J. Math. Anal. Appl. 484 (2020) 123666

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Journal of Mathematical Analysis and Applications

www.elsevier.com/locate/jmaa

Backstepping-based adaptive error feedback regulator design for

one-dimensional reaction-diffusion equation ✩

Wei-Wei Liu a, Wei Guo b,∗, Jun-Min Wang a

a Beijing Institute of Technology, Beijing 100081, Chinab University of International Business and Economics, Beijing 100029, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 17 June 2019Available online 18 November 2019Submitted by G. Chen

Keywords:Adaptive controlReaction-diffusion equationError feedbackOutput regulation

In this paper, we consider the error feedback regulator problem (EFRP) for one-dimensional reaction-diffusion equation with unknown harmonic boundary disturbance, where the regulated output is anti-collocated with the control. Some auxiliary systems are constructed in order to make the control and the disturbance become collocated, or make the measured tracking error become the output. Then the error feedback adaptive servomechanism is presented, in which the parameters of the disturbance and the reference signals are estimated. Our design is based on the motion planning and the backstepping approaches. In addition, we give a brief description that our approach is applicable to the heat equation with distributed disturbance. It is shown that the proposed adaptive control law regulates the tracking error to zero and keeps the states of all the internal loops bounded.

© 2019 Elsevier Inc. All rights reserved.

1. Introduction

In this paper, we are concerned with the error feedback regulator problem (EFRP) for an unstable heat equation. In contrast to the state feedback regulation problem (SFRP), in which the controller is designed with full information of the state of the plant and exo-system, the error feedback regulation problem is perhaps more realistic, for which the components of the tracking error are available for measurement.

The EFRP for finite-dimensional systems is studied extensively since 1970s, which can be found in [3,4,8–10,28], where the EFRP are solved via the resolution of algebra equations known as regulator equations. Some attempts have been made to solve the EFRP for the infinite dimensional case. For systems with bounded control and observation operators, many contributions on the EFRP have been developed in [25,2,1,29] in non-robust sense and [19,18] in the robust sense. For systems with unbounded operators and infinite-dimensional exosystems, we refer to relevant works on the EFRP in [21,23,22,24].

✩ This work was supported by the National Natural Science Foundation of China (No. 61973084, 61673061).* Corresponding author.

E-mail addresses: liuweiwei123@126.com (W.-W. Liu), gwei@amss.ac.cn, guowei74@126.com (W. Guo), jmwang@bit.edu.cn(J.-M. Wang).

https://doi.org/10.1016/j.jmaa.2019.1236660022-247X/© 2019 Elsevier Inc. All rights reserved.

2 W.-W. Liu et al. / J. Math. Anal. Appl. 484 (2020) 123666

Most of the mentioned research works about the EFRP focus on the extensions of the internal model principle theory from finite-dimensional setting to infinite-dimensional case. The solvability of the EFRP for these systems is characterized by the solvability of certain equations referred to as the regulator equations. Since the regulator equations are almost abstract operator equations in infinite-dimensional system setting, their solvability is not easy to verify when the considered infinite systems are described by, for example, partial differential equations (PDEs). Though the extensions can build some theoreti-cal frameworks which covers a large class of real control systems, many boundary control PDEs which are unstable or even anti-stable seem not to be included. Recently, some interesting efforts on com-bining the traditional regulation method and the backstepping approach for PDEs to solve the robust output regulation problem for boundary controlled parabolic PDEs or even coupled Partial Integro-Differential Equation (PIDEs), first-order hyperbolic PIDEs, anti-stable coupled wave equations and other PDEs have done in [5–7,30,12,11,13]. But the reference signals in [5–7,30,12,11,13] are all assumed to be known.

Recent works on solving the EFRP for PDEs by using adaptive control approach are reported first in [17]for wave equation and then in [16,15] for beam equation and heat equation respectively. The approaches presented in [17,16,15] to solve the EFRP, which are not based on traditional internal model principle and thus does not involve the regulator equations. However, the considered systems in [17,16,15] without control is conservative. In this paper, we will focus on solving the EFRP for unstable heat equations.

For simplicity, we omit the details of the motivation to solve the EFRP by making use of adaptive control approach which is referred to [17,16]. We extend an application of this approach to solve the EFRP for the following heat equation:⎧⎪⎪⎪⎨⎪⎪⎪⎩

wt(x, t) = wxx(x, t) + λw(x, t), 0 ≤ x ≤ 1, t ≥ 0,wx(0, t) = U(t), t ≥ 0,wx(1, t) = D(t), t ≥ 0,e(t) = yout(t) −R(t) = w(1, t) −R(t), t ≥ 0,

(1.1)

where w(x, t) is the state, and wx(x, t) (or w′(x, t)) denotes the derivative of w(x, t) with respect to x and wt(x, t) (or w(x, t)) the derivative of w(x, t) with respect to t, λ is a positive constant, which results in that system (1.1) is unstable. Here and in the rest of this paper, we omit the initial values for all systems. U(t)is the control input, yout(t) is the regulated output, D(t) is the disturbance which has the form

D(t) =m∑i=1

[ai sinαit + bi cosαit

], t ≥ 0, i ∈ J1 = {1, 2, · · · ,m}, (1.2)

and R(t) is the reference signal which has the form

R(t) =n∑

k=1

[ck sin βkt + dk cosβkt

], t ≥ 0, k ∈ J2 = {1, 2, · · · , n}. (1.3)

The amplitudes ai, bi, ck, dk are assumed to be unknown, but the frequencies αi, βk are given. e(t) is the tracking error which is can be measured:

e(t) = w(1, t) −R(t).

System (1.1) is a typical anti-collocated Neumann control problem: control is on one end and the reg-ulated output is on the other end. In addition, the disturbance D(t) and the control U(t) are also anti-collocated.

The objective of this paper is to design an adaptive controller for system (1.1) such that

W.-W. Liu et al. / J. Math. Anal. Appl. 484 (2020) 123666 3

(1) for ai = bi = ck = dk = 0 (i ∈ J1, k ∈ J2), the resulting closed-loop system without disturbance and reference signal is exponentially stable;(2) for any initial value w(x, 0) ∈ L2(0, 1), ∀ai, bi, ck, dk ∈ R (i ∈ J1, k ∈ J2), the tracking error is stable, that is, e(t) → 0(t → ∞).

The key characterization of our approach is to use motion planing in [20] to construct some auxiliary systems in which the measured error becomes output, or the control becomes collocated with the disturbance. In the controller design, we use the Backstepping method for PDEs to deal with unstable term in domain of the equation and adaptive control technology to estimate unknown parameters. The considered heat equation is unstable and with unbounded input and output operators, which is not included in the abstract frameworks in existing literatures.

The rest of this paper is organized as follows. In Section 2, we present the adaptive error feedback regulator design for system (1.1) and main results. Section 3 is devoted to the proof of the main Theorems. The numerical simulations are presented in Section 4 to illustrate theory results. The conclusion is given in Section 5.

2. Error feedback regulator design and main result

In this section, we are devoted to the adaptive controller design to solve the EFRP for system (1.1). As mentioned before that system (1.1) is a typical anti-collocated control problem, we have to construct two auxiliary systems, in which firstly the measured tracking error becomes the output, then the boundary disturbance becomes collocated with the control. We now construct the first auxiliary system, in which the measured tracking error becomes the output. To this end, consider the following heat equations:{

pkt(x, t) = pkxx(x, t) + λpk(x, t), 0 ≤ x ≤ 1, t ≥ 0,pkx(0, t) = 0, pk(1, t) = sin βkt, t ≥ 0,

(2.1-1)

{qkt(x, t) = qkxx(x, t) + λqk(x, t), 0 ≤ x ≤ 1, t ≥ 0,qkx(0, t) = 0, qk(1, t) = cosβkt, t ≥ 0,

(2.1-2)

where k ∈ J2. Following the steps as those in Chapter 12 in [20], we have the explicit solutions of (2.1-1)and (2.1-2) respectively:

pk(x, t) = Im{

cosh(√jβk − λx)

cosh(√jβk − λ)

ejβkt

}, qk(x, t) = Re

{cosh(

√jβk − λx)

cosh(√jβk − λ)

ejβkt

}. (2.2)

Hence, one has

pkx(1, t) = Im{√jβk − λ tanh(

√jβk − λ)ejβkt} = c1k sin βkt + c2k cosβkt, k ∈ J2,

qkx(1, t) = Re{√jβk − λ tanh(

√jβk − λ)ejβkt} = c1k cosβkt− c2k sin βkt, k ∈ J2,

(2.3)

where

c1k = a1k sinh(2a1k) − b1k sin(2b1k)2[(cosh a1k cos b1k)2 + (sinh a1k sin b1k)2]

, c2k = b1k sinh(2a1k) + a1k sin(2b1k)2[(cosh a1k cos b1k)2 + (sinh a1k sin b1k)2]

,

a1k =√

22

√√λ2 + β2

k − λ, b1k =√

22

√√λ2 + β2

k + λ, k ∈ J2.

Let

4 W.-W. Liu et al. / J. Math. Anal. Appl. 484 (2020) 123666

z(x, t) = w(x, t) −n∑

k=1

[ckpk(x, t) + dkqk(x, t)

], 0 ≤ x ≤ 1, t ≥ 0, (2.4)

then z yields

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩zt(x, t) = zxx(x, t) + λz(x, t),zx(0, t) = U(t),

zx(1, t) =m∑i=1

[ai sinαit + bi cosαit

]−

n∑k=1

[ck(c1k sin βkt + c2k cosβkt) + dk(c1k cosβkt

−c2k sin βkt)].

(2.5)

Moreover,

z(1, t) = w(1, t) −n∑

k=1

[ckpk(1, t) + dkqk(1, t)

]= w(1, t) −

n∑k=1

[ck sin βkt + dk cosβkt

]= e(t). (2.6)

By making use of the measured output z(1, t), we design an observer for system (2.5) which yields⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

zt(x, t) = zxx(x, t) + λz(x, t) + p1(x)[z(1, t) − z(1, t)],zx(0, t) = U(t),

zx(1, t) =m∑i=1

[ai(t) sinαit + bi(t) cosαit

]−

n∑k=1

[ck(t)(c1k sin βkt + c2k cosβkt)

+dk(t)(c1k cosβkt− c2k sin βkt)]+ p10[z(1, t) − z(1, t)],

˙ai(t) = r1i[z(1, t) − z(1, t)] sinαit, i ∈ J1,˙bi(t) = r2i[z(1, t) − z(1, t)] cosαit, i ∈ J1,˙ck(t) = −r3k[z(1, t) − z(1, t)](c1k sin βkt + c2k cosβkt), k ∈ J2,˙dk(t) = −r4k[z(1, t) − z(1, t)](c1k cosβkt− c2k sin βkt), k ∈ J2,

(2.7)

which is a copy of system (2.5) plus output injection terms with the gains p1(x), p10 to be determined. Here, r1i, r2i, r3k, r4k (i ∈ J1, k ∈ J2) are positive design parameters.

Define the error variable z(x, t) = z(x, t) − z(x, t), ai(t) = ai − ai(t), bi(t) = bi − bi(t), ck(t) = ck − ck(t)and dk(t) = dk − dk(t), then (z, ai, bi, ck, dk) satisfies⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

zt(x, t) = zxx(x, t) + λz(x, t) − p1(x)z(1, t),zx(0, t) = 0,

zx(1, t) =m∑i=1

[ai(t) sinαit + bi(t) cosαit

]−

n∑k=1

[ck(t)(c1k sin βkt + c2k cosβkt)

+dk(t)(c1k cosβkt− c2k sin βkt)]− p10z(1, t),

˙ai(t) = −r1iz(1, t) sinαit, i ∈ J1,˙bi(t) = −r2iz(1, t) cosαit, i ∈ J1,˙ck(t) = r3kz(1, t)(c1k sin βkt + c2k cosβkt), k ∈ J2,˙dk(t) = r4kz(1, t)(c1k cosβkt− c2k sin βkt), k ∈ J2.

(2.8)

Let

˜z(x, t) = z(1 − x, t), (2.9)

one has

W.-W. Liu et al. / J. Math. Anal. Appl. 484 (2020) 123666 5

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

˜zt(x, t) = ˜zxx(x, t) + λ˜z(x, t) − p1(x)˜z(0, t),˜zx(1, t) = 0,˜zx(0, t) = −m∑i=1

[ai(t) sinαit + bi(t) cosαit

]+

n∑k=1

[ck(t)(c1k sin βkt + c2k cosβkt)

+dk(t)(c1k cosβkt− c2k sin βkt)]+ p10˜z(0, t),

˙ai(t) = −r1i˜z(0, t) sinαit, i ∈ J1,˙bi(t) = −r2i˜z(0, t) cosαit, i ∈ J1,

˙ck(t) = r3k ˜z(0, t)(c1k sin βkt + c2k cosβkt), k ∈ J2,˙dk(t) = r4k ˜z(0, t)(c1k cosβkt− c2k sin βkt), k ∈ J2.

(2.10)

We introduce the following backstepping transformation ([20]):

˜z(x, t) = ε(x, t) −x∫

0

s(x, y)ε(y, t) dy, 0 ≤ x ≤ 1, t ≥ 0, (2.11)

where

s(x, y) = −λ(1 − y)I1(

√λ(x− y)(2 − x− y))√λ(x− y)(2 − x− y)

.

Its inverse transformation is

ε(x, t) = ˜z(x, t) +x∫

0

s(x, y)˜z(y, t) dy, 0 ≤ x ≤ 1, t ≥ 0,

where

s(x, y) = −λ(1 − y)J1(

√λ(x− y)(2 − x− y))√

λ(x− y)(2 − x− y).

Under the transformation (2.11), we transform the system (2.10) into the following equivalent system de-scribed by ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

εt(x, t) = εxx(x, t),

εx(0, t) = λ

2 ε(0, t) −m∑i=1

[ai(t) sinαit + bi(t) cosαit

]+

n∑k=1

[ck(t)(c1k sin βkt + c2k cosβkt) + dk(t)(c1k cosβkt− c2k sin βkt)

],

εx(1, t) = 0,˙ai(t) = −r1iε(0, t) sinαit, i ∈ J1,˙bi(t) = −r2iε(0, t) cosαit, i ∈ J1,˙ck(t) = r3kε(0, t)(c1k sin βkt + c2k cosβkt), k ∈ J2,˙dk(t) = r4kε(0, t)(c1k cosβkt− c2k sin βkt), k ∈ J2,

(2.12)

by which, we obtain p1(x) and p10 in (2.7) as follows:

p1(x) = sy(x, 0) − λs(x, 0), p10 = λ.

2

6 W.-W. Liu et al. / J. Math. Anal. Appl. 484 (2020) 123666

Define the following energy function for system (2.12):

Eε(t) = 12

1∫0

ε2(x, t) dx +m∑i=1

[ai

2(t)2r1i

+ bi2(t)

2r2i

]+

n∑k=1

[ck

2(t)2r3k

+ dk2(t)

2r4k

]. (2.13)

Differentiating Eε(t) with respect to t, we obtain

Eε(t) = −λ

2 ε2(0, t) −

1∫0

ε2x(x, t) dx ≤ 0, (2.14)

which gives the motivation of the update law for ai, bi, ck and dk, i ∈ J1, k ∈ J2 in (2.7). For system (2.12), define the operator A in L2(0, 1) by{

(Af)(x) = f ′′(x),∀f ∈ D(A),D(A) = {f ∈ H2(0, 1)|f ′(0) = f ′(1) = 0}. (2.15)

It is a routine exercise to obtain the eigenvalues σn and the corresponding eigenfunctions {φn} of A, which read as {

σ0 = 0, σn = −n2π2, n = 1, 2, · · · ,φ0(x) = 1, φn(x) =

√2 cos(nπx), n = 1, 2, · · · . (2.16)

Since the eigenfunctions {φn} of A is an orthonormal basis of L2(0, 1), we can construct the Galerkin scheme to show that there exits a unique classical solution to system (2.12) as similar procedure in [14].

Theorem 2.1. Suppose that (ε0, a1(0), · · · , am(0), b1(0), · · · , bm(0), c1(0), · · · , cn(0), d1(0), · · · , dn(0)) ∈D(A) ×R2m+2n satisfy the compatible condition as follows:

λ

2 ε0 −m∑i=1

bi(0) +n∑

k=1

[c2k ck(0) + c1kdk(0)

]= 0. (2.17)

Then system (2.12) admits a unique classical solution, that is, for any time T > 0, i ∈ J1, k ∈ J2,

ε ∈ L∞(0, T ;H2(0, 1)), ε ∈ L∞(0, T ;L2(0, 1)), ai, bi, ck, dk ∈ C1[0, T ],

and satisfy (2.12). From the Sobolev embedding theorem, it follows ε ∈ C([0, T ] × [0, 1]).

Now we are in a position to give the definition of the weak solution to system (2.12).

Definition 2.2. For any initial value (ε(x, 0), a1(0), · · · , am(0), b1(0), · · · , bm(0), c1(0), · · · , cn(0), d1(0), · · · ,dn(0)) ∈ H = L2(0, 1) ×R2m+2n, the weak solution (ε, a1, · · · , am, b1, · · · , bm, c1, · · · , cn, d1, · · · , dn) of equa-tion (2.12) is defined as the limit of any convergent subsequence of (εn, a1

n, · · · , amn, b1

n, · · · , bm

n, c1

n, · · · ,cn

n, d1n, · · · , dn

n) in L∞(0, ∞; H), where (εn, a1

n, · · · , amn, b1

n, · · · , bm

n, c1

n, · · · , cnn, d1n, · · · , dn

n) is the

classical solution (ensured by Theorem 2.1) with the initial condition (for all x ∈ (0, 1))(εn(x, 0), a1n(0),

· · · , amn(0), b1n(0), · · · , bm

n(0), c1n(0), · · · , cnn(0), d1

n(0), · · · , dn

n(0)) ∈ D(A) ×R2m+2n, which satisfies

limn→∞

‖(εn(x, 0), a1n(0), · · · , amn(0), b1

n(0), · · · , bm

n(0), c1n(0), · · · , cnn(0), d1

n(0), · · · , dn

n(0))

− (ε(x, 0), a (0), · · · , a (0), b (0), · · · , b (0), c (0), · · · , c (0), d (0), · · · , d (0))‖ = 0.

1 m 1 m 1 n 1 n H

W.-W. Liu et al. / J. Math. Anal. Appl. 484 (2020) 123666 7

By (2.13) and (2.14), the weak solution mentioned above is well defined, since it does not depend on

the choice of initial sequence (εn(x, 0), a1n(0), · · · , amn(0), b1

n(0), · · · , bm

n(0), c1n(0), · · · , cnn(0), d1

n(0),

· · · , dnn(0)). Consequently, (ε, a1, · · · , am, b1, · · · , bm, c1, · · · , cn, d1, · · · , dn) ∈ C(0, ∞; H). In addition, from

(2.13) and (2.14), we can see that this solution depends continuously on its initial value.

Theorem 2.3. Suppose that αi, βk, i ∈ J1, k ∈ J2 are distinct. For any initial state (ε(x, 0), a1(0), · · · ,am(0), b1(0), · · · , bm(0), c1(0), · · · , cn(0), d1(0), · · · , dn(0)) ∈ H, the (weak) solution to (2.12) is asymptoti-cally stable in the sense that

limt→∞

⎡⎣12

1∫0

ε2(x, t) dx +m∑i=1

(ai

2(t) + bi2(t)

)+

n∑k=1

(ck

2(t) + dk2(t)

)⎤⎦ = 0. (2.18)

Moreover,

limt→∞

ε(0, t) = 0.

By the (2.9), backstepping transformations (2.11) and its inverse transformation and Theorem 2.3, we can obtain the following theorem.

Theorem 2.4. Suppose that αi, βk, i ∈ J1, k ∈ J2 are distinct. For each initial value (z(x, 0), a1(0), · · · ,am(0), b1(0), · · · , bm(0), c1(0), · · · , cn(0), d1(0), · · · , dn(0)) ∈ L2(0, 1) × R2m+2n, there exists a unique so-lution to (2.8) such that (z, a1, · · · , am, b1, · · · , bm, c1, · · · , cn, d1, · · · , dn) ∈ C(0, ∞; L2(0, 1) × R2m+2n). Moreover, the solution of (2.8) is asymptotically stable in the sense that

limt→∞

⎡⎣12

1∫0

z2(x, t) dx +m∑i=1

(ai

2(t) + bi2(t)

)+

n∑k=1

(ck

2(t) + dk2(t)

)⎤⎦ = 0. (2.19)

Moreover,

limt→∞

z(1, t) = 0.

Now, we construct the second auxiliary system in which the disturbance signal and reference signal become collocated with the control. Let us introduce the heat equations as follows:{

uit(x, t) = uixx(x, t) + λui(x, t), 0 ≤ x ≤ 1, t ≥ 0,ui(1, t) = 0, uix(1, t) = sinαit, t ≥ 0,

(2.20-1){vit(x, t) = vixx(x, t) + λvi(x, t), 0 ≤ x ≤ 1, t ≥ 0,vi(1, t) = 0, vix(1, t) = cosαit, t ≥ 0,

(2.20-2)⎧⎪⎨⎪⎩ξkt(x, t) = ξkxx(x, t) + λξk(x, t), 0 ≤ x ≤ 1, t ≥ 0,ξk(1, t) = 0, t ≥ 0,ξkx(1, t) = c1k sin βkt + c2k cosβkt, t ≥ 0,

(2.20-3)

⎧⎪⎨⎪⎩ηkt(x, t) = ηkxx(x, t) + ληk(x, t), 0 ≤ x ≤ 1, t ≥ 0,ηk(1, t) = 0, t ≥ 0,ηkx(1, t) = c1k cosβkt− c2k sin βkt, t ≥ 0.

(2.20-4)

Here, i ∈ J1, k ∈ J2. We can solve ui, vi, ξk, ηk (i ∈ J1, k ∈ J2) using the similar processes in [20]:

8 W.-W. Liu et al. / J. Math. Anal. Appl. 484 (2020) 123666

ui(x, t) = −Im{

sinh(√jαi − λ(1 − x))√jαi − λ

ejαit

},

vi(x, t) = −Re{

sinh(√jαi − λ(1 − x))√jαi − λ

ejαit

},

ξk(x, t) = −Im{c1k sinh(

√jβk − λ(1 − x))√jβk − λ

ejβkt

}− Re

{c2k sinh(

√jβk − λ(1 − x))√jβk − λ

ejβkt

},

ηk(x, t) = −Re{c1k sinh(

√jβk − λ(1 − x))√jβk − λ

ejβkt

}+ Im

{c2k sinh(

√jβk − λ(1 − x))√jβk − λ

ejβkt

}.

(2.21)

Then straightforward computation gives

uix(0, t) = Im{cosh(√jαi − λ)ejαit} = c3i sinαit + c4i cosαit,

vix(0, t) = Re{cosh(√jαi − λ)ejαit}} = c3i cosαit− c4i sinαit,

ξkx(0, t) = Im{c1k cosh(√jβk − λ)ejβkt} + Re{c2k cosh(

√jβk − λ)ejβkt}

= (c1kc6k + c2kc5k) cosβkt + (c1kc5k − c2kc6k) sin βkt,

ηkx(0, t) = Re{c1k cosh(√jβk − λ)ejβkt} − Im{c2k cosh(

√jβk − λ)ejβkt}

= (c1kc5k − c2kc6k) cosβkt− (c1kc6k + c2kc5k) sin βkt,

(2.22)

where

c3i = cosh a2i cos b2i, c4i = sinh a2i sin b2i, a2i =√

22

√√λ2 + α2

i − λ, b2i =√

22

√√λ2 + α2

i + λ,

c5k = cosh a1k cos b1k, c6k = sinh a1k sin b1k.

Let

σ(x, t) = z(x, t) −m∑i=1

[ai(t)ui(x, t) + bi(t)vi(x, t)

]+

n∑k=1

[ck(t)ξk(x, t)

+dk(t)ηk(x, t)], (0 ≤ x ≤ 1, t ≥ 0),

(2.23)

then together with(2.7), (2.20-1) - (2.20-4), one has

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

σt(x, t) = σxx(x, t) + λσ(x, t) + L(x, t)[z(1, t) − z(1, t)],

σx(0, t) = U(t) −m∑i=1

[ai(t)(c3i sinαit + c4i cosαit) + bi(t)(c3i cosαit− c4i sinαit)

]+

n∑k=1

[ck(t)

((c1kc6k + c2kc5k) cosβkt + (c1kc5k − c2kc6k) sin βkt

)+dk(t)

((c1kc5k − c2kc6k) cosβkt− (c1kc6k + c2kc5k) sin βkt

)],

σx(1, t) = p10[z(1, t) − z(1, t)],

(2.24)

where L(x, t) = p1(x) −m∑i=1

[r1i(sinαit)ui(x, t) +r2i(cosαit)vi(x, t)

]−

n∑k=1

[r3k(c1k sin βkt +c2k cosβkt)ξk(x, t) +

r4k(c1k cosβkt − c2k sin βkt)ηk(x, t)]. Moreover,

σ(1, t) = z(1, t) −m∑i=1

[aiui(1, t) + bivi(1, t)

]+

n∑k=1

[ckξk(1, t) + dkηk(1, t)

]= z(1, t). (2.25)

We use the backstepping method to design the controller based on (2.24). Introduce the reversible transfor-mation ([20])

W.-W. Liu et al. / J. Math. Anal. Appl. 484 (2020) 123666 9

�(x, t) = σ(x, t) −1∫

x

K(x, y)σ(y, t) dy, 0 ≤ x ≤ 1, t ≥ 0, (2.26)

where

K(x, y) = −λ(1 − x)I1(

√λ(y − x)(2 − y − x))√λ(y − x)(2 − y − x)

.

Under the transformation (2.26), we can map the system (2.24) into the following system:

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩�t(x, t) = �xx(x, t) +

⎡⎣L(x, t) −K(x, 1)p10 −1∫

x

K(x, y)L(y, t) dy

⎤⎦ [z(1, t) − z(1, t)],

�x(0, t) = λ

2�(0, t),

�x(1, t) = p10[z(1, t) − z(1, t)].

(2.27)

From the boundary condition �x(0, t) = λ2�(0, t), the output feedback law can be given as follows:

U(t) =1∫

0

[Kx(0, y) −

λ

2K(0, y)]σ(y, t) dy + λσ(0, t)

+m∑i=1

[ai(t)(c3i sinαit + c4i cosαit) + bi(t)(c3i cosαit− c4i sinαit)

]−

n∑k=1

{ck(t)[(c1kc6k + c2kc5k) cosβkt + (c1kc5k − c2kc6k) sin βkt]

−dk(t)[(c1kc5k − c2kc6k) cosβkt− (c1kc6k + c2kc5k) sin βkt]}.

(2.28)

The closed-loop system of (2.24) corresponding to controller (2.28) is

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

σt(x, t) = σxx(x, t) + λσ(x, t) + L(x, t)[z(1, t) − z(1, t)],

σx(0, t) =1∫

0

[Kx(0, y) − λ

2K(0, y)]σ(y, t) dy + λσ(0, t),

σx(1, t) = p10[z(1, t) − z(1, t)].

(2.29)

Theorem 2.5. For each initial data σ(x, 0) ∈ L2(0, 1), there exists a unique solution to (2.29) such that σ ∈ C(0, ∞; L2(0, 1)). Moreover, the solution of (2.29) is asymptotically stable in the sense that

limt→∞

⎡⎣12

1∫0

σ2(x, t) dx

⎤⎦ = 0,

and

limt→∞

σ(1, t) = 0.

Noted that the controller (2.28) is expressed by variable σ. To obtain the closed-loop of (1.1), by (2.23), we can rewrite the controller by variable z to be

10 W.-W. Liu et al. / J. Math. Anal. Appl. 484 (2020) 123666

U(t) =1∫

0

[Kx(0, y) −

λ

2K(0, y)]{

z(y, t) −m∑i=1

[ai(t)ui(y, t) + bi(t)vi(y, t)

]+

n∑k=1

[ck(t)ξk(y, t) + dk(t)ηk(y, t)

]}dy + λ

{z(0, t) −

m∑i=1

[ai(t)ui(0, t)

+bi(t)vi(0, t)]+

n∑k=1

[ck(t)ξk(0, t) + d(t)ηk(0, t)

]}+

m∑i=1

[ai(t)(c3i sinαit + c4i cosαit) + bi(t)(c3i cosαit− c4i sinαit)

]−

n∑k=1

{ck(t)[(c1kc6k + c2kc5k) cosβkt + (c1kc5k − c2kc6k) sin βkt]

+d(t)[(c1kc5k − c2kc6k) cosβkt− (c1kc6k + c2kc5k) sin βkt]}.

(2.30)

Thus under controller (2.30), the resulting closed-loop of (1.1) is governed by⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

wt(x, t) = wxx(x, t) + λw(x, t),wx(0, t) = zx(0, t),

wx(1, t) =m∑i=1

[ai sinαit + bi cosαit

],

zt(x, t) = zxx(x, t) + λz(x, t) + p1(x)[e(t) − z(1, t)],

zx(0, t) =1∫

0

[Kx(0, y) −

λ

2K(0, y)]{

z(y, t) −m∑i=1

[ai(t)ui(y, t) + bi(t)vi(y, t)

]+

n∑k=1

[ck(t)ξk(y, t) + dk(t)ηk(y, t)

]}dy + λ

{z(0, t) −

m∑i=1

[ai(t)ui(0, t)

+bi(t)vi(0, t)]+

n∑k=1

[ck(t)ξk(0, t) + d(t)ηk(0, t)

]}+

m∑i=1

[ai(t)(c3i sinαit + c4i cosαit) + bi(t)(c3i cosαit− c4i sinαit)

]−

n∑k=1

{ck(t)[(c1kc6k + c2kc5k) cosβkt + (c1kc5k − c2kc6k) sin βkt]

+d(t)[(c1kc5k − c2kc6k) cosβkt− (c1kc6k + c2kc5k) sin βkt]},

zx(1, t) =m∑i=1

[ai(t) sinαit + bi(t) cosαit

]−

n∑k=1

ck(t)(c1k sin βkt + c2k cosβkt)

+dk(t)(c1k cosβkt− c2k sin βkt)]+ p10[e(t) − z(1, t)],

˙ai(t) = r1i[e(t) − z(1, t)] sinαit, i ∈ J1,˙bi(t) = r2i[e(t) − z(1, t)] cosαit, i ∈ J1,˙ck(t) = −r3k[e(t) − z(1, t)](c1k sin βkt + c2k cosβkt), k ∈ J2,˙dk(t) = −r4k[e(t) − z(1, t)](c1k cosβkt− c2k sin βkt), k ∈ J2,

e(t) = w(1, t) −n∑

k=1(ck sin βkt + dk cosβkt).

(2.31)

We consider system (2.31) in L2(0, 1) ×H = L2(0, 1) × L2(0, 1) ×R2m+2n.

Theorem 2.6. Suppose that αi, βk, i ∈ J1, k ∈ J2 are distinct. For any initial condition (w(x, 0), z(x, 0), a1(0),· · · , am(0), b1(0), · · · , bm(0), c1(0), · · · , cn(0), d1(0), · · · , dn(0)) ∈ L2(0, 1) × H, system (2.31) admits a unique solution (w, z, a1, · · · , am, b1, · · · , bm, c1, · · · , cn, d1, · · · , dn) ∈ C(0, ∞; L2(0, 1) ×H). And this closed-loop solution has the following properties:

(i) supt≥0

⎧⎨⎩1∫[w2(x, t) + z2(x, t)] dx +

m∑i=1

[ai

2(t) + bi2(t)

]+

n∑k=1

[ck

2(t) + dk2(t)

]⎫⎬⎭ < ∞.

0

W.-W. Liu et al. / J. Math. Anal. Appl. 484 (2020) 123666 11

(ii) limt→∞

ai(t) = ai, limt→∞

bi(t) = bi, limt→∞

ck(t) = ck, limt→∞

dk(t) = dk, i ∈ J1, k ∈ J2.

(iii) limt→∞

e(t) = w(1, t) −n∑

k=1(ck sin βkt + dk cosβkt) = 0.

(iv) When ai = bi = ck = dk = 0, i ∈ J1, k ∈ J2,

1∫0

[w2(x, t) + z2(x, t)] dx ≤ M4e−μ4t,

for some constants M4, μ4 > 0.

Remark 2.7. Our adaptive approach is also applicable for the heat equation with distributed disturbance as follows ⎧⎪⎪⎪⎨⎪⎪⎪⎩

wt(x, t) = wxx(x, t) + f(x)[a sin(αt) + b cos(αt)],wx(1, t) = 0,wx(0, t) = U(t),e(t) = w(1, t) − (c sin(βt) + d cos(βt)),

(2.32)

where f is known continuous functions, a, b are unknown constants.Introducing a transformation u(x, t) = w(x, t) − G1(x) sin(αt) − G2(x) cos(αt), where G1, G2 are to be

determined later, system (2.32) can be mapped to the system as follows⎧⎪⎪⎪⎨⎪⎪⎪⎩ut(x, t) = uxx(x, t),ux(1, t) = −G′

1(1) sin(αt) −G′2(1) cos(αt),

ux(0, t) = U(t),e(t) = u(1, t) − (c sin(βt) + d cos(βt)),

(2.33)

provided G1, G2 are chosen as ⎧⎪⎨⎪⎩G′′

1(x) + αG2(x) + af(x) = 0,G′′

2(x) − αG1(x) + bf(x) = 0,G′

1(0) = G′2(0) = G1(1) = G2(1) = 0.

(2.34)

Boundary value problem (2.34) has a unique classical solution by standard ODEs theory:

G1(x) = −1∫

x

C ′1(x) dxe

√jαx −

1∫x

C ′2(x) dxe−

√jαx −

1∫x

C ′3(x) dx cos(

√jαx)

−1∫

x

C ′4(x) dx sin(

√jαx) + C1(1)e

√jαx + C2(1)e−

√jαx + C3(1) cos(

√jαx)

+C4(1) sin(√jαx),

G2(x) = j

1∫x

C ′1(x) dxe

√jαx + j

1∫x

C ′2(x) dxe−

√jαx − j

1∫x

C ′3(x) dx cos(

√jαx)

−j

1∫x

C ′4(x) dx sin(

√jαx) − jC1(1)e

√jαx − jC2(1)e−

√jαx + jC3(1) cos(

√jαx)

+jC4(1) sin(√jαx) − a

f(x),

(2.35)

α

12 W.-W. Liu et al. / J. Math. Anal. Appl. 484 (2020) 123666

where

C ′1(x) = −C ′

2(x)e−√jαx, C ′

2(x) = −af ′′(x) + bαf(x)−2jα

√jα− 2jα

√jαe−

√jαx

,

C ′3(x) = −C ′

4(x) tan(√

jαx), C ′4(x) = −af ′′(x) + bαf(x)

−2jα√jα tan(

√jαx) sin(

√jαx) − 2jα

√jα cos(

√jαx)

.

C1(1) = −C2(1)e−2√jα − C3(1) sin

√jα− C4(1) sin

√jα,

C2(1) = −

1∫0

[C ′1(x) − C ′

2(x)] dx− af ′1(0)

2jα√jα

+ C3(1) cos√

jα + C4(1) cos√

jα

e−2√jα+1 ,

C3(1) = af(1) − 2jC4(1) sin√jα

2j cos√jα

,C4(1) =1∫

0

C ′4(x) dx + af ′(0)

2jα√jα

.

Since G′1(1), G′

2(1) contains unknown parameters a, b, the structure of system (2.33) is almost the same with (1.1). Thus, we can use our adaptive approach to solve EFRP for (2.33). That is to say, for systems with distributed disturbances, we can solve the EFRP by transforming the distributed disturbances into boundary disturbances.

3. Proof of the results

Proof of Theorem 2.3

Proof. We assume that the initial value (ε(x, 0), a1(0), · · · , am(0), b1(0), · · · , bm(0), c1(0), · · · , cn(0), d1(0),· · · , dn(0)) ∈ D(A) × R2m+2n and satisfies the compatible condition (2.17), which means that Theo-rem 2.1 assures the existence for the classical solution to system (2.12). We introduce some additional variable to transform the non-autonomous system (2.12) into the time invariant system. Obviously, the following equation with initial value (w10, w20, · · · , w(2m)0, w(2m+1)0, · · · , w(2m+2n−1)0, w(2m+2n)0) =(1, 0, · · · , 0, 1, · · · , 1, 0) has one solution (cosα1t, sinα1t, · · · , sinαmt, cosβ1t, · · · , cosβnt, sin βnt):

w(t) = Sw(t), w(0) = w0, (3.1)

where

w(t) = (w1(t), w2(t), · · · , w2m(t), w2m+1(t), · · · , w2m+2n−1(t), w2m+2n(t))�,

w0 = (w10, w20, · · · , w(2m)0, w(2m+1)0, · · · , w(2m+2n−1)0, w(2m+2n)0)�,

we define I =(

0 −11 0

), then S can be expressed as:

S = diag(α1I , · · · , αmI , β1I , · · · , β1I), i ∈ J1, k ∈ J2.

We consider (2.12) and (3.1) together in the energy state space H = L2(0, 1) ×R2m+2n,

d

dtZ(·, t) = AZ(·, t), Z(0, t) = Z0(·) ∈ H, (3.2)

where

W.-W. Liu et al. / J. Math. Anal. Appl. 484 (2020) 123666 13

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩Z(x, t) = (ε(x, t), a1(t), · · · , am(t), b1(t), · · · , bm(t), c1(t), · · · , cn(t), d1(t), · · · , dn(t),

θ1(t), ϑ1(t), · · · , θm(t), ϑm(t), θ1(t), ϑ1(t), · · · , θn(t), ϑn(t)),Z0(x) = (ε(·, 0), a1(0), · · · , am(0), b1(0), · · · , bm(0), c1(0), · · · , cn(0), d1(0), · · · , dn(0),

θ10, ϑ10, · · · , θm0, ϑm0, θ10, ϑ10, · · · , θn0, ϑn0),⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

A(f, a1, · · · , am, b1, · · · , bm, c1, · · · , cn, d1, · · · , dn, θ1, ϑ1, · · · , θm, ϑm, θ1, ϑ1, · · · , θn, ϑn)= (f ′′,−r11f(0)ϑ1, · · · ,−r1mf(0)ϑm,−r21f(0)θ1, · · · ,−r2mf(0)θm, r31f(0)(c11ϑ1 + c21θ1),· · · , r3nf(0)(c1nϑn + c2nθn), r41f(0)(c11θ1 − c21ϑ1), · · · , r4nf(0)(c1nθn − c2nϑn),−α1ϑ1,

α1θ1, · · · ,−αmϑm, αmθm,−α1ϑ1, α1θ1, · · · ,−αnϑn, αnθn),

D(A) ={

(f, a1, · · · , am, b1, · · · , bm, c1, · · · , cn, d1, · · · , dn, θ1, ϑ1, · · · , θm, ϑm, θ1, ϑ1, · · · ,

θn, ϑn) ∈ H2(0, 1) ×R2m+2n∣∣f ′(0) = λ

2 f(0) −m∑i=1

(aiϑi + biθi

)+

n∑k=1

[ck(c1kϑk + c2kθk)+

dk(c1kθk − c2kϑk)], f ′(1) = 0

}.

Define the Lyapunov function for system (3.2):

V (t) = 12

1∫0

ε2(x, t) dx +m∑i=1

[ai

2(t)2r1i

+ bi2(t)

2r2i

]+

n∑k=1

[ck

2(t)2r3k

+ dk2(t)

2r4k

]

+m∑i=1

[θ2i (t) + ϑ2

i (t)]+

n∑k=1

[θ2k(t) + ϑ2

k(t)].

(3.3)

Taking the time derivative of V (t):

V (t) = −λ

2 ε2(0, t) −

1∫0

ε2x(x, t) dx ≤ 0. (3.4)

Then, we have

V (t) ≤ V (0). (3.5)

Hence,

supt≥0

⎡⎣12

1∫0

ε2(x, t) dx +m∑i=1

(ai

2(t) + bi2(t)

)+

n∑k=1

(ck

2(t) + dk2(t)

)⎤⎦ ≤ ∞. (3.6)

In particular, one has

ε(0, t), ‖εx‖L2(0,1) ∈ L2(0,∞). (3.7)

Let εt = ε, then we have ⎧⎪⎪⎪⎨⎪⎪⎪⎩εt(x, t) = εxx(x, t),

εx(0, t) = λ

2 ε(0, t) + S(t) + T (t),

ε (1, t) = 0,

(3.8)

x

14 W.-W. Liu et al. / J. Math. Anal. Appl. 484 (2020) 123666

where

S(t) =m∑i=1

[r1iε(0, t)ai(t) sinα2

i t + r2iε(0, t)bi(t) cosα2i t]+

n∑k=1

[r3kε(0, t)ck(t)(c1k sin βkt

+c2k cosβkt)2 + r4kε(0, t)dk(t)(c1k cosβkt− c2k sin βkt)2],

T (t) = −m∑i=1

[ai(t)αi cosαit− bi(t)αi sinαit

]+

n∑k=1

[ck(t)βk(c1k cosβkt− c2k sin βkt) + dk(t)βk(−c1k sin βkt− c2k cosβkt)

].

Define an operator A to be⎧⎨⎩ (Af)(x) = f ′′(x),∀f ∈ D(A),

D(A) ={f ∈ H2(0, 1)

∣∣f ′(0) = λ

2 f(0), f ′(1) = 0}.

(3.9)

Then we can give the following abstract form for system (3.8):

˙ε = Aε + B[S(t) + T (t)],

where B = δ(x) (δ(·) is the Dirac distribution). It is easy to see that A generates a C0-semigroup eAt of contractions on L2(0, 1) and B is admissible for eAt by invoking Remark 2.6 in [27]. By (3.6) and (3.7), we can conclude that

S ∈ L2(0,∞), T ∈ L∞(0,∞).

By the Lemma 9.1 in [31], we have

supt≥0

‖ε(·, t)‖ = supt≥0

‖εxx(·, t)‖ < ∞.

Hence the trajectory of (2.12)

γ(Z0) = {(ε(x, t), a1(t), · · · , am(t), b1(t), · · · , bm(t), c1(t), · · · , cn(t), d1(t), · · · , dn(t),θ1(t), ϑ1(t), · · · , θm(t), ϑm(t), θ1(t), ϑ1(t), · · · , θn(t), ϑn(t)) | t ≥ 0}

is precompact in H. According to Lasalle’s invariance principle ([26]), any solution of system (3.2) tends to the following maximal invariant set:

S = {(ε(x, t), a1(t), · · · , am(t), b1(t), · · · , bm(t), c1(t), · · · , cn(t), d1(t), · · · , dn(t),θ1(t), ϑ1(t), · · · , θm(t), ϑm(t), θ1(t), ϑ1(t), · · · , θn(t), ϑn(t)) ∈ H | V (t) = 0}.

Since V (t) = 0, it follows that ε(0, t) = 0, 1∫

0

ε2x(x, t) dx = 0, ai ≡ ai0, bi ≡ bi0, ck ≡ ck0, dk ≡ dk0, i ∈ J1,

k ∈ J2. Then from Poincare’s inequality

1∫0

ε2(x, t) dx ≤ 2ε2(0, t) + 4

1∫0

ε2x(x, t) dx, we have

1∫ε2(x, t) dx = 0,

0

W.-W. Liu et al. / J. Math. Anal. Appl. 484 (2020) 123666 15

hence it is easy to obtain ε = 0. Thus,

εx(0, t) = 0 = −m∑i=1

[ai0(t) sinαit + bi0(t) cosαit

]+

n∑k=1

[ck0(c1k sin βkt + c2k cosβkt)

+dk0(c1k cosβkt− c2k sin βkt)].

(3.10)

It is a routine exercise from (3.10) to obtain ai0 = bi0 = ck0 = dk0 = 0. Hence, we have proved that S = {(0, 0, · · · , 0, 0, · · · , 0, 0, · · · , 0, 0, · · · , 0)} × {(θ1, ϑ1, · · · , θm, ϑm, θ1, ϑ1, · · · , θn, ϑn) ∈ R2m+2n|

m∑i=1

(θ2i +

ϑ2i ) +

n∑k=1

(θ2k + ϑ2

k) = m + n}, that is

limt→∞

⎧⎨⎩12

1∫0

ε2(x, t) dx +m∑i=1

[ai

2(t)2r1i

+ bi2(t)

2r2i

]+

n∑k=1

[ck

2(t)2r3k

+ dk2(t)

2r4k

]⎫⎬⎭ = 0. (3.11)

By Definition 2.2, (2.18) then follows from the density argument and conclusion (3.11) just justified for the classical solution.

Now, we claim that for any initial condition (ε(x, 0), a1(0), · · · , am(0), b1(0), · · · , bm(0), c1(0), · · · , cn(0),d1(0), · · · , dn(0)) ∈ H, one has

limt→∞

ε(0, t) = 0.

To this end, we write ε−part of (2.12) as follows:

ε(·, t) = Aε(·, t) + BP(t),

therein, A is given by (3.9), B is δ(x), and

P(t) = −m∑i=1

[ai(t) sinαit + bi(t) cosαit

]+

n∑k=1

[ck(t)(c1k sin βkt + c2k cosβkt)

+dk(t)(c1k cosβkt− c2k sin βkt)].

As mentioned before, we know that A generates an exponentially stable C0−semigroup eAt on L2(0, 1) and

B is admissible for eAt. Thus, for any t > 0, t∫

0

eA(t − s)BP(s)ds ∈ L2(0, 1). Assume that λm ∈ σ(A),

m = 0, 1, 2, · · · . We can get the following eigen-pairs {λm, gm(x)} of A defined by (3.9):

⎧⎪⎨⎪⎩λm = −λ− (mπ)2 + O( 1

m2 ),

gm(x) = cos mπx + O( 1m2 ).

(3.12)

We can see that λm < 0 and λm > λm+1, m = 0, 1, 2, · · · . Since A is self-adjoint, {gm(x)} forms an orthogonal basis for L2(0, 1) and there exist two positive constants m1, m2 independent of m such that

m1 < ‖gm(x)‖ < m2, m = 0, 1, 2, · · · .

From D(A) ⊂ L2(0, 1) ⊂ [D(A)]′, the operator δ(x) in [D(A)]′ can be written as follows:

16 W.-W. Liu et al. / J. Math. Anal. Appl. 484 (2020) 123666

B = δ(x) =∞∑

m=0cmgm(x), (3.13)

where cm = gm(0) = 1 +O( 1m2 ) and |cm| ≤ L1, m = 0, 1, 2, · · · . Thus, the weak solution of ε−part in (2.12)

can be written as

ε(x, t) = eAtε(x, 0) +t∫

0

eA(t−s)BP(s) ds

=∞∑

m=0eλmt〈ε(·, 0), gm(·)〉L2(0,1)gm(x) +

∞∑m=0

cmgm(x)‖gm‖2t∫

0

eλm(t−s)P(s) ds.

(3.14)

Then we can obtain

∣∣∣∣ε(0, t)∣∣∣∣ ≤ ∣∣∣∣ ∞∑m=0

eλmtgm(0)〈ε(·, 0), gm(·)〉L2(0,1)

∣∣∣∣ +∣∣∣∣ ∞∑m=0

c2m‖gm‖2t∫

0

eλm(t−s)P(s) ds∣∣∣∣. (3.15)

For the first term, we have the following estimation

∣∣∣∣ ∞∑m=0

eλmtgm(0)〈ε(·, 0), gm(·)〉L2(0,1)

∣∣∣∣≤[ ∞∑m=0

e2λmt

] 12[ ∞∑m=0

|〈ε(·, 0), gm(·)〉L2(0,1)|2g2m(0)

] 12

≤ L1‖ε(·, 0)‖eλ0t

[ ∞∑m=0

e2(λm−λ0)t1] 1

2

, 0 < t1 < t.

(3.16)

Thus,

limt→∞

∣∣∣∣ ∞∑m=0

eλmtgm(0)〈ε(·, 0), gm(·)〉L2(0,1)

∣∣∣∣ = 0. (3.17)

From (3.11), we have P(t) → 0(t → ∞). Hence for any given η > 0, there exists t2 > 0 large enough such that |P(t)| < η, t ≥ t2,

∣∣∣∣ ∞∑m=0

c2m‖gm‖2t∫

0

eλm(t−s)P(s) ds∣∣∣∣

≤ m22L

21

∞∑m=0

⎡⎣ t2∫0

eλm(t−s)P(s) ds +t∫

t2

eλm(t−s)P(s) ds

⎤⎦

≤ m22L

21

∞∑m=0

⎡⎢⎣⎛⎝ t2∫

0

e2λm(t−s) ds

⎞⎠12⎛⎝ t2∫

0

P(s) ds

⎞⎠12

+ η

t∫t2

eλm(t−s) ds

⎤⎥⎦≤ L2e

λ0(t−t2) + L3η,

(3.18)

where L2 = m22L

21√

−2λ0

⎛⎝ t2∫P(s) ds

⎞⎠12 ∞∑m=0

e(λm−λ0)(t−t2), L3 =∞∑

m=0

m22L

21

−λm. The arbitrariness of η implies

0

W.-W. Liu et al. / J. Math. Anal. Appl. 484 (2020) 123666 17

limt→∞

∣∣∣∣ ∞∑m=0

c2m‖gm‖2t∫

0

eλm(t−s)P(s) ds∣∣∣∣ = 0. (3.19)

By (3.17) and (3.19), we have

limt→∞

ε(0, t) = 0. � (3.20)

Proof of Theorem 2.5

Proof. Due to the equivalence between system (2.27) and system (2.24), we only need to prove that for any initial value �(x, 0) ∈ L2(0, 1) system (2.27) has a unique solution � ∈ C(0, ∞; L2(0, 1)) and asymptotically stabilization in the sense that

limt→∞

⎡⎣12

1∫0

�2(x, t) dx

⎤⎦ = 0. (3.21)

Define an operator A� : D(A�) → L2(0, 1) by⎧⎨⎩ (A�f)(x) = f ′′(x),∀f ∈ D(A�),

D(A�) ={f ∈ L2(0, 1)|f ′(0) = λ

2 f(0), f ′(1) = 0}.

(3.22)

Then we can give the following abstract form for system (2.27):

�(·, t) = A��(·, t) + F1(x)z(1, t) + B�p10z(1, t),

where F1(x) = L(x, t) −K(x, 1)p10 −1∫

x

K(x, y)L(y, t) dy, B� = −δ(x − 1) with δ(·) the Dirac distribution.

Clearly, we can get A� generates a C0-semigroup eA�t of contractions on L2(0, 1). There exists M3, μ3 > 0such that

‖eA�t‖ ≤ M3e−μ3t. (3.23)

It is a routine exercise to obtain that B1 and I are admissible for eA�t. For any initial value �0 ∈ L2(0, 1), system (2.27) has a unique solution � ∈ C(0, ∞; L2(0, 1)), which takes the form

�(·, t) = eA�t�(·, 0) +t∫

0

eA�(t−s)F1(x)z(1, s) ds + p10

t∫0

eA�(t−s)Bz(1, s) ds. (3.24)

The first part is proved. Then we are going to prove that system (2.27) is asymptotically stabilization. Since limt→∞

z(1, s) = 0, we have limt→∞

‖F1(x)z(1, s) ds‖ = 0. Thus from Lemma 9.1 in [31] we obtain

limt→∞

‖�(·, t)‖ = 0. (3.25)

Similar as the proof in Theorem 2.3, we have

lim �(1, t) = 0. (3.26)

t→∞

18 W.-W. Liu et al. / J. Math. Anal. Appl. 484 (2020) 123666

Fig. 1. The displacement of w(x, t) for system (2.31) and z(x, t) for system (2.8).

From backstepping transformation (2.26) and take x = 1, we get

σ(1, t) = �(1, t), (3.27)

thus, we can give

limt→∞

σ(1, t) = 0. � (3.28)

Proof of Theorem 2.6

Proof. From (2.4), (2.23) and z = z + z, one gets the following expression:

w(x, t) = z(x, t) + σ(x, t) +m∑i=1

[ai(t)ui(x, t) + bi(t)vi(x, t)

]−

n∑k=1

[ck(t)ξk(x, t) + dk(t)ηk(x, t)

]+

n∑k=1

[ckpk(x, t) + dkqk(x, t)

].

(3.29)

By (3.29), Theorem 2.4, Theorem 2.5, (2.2), (2.21) and (2.23), we know that system (2.31) has a unique solution. The existence and uniqueness of the solution to system (2.31) is proved. From Theorem 2.4, Theorem 2.5, (2.2), (2.21), (3.29) and (2.23), we can obtain properties (i) and (ii). From Theorem 2.4, Theorem 2.5, (2.6), (2.23) and z = z + z, property (iii) can be obtained. If ai = bi = ck = dk = 0, then ai(t) = bi(t) = ck(t) = dk(t) ≡ 0, ∀t ≥ 0, by the Exercise 2.2 and Section 5.2 in reference [20], it is easy to prove the property (iv). �4. Simulation results

In this section, we give some numerical simulations to show the effectiveness of the proposed adaptive feedback controller. The numerical results are obtained by the finite difference method. For simplicity, we choose i = 1, k = 1. We set the parameters to be that a1 = 1, b1 = 2, c1 = 3, d1 = −1, α1 = 1, β1 = 2, λ = 1, r11 = 1, r21 = 2, r31 = 2, r41 = 1. The initial values for (2.31) and (2.8) are taken as w(x, 0) = sin x, z(x, 0) = 3 cos(2x), z(x, 0) = 3 cosx, a1(0) = −3, b1(0) = 1, c1(0) = 2, d1(0) = −1, a1(0) = 4, b1(0) = −2, c1(0) = 1, d1(0) = 1. It can be seen that the state of w(x, t) is bounded and the error system (2.8) is converged to zero in Fig. 1(a) and Fig. 1(b) respectively. In Fig. 2(a), output signal w(1, t) asymptotically

W.-W. Liu et al. / J. Math. Anal. Appl. 484 (2020) 123666 19

Fig. 2. The output tracking performance and parameters tracking performance. (For interpretation of the colors in the figures, the reader is referred to the web version of this article.)

tracks the reference signal R(t) well. Fig. 2(b) shows that the parameters a1, b1, c1, d1 are approximated by a1(t), b1(t), c1(t), d1(t) respectively.

5. Conclusion

In this paper, the adaptive EFRP for 1D reaction-diffusion equation with general harmonic disturbance is solved. We firstly construct two auxiliary systems in order to make the disturbance become collocated with the control, or make the measured error become the output. Then, based on backstepping approach, the error feedback adaptive servomechanism for the system is constructed in which the parameters of the disturbance and the reference signal are estimated. Moreover, our approach can be applicable to the EFRP for the heat equation with distributed disturbance. The case when the disturbance and reference signals with unknown frequencies maybe the future work.

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