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

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 C 0 -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.


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 wellbehaved 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][4][5][6][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][9][10][11][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 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  1 (, ),  2 (, ) ∈ R are the deviations of substrate and biomass concentrations from steady-state values at the time  and at the point  ∈ (0, 1), respectively.And () ∈ R is the control input, () ∈ R is the measured output, ] > 0 is the fluid superficial velocity,  1 ,  2 ,  3 , and  4 are positive constants, and  ≥ 0 is the length of time delay.
Let us define the operator  ∈ (H) as and consider the properties of a semigroup generated by the operator  −1 .The operator  −1  is expressed as Firstly, we have the following result.
Theorem 1. Suppose that the feedback gain  is chosen such that 0 <  < 1.Then, the operator  defined by (6) generates a uniformly bounded  0 -semigroup () on H.
Proof.In order to prove that  generates a uniformly bounded  0 -semigroup, we introduce a new equivalent inner product in H: From the domain of the operator  −1  it follows that the identities hold for all  ∈ ( −1 ).If the feedback gain  satisfies 0 <  < 1, then it is easy to see Next, for all  ∈ ( −1 ) and  ∈ H, we have From the definition of the adjoint operator and the conditions in the domain of the  −1 , we know that because it is easily verified that  −1  is a closed and densely defined linear operator.Thus, by similar arguments as previously, we obtain that, for all  ∈ (( −1 ) * ), For all  ≥ 0, if we define () by () = () −1 , then the semigroups () and () are similar.This means that the  0 -semigroups () are uniformly bounded (i.e., a  0semigroup with the operator norm bound ‖ ()‖ L(H 1 ) ≤ , for some  > 0 and ∀ ≥ 0) and their generator is .Thus, the proof of the theorem is complete since the new inner product is equivalent to the original one.

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  1 (, ) and  2 (, ) being 2 (] + 1) . ( Proof.First, let us calculate the eigenvalues of the operator . It is easy to see that, for  ∈ C and  = ( 1 (),  2 (),  3 ()) ∈ (),  =  is equivalent to By using similar argument of the appendix of [14], it is easy to know that  = − 4 belongs to the continuous spectrum   () of .When  ̸ = −  4 , solving (21) and ( 22), we have 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  = +, ,  ∈ R, 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   () =  1 ∩  2 .But we remark that the resolvent set () of  is (  1 ∪    2 ) \ {− 4 } (see the Appendix).This means that When  =  +  ∈   (), from the definition of the set  2 ,  must satisfy the inequality which is equivalent to From the inequality and the definition of the (, ), it is obvious that sup {Re () :  ∈  ()} ≤  (, ) .
(41) Lemma 3. Suppose that the assumption of Theorem 1 is satisfied.Then, for any  > 0, 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  0 -semigroup () on H when the feedback gain  is chosen such that  2 < 1/(]) < 1.Then it follows from Theorem 5.3 and Remark 5.4 of [17] that, for any  > 0, there exists some constant  such that holds for all  ∈ C with Re() ≥ .Now, let the subset  1 of the complex domain C be given by In order to apply Theorem 1.1 of [14], it must be shown that First, for each  ∈  1 and each  = ( 1 ,  2 ,  3 )  ∈ H, 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  = ( 1 ,  2 ,  3 )  belongs to the domain () of ,  1 () and  3 () should satisfy the relations Putting  = 1 in ( 50) and ( 51), respectively, we obtain If substitute them into (50) and (51), then we have Noting that we have () for all  ∈ [0, 1].Putting  = 1 in (58), we have Moreover, the continuous function ℎ(, ) := The Cauchy-Schwarz inequality is applied in the last step.This means that in which   Proof.According to Theorem 1.1 of [14], Theorem 4 is direct consequence of Lemmas 2 and 3.

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  0 -semigroup of linear operators if the feedback gain  satisfies 0 <  < 1.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.