to get well-posedness and stability of the closed-loop system. One can write 34) 36) as d 43) dt v,t)=bv,t) on H, where the operator B is defined by 44) Bϕx) = jϕ x) cϕx) ϕ DB), DB) ={ϕ H, 1) ϕ ) = ϕ1) = }. A simple computation shows that eigenvalues of B are 45) μ n = c + jπ n 1 ), n =1,,..., and the corresponding eigenfunctions are 46) f n x) =cos n 1 ) πx, n =1,,..., which forms an orthogonal) Riesz basis for H. This shows that the spectrum-determined growth condition holds for 43). Therefore, there exists a constant M 1 > such that 47) e Bt M 1 e ct t. One can show by direct substitution that the transformation 48) ψx, t) =vx, t)+ with [, 3] lx, y) = cjx x cjx ) J 1 y ) cjx y ) lx, y)vy, t) dy =I P) 1 v, 49) = cjx cjx m=1 1) m cjx y )) m 4 m, y x 1, m!m +1)! where J 1 is the Bessel function, is inverse to the transformation 3). Using the same argument as for the kernel k, we conclude from the second equality in 49) that

8 1486 M. KRSTIC, B.-Z. GUO, AND A. SMYSHLYAEV the kernel l is C in both variables in the triangle region defined by inequalities y x 1. Therefore, the transformations defined by 3) and 48) are bounded on H. Let us rewrite these transformations in the form vx) =ψx) ψx) =vx)+ x x kx, y)ψy) dy =[I P)ψ]x), lx, y)vy) dy =[I P) 1 v]x), where both I P and I P) 1 are bounded on H. A simple calculation shows that AI P) =I P)B, thatis, A =I P)BI P) 1, where A is defined by 4). Hence, μ, φ) is an eigenpair of B if and only if μ, I P)φ) is an eigenpair of A. This fact, together with the spectral properties of B see 44) 46)) and the relation e At =I P)e Bt I P) 1, gives the properties of A stated in the theorem. Remark 3.. In the proof of Theorem 3.1, the well-known property 47) of the operator B is rederived using the Riesz basis approach so that the Riesz basis property can be established for the operator A. This allows us to strengthen the arbitrary decay property iii) with properties i) and ii). This remark is also relevant for Theorems 3and4. Remark 3.3. It is possible to combine the feedback law 38) with the design presented in section. In order to do this, one modifies the target system 34) 36) by replacing the boundary condition 36) with 5) v x 1,t)= jc 1 v1,t). Then the combined controller for the Schrödinger equation becomes c ) 51) ψ x 1,t)= j + c 1 ψ1,t)+ k x 1,y)+jc 1 k1,y))ψy, t) dy. 4. Observer design. The controller 38) relies on full-state measurements. Let us assume now that only boundary measurements are available so that ψ x 1) is measured and ψ1) is actuated). We are going to design the observer which closely follows the observer design presented in [1] for the heat equation. We denote the estimate of the state by ˆψ and use the observer 5) 53) 54) ˆψ t x, t) = j ˆψ xx x, t)+p 1 x)[ψ x 1,t) ˆψ x 1,t)], ˆψ x,t)=, ˆψ1,t)=ψ1,t), which is in a familiar form of the copy of the plant 7) 9) plus output injection with the observer gain p 1 x) to be designed. The observer error ψ = ψ ˆψ satisfies the PDE 55) 56) 57) ψ t x, t) = j ψ xx x, t) p 1 x) ψ x 1,t), ψ x,t)=, ψ1,t)=.

9 BOUNDARY CONTROLLERS FOR SCHRÖDINGER EQUATION 1487 Consider the transformation 58) ψx, t) = wx, t)+ px, y) wy, t) dy. Note that the integral runs from x to 1 here, which is the consequence of the collocated input-output architecture. Let us select the target system for the observer error as x 59) 6) 61) w t x, t) = j w xx x, t) c wx, t), w x,t)=, w1,t)=. The design parameter c allows to set the desired observer convergence rate which usually needs to be faster than the closed-loop system decay rate c). Substituting the transformation 58) into 55) 57) and matching the terms, we get the following conditions on px, y) which form a PDE: 6) 63) 64) p xx x, y) p yy x, y) = j cpx, y), p x,y)=, px, x) =j c x. The observer gain p 1 x) is computed from px, y) as follows: 65) p 1 x) =jpx, 1). The PDE 6) 64) has the following solution [1]: ) I 1 cjy x ) px, y) =j cy cjy x ) = j cy + j cy = cy c y x ) 66) m=1 [ 1 j)ber 1 c y x ) j cy x )) m 4 m m!m +1)! )] +1+j)bei 1 c y x ). From the second equality in 66) it follows that p is C in both variables in the triangle region defined by inequalities x y 1. Using 65) we get the observer gain 67) p 1 x) = c 1 x ) [ ) 1 + j)ber 1 c 1 x ) )] +j 1)bei 1 c 1 x ), which is C in x. Note that when c = c, wehavep 1 x) =jk1,x), which is the well-known duality property between observer and control gains. The result of this section is summarized in the following theorem. Theorem 4.1. Consider the system ) 68) d ψ,t)=ã ψ,t) dt

10 1488 M. KRSTIC, B.-Z. GUO, AND A. SMYSHLYAEV in H = L, 1), where the operator à is defined as 69) Ãϕx) = jϕ x) p 1 x)ϕ 1) ϕ DÃ), DÃ) ={ ϕ H, 1) ϕ ) =, ϕ1) = } and p 1 x) is given by 67). Then, there exists a sequence of eigenfunctions of Ã, which forms a Riesz basis for H. Moreover, the following statements hold. i) The eigenvalues of à are 7) σ = c + jπ n 1 ), n =1,,... ii) The spectrum-determined growth condition holds true for the semigroup eãt. iii) The semigroup eãt is exponentially stable in the sense eãt M e ct t, where M > and c is an arbitrary positive design parameter. Proof. The proof closely follows the proof of Theorem 3.1. We define the operator B as 71) Bϕx) = jϕ x) cϕx) ϕ D B), D B) ={ϕ H, 1) ϕ ) = ϕ1) = } and write the system 59) 61) on H as 7) d dt w,t)= B w,t). Since the operator B is the same as B with c replaced by c), using the same arguments as in the proof of Theorem 3.1, we get that there exists a sequence of eigenfunctions of B, which forms a Riesz basis for H. Writing the transformation 58) in the form ψx) = wx)+ x px, y) wy) dy =[I P) 1 w]x), and using the fact that both I P and I P) 1 are bounded operators on H due to the C property of px, y), we obtain à =I P) 1 BI P). The results i) iii) follow immediately from the spectral properties of B and the relations 73) eãt =I P) 1 e Bt I P), 74) w,t) = e Bt w, ) Me ct w, ).

11 BOUNDARY CONTROLLERS FOR SCHRÖDINGER EQUATION Output-feedback compensator. In this section we combine the observer 5) 54) with the state feedback controller developed in section 3: 75) ψ1,t)= k1,y) ˆψy, t) dy. The result is an output-feedback compensator which is the alternative to 6). Under the feedback 75), the plant 7) 9) becomes 76) 77) 78) ψ t x, t) = jψ xx x, t), ψ x,t)=, ψ1,t)= k1,y)ψy, t) k1,y) ψy, t) dy, where ψ is the solution of 55) 57). Applying the transformation 3), we obtain 79) 8) 81) v t x, t) = jv xx x, t) cvx, t), v x,t)=, v1,t)= k1,y) ψy, t)dy. Now ψ in 78) can be expressed through w using the transformation 58): 8) v t x, t) = jv xx x, t) cvx, t), 83) v x,t)=, [ y ] 84) v1,t)= k1,y)+ k1,s)ps, y)ds wy, t)dy. The above PDE together with 59) 61) gives the following system v, w) inh H, equivalent to the closed-loop system ψ, ˆψ): 85) 86) 87) 88) 89) v t x, t) = jv xx cvx, t), v x,t)=, v1,t)= Lx) wx, t) dx ft), w t x, t) = j w xx c wx, t), w x,t)= w1,t)=, where Lx) = k1,x) x k1,s)ps, x)ds C, 1). In order to discuss the well-posedness of system 85) 89) in a suitable state space, we introduce an operator 9) Δfx) = f x) f DΔ), DΔ) = {f H, 1) f ) =,f1) = }. The operator Δ is self-adjoint and positive definite in H. analytic expression of Δ 1/ : 91) Δ 1/ fx) =f 1 x), DΔ 1/ )={f H 1, 1) f1) = }, Δ 1/ fx) = x We can easily find the fτ)dτ, DΔ 1/ )={f Δ 1/ f H}.

12 149 M. KRSTIC, B.-Z. GUO, AND A. SMYSHLYAEV Since Δ has eigenpairs {λ n = n 1 ) π, cos n 1 ) πx)} and { cos n 1 ) πx)} form an orthonormal basis for H, we can also use Fourier series to characterize DΔ 1/ ): Δ 1/ a n fx) = cos n 1 ) πx, λ n = n 1 π λn ) f DΔ 1/ ), { DΔ 1/ )= fx) = a n cos n 1 ) } πx a n 9) <, λ n Δ 1/ f H = a n λ n f DΔ 1/ ). Let H 1 =Δ 1/ H =[DΔ 1/ )], where [DΔ 1/ )] is the graph space of Δ 1/ in H with norm f H 1 = Δ 1/ f H. We consider the system 85) 89) in the state space X = H 1 H. Define the system operator A : DA) X) X for 85) 89) as follows: 93) Af,g) = jf cf, jg cg) f,g) DA), { } DA) = f,g) H, 1)) f ) =,f1) = Lx)gx)dx, g ) = g1) =. Then the system 85) 89) can be written as an evolutionary equation in X: d 94) v,t), w,t)) = Av,t), w,t)). dt Theorem 5.1. Let A be defined by 93) and suppose that c c. Then, there exists a sequence of eigenfunctions of A, which forms a Riesz basis for X. Moreover, the eigenvalues of A are 95) λ n1 = c + j n 1 π ), λ n = c + j n 1 ) π,,,..., and the following statements hold: i) The spectral growth condition holds for the semigroup e At generated by A in X. ii) The solution of 94) satisfies 96) v,t), w,t)) X Ke min {c, c}t v, ), w, )) X for some constant K>. Proof. Consider the eigenvalue problem of A, Af,g) = λf,g), that is, f x) jcfx) =jλfx), 97) f ) =,f1) = g x) j cgx) =jλgx), g ) = g1) =. Lx)gx)dx,

13 BOUNDARY CONTROLLERS FOR SCHRÖDINGER EQUATION 1491 Since c c, wehavecosh jc + λ n ) foranyn 1. Using 44) 46), we get two families of eigenpairs {λ n1,f n1 ), λ n,f n )} of A as follows: 98) λ n1 = c + j n 1 ) λn π,f n1 x) = cos n 1 ) ) πx,, λ n = c + j n 1 ) π cosh jc + λ n )x,f n x) = b n = Lx)cos n 1 ) πxdx, n =1,,... cosh jc + λ n ) b n, cos n 1 ) ) πx, Note that since {b n } are the Fourier coefficients of Lx) under the orthonormal basis { cosn 1/)πx} in H, wehave 99) b n <. For any n 1wehave jc + λn )=j 1) 11) n 1 ) [ π 1 = j n 1 ) π + cosh jc + λ n )=jsin n 1 ) From 91) it follows that ] jc c) n 1 ) π + On 4 ) c c ) n 1 + On 3 ), π c c π ) n 1 + On ). π 1) Δ 1/ cosh jc + λ n )x cosh jc + λ n ) = 1 sinh jc + λ n )1 x) j jc + λn ) cosh. jc + λ n ) Using 1) 1), we get 13) cosh jc + λ n )x cosh jc + λ n ) b cosh jc + λn )x n = Δ 1/ H 1 cosh jc + λ n ) b n C b n,,,..., H uniformly for x [, 1], where C > is a constant independent of n. Define 14) G n1 x) =F n1 x), G n x) =, cos n 1 ) ) πx,,,... Then {G n1,g n } forms an orthogonal) Riesz basis for X, and it follows from 98) 13) that [ Fn1 G n1 X + F n G n X] C b n <. By Bari s theorem [, Theorem.3, p. 317], we get that {F n1,f n } forms a Riesz basis for X. This, together with the expression of eigenvalues in 98), gives the required results.

14 149 M. KRSTIC, B.-Z. GUO, AND A. SMYSHLYAEV For c = c, it is not easy to find out whether the root subspace of A is complete in X. For this case, the Riesz basis approach cannot be applied, and we address it differently. Theorem 5.. Let A be defined by 93). Then A generates a C -semigroup on X. Moreover, for any <ε<min{c, c}, there exists a constant K ε >, such that 15) v,t), w,t)) X K ε e min{c, c}+ε)t v, ), w, )) X. Proof. Using the definitions of operators B and B 44), 71), let us write the system 85) 89) as follows see, e.g., [6, 11]): 16) v t,t)=bv,t)+bft), ft) = w t,t)= B w,t), b,u = i ux) dx u H. Lx) wx, t)dx, Itcanbeeasilyshownthatb B 1 is a bounded operator from H 1 to C, the complex number field. Consider the adjoint system that is, 17) z t,t)=b v,t), yt) = b,z,t) = i zx, t)dx, z t x, t) =jz xx czx, t), z x,t)=z1,t)=, yt) =i zx, t)dx. We can write the solution z explicitly using the orthonormal basis { λn cosn 1/)πx} in H 1 : Therefore, zx, t) = zx, ) = e νnt d n cos n 1 ) πx, d n cos n 1 ) πx. μ n = c j n 1 ) π, sin n yt) =i ) 1 π ) n 1 a n e μnt. π By Ingham s theorem [7, Theorem 4.3, p. 59] and 9), for any T>, there exists a constant C T > such that yt) dt C T d n ) n = C 1 T z, ) H π 1.

15 BOUNDARY CONTROLLERS FOR SCHRÖDINGER EQUATION 1493 This shows that b is admissible for the semigroup e Bt generated by B on H 1 see [5]). Therefore, for any v, ), w, )) X, there exists a unique solution v, w) C, ; X) to system 16) such that v,t)=e Bt v, ) + t e Bt s) bfs)ds, w,t)=e Bt w, ), where e Bt is now understood to be the C -semigroup generated by B on H 1.Therefore, A generates a C -semigroup e At on X. For any <ε<min{c, c}, 18) t e εt v,t)=e B+ε)t v, ) + e B+ε)t s) be εs fs)ds, e εt w,t)=e B+ε)t w, ). Since c + ε<, c + ε<, and e εs fs) L, ; C), it follows from 47), 73), and ) of [14, Theorem , p. 654] note that we use 7.4.) of [14] only for the ɛ = there) that t e B+ε)t s) be εs fs)ds L, ; H 1 ). This, together with 18) and the assumptions c + ε<, c + ε<, proves that e εt e At v, w ) Xdt < v, w ) X. By Theorem 4.1 in [18], there exist constants K,δ > such that e εt e At X K e δt. The proof is completed. Remark 5.3. A comparison of Theorems 5.1 and 5. shows that the Riesz basis approach leads to more profound results in the case of c c; in particular, the spectral growth condition and the exact decay rate are established. 6. Simulation. The results of numerical simulation of two designs proposed in the paper using finite-difference approximations) are presented in Figures 4. We present only state-feedback simulations since an observer design with typically desirable) c c would result in an output-feedback controller performing essentially the same as a full-state controller. In Figure top) one can see the uncontrolled oscillations of the open-loop system. The closed-loop response for the backstepping control design with c = 6 is shown in the bottom picture of Figure only for the real part of the state; the imaginary part is similar). In Figures 3 and 4 we compare the performance of two different control designs 6) and 38) for c 1 =3andc = 6, respectively. The coefficients c 1 and c are chosen so that all the modes of the closed-loop system except for first one have approximately the same decay rate in both designs. From the eigenvalue formulae 4) and 37), it follows that the real parts of all eigenvalue pairs except for the first one are 6. The difference is in the first eigenvalue pair: in the backstepping design, its real part is 6, whereas in design 6), it has a real part of about 1.4 as discussed in section, it cannot be moved further than that to the left in the complex plane; see Figure 1). In Figure 3 right) the real part of the control input ψ1,t) is shown. One can see that the backstepping controller is much less aggressive; its peak value is about three

16 1494 M. KRSTIC, B.-Z. GUO, AND A. SMYSHLYAEV 4 Reψ) 4 1 x.5. t Reψ) 4 1 x.5. t.4.6 Fig.. The open-loop top) and closed-loop bottom) response of the Schrödinger equation. Only the real part of the state is shown. times less than that of the control 6). From Figure 3 left), we see that the output performance is approximately the same in both designs; the backstepping controller achieves slightly faster convergence to zero. In Figure 4 we plot the logarithm of the L -norm of the closed-loop state. Up to

17 BOUNDARY CONTROLLERS FOR SCHRÖDINGER EQUATION design 38) design 6) Reψ,t)) t Reψ1,t)) 3 1 design 38) design 6) Fig. 3. Top: The real part of the output ψ,t). Bottom: The real part of the control effort ψ1,t). t some point, the energy decays with the same rate in both designs slope of 6). At t.45, higher modes die out, and the difference in the decay rate of the first mode in the two designs becomes obvious. To summarize, both control designs stabilize the plant successfully; however, the more complicated backstepping controller achieves slightly faster convergence with a much smaller control effort. 7. Future work. We have presented two boundary control designs for the Schrödinger equation. It is well known [17] that the linearized Schrödinger equation is equivalent to the Euler Bernoulli beam model both real and imaginary parts of the solution to the Schrödinger equation satisfy separate Euler Bernoulli beam equations). Therefore, the next natural step is to design the stabilizing controllers for the Euler Bernoulli beam based on the controllers presented in this paper. We should note, however, that the design does not carry over trivially from one system to another because the boundary conditions of the particular setup of the beam free, hinged, clamped, etc.) do not directly correspond to the boundary conditions of the

18 1496 M. KRSTIC, B.-Z. GUO, AND A. SMYSHLYAEV log ψ,t) 4 design 38) design 6) Fig. 4. Logarithm of the energy of the closed-loop system. t Schrödinger equation considered here. This result would be significant since even though there is an extensive literature on the control of the Euler Bernoulli beam, including, among others, [1, 9, 4, 4, 3, 5, 15], to the best of our knowledge, there are no boundary stabilization results that achieve a prescribed decay rate of the closed-loop system. REFERENCES [1] G.Chen,M.C.Delfour,A.M.Krall,andG.Payre, Modeling, stabilization and control of serially connected beams, SIAM J. Control Optim., ), pp [] I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Trans. of Math. Monographs 18, AMS, Providence, [3] B.-Z. Guo and R. Yu, The Riesz basis property of discrete operators and application to a Euler Bernoulli beam equation with boundary linear feedback control, IMA J. Math. Control Inform., 18 1), pp [4] B.-Z. Guo, Riesz basis approach to the stabilization of a flexible beam with a tip mass, SIAM J. Control Optim., 39 1), pp [5] B.-Z. Guo, Riesz basis property and exponential stability of controlled Euler Bernoulli beam equations with variable coefficients, SIAM J. Control Optim., 4 ), pp [6] B.-Z. Guo and Z. C. Shao, Regularity of a Schrödinger equation with Dirichlet control and colocated observation, Systems Control Lett., 54 5), pp [7] V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer, New York, 5. [8] M. Krstic, B. Z. Guo, and A. Smyshlyaev, Boundary controllers and observers for Schrodinger equation, in Proceedings of the 46th IEEE Conference on Decision and Control, IEEE, Washington, DC, 7, pp [9] K. Liu and Z. Liu, Exponential decay of energy of the Euler Bernoulli beam with locally distributed Kelvin Voigt damping, SIAM J. Control Optim., ), pp [1] I. Lasiecka and R. Triggiani, Optimal regularity, exact controllability and uniform stabilisation of Schrödinger equations with Dirichlet control, Differential Integral Equations, 5 199), pp [11] I. Lasiecka, Exponential decay rates for the solutions of Euler-Bernoulli equations with boundary dissipation occurring in the moments only, J. Differential Equations, ), pp [1] I. Lasiecka, R. Triggiani, and X. Zhang, Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates. Part II: L Ω)- estimates, J. Inverse Ill-Posed Probl., 1 4), pp

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