ENGEngineering1947-3931Scientific Research Publishing10.4236/eng.2020.129045ENG-103014ArticlesEngineering Boundary Exponential Stabilization of a One-Dimensional Anti-Stable Wave Equation with Control Matched Disturbance RuichengLi1*School of Mathematics and Statistics, Shandong Normal University, Jinan, China0709202012096406511, September 202019, September 2020 22, September 2020© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

In this paper, we are concerned with output feedback stabilization for a one-dimensional anti-stable wave equation with disturbance. First, we design a disturbance estimator for the original system. Then, we propose an output feedback controller for the original system. By calculation, the closed-loop of original system is proved to be exponentially stable and well-posed. Finally, this paper is summarized.

Anti-Stable Wave Equation Disturbance Estimator Exponential Stabilization Output Feedback
1. Introduction

Our main focus is on stabilization for the following anti-stable one-dimensional wave equation with general boundary disturbance:

{ u t t ( x , t ) = u x x ( x , t ) , x ∈ ( 0 , 1 ) , t > 0 u x ( 0 , t ) = − q u t ( 0 , t ) , t ≥ 0 , u x ( 1 , t ) = d ( t ) + U ( t ) , t ≥ 0 , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , x ∈ [ 0 , 1 ] , y 0 = { u ( 0 , t ) , u ( 1 , t ) , u t ( 0 , t ) } , t ≥ 0 , (1.1)

where y 0 is the output (measurement), U ( t ) the input (control), ( u 0 , u 1 ) the initial value, q > 0 , q ≠ 1 , and d ∈ L ∞ ( 0 , ∞ ) or d ∈ L 2 ( 0 , ∞ ) which generally represents an unknown external disturbance. The observer design for system (1.1) can describe the measurement of knocking in the combustion process of automotive engine . For the presence of external disturbance, this system became unstable. A state feedback control was designed in . Next, we consider system (1.1) in the state space H = H 1 ( 0 , 1 ) × L 2 ( 0 , 1 ) . This is enough to show in Feng  (2017), it is obvious that using three output signals in his paper. Besides, five equations are used in his study of stabilization and tracking problems, including three-wave equations and two transport equations in the recent work Zhou  (2017). There exists the approach of “backstepping” in designing controller. Compared with the method he studied, I simplify it. In this paper, I reduce the number of applying dynamic compensators, using three-wave equations and a transport equation to solve the problem. In this paper, we use a new approach to disturbance estimation by directly designing an infinite-dimensional disturbance estimator with using two measurements. Specifically, the stability of closed-loop system is proved by the method of matrix change, which is in the process of innovation.

The paper is organized as follows. In Section 2, we design a disturbance estimator for original system (1.1). The new system is constructed by the known system, and then the disturbance estimator is designed. In Section 3, we are devoted to designing the output feedback control, and proving the exponential stability of the resulting closed-loop system. Finally, gives the concluding remarks.

2. Disturbance Estimator Design

In this section, we devote to design a disturbance estimator for system (1.1). To this end, we first introduce the following coupled system consisting of transport equation and wave equation

{ w t ( x , t ) + w x ( x , t ) = 0 , w ( 0 , t ) = − q + c 0 1 + c 0 u ( 0 , t ) , β t t ( x , t ) = β x x ( x , t ) , β x ( 0 , t ) = c 0 β t ( 0 , t ) + c 1 β ( 0 , t ) + c 1 u ( 0 , t ) + c 1 w ( 0 , t ) , β x ( 1 , t ) = 0 , w ( x , 0 ) = w 0 ( x ) , β ( x , 0 ) = β 0 ( x ) , β t ( x , 0 ) = β 1 ( x ) (2.1)

where c 0 , c 1 > 0 are tuning parameters.

System (2.1) is known since the injection u ( 0 , t ) is one of the original system (1.1).

Let z ( x , t ) = u ( x , t ) + w ( x , t ) + β ( x , t ) . Then z is governed by the following wave equation.

{ z t t ( x , t ) = z x x ( x , t ) , x ∈ ( 0 , 1 ) , t > 0 z x ( 0 , t ) = c 0 z t ( 0 , t ) + c 1 z ( 0 , t ) , t ≥ 0 , z x ( 1 , t ) = U ( t ) + d ( t ) − w t ( 1 , t ) , t ≥ 0 , z ( x , 0 ) = z 0 ( x ) = u 0 ( x ) + w 0 ( x ) + β 0 ( x ) , x ∈ [ 0 , 1 ] , z t ( x , 0 ) = z 1 ( x ) = u 1 ( x ) + w ′ 0 ( x ) + β 1 ( x ) , x ∈ [ 0 , 1 ] , (2.2)

We construct an auxiliary dynamic system as follows:

{ z ^ t t ( x , t ) = z ^ x x ( x , t ) , x ∈ ( 0 , 1 ) , t > 0 z ^ x ( 0 , t ) = c 0 z ^ t ( 0 , t ) + c 1 z ^ ( 0 , t ) , t ≥ 0 , z ^ x ( 1 , t ) = U ( t ) − w t ( 1 , t ) , t ≥ 0 , z ^ ( x , 0 ) = z ^ 0 ( x ) , z ^ t ( x , 0 ) = z ^ 1 ( x ) , x ∈ [ 0 , 1 ] , (2.3)

The error between systems (2.2) and (2.3), z ˜ ( x , t ) = z ( x , t ) − z ^ ( x , t ) , is governed by

{ z ˜ t t ( x , t ) = z ˜ x x ( x , t ) , x ∈ ( 0 , 1 ) , t > 0 z ˜ x ( 0 , t ) = c 0 z ˜ t ( 0 , t ) + c 1 z ˜ ( 0 , t ) , t ≥ 0 , z ˜ x ( 1 , t ) = d ( t ) , t ≥ 0 , z ˜ ( x , 0 ) = z ˜ 0 ( x ) , z ˜ t ( x , 0 ) = z ˜ 1 ( x ) , x ∈ [ 0 , 1 ] , (2.4)

Define the state space H = H 1 ( 0 , 1 ) × L 2 ( 0 , 1 ) for system (2.4) with the norm

‖ ( f , g ) ‖ H ; ‖ · ‖ 1 2 = ∫ 0 1 ( | f ′ ( x ) | 2 + | g ( x ) | 2 ) d x + c 1 | f ( 0 ) | 2 , ∀ ( f , g ) ∈ H , (2.5)

or

‖ ( f , g ) ‖ H ; ‖ · ‖ 2 2 = ∫ 0 1 ( | f ′ ( x ) | 2 + | g ( x ) | 2 ) d x + c 3 | f ( 0 ) | 2 , ∀ ( f , g ) ∈ H , (2.6)

where c 3 > 0 . In the rest of this paper, we write norm ‖ · ‖ H without discrimination.

Define the system operator A : D ( A ) → H

{ A ( f , g ) = ( g , f ″ ) , ∀ ( f , g ) ∈ D ( A ) , D ( A ) = { ( f , g ) ∈ ( H 2 ( 0 , 1 ) × H 1 ( 0 , 1 ) ) | A ( f , g ) ∈ H , f ′ ( 0 ) = c 0 g ( 0 ) + c 1 f ( 0 ) , f ′ ( 1 ) = 0 } (2.7)

Then system (2.4) can be written as an abstract evolutionary equation in H.

d d t ( z ˜ ( · , t ) , z ˜ t ( · , t ) ) = A ( z ˜ ( · , t ) , z ˜ t ( · , t ) ) + B d ( t ) , (2.8)

where B ( 0 , δ ( x − 1 ) ) with δ ( · ) being the Dirac distribution.

The following lemma is Lemma 2.1 in .

Lemma 2.1. Assume that d ∈ L ∞ ( 0 , ∞ ) or d ∈ L 2 ( 0 , ∞ ) . For any initial value ( z ˜ 0 , z ˜ 1 ) ∈ H , system (2.4) admits a unique bounded solution ( z ˜ ( · , t ) , z ˜ t ( · , t ) ) ∈ C ( 0 , ∞ ; H ) .

Next, we design an observer for system (2.4)

{ θ t t ( x , t ) = θ x x ( x , t ) , x ∈ ( 0 , 1 ) , t > 0 θ x ( 0 , t ) = c 0 θ t ( 0 , t ) + c 1 θ ( 0 , t ) , t ≥ 0 , θ x ( 1 , t ) = z ˜ ( 1 , t ) , t ≥ 0 , θ ( x , 0 ) = θ 0 ( x ) , θ t ( x , 0 ) = θ 1 ( x ) , x ∈ [ 0 , 1 ] , (2.9)

Notice that z ˜ ( 1 , t ) = u ( 1 , t ) + w ( 1 , t ) + β ( 1 , t ) − z ^ ( 1 , t ) is known.

Let r ( x , t ) = θ ( x , t ) − z ˜ ( x , t ) = θ ( x , t ) − z ( x , t ) + z ^ ( x , t ) = θ ( x , t ) − u ( x , t ) − w ( x , t ) − β ( x , t ) + z ^ ( x , t ) .

Then r ( x , t ) satisfies

{ r t t ( x , t ) = r x x ( x , t ) , x ∈ ( 0 , 1 ) , t > 0 r x ( 0 , t ) = c 0 r t ( 0 , t ) + c 1 r ( 0 , t ) , t ≥ 0 , r ( 1 , t ) = 0 , t ≥ 0 , r ( x , 0 ) = r 0 ( x ) , r t ( x , 0 ) = r 1 ( x ) , x ∈ [ 0 , 1 ] , (2.10)

Define the space H E 1 ( 0 , 1 ) = { f ∈ H 1 ( 0 , 1 ) | f ( 1 ) = 0 } . Consider system (2.10) in the energy state space H 0 = H E 1 ( 0 , 1 ) × L 2 ( 0 , 1 ) with the norm

‖ ( f , g ) ‖ H 0 2 = ∫ 0 1 ( | f ′ ( x ) | 2 + | g ( x ) | 2 ) d x , ∀ ( f , g ) ∈ H 0 , (2.11)

Define the system operator A 0 : D ( A 0 ) → H 0 for system (2.10) by

{ A 0 ( f , g ) = ( g , f ″ ) , ∀ ( f , g ) ∈ D ( A 0 ) , D ( A 0 ) = { ( f , g ) ∈ ( H 2 ( 0 , 1 ) × H 1 ( 0 , 1 ) ) ∩ H 0 | A 0 ( f , g ) ∈ H 0 , f ′ ( 0 ) = c 0 g ( 0 ) + c 1 f ( 0 ) } (2.12)

Hence, system (2.10) can be written as an evolutionary equation in H 0 :

d d t ( r ( · , t ) , r t ( · , t ) ) = A 0 ( r ( · , t ) , r t ( · , t ) ) , (2.13)

Lemma 2.2. For any initial value ( r 0 , r 1 ) ∈ H , system (2.10) admits a unique solution ( r ( · , t ) , r t ( · , t ) ) ∈ C ( 0 , ∞ ; H 0 ) , which is exponentially stable in the sense that there exist two positive constants L , δ such that

‖ ( r ( · , t ) , r t ( · , t ) ) ‖ H 0 ≤ L e − δ t ‖ ( r 0 , r 1 ) ‖ H 0 , t ≥ 0 (2.14)

Moreover, r x ( 1 , t ) ∈ L 2 ( 0 , ∞ ) and there exists M ′ 1 > 0 such that

∫ t 1 t 2 r x 2 ( 1 , t ) d t ≤ M ′ 1 ( e − 2 δ t 1 + e − 2 δ t 2 ) , ∀ t 2 > t 1 > 0. (2.15)

when the initial value ( r 0 , r 1 ) ∈ D ( A 0 ) , the unique solution of system (2.10) satisfies ( r ( · , t ) , r t ( · , t ) ) ∈ C ( 0 , ∞ ; D ( A 0 ) ) and r x ( 1 , t ) decays exponentially.

Proof. From  and , operator A 0 defined by (2.12) generates an exponentially stable C 0 -semigroup e A 0 t on H 0 . Hence for any initial value ( r 0 , r 1 ) ∈ H , system (2.10) admits a unique solution ( r ( · , t ) , r t ( · , t ) ) ∈ C ( 0 , ∞ ; H 0 ) and (2.14) holds.

Define Lyapunov functions for system (2.10)

E v ( t ) = 1 2 ∫ 0 1 ( r t 2 ( x , t ) + r x 2 ( x , t ) ) d x , p v ( t ) = ∫ 0 1 ( x r t ( x , t ) r x ( x , t ) ) d x . (2.16)

Notice that 2 E v ( t ) = ‖ ( r ( · , t ) , r t ( · , t ) ) ‖ H 0 2 decays exponentially and | p v ( t ) | ≤ E v ( t ) . Differentiate p v ( t ) along the solution of system (2.10) to give

p ˙ v ( t ) = x 2 ( r t 2 ( x , t ) + r x 2 ( x , t ) ) | 0 1 − 1 2 ∫ 0 1 ( r t 2 ( x , t ) + r x 2 ( x , t ) ) d x = 1 2 r x 2 ( 1 , t ) − E v ( t ) (2.17)

Integrating above equation from t 1 to t 2 with respect to t, we have

∫ t 1 t 2 r x 2 ( 1 , t ) d t = 2 ∫ t 1 t 2 E v ( t ) d t + 2 p v ( t 2 ) − 2 p v ( t 1 ) ≤ 2 L 2 E v ( 0 ) 2 δ ( e − 2 δ t 1 − e − 2 δ t 2 ) + 2 E v ( t 2 ) + 2 E v ( t 1 ) ≤ [ L 2 E v ( 0 ) δ + 2 L E v ( 0 ) ] ( e − 2 δ t 1 − e − 2 δ t 2 ) , (2.18)

which implies (2.15) holds.

When the initial value ( r 0 , r 1 ) ∈ D ( A 0 ) , we define a new variable r ¯ ( x , t ) = r t ( x , t ) . Then r ¯ ( x , t ) is governed by the following PDEs

{ r ¯ t t ( x , t ) = r ¯ x x ( x , t ) , x ∈ ( 0 , 1 ) , t > 0 , r ¯ x ( 0 , t ) = c 0 r ¯ t ( 0 , t ) + c 1 r ¯ ( 0 , t ) , t ≥ 0 , r ¯ ( 1 , t ) = 0 , t ≥ 0 , r ¯ ( x , 0 ) = r 1 ( x ) , r ¯ t ( x , 0 ) = r ″ 0 ( x ) , x ∈ [ 0 , 1 ] . (2.19)

Notice that the initial value ( r 1 , r ″ 0 ) ∈ H 0 . System (2.19) is the same as (2.10) except the initial value. From the first assertion, we conclude that

‖ ( r ¯ ( · , t ) , r ¯ t ( · , t ) ) ‖ H 0 ≤ L e − δ t ‖ ( r 1 , r ″ 0 ) ‖ H 0 , t ≥ 0 (2.20)

By Poincaré’s inequality, it follows from the boundary condition of (2.10) at x = 0 that

| r x ( 1 , t ) | 2 ≤ 2 | r x ( 0 , t ) | 2 + 2 ∫ 0 1 | r x x ( x , t ) | 2 d x = 2 | c 0 r ¯ ( 0 , t ) + c 1 r ( 0 , t ) | 2 + 2 ∫ 0 1 | r ¯ t ( x , t ) | 2 d x ≤ 4 c 0 2 ∫ 0 1 | r ¯ x ( x , t ) | 2 d x + 4 c 1 2 r 0 2 + 2 ∫ 0 1 | r ¯ t ( x , t ) | 2 d x ≤ C ′ ( ‖ ( r ( · , t ) , r t ( · , t ) ) ‖ H 2 + ‖ ( r ¯ ( · , t ) , r ¯ t ( · , t ) ) ‖ H 0 2 ) (2.21)

for some constant C ′ > 0 . Hence r x ( 1 , t ) decays exponentially.

Remark 2.1. From Lemma 2.1, observer (2.9) is valid for system (2.4). Moreover, when ( r 0 , r 1 ) ∈ D ( A ) , we can obtain that r x ( 1 , t ) → 0 exponentially from Lemma 2.1. It equals to that θ x ( 1 , t ) → z ˜ x ( 1 , t ) = d ( t ) . Therefore, observer (2.9) can recover state and disturbance simultaneously for (2.4).

3. Output Feedback Controller Design

We design the output feedback of control for system (1.1) as follows

U ( t ) = { − θ x ( 1 , t ) + q + c 0 1 + c 0 u t ( 0 , t − 1 ) − c 3 u ( 1 , t ) + c 3 ( q + c 0 ) 1 + c 0 u ( 0 , t − 1 ) , t > 1 − θ x ( 1 , t ) − c 3 u ( 1 , t ) , t < 1 (3.1)

where c 3 > 0 is design parameter, θ x ( 1 , t ) , u t ( 0 , t − 1 ) , u ( 1 , t ) are given by the system (2.9) and (1.1) respectively.

Under the controller (3.1), next considering the case of t > 1 , the closed-loop system (1.1) can be written as

{ u t t ( x , t ) = u x x ( x , t ) , u x ( 0 , t ) = − q u t ( 0 , t ) , u x ( 1 , t ) = d ( t ) − θ x ( 1 , t ) + q + c 0 1 + c 0 u t ( 0 , t − 1 ) − c 3 u ( 1 , t ) + c 3 ( q + c 0 ) 1 + c 0 u ( 0 , t − 1 ) , w x ( x , t ) + w x ( x , t ) = 0 , w ( 0 , t ) = − q + c 0 1 + c 0 u ( 0 , t ) , β t t ( x , t ) = β x x ( x , t ) , β x ( 0 , t ) = c 0 β t ( 0 , t ) + c 1 β ( 0 , t ) + c 1 u ( 0 , t ) + c 1 w ( 0 , t ) , β x ( 1 , t ) = 0 , z ^ t t ( x , t ) = z ^ x x ( x , t ) , z ^ x ( 0 , t ) = c 0 z ^ t ( 0 , t ) + c 1 z ^ ( 0 , t ) , z ^ x ( 1 , t ) = − θ x ( 1 , t ) − c 3 u ( 1 , t ) + c 3 ( q + c 0 ) 1 + c 0 u ( 0 , t − 1 ) , θ t t ( x , t ) = θ x x ( x , t ) , θ x ( 0 , t ) = c 0 θ t ( 0 , t ) + c 1 θ ( 0 , t ) , θ ( 1 , t ) = u ( 1 , t ) − c 3 ( q + c 0 ) 1 + c 0 u ( 0 , t − 1 ) + β ( 1 , t ) − z ^ ( 1 , t ) , (3.2)

where the initial value of system (3.2) is ( u 0 , u 1 , w 0 , β 0 , β 1 , z ^ 0 , z ^ 1 , θ 0 , θ 1 ) .

Consider system (3.2) in the state space

H ′ = { ( f 1 , g 1 , f 2 , f 3 , g 3 , f 4 , g 4 , f 5 , g 5 ) ∈ H 1 ( 0 , 1 ) × L 2 ( 0 , 1 ) × H 1 ( 0 , 1 )                   × ( H 1 ( 0 , 1 ) × L 2 ( 0 , 1 ) ) 3 | f 2 ( 0 ) = q + c 0 1 + c 0 f 1 ( 0 ) ,                   f 5 ( 1 ) = f 1 ( 1 ) + f 2 ( 1 ) + f 3 ( 1 ) − f 4 ( 1 ) } (3.3)

with the norm

‖ ( f 1 , g 1 , f 2 , f 3 , g 3 , f 4 , g 4 , f 5 , g 5 ) ‖ H ′ 2 = ∫ 0 1 ( | f ′ 1 ( x ) | 2 + | g 1 ( x ) | 2 + | f ′ 2 ( x ) | 2 + | f ′ 3 ( x ) | 2 + | g 3 ( x ) | 2         + | f ′ 4 ( x ) | 2 + | g 4 ( x ) | 2 + | f ′ 5 ( x ) | 2 + | g 5 ( x ) | 2 ) d x         + c 3 | f 1 ( 1 ) | 2 + | f 2 ( 0 ) | 2 + c 1 | f 3 ( 0 ) | 2 + c 1 | f 4 ( 0 ) | 2 (3.4)

Theorem 3.1. Assume that c 0 , c 1 , c 3 > 0 , For any initial value ( u 0 , u 1 , w 0 , β 0 , β 1 , z ^ 0 , z ^ 1 , θ 0 , θ 1 ) ∈ H ′ , System (3.2) admits a unique bounded solution

( u ( · , t ) , u t ( · , t ) , w ( · , t ) , β ( · , t ) , β t ( · , t ) , z ^ ( · , t ) , z t ( · , t ) , θ ( · , t ) , θ t ( · , t ) ) ∈ C ( 0 , ∞ ; H ′ ) .

Moreover, the solution of u-subsystem is exponentially stable in the sense that there exist two positive constants M , ε , such that

‖ u ( · , t ) , u t ( · , t ) ‖ H ≤ M e − ε t ‖ ( u 0 , u 1 , w 0 , β 0 , β 1 , z ^ 0 , z ^ 1 , θ 0 , θ 1 ) ‖ H , ∀ t ≥ 0 (3.5)

Proof. Introducing a new variable γ ( x , t ) = u ( x , t ) + w ( x , t ) and the variables z ˜ ( x , t ) , r ( x , t ) in Section 2, that is,

( γ w β z ˜ r ) = ( 1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 1 1 − 1 0 − 1 − 1 − 1 1 1 ) ( u w β z ^ θ ) , (3.6)

we can transform system (3.2) into the following equivalent system

{ γ t t ( x , t ) = γ x x ( x , t ) , γ x ( 0 , t ) = c 0 γ t ( 0 , t ) , γ x ( 1 , t ) = − c 3 γ ( 1 , t ) − r x ( 1 , t ) , w x ( x , t ) + w x ( x , t ) = 0 , w ( 0 , t ) = − q + c 0 1 − q γ ( 0 , t ) , β t t ( x , t ) = β x x ( x , t ) , β x ( 0 , t ) = c 0 β t ( 0 , t ) + c 1 β ( 0 , t ) + c 1 γ ( 0 , t ) , β x ( 1 , t ) = 0 , z ˜ t t ( x , t ) = z ˜ x x ( x , t ) , z ˜ x ( 0 , t ) = c 0 z ˜ t ( 0 , t ) + c 1 z ˜ ( 0 , t ) , z ˜ x ( 1 , t ) = d ( t ) , r t t ( x , t ) = r x x ( x , t ) , r x ( 0 , t ) = c 0 r t ( 0 , t ) + c 1 r ( 0 , t ) , r ( 1 , t ) = 0. (3.7)

where the initial value

{ γ ( x , 0 ) = γ 0 ( x ) = u 0 ( x ) + w 0 ( x ) , γ t ( x , 0 ) = γ 1 ( x ) = u 1 ( x ) − w ′ 0 ( x ) , w ( x , 0 ) = w 0 ( x ) , β ( x , 0 ) = β 0 ( x ) , β t ( x , 0 ) = β 1 ( x ) , z ˜ ( x , 0 ) = z ˜ 0 ( x ) = u 0 ( x ) + w 0 ( x ) + β 0 ( x ) − z ^ 0 ( x ) , z ˜ t ( x , 0 ) = z ˜ 1 ( x ) = u 1 ( x ) + w ′ 0 ( x ) + β 1 ( x ) − z ^ 1 ( x ) , r ( x , 0 ) = r 0 ( x ) = θ 0 ( x ) + z ^ 0 ( x ) − u 0 ( x ) − w 0 ( x ) − β 0 ( x ) , r t ( x , 0 ) = r 1 ( x ) = θ 1 ( x ) + z ^ 1 ( x ) − u 1 ( x ) − w ′ 0 ( x ) − β 1 ( x ) . (3.8)

We will consider the r-subsystems of (3.7) respectively. For any initial value ( u 0 , u 1 , w 0 , β 0 , β 1 , z ^ 0 , z ^ 1 , θ 0 , θ 1 ) ∈ H ′ , we have ( γ 0 , γ 1 , w 0 , β 0 , β 1 , z ^ 0 , z ^ 1 , r 0 , r 1 ) ∈ H × H 1 ( 0 , 1 ) × H 2 × H 0 .

Since ( r 0 , r 1 ) ∈ H 0 , the z ˜ -subsystem of (3.7) admits a unique exponentially stable solution ( r ( · , t ) , r t ( · , t ) ) ∈ C ( 0 , ∞ ; H 0 ) guaranteed by Lemma 2.2.

Since ( z ˜ 0 , z ˜ 1 ) ∈ H , the -subsystem of (3.7) admits a unique exponentially stable solution ( z ˜ ( · , t ) , z ˜ t ( · , t ) ) ∈ C ( 0 , ∞ ; H ) guaranteed by Lemma 2.1.

Define the system operator A 1 : D ( A 1 ) → H

{ A 1 ( f , g ) = ( g , f ″ ) , ∀ ( f , g ) ∈ D ( A 1 ) , D ( A 1 ) = { ( f , g ) ∈ ( H 2 ( 0 , 1 ) × H 1 ( 0 , 1 ) ) | A ( f , g ) ∈ H , f ′ ( 0 ) = c 0 g ( 0 ) , f ′ ( 1 ) = − c 3 f ( 1 ) } . (3.9)

Then the γ-subsystem can be written as an abstract evolutionary equation in H

d d t ( γ ( · , t ) , γ t ( · , t ) ) = A 1 ( γ ( · , t ) , γ t ( · , t ) ) + B 1 ( − r x ( 1 , t ) ) , (3.10)

where B 1 = ( 0 , δ ( x − 1 ) ) .

It is well-known that operator A 1 generates an exponentially stable C 0 -semigroup e A 1 t on H. It is a routine exercise that B 1 is admissible for e A 1 t (  ). Then the solution to system (3.9) can be written as

( γ ( · , t ) , γ t ( · , t ) ) = e A 1 t ( γ 0 , γ 1 ) + ∫ 0 t e A 1 ( t − s ) B 1 ( − γ x ( 1 , s ) ) d s . (3.11)

From (2.15) in Lemma 2.2 and the estimation in

‖ ∫ 0 t e A ( t − s ) B 1 ε s ( 0 , s ) d s ‖ H 2 ≤ 2 M 1 2 e − β t K t M 2 k 1 E ε ( 0 ) ( 1 + e − α 2 t 2 ) + 4 K 1 M 2 k 1 E ε ( 0 ) e − α 2 t 2

(61) of , we obtain the exponential stability of γ-subsystem in the sense that there exist two positive constants M 2 , δ 2 such that

‖ γ ( · , t ) , γ t ( · , t ) ‖ H ≤ M 2 e − δ 2 t ‖ ( γ 0 , γ 1 ) ‖ H , ∀ t ≥ 0. (3.12)

Define Lyapunov functions for γ-subsystem

E γ ( t ) = 1 2 ∫ 0 1 ( γ t 2 ( x , t ) + γ x 2 ( x , t ) ) d x + c 3 2 | γ ( 1 , t ) | 2 , p γ ( t ) = ∫ 0 1 x γ t ( x , t ) γ x ( x , t ) d x . (3.13)

Notice that 2 E γ t = ‖ γ ( · , t ) , γ t ( · , t ) ‖ H 2 decays exponentially and | p γ ( t ) | ≤ E γ ( t ) . Differentiate p γ ( t ) along the solution of γ-subsystem to give

p ˙ γ ( t ) = x − 1 2 ( γ x 2 ( x , t ) + γ t 2 ( x , t ) ) | 0 1 − 1 2 ∫ 0 1 ( γ t 2 ( x , t ) + γ x 2 ( x , t ) ) d x = 1 2 ( γ x 2 ( 0 , t ) + γ t 2 ( 0 , t ) ) − 1 2 ∫ 0 1 ( γ t 2 ( x , t ) + γ x 2 ( x , t ) ) d x (3.14)

Integrating above equation from t − 1 to t, we have

∫ t − 1 t ( γ x 2 ( 0 , s ) + γ t 2 ( 0 , s ) ) d s ≤ 2 ∫ t − 1 t E γ ( s ) d s + 2 p γ ( t ) − 2 p γ ( t − 1 ) ≤ 2 M 2 2 E γ ( 0 ) 2 δ 2 ( e − 2 δ 2 ( t − 1 ) − e − 2 δ 2 t ) + 2 E γ ( t ) + 2 E γ ( t − 1 ) ≤ [ M 2 2 E γ ( 0 ) δ 2 e 2 δ 2 + 2 M 2 E γ ( 0 ) + 2 M 2 E γ ( 0 ) e 2 δ 2 ] e − 2 δ 2 t (3.15)

For the w-subsystem, the exact solution can be written as

w ( x , t ) = { w 0 ( x − t ) , x ≥ t , − q + c 0 1 − q γ ( 0 , t − x ) , t ≥ x . (3.16)

which has the estimation

∫ 0 1 ( | w t ( x , t ) | 2 + | w x ( x , t ) | 2 ) d x ≤ 2 ( q + c 0 ) 2 ( 1 − q ) 2 ∫ 0 1 | γ t ( 0 , t − x ) | 2 d x = 2 ( q + c 0 ) 2 ( 1 − q ) 2 ∫ t − 1 t | γ t ( 0 , s ) | 2 d s , t ≥ 1 (3.17)

| w ( 0 , t ) | 2 = ( q + c 0 ) 2 ( 1 − q ) 2 | γ ( 0 , t ) | 2

From (3.12) and (3.15), ‖ w ( · , t ) , w t ( · , t ) ‖ H decays exponentially.

For β-subsystem, we can write it into an abstract evolutionary equation in H.

d d t ( β ( · , t ) , β t ( · , t ) ) = A ( β ( · , t ) , β t ( · , t ) ) + B 3 ( − c 3 γ ( 0 , t ) ) , (3.18)

where A is defined by (2.7) and B 3 = ( 0 , δ ( x ) ) . Direct computations indicate that B 3 is admissible for e A t . Hence the solution to (3.18) can be written as

( β ( · , t ) , β t ( · , t ) ) = e A t ( β 0 , β 1 ) + ∫ 0 t e A ( t − s ) B 3 ( − c 3 γ ( 0 , s ) ) d s , (3.19)

which is exponentially stable because of the exponential decay of γ ( 0 , t ) . Hence we obtain that system (3.7) admits a unique bounded solution

By the bounded inverse transformation of (3.6), we obtain the solution to system (3.2) as follows

{ u ( x , t ) = γ ( x , t ) − w ( x , t ) , u t ( x , t ) = γ t ( x , t ) − w t ( x , t ) , w ( x , t ) = w ( x , t ) , β ( x , t ) = β ( x , t ) , β t ( x , t ) = β t ( x , t ) , z ^ ( x , t ) = γ ( x , t ) + β ( x , t ) − z ^ ( x , t ) , z ^ t ( x , t ) = γ t ( x , t ) + β t ( x , t ) − z ^ t ( x , t ) , θ ( x , t ) = r ( x , t ) + z ˜ ( x , t ) , θ t ( x , t ) = r t ( x , t ) + z ˜ t ( x , t ) . (3.20)

Hence system (3.2) admits a unique bounded solution. From the exponential stability of γ- and w-subsystems, u-subsystem is exponentially stable.

4. Conclusion Remarks

In this paper, the anti stable wave equation with disturbance is controlled. Firstly, the disturbance estimator is designed for the original system, and then the disturbance is estimated. Then, the controller is designed to realize the control. In addition, many references, especially the relevant data about ADRC, are used to verify the closed-loop system, the closed-loop system is simplified and the content of this paper is obtained.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

Cite this paper

Li, R.C. (2020) Boundary Exponential Stabilization of a One-Dimensional Anti-Stable Wave Equation with Control Matched Disturbance. Engineering, 12, 640-651. https://doi.org/10.4236/eng.2020.129045

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