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1 Systes & Control Letters 6 (3) 53 5 Contents lists available at SciVerse ScienceDirect Systes & Control Letters journal hoepage: Stabilization of an ODE Schrödinger Cascade Beibei Ren a,, Jun-Min Wang c,, Miroslav Krstic b a Departent of Mechanical Engineering, Texas Tech University, Lubbock, TX 7949, USA b Departent of Mechanical and Aerospace Engineering, University of California at San Diego, La Jolla, CA 993, USA c Departent of Matheatics, Beijing Institute of Technology, Beijing 8, PR China a r t i c l e i n f o a b s t r a c t Article history: Received 8 Septeber Received in revised for 3 February 3 Accepted 5 March 3 Keywords: Boundary control Partial differential equations Backstepping approach Schrödinger equation Riesz basis We consider the proble of stabilization of a linear ODE with input dynaics governed by the linearized Schrödinger equation. The interconnection between the ODE and Schrödinger equation is bi-directional at a single point. We construct an explicit feedback law that copensates the Schrödinger dynaics at the inputs of the ODE and stabilizes the overall syste. Our design is based on a two-step backstepping transforation by introducing an interediate syste and an interediate control. By adopting the Riesz basis approach, the exponential stability of the closed-loop syste is built with the pre-designed decay rate and the spectru-deterined growth condition is obtained. A nuerical siulation is provided to illustrate the effectiveness of the proposed design. 3 Elsevier B.V. All rights reserved.. Introduction The backstepping approach, which was originally developed for parabolic PDEs [,], has recently been applied to control probles for PDE ODE cascades [3 7] etc., with applications of interest including fluids, structures, theral, cheically-reacting, and plasa systes. This ethod uses an invertible Volterra integral transforation together with the boundary feedback to convert the unstable plant into a well-daped target syste. The kernel of this transforation satisfies a certain PDE and ODE, which turn out to be solvable in closed for. In [3], the backstepping ethod was applied to a syste which consists of the first-order hyperbolic PDE coupled with a second-order (in space) ODE. This syste resebles the Korteweg de Vries equation which describes shallow water waves and ion acoustic waves in plasa. In [5], predictor-like feedback laws and observers have been developed for diffusion PDE ODE cascades. In [6], an explicit feedback law was developed that copensates the wave PDE dynaics at the input of an linear tie-invariant ODE and stabilizes the overall syste. In [7], PDE ODE cascades considered were extended fro Dirichlet type interconnections to Neuann type interconnections. For PDE ODE cascades considered above, the interconnection between the PDE and the ODE is one directional. For exaple, in [5], the dynaics at Corresponding author. E-ail addresses: helenren.ac@gail.co (B. Ren), wangjc@graduate.hku.hk (J.-M. Wang), krstic@ucsd.edu (M. Krstic). The second author is supported by the National Natural Science Foundation of China. the input of the ODE is governed by a heat equation, whereas the ODE does not act on the PDE. In soe cases, the interconnection between the could be bi-directional, i.e., the PDE and ODE are coupled with each other as discussed in [8]. In this paper, we consider a proble of stabilization of an ODE Schrödinger equation cascade as in () (4), where the interconnection between the is bi-directional at a single point: Ẋ(t) AX(t) + Bu(, t), t >, () u t (x, t) iu xx (x, t), x (, ), t >, () u x (, t) CX(t), (3) u(, t) U(t) (4) where A C n n, B C n, C C n ; X(t) C n is the state of ordinary differential equation; u(x, t) C is the state of Schrödinger equation; and U(t) C is the control force to the entire syste. The whole syste is depicted in Fig.. Both states of ODE and Schrödinger equation are coplex valued. The otivation for this kind of proble can be provided in the context of various applications in quantu echanics, cheical process control, and other areas. The stabilization of the linearized Schrödinger equation using boundary control was investigated in [9 ]. For exact controllability and observability results, see [ 4]. The interconnection of Schrödinger heat equation with boundary coupling was considered in [5] and the Schrödinger equation has the Gevrey regularity under the copensation of the heat equation. As in [,6], the Schrödinger equation is usually considered as a coplex-valued 67-69/$ see front atter 3 Elsevier B.V. All rights reserved.

2 54 B. Ren et al. / Systes & Control Letters 6 (3) 53 5 Fig.. Block diagra for the coupled ODE PDE syste. heat equation such that the backstepping ethod developed for parabolic PDEs could be applied. Motivated by two-step backstepping transforations in [5], which was adopted to iprove perforance and achieve exponential stability with arbitrarily fast decay rate, we adopt two-step backstepping control design to ake the syste stable in this paper. Copared with our previous works using backstepping designs, the ain contributions of the results in this paper lie in: (i) Instead of one-step backstepping control, which results in difficulty in finding the kernels for the stabilization of the ODE Schrödinger cascade considered in this paper, the design developed in this paper is based on a two-step backstepping transforation by introducing an interediate syste and an interediate control. (ii) Instead of the Lyapunov synthesis, the Riesz basis approach is adopted in this paper, through which the exponential stability of the closed-loop syste was built with the pre-designed decay rate and the spectru-deterined growth condition was obtained. The paper is organized as follows. In Section, the two-step backstepping design is developed using the bounded and invertible operators. First, we design the coprehensive control to convert the original syste into the interediate syste. Secondly, we design the interediate control to convert the interediate syste into the final target syste. Section 3 is devoted to the spectral analysis of the target syste. In Section 4, Riesz spectral ethod is adopted for the stability analysis of the closed-loop syste. Finally, a siulation exaple and the concluding rearks are provided in Section 5. In this paper, we consider the following energy Hilbert space H C n L (, ) (5) with inner product f, f H X T X + z (x)z (x)dx, f i (X i, z i ) H, i,, (6) and H-nor f i H X i C n + z i L (,). Backstepping design. (7) The control objective is to ake the syste () (4) exponentially stable. To achieve this, a new cascaded ODE PDE target syste is introduced in the for: Ẋ(t) (A + BK)X(t) + Bz(, t), (8) z t (x, t) iz xx (x, t) cz(x, t), c >, (9) z x (, t), () z(, t) () Fig.. Block diagra for two-step backstepping control design. where z(x, t) C, c > is an arbitrary pre-defined decay rate, and we assue that the pair (A, B) is stabilizable and take K C n to be a known vector such that A + BK is Hurwitz. In this section, we seek the boundary controller U(t) in (4) to exponentially stabilize the syste (X, u) () (4) into the target syste (X, z) (8) () using the backstepping control design for parabolic PDEs [,]. The ethod uses invertible Volterra integral transforation together with the boundary feedback to convert the unstable plant (X, u) () (4) into a well-daped target syste (X, z) (8) (). For this we use a change of variables based on a Volterra series. However, the one-step backstepping transforation for (X, u) (X, z) results in difficulty in finding the kernels. To avoid this, we introduce an interediate syste (X, w): Ẋ(t) (A + BK)X(t) + Bw(, t), () w t (x, t) iw xx (x, t), (3) w x (, t), (4) w(, t) W(t) (5) where w(x, t) C, and W(t) C is the interediate control. The ain idea is using two-step backstepping control design as shown in Fig. : (i) design the coprehensive control U(t) to convert the original syste into the interediate syste; and (ii) design the interediate control W(t) to convert the interediate syste (X, w) into the final target syste (X, z)... Design for syste (X, u) to (X, w) In this subsection, we look for the backstepping control U(t) and the transforation (X, u) (X, w) between systes () (4) and () (5). We postulate the transforation (X, u) (X, w) in the for X(t) X(t) (6) w(x, t) u(x, t) x q(x, y)u(y, t)dy γ (x)x(t), y x (7) where kernels q(x, y) and γ (x) are to be derived. Differentiating (7) once and twice with respect to x, we get, respectively, w x (x, t) u x (x, t) q(x, x)u(x, t) x q x (x, y)u(y, t)dy γ (x)x(t) (8)

3 B. Ren et al. / Systes & Control Letters 6 (3) w xx (x, t) u xx (x, t) q(x, x)u x (x, t) q (x, x) + q x (x, x) u(x, t) x q xx (x, y)u(y, t)dy γ (x)x(t) (9) where q (x, x) q x (x, x) + q y (x, x). The first derivative of w(x, t) with respect to t is w t (x, t) u t (x, t) x q(x, y)u t (y, t)dy γ (x)(ax(t) + Bu(, t)) iu xx (x, t) + iq(x, x)u x (x, t) iq y (x, x)u(x, t) + i x q yy (x, y)u(y, t)dy + iq y (x, ) γ (x)b u(, t) (γ (x)a + iq(x, )C)X(t). () Let us now exaine the expressions: w(, t) u(, t) γ ()X(t) () w x (, t) (C γ ())X(t) q(, )u(, t) () w t (x, t) + iw xx (x, t) iq (x, x)u(x, t) (iq(x, )C + γ (x)a + iγ (x))x(t) + (iq y (x, ) γ (x)b)u(, t) + i x (q yy (x, y) q xx (x, y))u(y, t)dy. (3) A sufficient condition for () (5) to hold for any continuous functions u(x, t) and X(t) is that γ (x) and q(x, y) satisfy γ (x) iγ (x)a + q(x, )C, x (, ) γ () K, (4) γ () C, which represents a second-order ODE in x, and qxx (x, y) q yy (x, y), < y < x < q(x, x), q y (x, ) iγ (x)b, (5) which is a second-order hyperbolic PDE. A direct coputation gives the solution of (5): q(x, y) x y iγ (σ )Bdσ. (6) Substituting (6) into (4) leads to γ (x) iγ (x)a + x iγ (σ )Bdσ C. (7) Differentiating (7) once, we get the third order ODE γ (3) (x) iγ (x)a + iγ (x)bc, γ () ika, γ () C, γ () K. Define Γ (x) [γ (x) γ (x) γ (x)], D Then (8) is written into ibc I ia. I (8) Γ (x) Γ (x)d. (9) Its solution is Γ (x) Γ ()e Dx, Γ () [K C ika]. Hence, the solution to the ODE (4) is ibc I ia x I I γ (x) [K C ika]e. (3) The transforation (X, u) (X, w) (6) (7) is invertible, and the inverse transforation (X, w) (X, u) is postulated in the following for X(t) X(t) (3) u(x, t) w(x, t) + x ι(x, y)w(y, t)dy + ψ(x)x(t) (3) where kernels ι(x, y) and ψ(x) to be derived. In a siilar anner to finding the kernels q(x, y) and γ (x) of the direct transforation, the derivatives u x, u xx and u t are coputed, and a sufficient condition for () (3) to hold is that ι(x, y) and ψ(x) satisfy ψ (x) iψ(x)(a + BK), ψ() K, (33) ψ () C, and ιxx (x, y) ι yy (x, y), ι(x, x), ι y (x, ) iψ(x)b. (34) This cascade syste can be solved explicitly. The solution to the ODE (33) is i(a + BK) x I I ψ(x) [K C]e (35) and the explicit solution to the PDE (34) is ι(x, y) x y iψ(σ )Bdσ. (36) Thus, the control law is obtained by setting x in (7): U(t) u(, t) W(t) + q(, y)u(y, t)dy + γ ()X(t) (37) where q(x, y) and γ (x) are defined in (6) and (3) respectively, and the interediate control W(t) to be deterined in Section. later. Fro the above coputations, it is noted that the transforations (6) (7) and (3) (3) are invertible between (X, u) and (X, w) in H. For convenience, we introduce the following lea. Lea. Let F : H H be a linear operator defined by X X F (x) x, u u(x) q(x, y)u(y)dy γ (x)x (X, u) H, x (, ), (38) where q(x, y) and γ (x) are given by (6) and (3) respectively. Then F is bounded and invertible on H. Moreover, G (F ) has the following expression: X X G (x) x, w w(x) + ι(x, y)w(y)dy + ψ(x)x (X, w) H, x (, ), (39) where ψ(x) and ι(x, y) are given by (35) and (36) respectively.

4 56 B. Ren et al. / Systes & Control Letters 6 (3) 53 5 Proof. It is noted that the transforations (6) (7) and (3) (3) are invertible between (X, u) and (X, w) in H. We have that F G G F I and (F ) G. Let (X, w) F (X, u) be given by (38). Denote Θ(s) Ψ (s) s s γ (σ )Bdσ, (4) ψ(σ )Bdσ, (4) and the convolution operator (f g)(x) x f (x y)u(y)dy, (4) for functions f (x), g(x) L (, ). As Θ, γ L (, ), Θ L (,) B γ L (,) and (X, w) F (X, u) are given by (38), a direct calculation gives that w L (,) w(x)w(x)dx [u(x) (Θ u)(x) γ (x)x] [u(x) (Θ u)(x) γ (x)x]dx 3 u(x) + (Θ u)(x) + γ (x)x dx α u L (,) + α X C n where Cauchy Schwarz Inequality is used and α 3( + Θ L (,) ), α 3 γ L (,). (43) Hence, there is an M ax{α, α + } such that (X, w) H w L (,) + X C n α u L (,) + ( + α ) X C n M (X, u) H, which indicates that F is bounded on H. Siilarly we can show that G is bounded on H. The proof is coplete... Design for syste (X, w) to (X, z) In this subsection, we look for the interediate control W(t) and the transforation (X, w) (X, z) between the interediate syste () (5) and the final target syste (8) (). As in [,6], the Schrödinger equation is usually considered as a coplex-valued heat equation such that the backstepping ethod developed for parabolic PDEs [] could be applied. Taking the diffusion coefficient as i in the reaction advection diffusion equation [] (with the advection and reaction coefficients being zero) and following the control design developed there, we use the transforation (X, w) (X, z) in the for X(t) X(t) (44) z(x, t) w(x, t) x κ(x, y)w(y, t)dy (45) along with the interediate controller W(t) at x : W(t) w(, t) κ(, y)w(y, t)dy (46) where κ(x, y) is a coplex-valued control gain. Using the sae reasoning as above, recalling that z satisfies (8) (), we get the gain kernel PDE in the for κ xx (x, y) κ yy (x, y) ciκ(x, y), < y < x <, κ y (x, ), κ(x, x) ci (47) x. The explicit solution to the PDE (47) is given in [] by: κ(x, y) cix I ci(x y ) ci(x y ) (ci(x y )) cix cix, y x, (48) 4!( + )! where I ( ) is the odified Bessel function of the first kind. In a siilar anner to finding the kernel κ(x, y), the inverse of the transforation (44), (45) can be found as follows X(t) X(t) (49) w(x, t) z(x, t) + x p(x, y)z(y, t)dy (5) where p(x, y) cix J ci(x y ) ci(x y ) ( ) (ci(x y )) cix cix, 4!( + )! y x, (5) and J is a Bessel function of the first kind. Thus, the interediate controller W(t) is given by substituting (48) into (46) W(t) w(, t) ci κ(, y)w(y, t)dy I ci( y ) w(y, t)dy. (5) ci( y ) Siilar to the Lea, we have the following lea for the transforations (44) (45) and (49) (5) between (X, w) and (X, z) in H. Lea. Let F : H H be a linear operator defined by X X F (x) x, w w(x) κ(x, y)w(y)dy (X, w) H, x (, ), (53) where κ(x, y), y x, is given by (48). Then F is bounded and invertible on H. Moreover, G (F ) has the following expression: G X z (x) z(x) + x X, p(x, y)z(y)dy (X, z) H, x (, ), (54) where p(x, y), y x, is given by (5).

5 B. Ren et al. / Systes & Control Letters 6 (3) Stability analysis for the target syste In this section, we consider the target syste (8) (). Define the syste operator of (8) () by A z (X, z) (A + BK)X + Bz(), iz cz, (X, z) D(A z ), D(A z ) (X, z) C n H (, ) z() z () (55). Then (8) () can be written as an evolution equation in H: dyz (t) A z Y z (t), t >, dt (56) Y z () Y z. where Y z (t) (X(t), z(, t)). Theore. Let A z be given by (55). Then A z exists and is copact on H and hence σ (A z ), the spectru of A z, consists of isolated eigenvalues of finitely algebraic ultiplicity only. Proof. For any given (X, z ) H, solve A z (X, z) (A + BK)X + Bz(), iz cz (X, z ). We get (A + BK)X + Bz() X, iz (x) cz(x) z (x), z() z (), with the solution X (A + BK) (X Bz()), z(x) ϕ(x) ϕ() cosh( ci) cosh( cix), ϕ(x) c i x z() ϕ() cosh( ci). sinh( ci(x τ))z (τ)dτ, (57) Hence, we get the unique (X, z) D(A z ). Hence, A z exists and is copact on H by the Sobolev ebedding theore [7, p. 85]. Therefore, σ (A z ) consists of isolated eigenvalues of finite algebraic ultiplicity only. Now we are in a position to consider the eigenvalue proble of A z. Let A z Y z λy z, where Y z (X, z). Then we have (A + BK)X + Bz() λx, iz cz λz, (58) z () z(). Denote λ p c + + π i, N. (59) Theore. Let A z be given by (55), let λ o j, j,,..., n be the siple eigenvalue of A + BK with the corresponding eigenvector X j, and assue that λ o j λ p, N, j,,..., n, (6) where λ p is defined in (59). Then the eigenvalues of A z are λ o j, j,,..., n λ p,,,,..., (6) and the eigenfunctions corresponding to λ o j and λ p are respectively Z j (X j, ), j,,..., (6) and Z (x) [λ p (A + BK)] B, z (x), N, (63) where z (x) cos + πx, N, (64) which fors an orthogonal basis for L (, ). Proof. Since A + BK is Hurwitz, we have Reλ j <, j,,..., n. (65) A siple coputation shows that the eigenvalue proble iz cz λz, z () z() (66) has the nontrivial solutions λ p, z (x), N (67) where λ p and z (x) are given by (59) and (64) respectively. Next we look for the eigenvalues for (58). Let λ λ o j, j,,..., n. Since B and (A + BK)X j λ o j X j, (A + BK)X j + Bz() λ o j X j yields z(). Moreover, iz cz λ o j z, z() z () z() only has trivial solutions. So we get that λ o j, j,,..., n is the eigenvalues of (58) and has the corresponding eigenfunctions (X j, ), which is (6). On the other hand, when λ λ p, (λ p, z (x)) satisfies the second and third equations of (58) and z (). By the first equation of (58), we have X p [λp (A + BK)] B. So λ p, N, is the eigenvalue of (58) and has the corresponding eigenfunction [λ (A + BK)] B, cos + πx. This is (64). The proof is coplete. Since (A, B) is controllable, the eigenvalues of A + BK can be designed at any points in coplex plane. For siplicity, for each eigenvalue λ of A + BK, we assue that Re(λ) < c, which guarantee the condition (6). Theore 3. Let A z be given by (55), let λ o j, j,,..., n be the siple eigenvalue of A + BK with the corresponding eigenvector X j, and let Reλ o j < c. Then, there is a sequence of eigenfunctions of A z which fors a Riesz basis for H. Moreover, the following conclusions are true: (i) A z generates a C -seigroup e A z t on H. (ii) The spectru-deterined growth condition ω(a z ) s(a z ) holds true for e A z t, where ω(a z ) li t t ea z t is the growth bound of e A z t, and s(a z ) sup{re λ λ σ (A z )} is the spectral bound of A z [8]. (iii) The C -seigroup e A z t is exponentially stable in the sense e A z t M e ct, (68) where M > and c is an arbitrary pre-designed decay rate.

6 58 B. Ren et al. / Systes & Control Letters 6 (3) 53 5 Proof. It is noted that {X j, j,,..., n} is an orthogonal basis in C n and {z (x) cos + πx, N} given by (64) fors an orthogonal basis in L (, ). We have {F j, F (x), j,,..., n, N} which fors an orthogonal basis in H with F j (X j, ) and F (x) (, z (x)). It follows fro (6), (63) that n Z j F j + Z (x) F (x) j [λ p (A + BK)] B, (69) C n where C n denotes the nor in C n. A siple coputation gives [λ p (A + BK)] B C n ([λ p (A + BK)] B) T [λ p (A + BK)] B B T ([λ p (A + BK)] ) T [λ p (A + BK)] B I T λ p (A + BK) λ p BT I λ p (A + BK) B. It follows fro (59) that when, λ p. So there is a positive nuber N such that for > N, we have I λ p (A + BK) I + O, λ p > N. Thus, [λ p (A + BK)] B B C n C + O. n λ p λ p Hence, it follows fro (59) and (69) that n Z j F j + Z (x) F (x) j [λ p (A + BK)] B C n N [λ p (A + BK)] B C n + N+ λ p B C n + O <. λ p Therefore, by Bari s theore [9, p. 38], {Z j, Z (x), j,,..., n,,,...} fors a Riesz basis for H. Moreover, by Theore, A z generates a C -seigroup e A z t on H and the spectru-deterined growth condition ω(a z ) s(a z ) holds true for e A z t. Finally, by the eigenvalues of A z given by (6) and (ii) and (iii), there is a positive constant M > such that e A z t M e ct t. The proof is coplete. 4. Stability analysis for the closed-loop syste In this section, we will investigate the stability of the closedloop systes for both (X, w) and (X, u). 4.. Closed-loop syste for (X, w) In this subsection, we present the closed-loop syste for (X, w) and establish the exponential stability. Under the backstepping feedback control (5), we have the closed-loop syste for () (5): Ẋ(t) (A + BK)X(t) + Bw(, t), w t (x, t) iw xx (x, t), w x (, t), (7) w(, t) κ(, y)w(y, t)dy, where κ(x, y) is given by (48). Define a linear operator A w for the closed-loop syste (7) by: A w (X, w) (A + BK)X + Bw(), iw, (X, w) D(A w ), D(A w ) (X, w) C n H (, ) w (), w() κ(, y)w(y)dy. (7) With A w at hand, the closed-loop syste (7) can be written as an evolution equation in H: dyw (t) A w Y w (t), t >, dt (7) Y w () Y w. where Y w (t) (X(t), w(, t)). Theore 4. Let A w be given by (7), let λ o j, j,,..., n be the siple eigenvalue of A + BK with the corresponding eigenvector X j, and let Reλ o j < c. Then, there is a sequence of eigenfunctions of A w, which fors a Riesz basis for H. Moreover, the following conclusions are true: (i) The eigenvalues of A w are λ o j, j,,..., n λ p,,,,..., (73) where λ p is defined in (59). (ii) A w generates a C -seigroup e A wt on H. (iii) The spectru-deterined growth condition ω(a w ) s(a w ) holds true for e A wt, where ω(a w ) is the growth bound of e A wt, and s(a w ) is the spectral bound of A w. (iv) The C -seigroup e A wt is exponential in the sense e A wt M e ct, (74) where M > and c is an arbitrary pre-designed decay rate. Proof. Since the transforation defined by (44) (45) is invertible between syste (7) and the target syste (8) (), and its invertible transforation is given by X(t) X(t), w(x, t) z(x, t) + x p(x, y)z(y, t)dy, it is easy to see that the syste operators A w and A z defined by (55) and (7) respectively, have the relationship: A z F F A w, (75) where F is given by (53). Fro Lea, we know that F is bounded and invertible with F G given (54). Hence A w F A z F G A z F,

7 B. Ren et al. / Systes & Control Letters 6 (3) Fig. 3. The closed-loop response of the ODE (left) and Schrödinger equation (right). Only the real part of the state is shown. and (λ, Y z ) is an eigenpair of A z if and only if (λ, F Y z ) is an eigenpair of A w. So, fro the eigenvalues of A z given by (59) (6), we get the eigenvalues of A w as assertion (i). Moreover, since F is bounded in H, it follows that fro Theore 3, the eigenfunctions of A w fors a Riesz basis in H. Moreover, by Theore, A z generates a C -seigroup e A z t on H and the spectru-deterined growth condition ω(a z ) s(a z ) holds true for e A z t. Finally, by the relation With A u at hand, the closed-loop syste (76) can be written as an evolution equation in H: dyu (t) A u Y u (t), t >, dt Y u () Y u, where Y u (t) (X(t), u(, t)). (78) e Awt e F A z F t F e A z t F, the assertion (iv) is then concluded. 4.. Closed-loop syste for (X, u) Theore 5. Let A u be given by (77), let λ o j, j,,..., n be the siple eigenvalue of A + BK with the corresponding eigenvector X j, and let Reλ o j < c. Then, there is a sequence of eigenfunctions of A u, which fors a Riesz basis for H. Moreover, the following conclusions are true: In this subsection, we present the closed-loop syste for (X, u). By the controls (37) and (5), we have the backstepping feedback control U(t) u(, t) W(t) + + κ(, y) + q(, y) γ () q(, y)u(y, t)dy + γ ()X(t) y κ(, y)γ (y)dy X(t), κ(, ξ)q(ξ, y)dξ u(y, t)dy where q(x, y) is given by (6), κ(x, y) is given by (48) and γ (x) is given by (3). Then the closed-loop syste becoes Ẋ(t) AX(t) + Bu(, t), u t (x, t) iu xx (x, t), x (, ), u x (, t) CX(t), (76) u(, t) κ(, y) + q(, y) κ(, ξ)q(ξ, y)dξ u(y, t)dy y + γ () κ(, y)γ (y)dy X(t). Define a linear operator A u for the closed-loop syste (76) by: A u (X, u) AX + Bu(), iu, (X, u) D(A u ), D(A u ) (X, u) C n H (, ) u () CX u() γ () κ(, y)γ (y)dy + κ(, y) + q(, y) κ(, ξ)q(ξ, y)dξ u(y)dy. y (77) (i) The eigenvalues of A u are λ o j, j,,..., n λ p,,,,..., (79) where λ p is defined in (59). (ii) A u generates a C -seigroup e A ut on H. (iii) The spectru-deterined growth condition ω(a u ) s(a u ) holds true for e A ut, where ω(a u ) is the growth bound of e A ut, and s(a u ) is the spectral bound of A u. (iv) The C -seigroup e A ut is exponential in the sense e A ut Me ct, (8) where M > and c is an arbitrary pre-designed decay rate. Proof. The proof is siilar to that of Theore 3. Since the transforation defined by (6) (7) is invertible between syste (7) and the closed-loop syste (76), and its invertible transforation is given by X(t) X(t), u(x, t) w(x, t) + x ι(x, y)w(y, t)dy + ψ(x)x(t), it is easy to see that the syste operators A w and A u defined by (7) and (77) respectively, have the relationship: A w F F A u (8) where F is given by (38). Fro Lea, we know that F is bounded and invertible with F G given (39). Hence A u F A w F G A w F,

8 5 B. Ren et al. / Systes & Control Letters 6 (3) 53 5 References Fig. 4. Control effort. Only the real part is shown. and (λ, Y w ) is an eigenpair of A w if and only if (λ, F Y w ) is an eigenpair of A u. The rest of the proof is siilar to Theore 3. The details are oitted. 5. Siulation results and concluding rearks In this section, we present the results of nuerical siulations for the syste () (4) with boundary control (37) and (5). The scalar case is considered here, with the paraeters of the syste taken as A, B, C, K 5, c 3. The response of the closed-loop of the plant () (4) are shown in Fig. 3. Both the ODE and Schrödinger equation are stabilized. Fig. 4 shows the control effort (only the real part is shown). In conclusion, we have designed an explicit feedback law for a linear ODE coupled with a linearized Schrödinger equation. Our design is based on two-step backstepping transforation by introducing an interediate syste and an interediate control. Using the Riesz spectral ethod we proved exponential stability of the target syste and the closed-loop systes. [] A. Syshlyaev, M. Krstic, Closed-for boundary state feedbacks for a class of -D partial integro-differential equations, IEEE Transactions on Autoatic Control 49 () (4) 85. [] A. Syshlyaev, M. Krstic, Backstepping observers for a class of parabolic PDEs, Systes & Control Letters 54 (5) [3] A. Syshlyaev, M. Krstic, Backstepping boundary control for first-order hyperbolic PDEs and application to systes with actuator and sensor delays, Systes & Control Letters 57 (8) [4] M. Krstic, Delay Copensation for Nonlinear, Adaptive, and PDE Systes, Birkhauser, 9. [5] M. Krstic, Copensating actuator and sensor dynaics governed by diffusion PDEs, Systes & Control Letters 58 (9) [6] M. Krstic, Copensating a string PDE in the actuation or sensing path of an unstable ODE, IEEE Transactions on Autoatic Control 54 (6) (9) [7] G.A. Susto, M. Krstic, Control of PDE ODE cascades with Neuann interconnections, Journal of The Franklin Institute 347 () [8] S. Tang, C. Xie, State and output feedback boundary control for a coupled PDE ODE syste, Systes & Control Letters 6 () [9] B.-Z. Guo, Z.C. Shao, Regularity of a Schrödinger equation with Dirichlet control and colocated observation, Systes & Control Letters 54 (5) [] E. Machtyngier, E. Zuazua, Stabilization of the Schrödinger equation, Portugaliae Matheatica 5 (994) [] M. Krstic, B.-Z. Guo, A. Syshlyaev, Boundary controllers and observers for the linearized Schrödinger equation, SIAM Journal on Control and Optiization 49 (4) () [] I. Lasiecka, R. Triggiani, Optial regularity, exact controllability and unifor stabilization of Schrödinger equations with Dirichlet control, Differential Integral Equations 5 (99) [3] E. Machtyngier, Exact controllability for the Schrödinger equation, SIAM Journal on Control and Optiization 3 (994) [4] K.-D. Phung, Observability and control of Schrödinger equation, SIAM Journal on Control and Optiization 4 () 3. [5] J.-M. Wang, B. Ren, M. Krstic, Stabilization and Gevrey regularity of a Schrödinger equation in boundary feedback with a heat equation, IEEE Transactions on Autoatic Control 57 () [6] M. Krstic, A. Syshlyaev, Boundary Control of PDEs: A Course on Backstepping Designs, SIAM, 8. [7] R.A. Adas, J.J.F. Fournier, Sobolev Spaces, second ed., in: Pure and Applied Matheatics (Asterda), vol. 4, Elsevier/Acadeic Press, Asterda, 3. [8] Z.-H. Luo, B.-Z. Guo, O. Morgul, Stability and Stabilization of Infinite Diensional Systes with Applications, in: Counications and Control Engineering Series, Springer-Verlag London, Ltd., London, 999. [9] R.M. Young, An Introduction to Non-Haronic Fourier Series, Revised ed., Acadeic Press, Inc., San Diego, CA,.