Arbitrary decay for boundary stabilization of Schrödinger equation subject to unknown disturbance by Lyapunov approach

Accepted Manuscript Arbitrary decay for boundary stabilization of Schrödinger equation subject to unknown disturbance by Lyapunov approach Wen Kang, Bao-Zhu Guo

PII: DOI: Article number: Reference:

S2468-6018(18)30055-5 https://doi.org/10.1016/j.ifacsc.2019.100033 100033 IFACSC 100033

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IFAC Journal of Systems and Control

Received date : 16 April 2018 Revised date : 30 October 2018 Accepted date : 16 January 2019 Please cite this article as: W. Kang and B.-Z. Guo, Arbitrary decay for boundary stabilization of Schrödinger equation subject to unknown disturbance by Lyapunov approach. IFAC Journal of Systems and Control (2019), https://doi.org/10.1016/j.ifacsc.2019.100033 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Arbitrary Decay for boundary Stabilization of Schr¨ odinger Equation subject to Unknown Disturbance by Lyapunov Approach ? Wen Kang a,b , Bao-Zhu Guo c,d a b

School of Automation and Electrical Engineering, University of Science and Technology Beijing 100083, China

Key Laboratory of Knowledge Automation for Industrial Processes, Ministry of Education, Beijing 10083, China c

Department of Mathematics and Physics, North China Electric Power University, Beijing 102206, China d

School of Mathematics and Big Data, Foshan University, Foshan 528000, China

Abstract This paper deals with the design of boundary control to stabilize one-dimensional Schr¨ odinger equation with general external disturbance. The backstepping method is first applied to transform the anti-stability from the free end to the control end. A variable structure feedback stabilizing controller is then designed to achieve arbitrary assigned decay rate. The Galerkin approximation scheme is used to show the existence of the solution to the closed-loop system. The exponential stability of the closed-loop system is obtained by the Lyapunov functional method. A numerical example demonstrates the efficiency of the proposed control scheme. Key words: Distributed parameter system, Schr¨ odinger equation, Lyapunov function, disturbance, boundary control.

1

Introduction

In quantum mechanics, Schr¨odinger equation is a mathematical equation that describes the changes over time of a physical system in which quantum effects, such as wave-particle duality, are significant. Boundary control of linearized Schr¨odinger equation has been studied in [1]-[3]. Recent results for the Schr¨odinger equation include the multiplier technique (see e.g., [1]), the backstepping method ([2]), sliding mode control ([3]) and extension of backstepping to heat-Schr¨odinger equation cascade citejmwang and ODE-Schr¨odinger equation cascade [5]. Stabilization of linear Schr¨odinger equation with time delay by boundary or internal feedbacks has been dealt ? This work was support by National Natural Science Foundation of China (grant No. 61803026, 61873260), Fundamental Research Funds for the Central Universities (Grant No. FRFTP-18- 032A1), China Postdoctoral Science Foundation (Grant No. 2018M640065), Fundamental Research Funds for the China Central Universities of USTB (No. FRF-BD-17-002A), and the Project of Department of Education of Guangdong Province (No. 2017KZDXM087). Email addresses: [email protected] (Wen Kang), [email protected] (Bao-Zhu Guo).

Preprint submitted to IFAC Journal of Systems and Control

with in [6]. Regional boundary stabilization of nonlinear Schr¨ odinger equation with state delay and bounded internal disturbance can be found in [7]. Very recently, a stabilizing output feedback control for Schr¨odinger equation subject to internal unknown dynamic and external disturbance has been developed in [8]. Differently from [8], we focus on the design of state feedback to reject the unknown disturbance by using backstepping method and Lyapunov method directly, with arbitrarily assigned decay rate whereas in [8], only asymptotic stability can be achieved. Stabilization of systems described by partial differential equations (PDEs) subject to unknown disturbance has attracted considerable attention (see e.g., [9]-[13] and the references therein) in recent years. Many different approaches have been developed to deal with disturbance such as the internal model principle for output regulation, the robust control for systems with uncertainties from both internal and external disturbance, the adaptive control for systems with unknown parameters, to name just a few. In this paper, unknown bounded disturbances are modeled into the boundary control terms. The objective of this work is to design a stabilizing controller for perturbed Schr¨odinger equation by using backstepping transformation and appro-

20 January 2019

Since the feedback control is discontinuous with multivalue, we actually consider (2.1) as   wt (x, t) = −iwxx (x, t), x ∈ (0, 1), t > 0,      w (0, t) = −iqw(0, t), t > 0, x (2.1∗)  w  x (1, t) ∈ U (t) + d(t), t > 0,     w(x, 0) = w (x), x ∈ [0, 1], 0

priate Lyapunov-Krasovskii functional. The paper is organized as follows. In the following section, the problem statement is presented and the backstepping transformation is introduced. We transform original system into an equivalent target system where the internal instability terms are transformed into the control boundary. A state feedback is thus constructed to stabilize the system. In Section 3, a Filippov type solution of the closed-loop system is determined by a Galerkin approximation scheme. In Section 4, by using Lyapunov-Krasovskii method for the target system we show the closed-loop system to be exponentially stable. A numerical example is presented in Section 5 for illustration of the effectiveness of the method. Some concluding remarks are presented in Section 6.

2.2

First we consider a transformation Z x

u(x, t) = w(x, t) −

2.1

l(x, y)w(y, t)dy,

(2.2)

0

where the kernel l(x, y) satisfies the following PDE:   l (x, y) − lyy (x, y) = icl(x, y),    xx ly (x, 0) = −iql(x, 0), (2.3)   ic   l(x, x) = − x − iq. 2 Lemma 2.1 ([3]). The problem (2.3) admits a unique solution that is twice continuously differentiable in 0 ≤ y ≤ x ≤ 1.

Notation. Throughout the paper, Re{·}, Im{·} stand for real and imaginary √ parts of associated complex variable respectively. i = −1 denotes the imaginary unit. The subscripts x and t of w(x, t) are partial derivatives with respect to x and t, i.e., wt (x, t) = ∂w(x, t)/∂t, wx (x, t) = ∂w(x, t)/∂x, and wxx (x, t) = ∂ 2 w(x, t)/∂x2 , respectively. L2 (0, 1) is the Hilbert space of square integrable scalar functions f (x), x ∈ (0, 1) with the corresponding norm R1 kf k2L2 (0,1) = 0 |f (x)|2 dx. H 1 (0, 1) denotes the Sobolev space of continuously differentiable functions on [0, 1] with measurable, square integrable first derivative.

2

Backstepping transformation

The transformation (2.2) transforms the system (2.1*) into the following target system:   ut (x, t) = −iuxx (x, t) − cu(x, t),       ux (0, t) = 0,    ux (1, t) ∈ U (t) + d(t) − l(1, 1)u(1, t) (2.4) Z 1     − lx (1, y)u(y, t)dy,    0    u(x, 0) = u (x), 0

State feedback controller design Problem formulation

Consider the following anti-stable one dimensional Schr¨odinger equation with an external disturbance:   wt (x, t) = −iwxx (x, t), x ∈ (0, 1), t > 0,      w (0, t) = −iqw(0, t), t > 0, x (2.1)  wx (1, t) = U (t) + d(t), t > 0,      w(x, 0) = w (x), x ∈ [0, 1], 0

where c is an arbitrary positive constant. By selecting the feedback controller ! u(1, t) − u(1, t) U (t) = −ic1 u(1, t) + M sign 2i ! u(1, t) + u(1, t) (2.5) −iM sign 2 Z 1 +l(1, 1)u(1, t) + lx (1, y)u(y, t)dy,

where q > 0, w(x, t) is the complex-valued state, and U (t) is the control input. The unknown disturbance d(t) ∈ C is supposed to be uniformly bounded measurable and satisfies 1 d ∈ Hloc (0, ∞), |d| ≤ M for some constant M > 0. The open-loop system (2.1) (subject to U (t) = d(t) ≡ 0) is anti-stable, which has all of its eigenvalues in the right half-plane. In the rest of the paper, we drop the obvious domains for both x and t.

0

one arrives at the target system    ut (x, t) = −iuxx (x, t) − cu(x, t), x ∈ (0, 1), t > 0,     ux (0, t) = 0,   !      u (1, t) ∈ −ic u(1, t) + M sign u(1, t) − u(1, t) x 1 2i !     u(1, t) + u(1, t)   −iM sign + d(t),   2      u(x, 0) = u (x),

The objective of this work is to find a control law such that the resulting closed-loop system with disturbance is exponentially stable with an arbitrary decay rate by using backstepping transformation and Lyapunov approach.

0

(2.6)

2

initial value. Set p = u − u ˜. Then p satisfies

where c1 is a positive constant, and the symbolic function is a set-valued function defined by  z  , z 6= 0, sign(z) = |z|  {˜ z ∈ C : |˜ z | ≤ 1}, z = 0.

(2.7)

  pt (x, t) = −ipxx (x, t) − cp(x, t),       px (0, t) = 0,  !     u(1, t) − u(1, t)   px (1, t) = −ic1 p(1, t) + M sign   2i   !     ˜(1, t) − u ˜(1, t)  −M sign u 2i !     u(1, t) + u(1, t)   −iM sign   2   !     ˜(1, t) u ˜(1, t) + u   +iM sign ,   2      p(·, 0) = 0.

In order the function on the right-hand side of (2.5) to be measurable, for any T > 0 and f ∈ L2 (0, T ), we restrict the set-valued composition of the symbolic function as follows: sign(f (t)) = ( (

sign(f (t)), g(t)| g(t) = f˜ ∈ L2 ({t| f (t) = 0}), |f˜| ≤ 1,

f (t) 6= 0

)

f (t) = 0 (2.8) Remark 2.1. Our control law (2.5) is different from the one introduced by [3], where SMC(slide mode control) and ADRC (active disturbance rejection control) method were applied to deal with the disturbance. Here we focus on designing a feedback via backstepping approach and Lyapunov approach directly.

3

.

(3.2)

Define a Lyapunov functional as follows

V1 (t) =

Well-posedness of closed-loop system

In this section, we show that the existence and uniqueness of solution to closed-loop system (2.6) which will be accomplished in Theorem 3.1. To this purpose, we define 2

1 1 kp(·, t)k2L2 (0,1) = 2 2

Z

0

1

|p(x, t)|2 dx.

(3.3)

Taking the time derivative of V1 along (3.2), we obtain

0

H = {f ∈ H (0, 1)|f (0) = 0}. It is obvious that H is a closed subspace of Sobolev space H 2 (0, 1). Theorem 3.1. Given T > 0. For any initial value u0 ∈ H satisfying the compatible conditions:   u00 (0) = 0,  !      u0 (1) ∈ −ic u (1) + M sign u0 (1) − u0 (1) 1 0 0 2i !     u0 (1) + u0 (1)   + d(0), −iM sign  2

V˙ 1 (t) = Rehpt (·, t), p(·, t)i = −c1 |p(1, t)|2 − ckp(·, t)k2L2 (0,1) n +Re i(u(1, t) − u ˜(1, t)) ! " u(1, t) − u(1, t) ×M sign 2i ! u ˜(1, t) − u ˜(1, t) −sign 2i ! u(1, t) + u(1, t) −i · sign 2 !#) u ˜(1, t) + u ˜(1, t) +i · sign . 2

(3.1)

Eq.(2.6) admits a (Filippov type) solution.

Proof. We start by showing that the classical solution must be locally unique if u(1, t) 6= 0 locally. Otherwise, there are two different solutions u and u ˜ to (2.6) with the same

(3.4)

Since u(1, t) = Re{u(1, t)} + iIm{u(1, t)}, and u ˜(1, t) =

3

which satisfies  N N  huN  t (·, t), φi − ihux (·, t), φx i + chu (·, t), φi      ∈ −iφ(1)[−ic1 uN (1, t)  !    N   +M sign u (1, t) − uN (1, t) 2i !     uN (1, t) + uN (1, t)   + d(t)], ∀ φ ∈ VN , −iM sign   2      uN (·, 0) = u (·), uN (·, 0) ∈ V . 0 N (3.9) Equation (3.9) is equivalent to  N 0 N  huN  t (·, t), φm i − ihux (·, t), φm i + chu (·, t), φm i      ∈ −iφm (1)[−ic1 uN (1, t)   !   N N (1, t)  u u (1, t) −     +M sign 2i ! N  u (1, t) + uN (1, t)   −iM sign + d(t)],    2      m = 1, 2, . . . , N,      uN (·, 0) = u (·). 0 (3.10) Set X(t) = (a1N (t), . . . , aN N (t))> . Then (3.10) can be written as a system of ODEs as

Re{˜ u(1, t)} + iIm{˜ u(1, t)}, we obtain V˙ 1 (t) = −c1 |p(1, t)|2 − ckp(·, t)k2L2 (0,1) (" u(1, t) − u(1, t) u ˜(1, t) ˜(1, t) − u −Re − 2i 2i # u ˜(1, t) + u u(1, t) + u(1, t) ˜(1, t) +i −i 2 2 " ! u(1, t) − u(1, t) ×M sign 2i ! u ˜(1, t) − u ˜(1, t) −sign 2i ! u(1, t) + u(1, t) −i · sign 2 !#) u ˜(1, t) + u ˜(1, t) +i · sign 2

(3.5)

≤ −c1 |p(1, t)|2 − ckp(·, t)k2L2 (0,1) , where we used the following facts: Re{u(1, t)} =

u(1, t) + u(1, t) , 2

Im{u(1, t)} =

u(1, t) − u(1, t) , 2i

> ˙ A1 X(t) ∈ A2 X(t) + A3 X(t) − k3 c1 k3 X(t)

and Re([sign(a) − sign(b)][a − b]) > 0, ∀ a, b ∈ C.

+k1 sign(hC1 , X(t)i) + k2 sign(hC2 , X(t)i) + k3 d(t). (3.11) where A1 , A2 , A3 are appropriate matrices, ki (i = 1, 2, 3) are some vectors in CN , and     0 0 N A1 = {hφi , φj i}N , A = {ihφ , φ i} 2 i,j=1 i j i,j=1 ,   A3 = {−chφi , φj i}N i,j=1 , !> φN (1) − φN (1) φ1 (1) − φ1 (1) C1 = ,··· , , 2i 2i !> φ1 (1) + φ1 (1) φN (1) + φN (1) C2 = ,··· , , 2 2

(3.6)

Hence, for t ≥ 0, V1 (t) ≡ V1 (0) = 0 or u = u ˜. Next we prove the existence of a Filippov type solution to (2.6) by a Galerkin approximation scheme. Multiplying the first equation of (2.6) by φ ∈ H 1 (0, 1) and integrating from 0 to 1 with respect to x, we obtain hut (·, t), φi − ihux (·, t), φx i + chu(·, t), φi " ! u(1, t) − u(1, t) ∈ iφ(1) −ic1 u(1, t) + M sign 2i ! # u(1, t) + u(1, t) −iM sign + d(t) . 2

(3.7)

k1 = (iM φ1 (1), · · · , iM φN (1))> , k2 = (M φ1 (1), · · · , M φN (1))> ,

k3 = (−iφ1 (1)), · · · , −iφN (1))> .

{φn (x)}∞ 1

Suppose that is an orthonormal basis for H 1 (0, 1) and for any N ∈ Z+ define a finite-dimensional subspace of H by VN = span{φ1 , φ2 , · · · , φN }. Since u0 ∈ H, we may assume without loss of generality that u0 ∈ span{φ1 }. A Galerkin approximation solution to (2.6) is constructed as follows: uN (x, t) =

N X

anN (t)φn (x),

Since {hφi , φj i}N i,j=1 is the Gramian matrix composed of 2 the basis {φi (x)}N i=1 for L (0, 1) owing to the density of H 2 in L (0, 1), we can rewrite (3.11) as > ˙ X(t) ∈ A−1 1 (A2 + A3 − k3 c1 k3 )X(t)

−1 +sign(hC1 , X(t)i)A−1 1 k1 + sign(hC2 , X(t)i)A1 k2

+d(t)A−1 1 k3 .

(3.8)

(3.12)

n=1

4

Proof. Substituting φ = uN (·, t) into (3.9), we have

Because of the multi-value sign function, above equation is understood in the sense of Filippov ([14], p48) by

1 d N ku (·, t)k2 ∈ −c1 |uN (1, t)|2 − ckuN (·, t)k2 2 dt (" # uN (1, t) − uN (1, t) uN (1, t) + uN (1, t) −Re −i 2i 2 " ! N N u (1, t) − u (1, t) × M sign 2i ! #) N u (1, t) + uN (1, t) −iM sign + d(t) 2

−1 > ˙ X(t) ∈ A−1 1 (A2 + A3 − k3 c1 k3 )X(t) + U1 (X)A1 k1

+U2 (X)A1−1 k2 + d(t)A−1 1 k3 ,

(3.13) where U1 (X) = sign(hC1 , X(t)i), U2 (X) = sign(hC2 , X(t)i). When k3 = 0, (3.13) admits only classical solution:

≤ −c1 |uN (1, t)|2 − ckuN (·, t)k2 ≤ 0.

(3.17) Therefore, for either classical solution or the Filippov solution, we can always draw the conclusion (3.16).

˙ X(t) = A−1 1 (A2 + A3 )X(t). When k3 6= 0, set

Lemma 3.2. For either the classical solution or the Filippov solution of Equation (3.9),

S1 (X) = hC1 , X(t)i, S2 (X) = hC2 , X(t)i.

sup ku˙ N (·, 0)k < ∞.

Then Ui is discontinuous only on Si = 0 (i = 1, 2). Now we use the equivalent control method to find the sliding mode solution (locally) for (3.13). That is, we need to find a continuous function Uieq (i = 1, 2) that is called the equivalent control, so that

Proof. Setting t = 0 in (3.9), and taking the compatible conditions (3.1) into account, we obtain hu˙ N (·, 0), φi − ihu00 , φx i + chu0 , φi

eq −1 > ˙ X(t) = A−1 1 (A2 + A3 − k3 c1 k3 )X(t) + U1 (t)A1 k1

+U2eq (t)A−1 1 k2

+

u0 (1) − u0 (1) 2i

d(t)A−1 1 k3 .

∈ −iφ(1)[−ic1 u0 (1) + M sign ! u0 (1) + u0 (1) −iM sign + d(0)]. 2

(3.14) Since the equivalent control is needed only when Si (t) = ˙ 0(i = 1, 2) and hence S˙1 (t) = hC1 , X(t)i = 0, S˙2 (t) = eq eq ˙ hC2 , X(t)i = 0, we can find U1 and U2 directly. Returning back to (3.9), the sliding mode is S1 (t) = S2 (t) = 0 on which we have  N N  huN  t (·, t), φi − ihux (·, t), φx i + chu (·, t), φi      ∈ −iφ(1)[−ic1 uN (1, t) + M U1eq (t) − iM U2eq + d(t)],    ∀ φ ∈ VN ,    uN (1, t) = Re{uN (1, t)} + iIm{uN (1, t)}     N N N N    = u (1, t) − u (1, t) + i u (1, t) + u (1, t) = 0. 2i 2 (3.15)

−iu˙ N (1, 0)[−ic1 u0 (1) +M sign

−iM sign

u0 (1) − u0 (1) 2i

!

u0 (1) + u0 (1) 2

!

(3.20) + d(0)]

= −chu0 , u˙ N (x, 0)i. By compatible condition (3.1), we have ku˙ N (·, 0)k2 = −chu0 , u˙ N (x, 0)i. ku˙ N (·, 0)k 6 cku0 k.

Lemma 3.1. For either the classical solution or the Filippov solution of Equation (3.9),

t>0

(3.19)

ku˙ N (x, 0)k2 ∈ ihu00 , u˙ N ˙ N (x, 0)i x (·, 0)i − chu0 , u

Therefore,

 max sup kuN (·, t)k < ∞.

!

Substituting φ(x) = u˙ N (x, 0) into the latter equation we obtain

The remaining proof will be split into several lemmas.

  uN (1, t) ∈ L2 (0, ∞),

(3.18)

N

Lemma 3.3. For either the classical solution or the Filippov solution of (3.9), (3.16)

sup ku˙ N (·, t)k < ∞, ∀ t ∈ [0, T ] a.e.

N

N

5

(3.21)

Proof. For fixed t, ξ > 0 with ξ < T − t, replace t by t + ξ in (3.9) and subtract the first equation of (3.9) to have N 0 hu˙ N (·, t + ξ) − u˙ N (·, t), φi − ihuN x (·, t + ξ) − ux (·, t), φ i

We also need the following Lemma 3.4 from[12]. For the sake of complete, the simple proof is given.

+chuN (·, t + ξ) − uN (·, t), φi ∈ −iφ(1)[−ic1 uN (1, t + ξ)

!

+M sign

uN (1, t + ξ) − uN (1, t + ξ) 2i

−iM sign

uN (1, t + ξ) + uN (1, t + ξ) 2

+iφ(1)[−ic1 uN (1, t) +M sign −iM sign

uN (1, t) − uN (1, t) 2i

!

Lemma 3.4. For any T > 0, suppose that fi → f in L2 (0, T ) strongly as i → ∞ and g ∈ L2 (0, T ). Then under definition (2.8) for sign(f (t)), there exists a subsequence ∞ {fij }∞ j=1 of {fi }i=1 such that Z T Z T sign(fij (t))g(t)dt ⊂ sign(f (t))g(t)dt, (3.27) lim

+ d(t + ξ)]

j→∞

!

uN (1, t) + uN (1, t) 2

!

(3.22) Substituting φ(x) = uN (·, t + ξ) − uN (·, t) into (3.22) we have 1 d N (3.23) ku (·, t + ξ) − uN (·, t)k2 = I1 + I2 , 2 dt where

(3.28)

where

−c1 |u (1, t + ξ) − u (1, t)| 6 0, N

N

0

which is understood in the following sense: Z Z   lim sign(f (t))g(t)dt = sign(f (t))g(t)dt,  ij  j→∞  Z P P      sign(fij (t))g(t)dt    Qn o R λ(t) ∈ sign(fij (t))|Q , j = 1, 2, . . . , = λ(t)g(t)dt Q  Z    ⊂  sign(f (t))g(t)dt    nQ o  R  = λ(t) ∈ sign(f (t))|Q , λ(t)g(t)dt Q

+ d(t)].

I1 = −ckuN (·, t + ξ) − uN (·, t)k2

0

2

P = {t ∈ [0, T ]| f (t) 6= 0},

I2 = Re{−i[uN (1, t + ξ) − uN (1, t)] " ! uN (1, t + ξ) − uN (1, t + ξ) × M sign 2i ! N u (1, t + ξ) + uN (1, t + ξ) −iM sign + d(t + ξ) 2 ! uN (1, t) − uN (1, t) −M sign 2i ! #) N u (1, t) + uN (1, t) +iM sign − d(t) 2 h i ≤ Re{−i uN (1, t + ξ) − uN (1, t)][d(t + ξ) − d(t) } c1 1 ≤ |uN (1, t + ξ) − uN (1, t)|2 + |d(t + ξ) − d(t)|2 . 2 c1 (3.24) Integrating (3.23) over [0, t] with respect to t, we obtain

Q = {t ∈ [0, T ]| f (t) = 0}. Proof. From (2.8) and ∪∞ j=1 sign(fij ) ⊂ sign(f ), the second part is obtained. We need only show the first part of (3.28). Let P + = {t ∈ [0, T ]| f (t) > 0}, and P − = {t ∈ [0, T ]| f (t) < 0}; P = P + ∪ P − . Since fi → f in L2 (0, T ) strongly, by the Riesz-Fischer theorem ([15, p. 148]), there exists {fij }∞ j=1 such that fij (t) → f (t), t ∈ [0, T ] a.e. as j → ∞. By Lusin’s theorem ([15, p. 66]), there exist closed sets + − + P1ε ⊂ P + and P1ε ⊂ P − with meas(P + \ P1ε ) < ε, − + − − meas(P \P1ε ) < ε such that f is continuous over P1ε ∪P1ε . By Egoroff’s theorem ([15, p. 64]), there exist closed sets + + − − + + P2ε ⊂ P1ε and P2ε ⊂ P1ε satisfying meas(P1ε \ P2ε ) < ε, − − meas(P1ε \ P2ε ) < ε such that

kuN (·, t + ξ) − uN (·, t)k2 − kuN (·, ξ) − uN (·, 0)k2 Z 2 t ≤ |d(s + ξ) − d(s)|2 ds c1 0 (3.25) in which we choose ξ sufficiently small. Dividing ξ on both sides (3.25) and passing to the limit as ξ → 0, we have Z 2 t ˙ 2 ku˙ N (·, t)k2 ≤ ku˙ N (·, 0)k2 + |d(s)| ds. (3.26) c1 0 By Lemma 3.2, we obtain ku˙ N (·, t)k < ∞, ∀ t ∈ [0, T ].

+ − fij (t) → f (t) uniformly for t ∈ P2ε ∪ P2ε .

(3.29)

+ − + Since P2ε and P2ε are closed and f is continuous on P2ε ∪ − P2ε , mint∈P + f > 0 and maxt∈P − f < 0. By (3.29), there 2ε 2ε exists a positive integer N > 0 such that for all j ≥ N ,  1 1 +   fij > f − min f ≥ min f > 0 on P2ε ,  + + 2 x∈P2ε 2 t∈P2ε (3.30) 1 1 −   f ≤ max f < 0 on P2ε .  fij < f − max + + 2 t∈P2ε 2 t∈P2ε

6

Therefore, for j ≥ N ,

Now, for any ψ ∈ D(0, T ) and φ ∈ H,

Z Z sign(fi (t))g(t)dt − sign(f (t))g(t)dt j P P Z = (sign(fij (t)) − sign(f (t)))g(t)dt (P + \P + )∪(P − \P − ) 2ε 2ε √ √ ≤ 4 εkgkL2 ((P + \P + )∪(P − \P − )) ≤ 4 εkgkL2 (0,T ) . 2ε 2ε (3.31) This proves the first part of (3.28). Notice that in derivation + of (3.31), we used the estimation meas((P + \ P2ε ) ∪ (P − \ − P2ε )) ≤ 4ε.

Z

0Z

uN * u in L∞ (0, T ; H 1 (0, 1)) weak∗ , u˙ N * u˙ in L∞ (0, T ; L2 (0, 1)) weak∗ .

T

−

Z

0

T

ihuN x (·, t), φx iψ(t)dt

chuN (·, t), φiψ(t)dt Z T [−ic1 uN (1, t) ∈ −iφ(1) 0

0

N

(3.35)

!

+M sign

u (1, t) − uN (1, t) 2i

−iM sign

uN (1, t) + uN (1, t) 2

!

+ d(t)]ψ(t)dt.

By Lemma 3.4, there exists a sequence of integer {Nk }∞ k=1 such that ! Z T uNk (1, t) + uNk (1, t) lim sign ψ(t)dt k→∞ 0 2 ! RT u(1, t) + u(1, t) ψ(t)dt, ⊂ 0 sign 2 ! (3.36) Z T uNk (1, t) − uNk (1, t) lim sign ψ(t)dt k→∞ 0 2i ! RT u(1, t) − u(1, t) ⊂ 0 sign ψ(t)dt. 2i

(3.32)

We first show that there exists a subsequence of uN (1, t), still denote by itself without confusion, such that uN (1, t) → u(1, t) in L2 (0, T ). To this end, by([16, Theorem 2.32]), we only need to show: (i) {uN (1, t)}∞ N =1 is bounded in L2 (0, T ); (ii) {uN (1, t)}∞ N =1 is equicontinuous, that is, kuN (t + ξ) − uN (t)kL2 (0,T ) → 0 as ξ → 0 uniformly with respect to N . The uniform boundedness of {uN (1, t)}∞ N =1 is trivial fact from Lemma 3.1. Now we 2 show that {uN (1, t)}∞ N =1 is equicontinuous in L (0, T ). From the proof of Lemma 3.3 in (3.23) and (3.24), we have

Due to (3.32), passing to the limit as k → ∞ in (3.35) for the subsequence {Nk }∞ k=1 , we obtain hut , φi − ihux , φ0 i + chu, φi

u(1, t) − u(1, t) 2i

!

∈ −iφ(1)[−ic1 u(1, t) + M sign ! u(1, t) + u(1, t) −iM sign + d(t)], t ∈ [0, T ] a.e. 2 (3.37) Taking φ ∈ D(0, 1) in (3.37), we obtain

Z c1 T N |u (1, t + ξ) − uN (1, t)|2 dt 2 0 ≤ −kuN (·, T + ξ) − uN (·, T )k2 + kuN (·, ξ) − uN (·, 0)k2 Z 2 T |d(t + ξ) − d(t)|2 dt. + c1 0 (3.33) Since d(t) is a continuous function, it is sufficient to show that, for τ = 0, T , kuN (·, τ + ξ) − uN (·, τ )k2 → 0, as ξ → 0,

huN t (·, t), φiψ(t)dt

+

Continuation of proof of Theorem 3.1. From Lemmas 3.1 to 3.3, we can extract a subsequence Nk , still denoted by N without confusion, such that (

T

hut , φi − ihux , φ0 i + chu, φi = 0, ∀ φ ∈ D(0, 1).

(3.38)

This shows that the generalized derivative uxx exists, and ut = −iuxx −cu holds for almost all t ∈ [0, T ]. In particular, u(·, t) ∈ H 2 (0, 1). Integration by parts of (3.37) over [0, 1] with respect to x leads to

(3.34)

uniformly with respect to N . It follows from Lemma 3.3 that for τ = 0, T , and for some C3 > 0

hut , φi − iux (1, t)φ(1) + iux (0, t)φ(0) + ihuxx , φi + chu, φi " ! u(1, t) − u(1, t) ∈ −iφ(1) −ic1 u(1, t) + M sign 2i ! # u(1, t) + u(1, t) −iM sign + d(t) , 2 (3.39)

kuN (·, τ + ξ) − uN (·, τ )k2 ≤ ku˙ N (·, τ1ξ )k2 |ξ| ≤ C3 |ξ|. Hence, (3.34) holds. As a result, there exists a subsequence uNj (x, t) such that uNj (x, t) → u(x, t) in L2 (0, T ) strongly. 7

which implies   ux (0, t) = 0,  !      u (1, t) ∈ −ic u(1, t) + M sign u(1, t) − u(1, t) x 1 2i !     u(1, t) + u(1, t)   −iM sign + d(t).  2

In order to prove the stability of (4.5), we employ the following Lyapunov-Krasovskii function: Z Z 1 1 1 1 2 2 V (t) = |u(x, t)| dx = [p (x, t) + κ2 (x, t)]dx. 2 0 2 0 (4.6) Differentiating (4.6) along (2.6), we have Z 1 Z 1 ˙ κ(x, t)κt (x, t)dx p(x, t)pt (x, t)dx + V (t) = 0Z Z0 1 1 = p(x, t)κxx (x, t)dx − κ(x, t)pxx (x, t)dx 0 Z0 1 −c [p2 (x, t) + κ2 (x, t)]dx 0 x=1 x=1 − κ(x, t)px (x, t) = p(x, t)κx (x, t) x=0 x=0 Z 1 2 2 −c [p (x, t) + κ (x, t)]dx

Thus, we obtain   ut = −iuxx − cu,      ux (0, t) = 0,   !   u(1, t) − u(1, t) ux (1, t) ∈ −ic1 u(1, t) + M sign  2i   !     u(1, t) + u(1, t)   −iM sign + d(t).  2

0

= −2cV (t) + p(1, t)κx (1, t) − κ(1, t)px (1, t)

≤ −2cV (t) − c1 p2 (1, t) − c1 κ2 (1, t)

It should be noted that we do not use Filippov’s concepts to system (2.6). Indeed, all Filippov’s concepts are only limited to approximated system (3.11). Although Filippov’s concepts [14] were used for ODEs rather than PDEs, the extension of Filippov’s concepts have been generalized to evolution equations in Hilbert spaces (see e.g. [18, p.30] and [19]). 4

−[M − |d(t)|]|p(1, t)| − [M − |d(t)|]|κ(1, t)|

≤ −2cV (t).

(4.7) From (4.7) it follows that the norm of this system decays exponentially at the arbitrarily defined rate c, which implies the target system (2.6) is exponentially stable in L2 (0, 1). This completes the proof of the theorem.

Stability Analysis Returning to the original system by the transformation (2.2), we obtain the following result: Theorem 4.2. Consider the system (2.1*) with the following control law ! g(t) − g(t) U (t) = −ic1 g(t) + M sign 2i ! g(t) + g(t) −iM sign + l(1, 1)g(t) 2   Z 1 Z y + lx (1, y) w(y, t) − l(y, z)w(z, t)dz dy,

In this section, we study the global stability of the closed-loop system (2.6) as in [17]. Theorem 4.1. The closed-loop system (2.6) is exponentially stable with the decay rate c: ku(·, t)k2L2 (0,1) ≤ e−2ct ku0 k2L2 (0,1) , ∀ t ≥ 0. (4.1) Proof. Set u(x, t) = p(x, t) + iκ(x, t), (4.2) u(x, t) + u(x, t) p(x, t) = Re{u(x, t)} = , (4.3) 2 u(x, t) − u(x, t) κ(x, t) = Im{u(x, t)} = , (4.4) 2i where p(x, t) is the real part of u(x, t), and κ(x, t) the imaginary part of u(x, t). Then, p and κ satisfy the following coupled PDEs:   pt (x, t) = κxx (x, t) − cp(x, t), x ∈ (0, 1), t > 0,      κt (x, t) = −pxx (x, t) − cκ(x, t), x ∈ (0, 1), t > 0,      p (0, t) = 0, x  κ  x (0, t) = 0,        px (1, t) = c1 κ(1, t) + M sign(κ(1, t)) + Re{d(t)},   κx (x, t) = −c1 p(1, t) − M sign(p(1, t)) + Im{d(t)}. (4.5)

0

where g(t) = w(1, t) −

0

Z

(4.8)

1

l(1, y)w(y, t)dy.

0

Then the closed-loop system (2.1*) under the feedback control (4.8) is exponentially stable with the decay rate c. 5

Example

To illustrate the satisfactory and better performance of the proposed design method, this section considers an anti-stable Schr¨ odinger equation (2.1) with the parameters q = 3 and the initial condition w0 (x) = sin x, x ∈ [0, 1]. Figure 1 provides open-loop numerical simulation result:

8

(a) State Re{w(x, t)}

(a) State Re{u(x, t)}

(b) State Im{w(x, t)}

(b) State Im{u(x, t)}

Fig. 1. Open-loop system (without control input)

profile of evolution of w(·, t). It is seen that the open-loop system is unstable. Consider next the closed-loop (2.6) system under the feedback controller (2.5) with the parameters c = 10, c1 = 5, and M = 1. Here, we simply specify sign(0) = 0. The disturbance is chosen as d(t) = e−t . A finite difference method is applied to compute the displacement of the closed-loop system (2.6). We choose initial condition u(x, 0) = sin x, 0 ≤ x ≤ 1. The steps of space and time are taken as 0.1 and 0.0001, respectively. Figure 2 shows that the closedloop system is exponentially stable. The simulations of the solutions confirm the theoretical results. 6

(c) The input signal Re{U (t)}

Conclusion

This paper develops a boundary controller combined with a backstepping technique for Schr¨odinger equation subject to unknown bounded disturbances on the controlled boundary. The technique introduces a new variable transformation that makes it possible to apply the backstepping technique to provide arbitrary exponential de(d) The input signal Im{U (t)}

9

Fig. 2. Closed-loop system

cay of the system to its origin in a way that is robust to bounded disturbances. Observer-based boundary control of perturbed Schr¨odinger equation to achieve arbitrary decay rate, which is different to asymptotic stability obtained in [8], may be a topic for future research.

[19] L. Levaggi, Sliding modes in Banach spaces, Differential Integral Equations, 15(2002), 167-189.

References [1] E. Machtyngier and E. Zuazua, Stabilization of the Schr¨ odinger equation, Portugal. Math., 51(1994), 243-256. [2] M. Krstic, B.Z. Guo, and A. Smyshlyaev, Boundary controllers and observers for the linearized Schr¨ odinger equation, SIAM J. Control Optim., 49(2011), 1479-1497. [3] B.Z. Guo and J.J. Liu, Sliding mode control and active disturbance rejection control to the stabilization of onedimensional Schr¨ odinger equation subject to boundary control matched disturbance, Internat. J. Robust Nonlinear Control 24(2014), 2194-2212. [4] J.M. Wang, B.B. Ren, and M. Krstic, Stabilization and Gevrey regularity of a Schr¨ odinger equation in boundary feedback with a heat equation, IEEE Trans. Automat. Control, 57(2012), 179185. [5] B.B. Ren, J.M. Wang, and M. Krstic, Stabilization of an ODESchr¨ odinger Cascade, Systems Control Lett., 62(2013), 503-510. [6] S. Nicaise and S. Rebiai, Stabilization of the Schr¨ odinger equation with a delay term in boundary feedback or internal feedback, Portugal. Math, 68(2011), 19-39. [7] W. Kang and E. Fridman, Boundary constrained control of delayed nonlinear Schr¨ odinger equation, IEEE Trans. Automat. Control, 63(2018) 3873-3880. [8] X. Zhang, H. Feng, S.G. Chai, Output feedback stabilization for an anti-stable Schr¨ odinger equation with internal unknown dynamic and external disturbance, J. Franklin Inst., 355(2018), 5632-5648. [9] A. Pisano and Y. Orlov, Boundary second-order sliding-mode control of an uncertain heat process with unbounded matched perturbation, Automatica, 48(2012), 1768-1775. [10] B.Z. Guo and W. Kang, The Lyapunov approach to boundary stabilization of an anti-stable one-dimensional wave equation with boundary disturbance, Internat. J. Robust Nonlinear Control, 24(2014), 54-69. [11] W. Kang and E. Fridman, Boundary control of delayed ODEHeat cascade under actuator saturation, Automatica, 83(2017), 252-261. [12] F. F. Jin and B.Z. Guo, Lyapunov approach to output feedback stabilization of Euler-Bernoulli beam equation with boundary input disturbance, Automatica, 52(2015), 95-102. [13] W. Kang and E. Fridman, Sliding mode control of Schr¨ odinger equation-ODE in the presence of unmatched disturbances, Systems Control Lett., 98(2016), 65-73. [14] A.F. Filippov, Differential Equations with Discontinuous RightHand Sides, Kluwer Academic Publishers, Dordrecht, 1988. [15] H.L. Royden and P. M. Fitzpatrick, Real Analysis, 4ed, Prentice Hall, Boston, 2010. [16] R.A. Adams and John J.F. Fournier, Sobolev Spaces, Academic Press, 2nd ed, 2003. [17] M. Krstic, A. Smyshlyaev, Boundary Control of PDEs: A Course on Backstepping Designs, Philadelphia, PA:SIAM, 2008. [18] Y. Orlov, Discontinuous Systems Lyapunov Analysis and Robust Synthesis under Uncertainty Conditions, SpringerVerlag, Springer, 2009.

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