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Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 274857, 9 pages

http://dx.doi.org/10.1155/2013/274857

## Exponential Stability of a Linear Distributed Parameter Bioprocess with Input Delay in Boundary Control

^{1}Department of Mathematics, Bohai University, Jinzhou 121013, China^{2}Beijing Institute of Information and Control, Beijing 100037, China

Received 9 April 2013; Accepted 20 July 2013

Academic Editor: Gen-Qi Xu

Copyright © 2013 Fu Zheng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider a linear distributed parameter bioprocess with boundary control input possessing a time delay. Using a simple boundary feedback law, we show that the closed-loop system generates a uniformly bounded -semigroup of linear operators under a certain condition with respect to the feedback gain. After analyzing the spectrum configuration of closed-loop system and verifying the spectrum determined growth assumption, we show that the closed-loop system is exponentially stable. Thus, we demonstrate that the linear distributed parameter bioprocess preserves the exponential stability for arbitrary time delays.

#### 1. Introduction

In a practical control system, there is often a time delay between the controller to be implemented and the information via the observation of the system. These hereditary effects are sometime unavoidable because they might turn a well-behaved system into a wild one. A simple example can be found in Gumowski and Mira [1], where they demonstrated that the occurrence of delays could destroy the stability and cause periodic oscillations in a system governed by differential equation. Datko [2, 3] illustrated that an arbitrary small time delay in the control could destabilize a boundary feedback hyperbolic control system as well. On the other side, the inclusion of an appropriate time delay effect can sometime improve the performance of the system (e.g., see [3–7]). When the time delay appears, redesigning a stabilizing controller becomes thereby sometimes necessary because the stabilization by the PI output feedback becomes defective or the stabilization is not robust to time delay. The stabilization with time delay in observation or control represents difficult mathematical challenges in the control of distributed parameter systems. However, this does not mean that there is no stabilizing controller in the presence of time delay. You can refer to [8–12] for some successful examples.

Motivated by these works, we will introduce time delays to a linear distributed parameter bioprocess and investigate the effect of time delays on exponential stability of the system. The linear distributed parameter bioprocess treated here was firstly discussed by Bourrel and Dochain in [13]. They showed that the system with zero boundary input is exponentially stable. Following [13], Sano considered the linear distributed parameter bioprocess from the feedback control point of view in [14]. Namely, the control input and the measured output were imposed on the boundaries, and a simply proportional feedback controller was designed. By using Huang’s result in [15], He showed that the closed-loop system is exponentially stable under a certain condition with respect to the feedback gain and further that the exponential decay rate of the system with zero input was derived by letting the feedback gain tend to zero. However, if time delays in the boundary input arise in this linear distributed parameter bioprocess, we want to pose a question. Is the stabilization robust to time delays for the proportional feedback controller? The present paper is devoted to answering this question.

The content of this paper is organized as follows. In Section 2 we will introduce the linear distributed parameter bioprocess mentioned previously and formulate our problem in a suitable Hilbert space. We show that the closed-loop system generates a uniformly bounded -semigroup of linear operators and obtain the wellposedness of the system as well. In Section 3, we carry out a spectral analysis and obtain the spectrum configuration of the closed-loop system. From verifying the spectrum determined growth assumption, we show that the closed-loop system is exponentially stable. In the last section, a concise conclusion is given.

#### 2. System Description and Wellposedness of the System

We will consider the following type of linear distributed parameter bioprocess model in which time delays occur in boundary control input: where , are the deviations of substrate and biomass concentrations from steady-state values at the time and at the point , respectively. And is the control input, is the measured output, is the fluid superficial velocity, , , , and are positive constants, and is the length of time delay.

As usual, we adopt the simple feedback control law with which results in the following closed-loop system:

Setting , (2) is equivalent to

We take the state Hilbert space , equipped with inner product Define the operator as Then the system (3) can be written as where , , . Therefore, if the operator generates a -semigroup on , then a unique solution of (7) is expressed as which means that the unique solution to (2) or (3) exists.

Let us define the operator as and consider the properties of a semigroup generated by the operator . The operator is expressed as

Firstly, we have the following result.

Theorem 1. *Suppose that the feedback gain is chosen such that . Then, the operator defined by (6) generates a uniformly bounded -semigroup on . *

*Proof. *In order to prove that generates a uniformly bounded -semigroup, we introduce a new equivalent inner product in :
From the domain of the operator it follows that the identities
hold for all . If the feedback gain satisfies , then it is easy to see

Next, for all and , we have
From the definition of the adjoint operator and the conditions in the domain of the , we know that
because it is easily verified that is a closed and densely defined linear operator. Thus, by similar arguments as previously, we obtain that, for all ,

It follows from and that the operators and are dissipative. According to Proposition 3.1.11 of [16], the closed operator is m-dissipative. Therefore, Lumer-Philips theorem implies that generates contraction semigroups on state space . For all , if we define by , then the semigroups and are similar. This means that the -semigroups are uniformly bounded (i.e., a -semigroup with the operator norm bound , for some and ) and their generator is . Thus, the proof of the theorem is complete since the new inner product is equivalent to the original one.

#### 3. Exponential Stability of the System (7)

In order to show the exponential stability of the system (7), we will verify that the operator satisfies the conditions of Theorem 1.1 of [14], which is a summarized edition of Huang’s result on the spectrum determined growth assumption in [15]. To this end, we should analyze the spectrum configuration of the operator and show that the norm of the resolvent is uniformly bounded in any given right half-plane. All these results are collected in the following two lemmas.

Lemma 2. *Suppose that the assumption of Theorem 1 is satisfied. Then, the following inequality holds:
**
where is defined by
**
with and being*

*Proof. *First, let us calculate the eigenvalues of the operator . It is easy to see that, for and , is equivalent to
By using similar argument of the appendix of [14], it is easy to know that belongs to the continuous spectrum of . When , solving (21) and (22), we have
Set
Substituting (24) to (20), we have
It follows from (23), (25), and (27) that

In order to solve (28) with respect to , let us set , , and
Then, (28) becomes
which is equivalent to
Thus, it follows from the previous equations that
which are equivalent to
Combining (33) with (34), we get
On the other hand, solving (33) with respect to , we have
As a result, introducing two sets
we see that the point spectrum of is given by . But we remark that the resolvent set of is (see the Appendix). This means that

When , from the definition of the set , must satisfy the inequality
which is equivalent to
From the inequality and the definition of the , it is obvious that

Lemma 3. *Suppose that the assumption of Theorem 1 is satisfied. Then, for any , the following holds:
**
in which is the number defined in Lemma 2. *

*Proof. *In Theorem 1, it is shown that the operator generates a uniformly bounded -semigroup on when the feedback gain is chosen such that . Then it follows from Theorem 5.3 and Remark 5.4 of [17] that, for any , there exists some constant such that
holds for all with .

Now, let the subset of the complex domain be given by
In order to apply Theorem 1.1 of [14], it must be shown that
First, for each and each , we consider the resolvent equation , which is equivalent to
Solving (47) and (48), we have
Substituting (49) with (46) and solving it, we have
Since belongs to the domain of , and should satisfy the relations
Putting in (50) and (51), respectively, we obtain
If substitute them into (50) and (51), then we have
Also, can be obtained from the previous equations and (49).

Next, we will estimate a bound of norm of . It follows from
that
Noting that
we have
for all . Putting in (58), we have
Moreover, the continuous function defined on the compact set has absolute maximum and absolute minimum, which are denoted by and , respectively. Thus, we have
The Cauchy-Schwarz inequality is applied in the last step. This means that
in which
Similarly, we have
This means that
in which
It follows from (49) and (62) that
This implies that
in which

Therefore, by using (62), (65), and (68), we can estimate a bound of norm of as follows:
which is equivalent to
Since the inequality holds for all and for all , we have
This shows that (45) holds. In this way, we finally obtain

Theorem 4. *Suppose that the assumption of Theorem 1 is satisfied. Then, for any , there exists a constant such that
**
in which is defined in Lemma 2.*

*Proof. *According to Theorem 1.1 of [14], Theorem 4 is direct consequence of Lemmas 2 and 3.

#### 4. Conclusion

In the present paper, we have considered a linear distributed parameter bioprocess with boundary control input possessing a time delay. Using a simple boundary feedback law, we have shown that the closed-loop system generates a uniformly bounded -semigroup of linear operators if the feedback gain satisfies . After analyzing the spectrum configuration of the closed-loop system and verifying the spectrum determined growth assumption, we have demonstrated that the closed-loop system becomes exponentially stable. Our main result implies that the linear distributed parameter bioprocess preserves the exponential stability for arbitrary time delay. This means that the answer to the question posed in Section 1 is positive.

#### Appendix

Let and be the sets defined in the proof of Lemma 2. To show that the resolvent set of is , we have to prove that the operator is bijective for each . Thus, for each and each , we consider the resolvent equation , which is equivalent to It follows from the proof of Lemma 3 that It is easy to see that is bijective if and only if . From the proof of Lemma 3, we know that for . This implies that .

#### Acknowledgment

This work was supported by the National Natural Science Foundation of China (11201037).

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