Received: 3 July 2018 Revised: 3 November 2018 Accepted: 27 November 2018

DOI: 10.1002/rnc.4448

R E S E A R C H A R T I C L E

Sliding mode control for N-coupled reaction-diffusion PDEswith boundary input disturbances

Jian-Jun Gu1 Jun-Min Wang2

1School of Mathematics and Statistics,Changshu Institute of Technology,Changshu, China2School of Mathematics and Statistics,Beijing Institute of Technology, Beijing,China

CorrespondenceJun-Min Wang, School of Mathematicsand Statistics, Beijing Institute ofTechnology, Beijing 100081, China.Email: jmwang@bit.edu.cn

Funding informationNational Natural Science Foundation ofChina, Grant/Award Number: 61673061

Summary

This paper develops the sliding mode control (SMC) design for N-coupledreaction-diffusion parabolic PDEs with boundary input disturbances. In orderto reject the disturbances, the backstepping-based boundary SMC law is con-structed to steer the system trajectory to a suitable sliding surface and thenmaintain sliding motion on the surface thereafter, resulting in the exponentialconvergence to the zero equilibrium state. The well-posedness of the closed-loopsystem is established based on a detailed spectral analysis and Riesz basis gen-eration. Finally, a simulation example is provided to illustrate the effectivenessof the SMC design.

KEYWORDS

backstepping, disturbance rejection, N-coupled reaction-diffusion PDEs, Riesz basis, sliding modecontrol

1 INTRODUCTION

The purpose of this paper is to investigate the boundary stabilization for N-coupled reaction-diffusion parabolic PDEs withthe external disturbances flowed into the control end. The coupled parabolic system can describe the dynamics involvedin diffusion-reaction phenomenon, such as the evolution of interacting populations,1,2 chemical tubular reactor,3 andthermal processing,4 to name a few. A considerable amount of attention has been paid to the boundary control design forthe unperturbed coupled system class. In the work of Baccoli et al,5 a state feedback control law was designed to stabilize aset of coupled parabolic PDEs with constant parameters, and the design procedure is based on the backstepping approach.6

The approach has also been applied in the works of Liu et al7 and Orlov et al8 for the output feedback stabilization of thecoupled parabolic PDEs with different diffusions. More recently, the problems of boundary stabilization9 and observerdesign10 have been solved for the spatially varying coefficients case.

In practical engineering, the control systems inevitably suffer from the external disturbances and may become unstable.Hence, the control laws need to be designed to stabilize the systems in the time of rejecting the disturbances. Severalcontrol approaches have been developed for the disturbance rejection, including output regulation,11-14 active disturbancerejection control,15,16 disturbance observer-based control,17,18 and sliding mode control (SMC).19-23 The SMC approachis based on designing a unit control,24,25 the norm of which is equal to one when beyond a suitable sliding manifold.The closed-loop system enforced by the unit control is guaranteed to exhibit robustness properties against the matcheddisturbance. Therefore, the SMC has been widely applied for the control design because of its robustness properties andsimplicity of implementation.26,27 In the work of Orlov et al,19 an integral form of distributed control law incorporatingboth feedforward and feedback parts has been proposed for the state-tracking and disturbance rejection of a heat diffusionprocess. The tracking control of a wave equation with distributed control and disturbances has been investigated in the

Int J Robust Nonlinear Control. 2019;29:1437–1461. wileyonlinelibrary.com/journal/rnc © 2018 John Wiley & Sons, Ltd. 1437

1438 GU AND WANG

work of Pisano et al,20 wherein the designed controller contains a second-order SMC law term. Based on backsteppingapproach, the boundary SMC laws have also been designed for parabolic,21 antistable wave,22 and Schrödinger equation,23

respectively. In these studies, the exponential stability of the corresponding closed-loop systems was achieved by thedesigned control laws on the suitable sliding surfaces.

Extensions of boundary SMC to the case of PDE-ODE cascades have been proposed in several studies.28-30 In thework of Wang et al,28 a novel integral sliding surface was presented for a cascade of perturbed heat-ODE system withDirichlet/Neumann interconnection, and then, a systematic procedure for SMC design has been shown to be effec-tive to deal with the boundary input disturbance. In the work of Kang and Fridman,29 SMC has been applied to solvethe input-to-state stabilization (ISS) problem for a cascade of Schrödinger-ODE system with small unmatched distur-bance, which flows into the ODE. The stabilization of wave-ODE with boundary input disturbance can be found in thework of Liu and Wang.30 A recent progress is made in the work of Gu and Wang,31 where a boundary SMC has beendesigned for the Orr-Sommerfeld equation cascaded by both Squire equation and ODE, in which Orr-Sommerfeld andSquire equations are subject to boundary input disturbances, respectively. However, to the best of the authors' knowledge,the backstepping-based boundary SMC is still limited to two types of systems: (i) single PDE and (ii) cascaded systemcontaining single or two PDEs.

We consider the following N-coupled reaction-diffusion PDEs with the Dirichlet boundary conditions and the Neumannactuation: ⎧⎪⎨⎪⎩

Ut(x, t) = ΘUxx(x, t) + ΛU(x, t), x ∈ (0, 1), t > 0,U(0, t) = 0N×1, t ≥ 0,Ux(1, t) = Uc(t) + D(t), t ≥ 0,

(1)

where

• U(x, t) = [u1(x, t),u2(x, t), … ,uN(x, t)]⊤ ∈ ℝN is the state vector;• Θ = diag(𝜃𝑗) ∈ ℝN×N is a positive diagonal matrix, and the elements 𝜃j for j = 1, 2, … ,N, represent the diffusions of

the system;• Λ ∈ ℝN×N is a real-valued matrix denoting the reaction term;• Uc(t) = [uc1(t),uc2(t), … ,ucN(t)]⊤ ∈ ℝN is the control input vector;• D(t) = [d1(t), d2(t), … , dN(t)]⊤ ∈ ℝN is the unknown disturbance vector at the control end. Let 𝜃min and 𝜃max denote

the minimum and maximum elements of {𝜃𝑗}N𝑗=1, respectively. The unknown disturbance vector D(t) is supposed to be

bounded, ie, ||D(t)||ℝN ≤ M = M𝜃min∕𝜃max for some M > 0 and all t > 0. The Euclidian norm || · ||ℝN is defined by

‖D(t)‖ℝN =

√√√√ N∑𝑗=1

||d𝑗(t)||2. (2)

The main contribution of the present work is the extension of the backstepping-based boundary SMC to N-coupledsystem (1). By backstepping transformation, the original system (1) is converted into a target system, which is driven bya designed SMC law to arrive at a properly chosen sliding surface and then maintain an ideal sliding motion. Since theSMC law including the unit-vector control structure undergoes discontinuities on the sliding surface, the precise mean-ing of the generalized solutions of the targeted closed-loop system can be defined as a limiting result obtained through aregularization procedure.32(p30) An important problem in the proofs of existence and uniqueness for the generalized solu-tions is to verify that the boundary control law is admissible for the semigroup associated with the resulting closed-loopsystem. We try to solve the problem by developing the Riesz basis generation for the adjoint system of the closed-loopsystem based on the spectral analysis. However, due to its coupled property, the difficulty for the adjoint system occursin analyzing the spectrum distribution. The matrix operator pencil method, developed by Tretter33 and Wang et al,34 isapplied to overcome the difficulty.

Throughout this paper, we will use the symbol to denote the Hilbert space (L2(0, 1))N, which is the Cartesian productof L2(0, 1) and equipped with the norm

||Z(·)|| =

√√√√ N∑𝑗=1

‖‖z𝑗(·)‖‖2L2 , with Z(·) = [z1(·), z2(·), … , zN(·)]⊤ ∈ , (3)

GU AND WANG 1439

where zj(·) ∈ L2(0, 1), j = 1, 2, … ,N. We follow the notation of Baccoli et al,5 and the symmetric part of a square matrixΛ is denoted by S[Λ] ∶= (Λ + Λ⊤)∕2. The smallest and largest eigenvalues of a symmetric and positive definite matrixΛ are denoted by 𝜎m(Λ) and 𝜎M(Λ), respectively.

The rest of this paper is organized as follows. In Section 2, the backstepping-based SMC law is designed to guarantee theexistence of the stable sliding motion, which is governed by a sliding mode equation. The precise meaning of the solutionto the targeted closed-loop is also defined. In Section 3, we first show that the adjoint system to the resulting closed-loopsystem has the Riesz basis property, and then establish the regularization for the closed-loop system. The validity of thesliding mode equation is confirmed. Section 4 presents the numerical simulations to illustrate the effectiveness of theproposed method, followed by the conclusion of this paper in Section 5. Finally, we include appendix with some technicalproofs on the spectral analysis.

2 SLIDING MODE CONTROL DESIGN

This section is devoted to the design of control vector Uc(t) based on backstepping method and SMC approach. First, weconsider the backstepping transformation of the form5

W(x, t) = U(x, t) −

x

∫0

K(x, 𝑦)U(𝑦, t)d𝑦, (4)

with the matrix kernel K(x, 𝑦) ∈ ℝN×N satisfying the following equations: (x, 𝑦) ∈ 𝔗 = {(x, 𝑦)|0 ≤ 𝑦 ≤ x ≤ 1},

⎧⎪⎪⎨⎪⎪⎩

ΘKxx(x, 𝑦) − K𝑦𝑦(x, 𝑦)Θ − K(x, 𝑦)Λ − CK(x, 𝑦) = 0N×N ,

ΘKx(x, x) + K𝑦(x, x)Θ + ΘK′(x, x) + C + Λ = 0N×N ,

K(x, 0)Θ = 0N×N ,

ΘK(x, x) − K(x, x)Θ = 0N×N ,

(5)

where the real-valued matrix C = [ci,𝑗]N ∈ ℝN×N satisfying

𝜎m(S[C]) > −𝜃min∕4. (6)

In order to fulfill the fourth equation of (5), we impose the following constraint on the kernel matrix K(x, y):

K(x, 𝑦) = k(x, 𝑦)IN×N , (7)

where k(x, 𝑦) ∈ ℝ. The constraint (7) implies that Θ−1C = 𝜅IN×N − Θ−1Λ, where 𝜅 is chosen such that7

−S[Λ] + 𝜃min

4IN×N + 𝜅Θ ≻ 0.

The spectrum of Θ−1C is supposed to satisfy

𝜎(Θ−1C

)⋂{(m + 1

2

)2𝜋2, m ∈ ℕ

}= ∅. (8)

By using a similar argument in the works of Baccoli et al5,35 (wherein Neumann left-hand boundary condition has beenconsidered), we conclude that (5) possesses a unique solution

K(x, 𝑦) = k(x, 𝑦)IN×N = −𝜆𝑦I1

[√𝜆(

x2 − 𝑦2)]

√𝜆(

x2 − 𝑦2) IN×N

with the modified Bessel function6(p35)

I1(·) =∞∑

n=0

(x∕2)2n+1

n!(n + 1)!.

1440 GU AND WANG

It is straightforward to show that (4) maps system (1) into the target system

⎧⎪⎨⎪⎩Wt(x, t) = ΘWxx(x, t) − CW(x, t), x ∈ (0, 1), t > 0,W(0, t) = 0N×1, t ≥ 0,Wx(1, t) = [Uc(t) + D(t)] − K(1, 1)U(1, t) − ∫ 1

0 Kx(1, 𝑦)U(𝑦, t)d𝑦, t ≥ 0,(9)

where W(x, t) = [w1(x, t),w2(x, t), … ,wN(x, t)]⊤ ∈ ℝN is the target state vector. We will design a state feedback controllerto drive (9) to arrive at a suitable sliding surface, on which the system state vector slides to zero. The precise meaning ofthe solutions of the resulting closed-loop system of (9) is then defined through a regularization procedure. Therefore, theremainder of this section is divided into two parts.

2.1 Sliding surface designThe state space for the target system (9) (or equivalently the original system (1)) is naturally with norm given by (3).We choose a closed subspace of as the sliding surface for (9)

H𝛿 =⎧⎪⎨⎪⎩W ∈

|||||||1

∫0

W(x) sin(𝜋2

x)

dx = 0N×1

⎫⎪⎬⎪⎭ . (10)

The corresponding sliding mode function is defined by

SW (t) ≜⎡⎢⎢⎣

1

∫0

w1(x, t) sin(𝜋2

x)

dx,

1

∫0

w2(x, t) sin(𝜋2

x)

dx, … ,

1

∫0

wN(x, t) sin(𝜋2

x)

dx⎤⎥⎥⎦⊤

=

1

∫0

W(x, t) sin(𝜋2

x)

dx.

(11)

By differentiating (11) with respect to t and combining (9), we have

SW (t) =

1

∫0

Wt(x, t) sin(𝜋2

x)

dx =

1

∫0

(ΘWxx(x, t) − CW(x, t)) sin(𝜋2

x)

dx

= Θ

1

∫0

Wxx(x, t) sin(𝜋2

x)

dx − CSW (t) (12)

= ΘWx(1, t) − Θ𝜋2

1

∫0

Wx(x, t) cos(𝜋2

x)

dx − CSW (t)

= ΘWx(1, t) −(

C + Θ𝜋2∕4)

SW (t)

= Θ [Uc(t) + D(t)] − ΘK(1, 1)U(1, t) − Θ

1

∫0

Kx(1, 𝑦)U(𝑦, t)d𝑦

−(

C + Θ𝜋2∕4)

SW (t).

The equivalent control value Ueqc (t) is obtained by setting SW (t) = 0 on the sliding surface H𝛿

Ueqc (t) = K(1, 1)U(1, t) +

1

∫0

Kx(1, 𝑦)U(𝑦, t)d𝑦 − D(t). (13)

GU AND WANG 1441

According to the equivalent control method,32 by substituting Ueqc (t) into (9) for Uc(t), we get the following sliding mode

equation: {Wt(x, t) = ΘWxx(x, t) − CW(x, t), x ∈ (0, 1), t > 0,W(0, t) = Wx(1, t) = 0N×1, t ≥ 0.

(14)

Define a linear operator w ∶ D(w)(⊂ H𝛿) → H𝛿 as follows:{wW = ΘW ′′ − CW ,D(w) =

{W ∈

(H2(0, 1)

)N ∩ H𝛿 |W(0) = 0N×1

}.

(15)

It is noted that, for any W ∈ D(w),W ′(1) = 0N×1 if and only if wW ∈ H𝛿 if and only if ∫ 10 W(x, t) sin

(𝜋

2x)

dx = 0N×1.Hence, the sliding mode Equation (14), governing the system motion of (9) on the sliding surface H𝛿 , is rewritten as

⎧⎪⎨⎪⎩Wt(x, t) = ΘWxx(x, t) − CW(x, t), x ∈ (0, 1), t > 0,

W(0, t) = ∫1

0W(x, t) sin

(𝜋

2x)

dx = 0N×1, t ≥ 0.(16)

The validity of the sliding mode Equation (16) for (9) will be confirmed in Section 3.

2.2 Stability analysisWrite Equation (16) as the following evolution equation:

ddt

W(·, t) = wW(·, t), W(·, 0) = W0 ∈ H𝛿. (17)

We now establish the existence and uniqueness, and the stability of (17) in H𝛿 .

Theorem 1. Let w be given by (15). Then, we have the following assertions:

1. −1w exists and is compact on H𝛿 , and consequently, 𝜎(w), the spectrum of w, consists of isolated eigenvalues of

finite algebraic multiplicity only.2. Equation (17) generates a C0-semigroup of contractions ewt on H𝛿 , and the semigroup is exponentially stable in

H𝛿 with decay rate −[𝜃min∕4 + 𝜎m(S[C])].

Proof. Since for any W ∈ D(w), ∫ 10 W(x, t) sin( 𝜋

2x)dx = 0N×1 if and only if W′(1) = 0N× 1. We can rewrite w as{wW = ΘW ′′ − CW ,

D(w) ={

W ∈(

H2(0, 1))N ∩ H𝛿||W(0) = 0N×1, W ′(1) = 0N×1

}.

Now, we will prove that −1w exists and is compact on H𝛿 . For any given F ∈ H𝛿 , solve F = wW to yield

ΘW ′′ − CW = F, ie, W ′′ − Θ−1CW = Θ−1F. (18)

For the matrix Θ−1C, there exists an invertible matrix T such that

J ∶= T−1 (Θ−1C)

T =

[ J1 0⋱

0 Jr

], (19)

where Jk is Jordan block corresponding to eigenvalue 𝜆k, and given by

Jk ∶=

⎡⎢⎢⎢⎢⎣𝜆k 0 · · · 0 01 𝜆k · · · 0 0⋮ ⋮ ⋱ ⋮ ⋮0 0 · · · 𝜆k 00 0 · · · 1 𝜆k

⎤⎥⎥⎥⎥⎦Nk×Nk

, k = 1, 2, … , r, and 1 ≤ Nk ≤ N.

1442 GU AND WANG

Let us define the new variable W = [W1, W2, … , WN]⊤ by W = T−1W . It follows from (18) that

T−1W ′′ − T−1 (Θ−1C)

T · T−1W = T−1Θ−1F, ie, W ′′ − JW = F,

where F = T−1Θ−1F. Define a linear operator w ∶ D(w)(⊂ H𝛿) → H𝛿 as follows:{wW = W ′′ − JW ,D(w) =

{W ∈

(H2(0, 1)

)N ∩ H𝛿||| W(0) = 0N×1, W ′(1) = 0N×1

}with J given by (19). Thus, w can be written as the diagonal form w = diag(Bk), where the element Bk, k =1, 2, … , r, is given by {

BkWk = W ′′k − JkWk, Wk =

[w1, w2, … , wNk

]⊤,

D(Bk) ={

Wk ∈(

H2(0, 1))Nk ∩ H𝛿k

||Wk(0) = 0Nk×1, W ′k(1) = 0Nk×1

}with

H𝛿k =

{Wk ∈

(L2(0, 1)

)Nk||||| ∫

1

0Wk(x) sin

(𝜋2

x)

dx = 0Nk×1

}.

Obviously, −1w exists if and only if −1

w exists if and only if B−1k exists for k = 1, 2, … , r. Therefore, for any Fk =

[𝑓1, 𝑓2, … , 𝑓Nk ]⊤ ∈ H𝛿k , solve Fk = BkWk to yield

⎧⎪⎪⎨⎪⎪⎩

w′′1 − 𝜆kw1 = 𝑓1,

w′′2 − 𝜆kw2 − w1 = 𝑓2,

⋮ ⋮ ⋮

w′′Nk

− 𝜆kwNk − wNk−1 = 𝑓Nk

(20a)

with the boundary conditions

w𝑗(0) = w′𝑗(1) = 0, 𝑗 = 1, 2, … ,Nk. (20b)

If 𝜆k = 0, then we obtain the solution of (20a),(20b)

⎧⎪⎪⎨⎪⎪⎩w1(x) = ∫

x

0

[∫

1

𝜉

𝑓1(𝜏)d𝜏]

d𝜉,

w𝑗(x) = ∫x

0

[∫

1

𝜉

(𝑓1(𝜏) − w𝑗−1(𝜏)

)d𝜏

]d𝜉, 𝑗 = 2, 3, … ,Nk.

If 𝜆k ≠ 0, then recalling (8), 𝜎(Θ−1C) ∩ {(m + 1∕2)2𝜋2, m ∈ ℕ} = ∅, we obtain the solution of (20a),(20b)

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

w1(x) = 𝜌(x) ∫ 10 cos

[√𝜆k(1 − 𝜏)

]𝑓1(𝜏)d𝜏 + 1√

𝜆k∫ x

0 sin[√𝜆k(x − 𝜏)

]𝑓1(𝜏)d𝜏,

w𝑗(x) = 𝜌(x) ∫ 10 cos

[√𝜆k(1 − 𝜏)

] (𝑓𝑗(𝜏) − w𝑗−1(𝜏)

)d𝜏

+ 1√𝜆k∫ x

0 sin[√𝜆k(x − 𝜏)

] (𝑓𝑗(𝜏) − w𝑗−1(𝜏)

)d𝜏,

𝜌(x) =sin

(√𝜆kx

)𝜆k cos

(√𝜆k

) , 𝑗 = 2, 3, … ,Nk,

which implies that B−1k exists, ie, −1

w exists and is compact on H𝛿 by Sobolev embedding theorem.36(p85) Hence, 𝜎(w)consists of isolated eigenvalues of finite algebraic multiplicity only.

GU AND WANG 1443

We are left with the task of confirming the second assertion. Given any W ∈ D(w) and recalling (6), we have

⟨wW , W⟩ =⟨ΘW ′′ − CW , W

⟩

=

1

∫0

W ′′⊤(x)ΘW(x)dx −

1

∫0

W⊤(x)C⊤W(x)dx

=[

W ′⊤(x)ΘW(x)]1

0−

1

∫0

W ′⊤(x)ΘW ′(x)dx −

1

∫0

W⊤(x)C⊤W(x)dx

≤ −14

1

∫0

W⊤(x)ΘW(x)dx −

1

∫0

W⊤(x)C⊤W(x)dx

≤ −[𝜃min

4+ 𝜎m(S[C])

] 1

∫0

W⊤(x)W(x)dx

≤ 0.

Consequently, w + [𝜃min∕4+ 𝜎m(S[C])] IN×N is dissipative and so is for w. By the Lumer-Phillips theorem,37(p14) wgenerates a C0-semigroup of contractions ewt on H𝛿 , and the semigroup is exponentially stable with the decay rate−[𝜃min∕4 + 𝜎m(S[C])], which completes the proof.

By (4), we have

SW (t) =

1

∫0

W(x, t) sin(𝜋2

x)

dx

=

1

∫0

U(x, t) sin(𝜋2

x)

dx −

1

∫0

sin(𝜋2

x) ⎡⎢⎢⎣

x

∫0

K(x, 𝑦)U(𝑦, t)d𝑦⎤⎥⎥⎦ dx.

Consequently, the original system (1) on the surface H𝛿 becomes

⎧⎪⎪⎪⎨⎪⎪⎪⎩

Ut(x, t) = ΘUxx(x, t) + ΛU(x, t), x ∈ (0, 1), t > 0,U(0, t) = 0N×1, t ≥ 0,

1

∫0

U(x, t) sin(𝜋

2x)

dx =

1

∫0

sin(𝜋

2x) ⎡⎢⎢⎣

x

∫0

K(x, 𝑦)U(𝑦, t)d𝑦⎤⎥⎥⎦ dx, t ≥ 0,

(21)

which is exponentially stable according to Theorem 1 and the equivalence between (16) and (21).

2.3 Controller designThe SMC law Uc(t) will be designed to guarantee the “𝜂-reaching condition,”21 which means that the trajectory of system(9) is driven toward H𝛿 and remains on it thereafter. In view of (13), we construct the controller as

Uc(t) = K(1, 1)U(1, t) +

1

∫0

Kx(1, 𝑦)U(𝑦, t)d𝑦 + U0(t), (22)

where U0(t) is an auxiliary control to reject disturbances. Substitute (22) into (12) to get

SW (t) = Θ (U0(t) + D(t)) −(

C + 𝜋2

4Θ)

SW (t). (23)

1444 GU AND WANG

Let us choose

U0(t) = −(

M + 𝜂

𝜃min

)SW (t)||SW (t)||ℝN

,

where the norm || · ||ℝN is defined by (2). Substituting the previous expression of U0(t) into (23), we get the following ODEsfor SW(t):

SW (t) = Θ[−(

M + 𝜂

𝜃min

)SW (t)‖SW (t)‖ℝN

+ D(t)]−

(C + 𝜋

2

4Θ)

SW (t). (24)

Therefore, we have

ddt

12‖SW (t)‖2

ℝN = S⊤W (t)[−(

C + 𝜋2

4Θ)

SW (t) − Θ(

M + 𝜂

𝜃min

)SW (t)‖SW (t)‖ℝN

+ ΘD(t)]

= −S⊤W (t)CSW (t) − 𝜋2

4S⊤W (t)ΘSW (t) −

(M + 𝜂

𝜃min

)S⊤W (t)Θ SW (t)‖SW (t)‖ℝN

+ S⊤W (t)ΘD(t)

≤ −𝜎m(S[C]) ‖SW (t)‖2ℝN − 𝜋

2

4𝜃min ‖SW (t)‖2

ℝN −(

M + 𝜂

𝜃min

)𝜃min‖SW (t)‖ℝN (25)

+ 𝜃max‖SW (t)‖ℝN‖D(t)‖ℝN

≤ 𝜃min

4‖SW (t)‖2

ℝN − 𝜋2

4𝜃min ‖SW (t)‖2

ℝN −(

M + 𝜂

𝜃min

)𝜃min‖SW (t)‖ℝN + 𝜃max‖SW (t)‖ℝN · M 𝜃min

𝜃max

≤ −𝜂‖SW (t)‖ℝN ,

which is just the “𝜂-reaching condition”. Hence, we obtain the following SMC law:

Uc(t) = K(1, 1)U(1, t) +

1

∫0

Kx(1, 𝑦)U(𝑦, t)d𝑦 −(

M + 𝜂

𝜃min

)SW (t)‖SW (t)‖ℝN

. (26)

It is noted that the boundary controller Uc(t) given by (26) is vector valued, and it undergoes discontinuity on H𝛿 andis continuous whenever it is beyond H𝛿 . The precise meaning of the solution of system (9) under (26) can be defined inthe mild sense of Orlov32(p30) as a limiting result obtained through a certain regularization procedure.

Definition 1. A continuous function W 𝛿(·, t) ∈ , defined on some time interval [0,T), is said to be an approximate𝛿-solution of (9) if it is a mild solution of the corresponding boundary value problem with some U𝛿c (t) such that‖‖‖U𝛿c (t) − Uc(t)

‖‖‖(L2(0,T))N≤ 𝛿

for almost all t ∈ [0,T), and W(·, t) ∈ with ||SW (t)||ℝN ≥ 𝛿, where 𝛿 > 0.

Definition 2. A continuous function W(·, t) ∈ , defined on some time interval [0,T), is said to be a generalizedsolution of (9) if there exists a family of approximate mild 𝛿-solution of the system such that

lim𝛿→0

‖‖‖W 𝛿(·, t) − W(·, t)‖‖‖ = 0

for all t ∈ [0,T).

Clearly, when beyond the discontinuity surface H𝛿 , the closed-loop system of (9) under (26) is written as

⎧⎪⎨⎪⎩Wt(x, t) = ΘWxx(x, t) − CW(x, t), x ∈ (0, 1), t > 0,W(0, t) = 0N×1, t ≥ 0,Wx(1, t) = −(M + 𝜂∕𝜃min)SW (t)∕‖SW (t)‖ℝN + D(t) ≜ D(t), t ≥ 0,

(27)

which is well posed in mild sense, whereas in a vicinity of H𝛿 , system (9) under (26) is shown to possess a unique general-ized solution through a limiting procedure that mentioned in Definition 2. Moreover, the generalized solution is governedby the sliding mode Equation (16). The previous results are presented in Section 3.

GU AND WANG 1445

3 WELL-POSEDNESS OF THE CLOSED-LOOP SYSTEM

This section is devoted to the regularization of system (9) under the control (26). We first define a linear operator ∶D()(⊂ ) → as follows:{W = ΘW ′′ − CW ,

D() ={

W ∈(

H2(0, 1))N |||W(0) = 0N×1, W ′(1) = 0N×1

}.

(28)

A straightforward calculation shows that the adjoint operator of is defined by{∗Z = ΘZ′′ − C⊤Z,D(∗) =

{Z ∈

(H2(0, 1)

)N |||Z(0) = 0N×1, Z′(1) = 0N×1

},

(29)

where Z = [z1, z2, z3, … , zN]⊤ ∈ D(∗). Take the inner product on the both sides of (27) with Z ∈ D(∗) to get

ddt

⟨W , Z⟩ =⟨ΘW ′′ − CW , Z

⟩ =

1

∫0

(ΘW ′′(x) − CW(x)

)⊤Z(x)dx

=

1

∫0

W ′′⊤(x)ΘZ(x)dx −

1

∫0

W⊤(x)C⊤Z(x)dx

=

1

∫0

W⊤(x)ΘZ′′(x)dx −

1

∫0

W⊤(x)C⊤Z(x)dx + D⊤(t)ΘZ(1)

=

1

∫0

W⊤(x)(ΘZ′′(x) − C⊤Z(x)

)dx + D⊤(t)ΘZ(1)

= ⟨W , ∗Z⟩ +⟨Θ𝛿(x − 1)D(t), Z

⟩D(∗)′×D(∗),

where D(∗)′ is the dual of D(∗) with the pivot space . Then, (27) can be reformulated as the following evolutionequation:

ddt

W(·, t) = W(·, t) + D(t), W(·, 0) = W0, (30)

where = Θ𝛿(x − 1) and D(t) is given in (27). The adjoint system of (27) is hence written as

⎧⎪⎨⎪⎩Zt(x, t) = ΘZxx(x, t) − C⊤Z(x, t), x ∈ (0, 1), t > 0,Z(0, t) = 0N×1, Zx(1, t) = 0N×1, t ≥ 0,Y (t) = ∗Z(x, t) = ΘZ(1, t).

(31)

In order to investigate the regularization of system (9) under the control (26), we first need to prove the existence anduniqueness of the mild solution to the closed-loop system (30) when (26) is beyond the discontinuity surface H𝛿 . Themain difficulty in dealing with (31) is the general coefficient matrix C appearing in the corresponding coupled term. Thisdoes not permit to easily prove that the control operator is admissible for et. Nevertheless, through a detailed spectralanalysis for the adjoint system (31), the asymptotic behavior of eigenfunctions of ∗ is obtained in Section 3.1. Then,using the Riesz basis approach, we shall show ∗ is admissible for e∗t. Based on the results, the regularization of system(9) under the control (26) is established in Section 3.2.

3.1 Asymptotic behavior of the eigenfunctionsWe consider the eigenvalue problem of ∗ with constant coefficients. For the asymptotic behavior of eigenfunctions of aSturm-Liouville operator, we refer to Orlov,38 where the ISS for single parabolic PDE with variable coefficients has been

1446 GU AND WANG

investigated. Let ∗Z = 𝜆Z. Then, we have ΘZ′′ − C⊤Z = 𝜆Z, that is,{Z′′ −

(Θ−1C⊤ + Θ−1𝜆

)Z = 0,

Z(0) = Z′(1) = 0(32)

with Θ−1C⊤ = [aij]N×N. Before going on, we present the following lemma concerning the eigenvalues of ∗.

Lemma 1. Let ∗ be given by (29). Then, for each 𝜆 ∈ 𝜎(∗), we have Re𝜆 < 0.

Proof. Analysis similar to that in the proof of Theorem 1 shows that ∗ is dissipative, which implies Re𝜆 ≤ 0 for𝜆 ∈ 𝜎(∗). Therefore, we only need to prove that there are no eigenvalues on the imaginary axis. Let 𝜆 = i𝛼 ∈ 𝜎(∗)with 𝛼 ∈ R, and let Z ∈ D(∗) be its associated eigenfunction of ∗. Then, by (29), we have

0 = Re⟨∗Z, Z⟩ = Re⟨i𝛼Z, Z⟩=

1

∫0

(ΘZ′′(x))⊤Z(x)dx −

1

∫0

(C⊤Z(x))⊤Z(x)dx

= −

1

∫0

Z′⊤(x)Θ⊤Z′(x)dx −

1

∫0

Z⊤(x)CZ(x)dx

≤ −[𝜃min

4+ 𝜎m (S[C])

] 1

∫0

Z⊤(x)Z(x)dx

≤ 0.

It follows that Z = 0. Thus, there are no eigenvalues on the imaginary axis. The proof is complete.

We are now in a position to calculate the asymptotic expansion of the eigenfunctions of∗. We apply the matrix operatorpencil method to solve (32). Three steps follow as described below.

Step 1: Writing (32) in the form of first-order equations.Let

z1 ∶= z1, z2 ∶= z′1, z3 ∶= z2, z4 ∶= z′2, … , z2N−1 ∶= zN , z2N ∶= z′N , (33)

and

Φ ∶= [z1, z2, z3, … , z2N]⊤. (34)

Then, (31) is written as {TD(x, 𝜆)Φ ∶= Φ′(x) + M(𝜆)Φ(x) = 0,TR(x, 𝜆)Φ ∶= W 0Φ(0) + W 1Φ(1) = 0,

(35)

where

W 0 ∶=

⎡⎢⎢⎢⎢⎢⎣

1 0 0 0 0 0 · · · 0 00 0 1 0 0 0 · · · 0 00 0 0 0 1 0 · · · 0 0⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮0 0 0 0 0 0 · · · 1 0

ON×(2N)

⎤⎥⎥⎥⎥⎥⎦(2N)×(2N)

, W 1 ∶=

⎡⎢⎢⎢⎢⎢⎣

ON×(2N)0 1 0 0 0 0 · · · 0 00 0 0 1 0 0 · · · 0 00 0 0 0 0 1 · · · 0 0⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮0 0 0 0 0 0 · · · 0 1

⎤⎥⎥⎥⎥⎥⎦(2N)×(2N)

,

and M(𝜆) ∶ = D0 − 𝜆D1 with D0 and D1, respectively, given by

D0 ∶=

⎡⎢⎢⎢⎢⎢⎢⎣

0 −1 0 0 · · · 0 0a11 0 a12 0 · · · a1N 00 0 0 −1 · · · 0 0

a21 0 a22 0 · · · a2N 0⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮0 0 0 0 · · · 0 −1

aN1 0 aN2 0 · · · aNN 0

⎤⎥⎥⎥⎥⎥⎥⎦(2N)×(2N)

,

GU AND WANG 1447

and

D1 ∶=⎡⎢⎢⎢⎣𝜃1D21 O2×2 · · · O2×2O2×2 𝜃2D21 · · · O2×2⋮ ⋮ ⋱ ⋮

O2×2 O2×2 · · · 𝜃N D21

⎤⎥⎥⎥⎦(2N)×(2N)

, where D21 =[

0 01 0

].

With the previous argument, we have that 𝜆 ∈ 𝜎(∗) if and only if (35) has a nontrivial solution.Step 2: Finding the fundamental matrix solution of (35).We first need to diagonalize the leading term 𝜆D1. For this purpose, let us define an invertible matrix in 𝜆 by

P(𝜆) ∶=⎡⎢⎢⎢⎣

p1(𝜆)p2(𝜆)

⋱pN(𝜆)

⎤⎥⎥⎥⎦(2N)×(2N)

with pi(𝜆) ∶=[(𝜃i𝜆)

12 (𝜃i𝜆)

12

𝜃i𝜆 −𝜃i𝜆

], i = 1, 2, … ,N,

where 0 ≠ 𝜆 ∈ ℂ. A direct computation shows that the corresponding inverse matrix is given by

P−1(𝜆) ∶=⎡⎢⎢⎢⎣

p−11 (𝜆)

p−12 (𝜆)

⋱p−1

N (𝜆)

⎤⎥⎥⎥⎦(2N)×(2N)

with p−1i (𝜆) ∶=

[12(𝜃i𝜆)−

12

12(𝜃i𝜆)−1

12(𝜃i𝜆)−

12 − 1

2(𝜃i𝜆)−1

], i = 1, 2, … ,N.

Define Ψ(x) = P−1(𝜆)Φ(x) and TD(𝜆) = P−1(𝜆)TD(𝜆)P(𝜆). Then, we get

TD(𝜆)Ψ = Ψ′(x) −𝑀(𝜆)Ψ(x) = 0, (36)

where

𝑀(𝜆) ∶= −P−1(𝜆)M(𝜆)P(𝜆) = 𝜆12𝑀 1

2+ 𝜆−

12𝑀− 1

2(37)

with

𝑀 12∶= diag

[−𝜃

121 , 𝜃

121 ,−𝜃

122 , 𝜃

122 , … ,−𝜃

12

N , 𝜃12

N

], (38)

and

𝑀− 12∶=

⎡⎢⎢⎢⎣M1,1 M1,2 · · · M1,NM2,1 M2,2 · · · M2,N⋮ ⋮ ⋱ ⋮

MN,1 MN,2 · · · MN,N

⎤⎥⎥⎥⎦(2N)×(2N)

, Mi,𝑗 ∶=⎡⎢⎢⎣

ai,𝑗

2𝜃− 1

2i

ai,𝑗

2𝜃− 1

2i

− ai,𝑗

2𝜃− 1

2i − ai,𝑗

2𝜃− 1

2i

⎤⎥⎥⎦ , i, 𝑗 = 1, … ,N. (39)

Next, we present the asymptotic expression for the fundamental matrix solution of (36).

Theorem 2. Let 0 ≠ 𝜆 ∈ ℂ, and𝑀(𝜆) given by (37). For x ∈ [0, 1], set

E(x, 𝜆) ∶= diag[

e−√𝜃1𝜆·x, e

√𝜃1𝜆·x, e−

√𝜃2𝜆·x, e

√𝜃2𝜆·x, … , e−

√𝜃N𝜆·x, e

√𝜃N𝜆·x

]⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

2N

. (40)

Then, there exists a fundamental matrix solution Ψ(x, 𝜆) to system (36) such that, for large enough |𝜆|,Ψ(x, 𝜆) =

(Ψ0(x) + 𝜆−

12 Ψ1(x) +

Θ(x, 𝜆)𝜆

)E(x, 𝜆), (41)

1448 GU AND WANG

where

Ψ0(x) ∶= I2N , Ψ1(x) ∶= diag[

a1,1

2𝜃− 1

21 x,−

a1,1

2𝜃− 1

21 x,

a2,2

2𝜃− 1

22 x,−

a2,2

2𝜃− 1

22 x, … ,

aN,N

2𝜃− 1

2N x,−

aN,N

2𝜃− 1

2N x

]⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

2N

, (42)

with all entries uniformly bounded in [0, 1], and

Θ(x, 𝜆) ∶= Ψ2(x) + 𝜆−12 Ψ3(x) + · · · .

The proof of Theorem 2 is given in Appendix A.1. The following corollary is a direct consequence of the previoustheorem.

Corollary 1. Let 0 ≠ 𝜆 ∈ ℂ, and let Ψ(x, 𝜆) given by (41) be a fundamental matrix solution of (36). Then,

Φ(x, 𝜆) ∶= P(𝜆)Ψ(x, 𝜆) (43)

is a fundamental matrix solution of the first-order linear system (35).

Step 3: Asymptotic behavior of eigenfunctions of ∗.We introduce the notation [b]s ∶= b + (𝜆− s

2 ). It follows from (35) that the eigenvalues of ∗ are given by the zeros ofthe following characteristic determinant:

Δ(𝜆) ∶= det(

TR(𝜆)Φ), 𝜆 ∈ ℂ,

where TR(𝜆) is given by (35), and Φ is a fundamental matrix solution of TD(x, 𝜆)Φ = 0. From (35) and (43), we have

TR(𝜆)Φ = W 0P(𝜆)Ψ(0, 𝜆) + W 1P(𝜆)Ψ(1, 𝜆), (44)

where W 0P(𝜆) and W 1P(𝜆) are given, respectively, by

W 0P(𝜆) =

⎡⎢⎢⎢⎢⎢⎢⎣

(𝜃1𝜆)12 (𝜃1𝜆)

12 0 0 · · · 0 0

0 0 (𝜃2𝜆)12 (𝜃2𝜆)

12 · · · 0 0

0 0 0 0 · · · 0 0⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮0 0 0 0 · · · (𝜃N𝜆)

12 (𝜃N𝜆)

12

ON×(2N)

⎤⎥⎥⎥⎥⎥⎥⎦(2N)×(2N)

,

and

W 1P(𝜆) =

⎡⎢⎢⎢⎢⎣ON×(2N)

𝜃1𝜆 −𝜃1𝜆 0 0 · · · 0 00 0 𝜃2𝜆 −𝜃2𝜆 · · · 0 0⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮0 0 0 0 · · · 𝜃N𝜆 −𝜃N𝜆

⎤⎥⎥⎥⎥⎦(2N)×(2N)

.

Using E(0, 𝜆) = I2N and Ψ0(x) = I2N , we immediately find

W 0P(𝜆)Ψ(0, 𝜆) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

(𝜆)12

[𝜃

121

]2(𝜆)

12

[𝜃

121

]2

0 0 · · · 0 0

0 0 (𝜆)12

[𝜃

122

]2(𝜆)

12

[𝜃

122

]2· · · 0 0

⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮

0 0 0 0 · · · (𝜆)12

[𝜃

12

N

]2(𝜆)

12

[𝜃

12

N

]2

ON×(2N)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(2N)×(2N)

,

GU AND WANG 1449

and

W 1P(𝜆)Ψ(1, 𝜆) =

⎡⎢⎢⎢⎢⎢⎣

ON×(2N)

𝜆E1[Θ1]2 −𝜆E2[Θ2]2 0 0 · · · 0 00 0 𝜆E3[Θ3]2 −𝜆E4[Θ4]2 · · · 0 0⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮0 0 0 0 · · · 𝜆E2N−1[Θ2N−1]2 −𝜆E2N[Θ2N]2

⎤⎥⎥⎥⎥⎥⎦(2N)×(2N)

,

with

Θ2i−1 = 𝜃i +ai,i

2𝜃

12

i 𝜆− 1

2 , Θ2i = 𝜃i −ai,i

2𝜃

12

i 𝜆− 1

2 , where i = 1, 2, … ,N,

and

E1 ∶= e−√𝜃1𝜆,E2 ∶= e

√𝜃1𝜆,E3 ∶= e−

√𝜃2𝜆,E4 ∶= e

√𝜃2𝜆, … ,E2N−1 ∶= e−

√𝜃N𝜆,E2N ∶= e

√𝜃N𝜆.

Therefore, we get

TR(𝜆)Φ = W 0P(𝜆)Ψ(0, 𝜆) + W 1P(𝜆)Ψ(1, 𝜆)

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

𝜆12

[𝜃

121

]2𝜆

12

[𝜃

121

]2

0 0 · · · 0 0

0 0 𝜆12

[𝜃

122

]2𝜆

12

[𝜃

122

]2

· · · 0 0

⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮

0 0 0 0 · · · 𝜆12

[𝜃

12

N

]2

𝜆12

[𝜃

12

N

]2

𝜆E1[Θ1]2 −𝜆E2[Θ2]2 0 0 · · · 0 00 0 𝜆E1[Θ3]2 −𝜆E2[Θ4]2 · · · 0 0⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮0 0 0 0 · · · 𝜆E1[Θ2N−1]2 −𝜆E2[Θ2N]2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(2N)×(2N)

.

Consequently,

Δ(𝜆) ∶= det(

TR(𝜆)Φ)

=(−1)𝜇𝜆3N2

( N∏k=1𝜃

32

k

)det

⎡⎢⎢⎣[1]2 [1]2

E1

[1 + c1,1

2𝜃− 1

21 𝜆

− 12

]2−E2

[1 − c1,1

2𝜃− 1

21 𝜆

− 12

]2

⎤⎥⎥⎦× det

⎡⎢⎢⎣[1]2 [1]2

E3

[1 + c2,2

2𝜃− 1

22 𝜆

− 12

]2−E4

[1 − c2,2

2𝜃− 1

22 𝜆

− 12

]2

⎤⎥⎥⎦× · · · × det

⎡⎢⎢⎣[1]2 [1]2

E2N−1

[1 + cN,N

2𝜃− 1

2N 𝜆

− 12

]2−E2N

[1 − cN,N

2𝜃− 1

2N 𝜆

− 12

]2

⎤⎥⎥⎦=(−1)𝜇𝜆

3N2

( N∏k=1𝜃

32

k

)·

( N∏𝑗=1

Δ𝑗(𝜆)

),

where

Δ𝑗(𝜆) = det⎡⎢⎢⎣

[1]2 [1]2

E2𝑗−1

[1 + 1

2a𝑗,𝑗𝜃

− 12𝑗 𝜆

− 12

]2−E2𝑗

[1 − 1

2a𝑗,𝑗𝜃

− 12𝑗 𝜆

− 12

]2

⎤⎥⎥⎦= −E2𝑗 − E2𝑗−1 +

12

a𝑗,𝑗𝜃− 1

2𝑗 𝜆

− 12(

E2𝑗 − E2𝑗−1)+ (𝜆−1), 𝑗 = 1, 2, … ,N.

(45)

The estimation of the asymptotic eigenvalues of ∗ is given as follows with proofs in Appendix A.2.

1450 GU AND WANG

Lemma 2. Let Δ(𝜆) be the characteristic determinant of (35). Then, the following assertions hold:

1. The asymptotic expansion for Δ(𝜆) holds

Δ(𝜆) = (−1)𝜇𝜆3N2

( N∏k=1𝜃

32

k

)·

( N∏𝑗=1

Δ𝑗(𝜆)

)(46)

with Δj(𝜆) being given by (45).2. The eigenvalues 𝜆jm have the following asymptotic expansion:

𝜆𝑗m = −𝜃−1𝑗 (m − 1∕2)2𝜋2 − 𝜃−1

𝑗 a𝑗,𝑗 + (m−1) as m → ∞, (47)

where m ∈ ℕ and j = 1, 2, … ,N.3. The zeros of Δ(𝜆) are simple when their moduli are sufficiently large.

Remark 1. Since 𝜃𝑗1 ≠ 𝜃𝑗2 with j1 ≠ j2, and

|||−𝜃−1𝑗 a𝑗,𝑗 + (m−1)|||≪ |||𝜃−1

𝑗 (m − 1∕2)2𝜋2||| , for m > N1, 𝑗 = 1, 2, … ,N,

where N1 is a sufficiently large positive integer, we have

−𝜃−1𝑗1(m − 1∕2)2𝜋2 − 𝜃−1

𝑗1a𝑗1,𝑗1 + (m−1) ≠ −𝜃−1

𝑗2(k − 1∕2)2𝜋2 − 𝜃−1

𝑗2a𝑗2,𝑗2 + (k−1)

with j1 ≠ j2 and m, k > N1. The previous inequality implies that the spectrum 𝜎(∗) consists of N branches

𝜎(∗) =N⋃𝑗=1

{𝜆𝑗m, m ∈ ℕ

}.

Remark 2. For the case of system (1) with the same diffusion 𝜃, we can choose suitable matrix C in the target system(9) such that the spectrum 𝜎(∗) consists of N branches

𝜎(∗) =N⋃𝑗=1

{𝜆𝑗m, m ∈ ℕ

}.

In fact, the matrix C is chosen to satisfy 0 < cj, j < 2𝜃𝜋2 and c𝑗1,𝑗1 ≠ c𝑗2,𝑗2 for j1 ≠ j2, and due to

||(m−1)||≪ |||𝜃−1(m − 1∕2)2𝜋2 + 𝜃−1a𝑗,𝑗||| , for m > N2, 𝑗 = 1, 2, … ,N,

where N2 is a sufficiently large positive integer and aj, j = 𝜃−1cj, j, we have

−𝜃−1(m − 1∕2)2𝜋2 − 𝜃−1a𝑗1,𝑗1 + (m−1) ≠ −𝜃−1(k − 1∕2)2𝜋2 − 𝜃−1a𝑗2,𝑗2 + (k−1)

with j1 ≠ j2 and m, k > N2.

The following theorem, whose proof is given in Appendix A.3, provides an asymptotic expansion for the eigenfunctionsof ∗.

Theorem 3. Let 𝜎(∗) = {𝜆𝑗m, 𝑗 = 1, 2, … ,N, m ∈ ℕ} be the eigenvalues of ∗ with 𝜆jm being given by (47). Then,the corresponding eigenfunctions{[

z1,𝑗m, z2,𝑗m, … , zN,𝑗m]⊤, 𝑗 = 1, 2, … ,N, m ∈ ℕ

}have the following asymptotic expression for m → ∞, m ∈ ℕ:

zk,𝑗m =

{sinh

[(m − 1∕2)𝜋ix + (m−1)

]+ (m−1), if k = 𝑗,

(m−1), if k ≠ 𝑗 with 1 ≤ 𝑗 ≤ N, 1 ≤ k ≤ N.(48)

GU AND WANG 1451

Moreover, {[z1,𝑗m, z2,𝑗m, … , zN,𝑗m

]⊤, 𝑗 = 1, 2, … ,N, m ∈ ℕ

}are approximately normalized in in the sense that there exist positive constants b1 and b2 independent of m such that(k, j = 1, 2, … ,N)

b1 ≤ ‖‖zk,𝑗m‖‖L2 ≤ b2 (49)

for all sufficient large positive integers m.

3.2 Regularization for the closed-loop system (27)In this section, we first establish a lemma on the ODEs for sliding mode functions. It follows from (25) that ||SW (t)||ℝN

is a decreasing function. Hence, we can suppose that ||SW (t)||ℝN ≤ M, where M is a positive constant. For any givent0 ∈ [0, tmax), let

Ct0 =[√

Υ2 + 16𝜃max(M + 𝜂∕𝜃min)Ξ∕‖SW (t0)‖ℝN − Υ]· (4Ξ)−1,

and

△T ={Ξ + 2𝜃max(M + 𝜂∕𝜃min)∕

[Ct0‖SW (t0)‖ℝN

]}−1,

where

Υ = 4𝜃max(M + 𝜂∕𝜃min)∕||SW (t0)||ℝN −(𝜃max𝜋

2∕4 + C), Ξ =

(𝜃max𝜋

2∕4 + C)(1 + M∕||SW (t0)||ℝN ).

Now, we define a closed subspace of C[t0, t0 + △T]

Ω ={

SW (t) ∈ C[t0, t0 +△T

] |‖SW (t0)‖ℝN ≥ Ct0‖SW (t0)‖ℝN}.

According to (24), we obtain the following ODEs for the sliding mode function SW(t):

SW (t) = −(

C + 𝜋2

4Θ)

SW (t) − Θ(

M + 𝜂

𝜃min

)SW (t)‖SW (t)‖ℝN

+ ΘD(t), ∀t ∈ [0,T]a.e. (50)

By using the same arguments in the proof of Lemma 3.1 in the work of Gu and Wang,31 we have the following lemmadirectly.

Lemma 3. Let SW(t) be given by (11). Then, for any SW(t0) ≠ 0N× 1, the Equation (50) admits a unique solutionSW(t) ∈ Ω.

Lemma 4. Let us define

Ψ𝑗m ∶= [z1,𝑗m, z2,𝑗m, … , zN,𝑗m]⊤, 𝑗 = 1, 2, … ,N, m ∈ ℕ, (51)

where the entries are given by (48) corresponding to the eigenvalues 𝜆jm. Then, {Ψ1m,Ψ2m, … ,ΨNm,m ∈ ℕ}, thegeneralized eigenfunctions of ∗ forms a Riesz basis in Hilbert space .

1452 GU AND WANG

Proof. Set ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Φ1m ∶=⎡⎢⎢⎢⎣sinh

[(m − 1∕2)𝜋ix

]⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

1st

, 0, 0, … , 0, … , 0⎤⎥⎥⎥⎦⊤

,

Φ2m ∶=⎡⎢⎢⎢⎣0, sinh

[(m − 1∕2)𝜋ix

]⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

2th

, 0, … , 0, … , 0⎤⎥⎥⎥⎦⊤

,

⋮ ⋮

Φ𝑗m ∶=⎡⎢⎢⎢⎣0, … , 0, sinh

[(m − 1∕2)𝜋ix

]⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

𝑗th

, 0, … , 0⎤⎥⎥⎥⎦⊤

,

⋮ ⋮

ΦNm ∶=⎡⎢⎢⎢⎣0, 0, … , 0, … , 0, sinh

[(m − 1∕2)𝜋ix

]⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

Nth

⎤⎥⎥⎥⎦⊤

.

The sequence {Φ1m,Φ2m, … ,ΦNm, m ∈ ℕ} is chosen as a reference Riesz basis for . According to the asymptoticexpressions of eigenvalues (47) and eigenfunctions (48), we have‖‖‖z𝑗,𝑗m − sinh

[(m − 1∕2)𝜋ix

]‖‖‖L2= ‖‖‖2 cosh

[(m − 1∕2)𝜋ix + (m−1)

]· sinh

[(m−1)]+ (m−1)‖‖‖L2

= (m−1) with m > N1.

Therefore, we obtainN∑𝑗=1

∞∑m>N1

‖‖Ψ𝑗m − Φ𝑗m‖‖2 =

N∑𝑗=1

∞∑m>N1

(m−2) < ∞,

where Ψjm is given by (51). By using Bari's theorem, {Ψ1m,Ψ2m, … ,ΨNm, m ∈ ℕ} forms a Riesz basis for .

Under the previous preparation, we can now prove the well-posedness of the closed-loop system (30) when (26) isbeyond the discontinuity surface H𝛿 .

Theorem 4. For system (30), we have the following assertions:

(1). generates a C0-semigroup et on .(2). is an admissible control operator for the C0-semigroup et on .(3). For any W(·, 0) ∈ , Sw(0) ≠ 0, system (30) admits a unique mild solution W(·, t) ∈ C([0, tmax);).(4). For any t ∈ [0, tmax), the solution of (30) can be written as

W(·, t) = etW(·, 0) +

t

∫0

e(t−s)D(S)ds.

Proof. Lemma 4 implies that ∗ generates C0-semigroup e∗t in , and so does for (see Corollary 10.6 in the workof Pazy37(p41)). We are now in a position to prove assertion (2). For any Z(·, 0) = [z1(·, 0), z2(·, 0), … , zN(·, 0)]⊤ ∈ , weassume that

Z(x, 0) =N∑𝑗=1

N1∑m=1

km∑s=1

a𝑗m,sΨ𝑗m,s(x) +N∑𝑗=1

∞∑m=N1+1

a𝑗mΨ𝑗m(x),

and

‖Z(·, 0)‖2 ≍N∑𝑗=1

N1∑m=1

km∑s=1

||a𝑗m,s||2 +N∑𝑗=1

∞∑m=N1+1

||a𝑗m||2.

GU AND WANG 1453

Then, we obtain the solution of (31)

Z(x, t) =N∑𝑗=1

N1∑m=1

e𝜆𝑗mtkm∑s=1

a𝑗m,ss−1∑l=0

tl

l!Ψ𝑗m,s−l(x) +

N∑𝑗=1

∞∑m=N1+1

e𝜆𝑗mta𝑗mΨ𝑗m(x).

Consequently,

Y (t) = ΘZ(1, t)

= ΘN∑𝑗=1

N1∑m=1

e𝜆𝑗mtkm∑s=1

a𝑗m,ss−1∑l=0

tl

l!Ψ𝑗m,s−l(1) + Θ

N∑𝑗=1

∞∑m=N1+1

e𝜆𝑗mta𝑗mΨ𝑗m(1).

Through a straightforward computation, there exists T > 0, such thatT

∫0

|Y (t)|2dt ≤ 2

T

∫0

||||||ΘN∑𝑗=1

N1∑m=1

e𝜆𝑗mtkm∑s=1

a𝑗m,ss−1∑l=0

tl

l!Ψ𝑗m,s−l(1)

||||||2

dt

+ 2

T

∫0

||||||ΘN∑𝑗=1

∞∑m=N1+1

e𝜆𝑗mta𝑗mΨ𝑗m(1)||||||2

dt

≤ 2N𝜃maxC22CT max

1≤m≤N1{km}

N∑𝑗=1

N1∑m=1

km∑s=1

||a𝑗m,s||2

+ 2N𝜃maxC22DT

N∑𝑗=1

∞∑m=N1+1

||a𝑗m||2

≤ ET

( N∑𝑗=1

N1∑m=1

km∑s=1

||a𝑗m,s||2 +N∑𝑗=1

∞∑m=N1+1

||a𝑗m||2

)= ET ‖Z(·, 0)‖2 ,

for some positive constants CT,DT that depend on T, and

ET = 2N𝜃maxC22 · max

{CT · max

1≤m≤N1{km} ,DT

}.

Therefore, the observation operator ∗ of (31) is admissible, and so is for et by Proposition 4.4.1 “dualityprinciple.”39(p126) Assertions (3) and (4) are direct consequences of (1) and (2), and the proof is complete.

Finally, on the sliding time interval [tmax, 𝜏) with 𝜏 > 0, we show the existence and uniqueness of the generalizedsolution for system (9) under (26). Inspired by theorem 2.4 in the work of Orlov,32(p30) we have the following results forthe case of boundary controlled coupled system.

Theorem 5. For any initial data W(·, tmax) ∈ , S(tmax) = 0N× 1. Then, system (9) under (26) possesses a uniquegeneralized solution W(·, t), and on the time interval [tmax, 𝜏), the solution is governed by the sliding mode Equation (16)with the same initial data.

Proof. In view of Definition 2, we can assume that ||SW (t)||ℝN ≤ 𝛿 on the stage [tmax, 𝜏), and 𝛿 > 0 and the gainK ≠ Θ−1, then the generalized solution W(·, t) of system (9) under (26) is approximated by 𝛿-solutions W𝛿(·, t) under areal control vector U𝛿c (t) = Ueq

c (t) +SW (t), which incorporates imperfections by the value SW (t).32 The 𝛿-solutionsW𝛿(·, t) are governed by the following real sliding mode equation:⎧⎪⎨⎪⎩

W 𝛿t (x, t) = ΘW 𝛿

xx(x, t) − CW 𝛿(x, t), x ∈ (0, 1), t > 0,W 𝛿(0, t) = 0N×1, t ≥ 0,W 𝛿

x (1, t) = SW𝛿 (t), t ≥ 0,(52)

where SW𝛿 (t) is given bySW𝛿 (t) = ΘW 𝛿

x (1, t) −(

C + Θ𝜋2∕4)

SW𝛿 (t).

1454 GU AND WANG

Define the error variables

W(x, t) = W 𝛿(x, t) − W(x, t),

where W(x, t) is governed by (14) (or (16)). Then, the error system is obtained by subtracting (52) from (14)

⎧⎪⎨⎪⎩Wt(x, t) = ΘWxx(x, t) − CW(x, t), x ∈ (0, 1), t > 0,W(0, t) = 0N×1, t ≥ 0,Wx(1, t) = ΘW 𝛿

x (1, t) − (C + Θ𝜋2∕4

)SW𝛿 (t), t ≥ 0.

(53)

Since Wx(1, t) = W 𝛿x (1, t) − Wx(1, t) = W 𝛿

x (1, t), system (53) is rewritten as{Wt(x, t) = ΘWxx(x, t) − CW(x, t), x ∈ (0, 1), t > 0,W(0, t) = 0N×1, Wx(1, t) = 1SW𝛿 (t), t ≥ 0,

(54)

where 1 = (Θ − E)−1(C + Θ𝜋2∕4) and ||SW𝛿 (t)||ℝN ≤ 𝛿.We proceed to show that lim𝛿→0||W(·, t)|| = 0 holds for any t ∈ [tmax, 𝜏). Due to Theorem 4, the mild solution of

(54) is given by

W(·, t) = etW(·, tmax) +

t

∫tmax

e(t−s)1SW𝛿 (s)ds, t ∈ [tmax, 𝜏),

where is defined by (28). Recalling W(·, tmax) = 0N×1 and by Proposition 4.2.2 in the work of Tucsnak andWeiss,39(p116) we have

‖‖W(·, t)‖‖ =‖‖‖‖‖‖‖

t

∫tmax

e(t−s)1SW𝛿 (s)ds‖‖‖‖‖‖‖ = M‖SW𝛿 (t0)‖ℝN ≤ M𝛿,

where M is positive constant independent on 𝛿. It follows that lim𝛿→0||W(·, t)|| = 0.

The following is a direct consequence of the previous theorems.

Corollary 2. Let SU(t) be the sliding mode function defined by

SU(t) =

1

∫0

U(x, t) sin(𝜋2

x)

dx −

1

∫0

x

∫0

K(x, 𝑦)U(𝑦, t) sin(𝜋2

x)

d𝑦dx.

Then, for every U(·, 0) ∈ (L2(0, 1))N, and SU(t) ≠ 0N× 1, there exists tmax > 0 such that the closed-loop system of (9)under the feedback control (26) is

⎧⎪⎪⎨⎪⎪⎩

Ut(x, t) = ΘUxx(x, t) + ΛU(x, t), x ∈ (0, 1), t > 0,U(0, t) = 0N×1, t ≥ 0,

Ux(1, t) = K(1, 1)U(1, t) +

1

∫0

Kx(1, 𝑦)U(𝑦, t)d𝑦 −(

M + 𝜂

𝜃min

)SW (t)‖SW (t)‖ℝN

+ D(t), t ≥ 0,

which admits a unique solution U ∈ C ([0, tmax);) and SU(t) = 0N× 1 for all t ≥ tmax. On the sliding mode surfaceSU(t) = 0N× 1, system (9) becomes

⎧⎪⎪⎨⎪⎪⎩

Ut(x, t) = ΘUxx(x, t) + ΛU(x, t), x ∈ (0, 1), t > 0,U(0, t) = 0N×1, t ≥ 0,

1

∫0

U(x, t) sin(𝜋

2x)

dx =

1

∫0

x

∫0

K(x, 𝑦)U(𝑦, t) sin(𝜋

2x)

d𝑦dx,

which is equivalent to (16), and hence is exponentially stable in with the decay rate −[𝜃min∕4 + 𝜎m(S[C])].

GU AND WANG 1455

FIGURE 1 The closed-loop response of “u1(x, t)” (left) and “u2(x, t)” (right) [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 2 The closed-loop response of “u3(x, t)” (left) and sliding mode functions (right) [Colour figure can be viewed atwileyonlinelibrary.com]

4 SIMULATION RESULTS

To demonstrate the effectiveness of the proposed SMC law, we present here some numerical simulations for system (1).The finite difference method is adopted in discretization.

We take the steps of space and time as 0.1 and 0.001, respectively, and choose the following parameter values for threecoupled reaction-diffusion parabolic PDEs:

Θ =

[ 1 0 00 2 00 0 3

], Λ =

[ 2 −3 −16 4 8−2 −5 6

].

The initial values are set as u1(x, 0) = sin(2𝜋x)−2x,u2(x, 0) = x2+1∕5x, and u3(x, 0) = 2x 2 − sin(𝜋x). Moreover, the distur-bance is given by D(t) = [d1(t), d2(t), d3(t)], where d1(t) = sin t, d2(t) = sin(2t), and d3(t) = cos(2t). The smallest real partof eigenvalues of Λ is positive, and the original system in the open loop is unstable. By the backstepping transformation(4), system (1) is converted into (9) with the parameter values

C =

[ 1 3 1−6 2 −82 5 3

].

A direct calculation shows that 𝜎m (S[C]) = −0.1458 > −𝜃min∕4 and 𝜎(Θ−1C

)∩ {(m + 1∕2)2𝜋2, m ∈ ℕ} = ∅ hold.

Figure 1 and the left side of Figure 2 show the closed-loop results. Under the proposed SMC law (26) with M = 1 and𝜂 = 3∕2, the system states [u1(x, t),u2(x, t),u3(x, t)] are stabilized to zero when they reach the sliding mode surface (10)after some time, respectively. The finite-time convergence of the sliding mode function to zero is observed in the rightside of Figure 2.

5 CONCLUDING REMARKS

In this paper, the systematic SMC design has been proposed for N-coupled reaction-diffusion parabolic PDEs with bound-ary input disturbances. The backstepping transformation is applied to convert the coupled PDEs into the target system.

1456 GU AND WANG

The SMC law is then designed to guarantee the existence of the stable sliding motion and therefore reject the disturbances.The well-posedness of the resulting closed-loop system has been obtained based on the Riesz basis method, and the finitetime “reaching condition” has been also verified.

ACKNOWLEDGEMENT

This research was supported by the National Natural Science Foundation of China, Grant/Award Number: 61673061.

ORCID

Jun-Min Wang https://orcid.org/0000-0002-7482-9386

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How to cite this article: Gu J-J, Wang J-M. Sliding mode control for N-coupled reaction-diffusion PDEs withboundary input disturbances. Int J Robust Nonlinear Control. 2019;29:1437–1461. https://doi.org/10.1002/rnc.4448

APPENDIX

This section provides the proofs of Theorems 2 and 3 and Lemma 2 and therefore falls naturally into three parts.

A.1 Proof of Theorem 2A direct computation yields

E′(x, 𝜆) = 𝜆12𝑀 1

2E(x, 𝜆),

where 𝑀 12

and E(x, 𝜆) are given by (38) and (40), respectively. We are now in a position to find a fundamental matrixsolution of (36) in the form of

Ψ(x, 𝜆) =

(+∞∑i=0𝜆−

i2 Ψi(x)

)E(x, 𝜆).

For this purpose, we compute the both sides of (36) and obtain

Ψ′(x, 𝜆) =

(+∞∑i=0𝜆−

i2 Ψ′

i(x)

)E(x, 𝜆) + 𝜆

12

(+∞∑i=0𝜆−

i2 Ψi(x)

)𝑀 1

2E(x, 𝜆),

and

𝑀(𝜆)Ψ(x, 𝜆) =(𝜆

12𝑀 1

2+ 𝜆−

12𝑀− 1

2

)(+∞∑i=0𝜆−

i2 Ψi(x)

)E(x, 𝜆).

1458 GU AND WANG

According to the coefficients of 𝜆12 , 𝜆0, 𝜆−

12 , … , 𝜆−

n2 , … , we have

𝜆12 ∶ Ψ0(x)𝑀 1

2−𝑀 1

2Ψ0(x) = 0,

𝜆0 ∶ Ψ′0(x) + Ψ1(x)𝑀 1

2−𝑀 1

2Ψ1(x) = 0,

𝜆−12 ∶ Ψ′

1(x) −𝑀− 12Ψ0(x) + Ψ2(x)𝑀 1

2−𝑀 1

2Ψ2(x) = 0,

⋮ ⋮

𝜆−n2 ∶ Ψ′

N(x) −𝑀− 12Ψn−1(x) + Ψn+1(x)𝑀 1

2−𝑀 1

2Ψn+1(x) = 0,

⋮ ⋮

where𝑀− 12

is given by (39). By following similar arguments as in Wang et al,34 we conclude that there is an asymptotic

fundamental matrix solution Ψ(x, 𝜆) for (36). It remains to prove that the leading order term Ψ0(x) is given by (42). Indeed,if we solve Ψ0(x) from the following matrix equations:

Ψ0(x)𝑀 12−𝑀 1

2Ψ0(x) = 0,

andΨ′

0(x) + Ψ1(x)𝑀 12−𝑀 1

2Ψ1(x) = 0,

then the leading order term Ψ1(x) of Θ(x, 𝜆) can be determined from

𝜆−12 ∶ Ψ′

1(x) −𝑀− 12Ψ0(x) + Ψ2(x)𝑀 1

2−𝑀 1

2Ψ2(x) = 0.

Similarly, we obtain all the terms Ψ1, Ψ2, … , ΨN , … of Θ(x, 𝜆). Therefore, the proof will be accomplished if we wouldfind the leading order term Ψ0.

Let us denote by di, j(x), ei, j(x) the (i, j)-entry of the matrix Ψ0(x) and Ψ0(x), respectively, i, j = 1, 2, … , 2N. Since𝑀 12

isdiagonal, it follows that the entries di, j(x) and ei, j(x) satisfy{

di,𝑗(x) = 0, 1 ≤ i, 𝑗 ≤ 2N, i ≠ 𝑗,d′

i,i(x) = 0, i = 1, 2, … , 2N.

and ⎧⎪⎪⎨⎪⎪⎩ei,𝑗(x) = 0, 1 ≤ i, 𝑗 ≤ 2N, i ≠ 𝑗,e′(2i−1),(2i−1)(x) =

ci,i

2𝜃− 1

2i , i = 1, 2, … ,N.

e′(2i),(2i)(x) = − ci,i

2𝜃− 1

2i , i = 1, 2, … ,N.

Then, (41) follows from Ψ0(x) = I. The proof is complete.□

A.2 Proof of Lemma 2According to Δ(𝜆) = 0 and (46), we have

N∏𝑗=1

Δ𝑗(𝜆) = 0,

andΔ𝑗(𝜆) = 0, for 𝑗 = 1, 2, … ,N.

Set Δj(𝜆) = 0, j = 1, 2, … ,N. Then, we obtain

− e√𝜃𝑗𝜆 − e−

√𝜃𝑗𝜆 +

a𝑗,𝑗2𝜃− 1

2𝑗 𝜆

− 12

(e√𝜃𝑗𝜆 − e−

√𝜃𝑗𝜆

)+ (𝜆−1) = 0, (A1)

which is also rewritten as− e

√𝜃𝑗𝜆 − e−

√𝜃𝑗𝜆 + (𝜆− 1

2 ) = 0. (A2)

GU AND WANG 1459

Note that −e√𝜃𝑗𝜆 − e−

√𝜃𝑗𝜆 = 0 has solutions ��

12𝑗,m = 𝜃

− 12𝑗 (m − 1∕2)𝜋i for m ∈ ℕ. Apply the Rouché's theorem to (A1) to get

the solutions of (A2) as follows:

𝜆12𝑗m = ��

12𝑗m + 𝛼m = −𝜃−

12

(m − 1

2

)𝜋i + 𝛼m for 𝑗 = 1, 2, … ,N, and m = N2,N2 + 1, · · ·

with N2 large enough. Substituting 𝜆12𝑗m into (A2), and using the fact that −e

√𝜃��𝑗m = e−

√𝜃��𝑗m , we obtain

e√𝜃𝑗 ·𝛼m − e−

√𝜃𝑗 ·𝛼m −

a𝑗,𝑗2𝜃− 1

2𝑗 𝜆

− 12𝑗,m

(e√𝜃𝑗 ·𝛼m + e−

√𝜃𝑗 ·𝛼m

)+ (

𝜆−1𝑗m

)= 0.

Expanding the exponential function according to their Taylor series, we get

𝛼m = 12𝜃− 1

2𝑗 ·

a𝑗,𝑗(m − 1∕2)𝜋i

+ (m−2).

Hence,

𝜆12𝑗m = −𝜃

− 12𝑗

(m − 1

2

)𝜋i + 1

2𝜃− 1

2𝑗 ·

c𝑗,𝑗(m − 1∕2)𝜋i

+ (m−2).

Consequently, we obtain

𝜆𝑗m = −𝜃−1𝑗

(m − 1

2

)2𝜋2 − 𝜃−1

𝑗 c𝑗,𝑗 + (m−1) for 𝑗 = 1, 2, … ,N, and m = N2,N2 + 1, · · ·

with N2 large enough. This finishes the proof.□

A.3 Proof of Theorem 3Since

Φ(x, 𝜆) = P(𝜆)Ψ(x, 𝜆) =⎡⎢⎢⎢⎣Φ1,1(x, 𝜆) O2×2 · · · O2×2

O2×2 Φ2,2(x, 𝜆) · · · O2×2⋮ ⋮ ⋱ ⋮

O2×2 O2×2 · · · ΦN,N(x, 𝜆)

⎤⎥⎥⎥⎦(2N)×(2N)

,

where

Φk,k(x, 𝜆) ∶=

⎡⎢⎢⎢⎢⎣(𝜃k𝜆)

12 e−

√𝜃k𝜆·x

[1 + ck,k

2𝜃− 1

2k 𝜆

− 12 x]

2(𝜃k𝜆)

12 e

√𝜃k𝜆·x

[1 − ck,k

2𝜃− 1

2k 𝜆

− 12 x]

2

(𝜃k𝜆)e−√𝜃k𝜆·x

[1 + ck,k

2𝜃− 1

2k 𝜆

− 12 x]

2−(𝜃k𝜆)e

√𝜃k𝜆·x

[1 − ck,k

2𝜃− 1

2k 𝜆

− 12 x]

2

⎤⎥⎥⎥⎥⎦with k = 1, 2, … ,N.

The kth component of Φ(x) = [z1(x), z2(x), z3(x), z4(x), … , z2N−1(x), z2N(x)]⊤ with respect to the eigenvalue 𝜆 can beobtained by taking the determinant of the matrices, which are replaced one of the rows of TRΦ in (44) by e⊤k Φ(x, 𝜆) suchthat their determinants are not zero, and e⊤k is the kth column of the identity matrix. Thus, the odd components of Φ(x)

1460 GU AND WANG

are given by

z2k−1(x, 𝜆) = (−1)−𝜇(𝜃k𝜆)12(1−3N) · det

⎡⎢⎢⎣𝜆

12

[𝜃

121

]2𝜆

12

[𝜃

121

]2

𝜆E1[Θ1]2 −𝜆E2[Θ2]2

⎤⎥⎥⎦× · · · × det

⎡⎢⎢⎢⎢⎣𝜆

12

[𝜃

12

k

]2

𝜆12

[𝜃

12

k

]2

(𝜃k𝜆)12 e−

√𝜃k𝜆·x

[1 + ck,k

2𝜃− 1

2k 𝜆

− 12 x]

2(𝜃k𝜆)

12 e

√𝜃k𝜆·x

[1 − ck,k

2𝜃− 1

2k 𝜆

− 12 x]

2

⎤⎥⎥⎥⎥⎦× · · · × det

⎡⎢⎢⎣𝜆

12

[𝜃

12

N

]2

𝜆12

[𝜃

12

N

]2

𝜆E2N−1[Θ2N−1]2 −𝜆E2N[Θ2N]2

⎤⎥⎥⎦=

[−e

√𝜃1𝜆 − e−

√𝜃1𝜆 +

c1,1

2𝜃− 1

21 𝜆

− 12

(e√𝜃1𝜆 − e−

√𝜃1𝜆

)+ (𝜆−1)

]× · · · ×

[e√𝜃k𝜆·x − e−

√𝜃k𝜆·x −

ck,k

2𝜃− 1

2k 𝜆

− 12 x

(e√𝜃k𝜆·x + e−

√𝜃k𝜆·x

)+ (𝜆−1)

]× · · · ×

[−e

√𝜃N𝜆 − e−

√𝜃N𝜆 +

cN,N

2𝜃− 1

2N 𝜆

− 12

(e√𝜃N𝜆 − e−

√𝜃N𝜆

)+ (𝜆−1)

].

We conclude that

z2k−1(x, 𝜆𝑗m) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

rk(𝜆𝑗m)[

e√𝜃k𝜆𝑗m·x − e−

√𝜃k𝜆𝑗m·x − ck,k

2𝜃− 1

2k 𝜆

− 12𝑗m x

(e√𝜃k𝜆𝑗m·x + e−

√𝜃k𝜆𝑗m·x

)+ (

𝜆−1𝑗m

)], if 𝑗 = k,

(𝜆− 1

2𝑗m

), if 𝑗 ≠ k with 1 ≤ 𝑗 ≤ N, 1 ≤ k ≤ N,

(A3)

where rk(𝜆jm) is bounded in 𝜆jm, and has the form

rk(𝜆𝑗m) =∏

1≤l≤N, l≠k

[−e

√𝜃l𝜆𝑗m − e−

√𝜃l𝜆𝑗m +

cl,l

2𝜃− 1

2l 𝜆

− 12𝑗m

(e√𝜃l𝜆𝑗m − e−

√𝜃l𝜆𝑗m

)+ (

𝜆−1𝑗m

)].

Similarly, we obtain the following even components of Φ(x):

z2k(x, 𝜆) = (−1)−𝜇𝜆−32

N ·

( ∏1≤i≤N

𝜃− 3

2i

)· det

⎡⎢⎢⎣𝜆

12

[𝜃

121

]2𝜆

12

[𝜃

121

]2

𝜆E1[Θ1]2 −𝜆E2[Θ2]2

⎤⎥⎥⎦× · · · × det

⎡⎢⎢⎢⎢⎣𝜆

12

[𝜃

12

k

]2

𝜆12

[𝜃

12

k

]2

(𝜃k𝜆)e−√𝜃k𝜆·x

[1 + ck,k

2𝜃− 1

2k 𝜆

− 12 x]

2−(𝜃k𝜆)e

√𝜃k𝜆·x

[1 − ck,k

2𝜃− 1

2k 𝜆

− 12 x]

2

⎤⎥⎥⎥⎥⎦× · · · × det

⎡⎢⎢⎣𝜆

12

[𝜃

12

N

]2

𝜆12

[𝜃

12

N

]2

𝜆E2N−1[Θ2N−1]2 −𝜆E2N[Θ2N]2

⎤⎥⎥⎦=

[−e

√𝜃1𝜆 − e−

√𝜃1𝜆 +

c1,1

2𝜃− 1

21 𝜆

− 12

(e√𝜃1𝜆 − e−

√𝜃1𝜆

)+ (𝜆−1)

]× · · · ×

[−e

√𝜃k𝜆·x − e−

√𝜃k𝜆·x +

ck,k

2𝜃− 1

2k 𝜆

− 12 x

(e√𝜃k𝜆·x − e−

√𝜃k𝜆·x

)+ (𝜆−1)

]× · · · ×

[−e

√𝜃N𝜆 − e−

√𝜃N𝜆 +

cN,N

2𝜃− 1

2N 𝜆

− 12

(e√𝜃N𝜆 − e−

√𝜃N𝜆

)+ (𝜆−1)

].

GU AND WANG 1461

Therefore,

z2k(x, 𝜆𝑗m) =

⎧⎪⎪⎨⎪⎪⎩rk(𝜆𝑗m)

[−e

√𝜃k𝜆𝑗m·x − e−

√𝜃k𝜆𝑗m·x + ck,k

2𝜃− 1

2k 𝜆

− 12𝑗m x

(e√𝜃k𝜆𝑗m·x − e−

√𝜃k𝜆𝑗m·x

)+ (𝜆−1)

],

if 𝑗 = k,

(𝜆− 1

2𝑗m

), if 𝑗 ≠ k with 1 ≤ 𝑗 ≤ N, 1 ≤ k ≤ N,

(A4)

where√𝜃𝑗𝜆𝑗m = −(m − 1∕2)𝜋i + (m−1). On the basis of above computations, (48) can then be obtained by setting

zk,m(x) =zk(x, 𝜆)−2rk(𝜆)

in (A3) and (A4). Finally, it follows from (47) that‖‖‖e√𝜃𝑗𝜆𝑗m·x‖‖‖L2

= 1 + (m−1), ‖‖‖e−√𝜃𝑗𝜆𝑗m·x‖‖‖L2

= 1 + (m−1), 1 ≤ 𝑗 ≤ N,

which together with (48) yield (49). This completes the proof.□