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

Authors: Fu Zheng; LingLing Fu; MingYan Teng
Journal: Journal of Function Spaces and Applications
 (2013)

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## 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 boundedC0-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.


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## Body

## 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 Section2 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 C0-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:(1)∂z1∂t(t,x)=-ν∂z1∂x(t,x)-a1z1(t,x)-a2z2(t,x),(t,x)∈(0,∞)×(0,1),∂z2∂t(t,x)=a3z1(t,x)-a4z2(t,x),(t,x)∈(0,∞)×(0,1),z1(t,0)=u(t-τ),t∈(0,∞),z1(0,x)=z10(x),z2(0,x)=z20(x),x∈(0,1),y(t)=z1(t,1),t∈(0,∞),
where z1(t,x), z2(t,x)∈ℝ are the deviations of substrate and biomass concentrations from steady-state values at the time t and at the point x∈(0,1), respectively. And u(t)∈ℝ is the control input, y(t)∈ℝ is the measured output, ν>0 is the fluid superficial velocity, a1, a2, a3, and a4 are positive constants, and τ≥0 is the length of time delay.As usual, we adopt the simple feedback control lawu(t)=-ky(t) with k>0 which results in the following closed-loop system:
(2)∂z1∂t(t,x)=-ν∂z1∂x(t,x)-a1z1(t,x)-a2z2(t,x),(t,x)∈(0,∞)×(0,1),∂z2∂t(t,x)=a3z1(t,x)-a4z2(t,x),(t,x)∈(0,∞)×(0,1),z1(t,0)=-kz1(t-τ,1),t∈(0,∞),z1(0,x)=z10(x),z2(0,x)=z20(x),x∈(0,1).Settingz3(t,x)=z1(t-xτ,1), (2) is equivalent to
(3)∂z1∂t(t,x)=-ν∂z1∂x(t,x)-a1z1(t,x)-a2z2(t,x),(t,x)∈(0,∞)×(0,1),∂z2∂t(t,x)=a3z1(t,x)-a4z2(t,x),(t,x)∈(0,∞)×(0,1),∂z3∂t(t,x)=-τ-1∂z3∂x(t,x),(t,x)∈(0,∞)×(0,1),z3(t,0)=z1(t,1),z1(t,0)=-kz3(t,1),t∈(0,∞),z1(0,x)=z10(x),z2(0,x)=z20(x),z3(0,x)=z30(-τx),x∈(0,1).We take the state Hilbert spaceℋ,
(4)ℋ=L2[0,1]×L2[0,1]×L2[0,1]=(L2[0,1])3,
equipped with inner product
(5)〈f,g〉ℋ=∫01‍f1(x)g1(x)¯dx+∫01‍f2(x)g2(x)¯dx+∫01‍f3(x)g3(x)¯dx,f,g∈ℋ.
Define the operator A:D(A)⊂ℋ→ℋ as
(6)Af=(-νddx-a1-a20a3-a4000-τ-1ddx)(f1f2f3),f=(f1,f2,f3)⊥∈D(A),D(A)={f∈H1(0,1)×L2[0,1]×H1(0,1)∣f1(0)=-kf3(1),f1(1)=f3(0)}.
Then the system (3) can be written as
(7)ddt(z1(t)z2(t)z3(t))=A(z1(t)z2(t)z3(t)),(z1(0)z2(0)z3(0))=(z10z20z30),
where z10, z20, z30∈L2[0,1]. Therefore, if the operator A generates a C0-semigroup T(t) on ℋ, then a unique solution of (7) is expressed as
(8)(z1(t)z2(t)z3(t))=T(t)(z10z20z30),∀t≥0,
which means that the unique solution to (2) or (3) exists.Let us define the operatorM∈L(ℋ) as
(9)M=(a2000a30001)
and consider the properties of a semigroup generated by the operator M-1AM. The operator M-1AM is expressed as
(10)M-1AMf=(-νddx-a1-a2a30a2a3-a4000-τ-1ddx)(f1f2f3),f=(f1,f2,f3)⊥∈D(M-1AM),D(M-1AM)={f∈H1(0,1)×L2[0,1]×H1(0,1)∣f1(0)=-kf3(1),f1(1)=f3(0)f∈H1(0,1)×L2[0,1]×H1(0,1)}.Firstly, we have the following result.Theorem 1.
Suppose that the feedback gaink is chosen such that 0<k<1. Then, the operator A defined by (6) generates a uniformly bounded C0-semigroup T(t) on ℋ.Proof.
In order to prove thatA generates a uniformly bounded C0-semigroup, we introduce a new equivalent inner product in ℋ:
(11)〈f,g〉ℋ1=1ν∫01‍f1(x)g1(x)¯dx+1ν∫01‍f2(x)g2(x)¯dx+τ∫01‍f3(x)g3(x)¯dx,f,g∈ℋ.
From the domain of the operator M-1AM it follows that the identities
(12)Re〈M-1AMf,f〉ℋ1=-a1ν∥f1∥L2(0,1)-a4ν∥f2∥L2(0,1)-(|f1(1)|2-|f1(0)|2)-(|f3(1)|2-|f3(0)|2)=-a1ν∥f1∥L2(0,1)-a4ν∥f2∥L2(0,1)-(1-k2)|f3(1)|2
hold for all f∈D(M-1AM). If the feedback gain k satisfies 0<k<1, then it is easy to see
(13)Re〈M-1AMf,f〉ℋ1≤0,∀f∈D(M-1AM).
Next, for allf∈D(M-1AM) and g∈ℋ, we have
(14)〈M-1AMf,g〉ℋ1=1ν∫01‍(-νf1′(x)-a1f1(x))g1(x)¯dx-a4ν∫01‍f2(x)g2(x)¯dx-a2a3ν∫01‍f2(x)g1(x)¯dx+a2a3ν∫01‍f1(x)g2(x)¯dx-∫01‍f3′(x)g3(x)¯dx=1ν∫01‍f1(x)(νg1′(x)¯-a1g1(x)¯+a2a3g2(x)¯)dx-1ν∫01‍f2(x)(a4g2(x)¯+a2a3g1(x)¯)dx+1τ∫01‍f3(x)g3′(x)¯dx-[f1(x)g1′(x)¯]01-1τ[f3(x)g3(x)¯]01.
From the definition of the adjoint operator and the conditions in the domain of the M-1AM, we know that
(15)(M-1AM)*f=(νddx-a1a2a30-a2a3-a4000τ-1ddx)(f1f2f3),f=(f1,f2,f3)⊥∈D((M-1AM)*),D((M-1AM)*)={f∈H1(0,1)×L2[0,1]×H1(0,1)∣f3(1)=-kf1(0),f3(0)=f1(1)f∈H1(0,1)×L2[0,1]×H1(0,1)}
because it is easily verified that M-1AM is a closed and densely defined linear operator. Thus, by similar arguments as previously, we obtain that, for all f∈D((M-1AM)*),
(16)Re〈(M-1AM)*f,f〉ℋ1=-a1ν∥f1∥L2(0,1)-a4ν∥f2∥L2(0,1)+(|f1(1)|2-|f1(0)|2)+(|f3(1)|2-|f3(0)|2)=-a1ν∥f1∥L2(0,1)-a4ν∥f2∥L2(0,1)-(1-k2)|f1(0)|2≤0.
It follows fromRe〈M-1AMf,f〉ℋ1≤0 and Re〈(M-1AM)*f,f〉ℋ1≤0 that the operators M-1AM and (M-1AM)* are dissipative. According to Proposition 3.1.11 of [16], the closed operator M-1AM is m-dissipative. Therefore, Lumer-Philips theorem implies that M-1AM generates contraction semigroups S(t) on state space ℋ. For all t≥0, if we define T(t) by T(t)=MS(t)M-1, then the semigroups T(t) and S(t) are similar. This means that the C0-semigroups T(t) are uniformly bounded (i.e., a C0-semigroup with the operator norm bound ∥T(t)∥ℒ(ℋ1)≤M, for some M>0 and ∀t≥0) and their generator is A. 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 A 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 A 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 Theorem1 is satisfied. Then, the following inequality holds:
(17)sup{Re(λ):λ∈σ(A)}≤μ(k,τ),
where μ(k,τ) is defined by
(18)μ(k,τ):={max{-a4,-a1-νlogkντ+1,β1(k,τ),β2(k,τ)}if(a1+a4-νlogk)2ντ+1>4[a2a3+a4(a1-νlogk)],max{-a4,-a1-νlogkντ+1,-(a1+a4-νlogk)22(ντ+1)}if(a1+a4-νlogk)2ντ+1=4[a2a3+a4(a1-νlogk)],μ(k,τ):=max{-a4,-a1+νlogkντ+1}μ(k,τ):=if(a1+a4-νlogk)2ντ+1<4[a2a3+a4(a1-νlogk)],
withβ1(k,τ) and β2(k,τ) being(19)β1(k,τ)=-(a1+a4-νlogk)+(a1+a4-νlogk)2-4(ντ+1)[a2a3+a4(a1-νlogk)]2(ντ+1),β2(k,τ)=-(a1+a4-νlogk)-(a1+a4-νlogk)2-4(ντ+1)[a2a3+a4(a1-νlogk)]2(ντ+1).Proof.
First, let us calculate the eigenvalues of the operatorA. It is easy to see that, for λ∈ℂ and f=(f1(x),f2(x),f3(x))∈D(A), Af=λf is equivalent to
(20)-νf1′(x)-a1f1(x)-a2f2(x)=λf1(x),(21)a3f1(x)(x)-a4f2(x)=λf2(x),(22)-τ-1f3′(x)=λf3(x),(23)f1(0)=-kf3(1),f3(0)=f1(1).
By using similar argument of the appendix of [14], it is easy to know that λ=-a4 belongs to the continuous spectrum σC(A) of A. When λ≠-a4, solving (21) and (22), we have
(24)f2(x)=a3λ+a4f1(x),(25)f3(x)=f3(0)e-τλx.
Set
(26)α(λ)=-1ν(λ+a1+a2a3λ+a4),β(λ)=τλ-α(λ).
Substituting (24) to (20), we have
(27)f1(x)=f1(0)eα(λ)x.
It follows from (23), (25), and (27) that
(28)eβ(λ)=-k.
In order to solve (28) with respect to λ, let us set λ=x+iy, x,y∈ℝ, and
(29)u(x,y)=(ντ+1)x+a1+a2a3(x+a4)(x+a4)2+y2,v(x,y)=y[1+ντ-a2a3(x+a4)2+y2].
Then, (28) becomes
(30)eβ(λ)=e(1/ν)[u(x,y)+iv(x,y)]=-k,
which is equivalent to
(31)e(1/ν)u(x,y)cos1νv(x,y)=-k,sin1νv(x,y)=0.
Thus, it follows from the previous equations that
(32)u(x,y)=νlogk,v(x,y)=(2n+1)νπ,n∈ℤ,
which are equivalent to
(33)(ντ+1)x+a1+a2a3(x+a4)(x+a4)2+y2=νlogk,(34)y[1+ντ-a2a3(x+a4)2+y2]=(2n+1)νπ.
Combining (33) with (34), we get
(35)y=(2n+1)νπ(x+a4)2(ντ+1)x+a1+a4(ντ+1)-νlogk,n∈ℤ.
On the other hand, solving (33) with respect to y, we have
(36)y=±a2a3(x+a4)νlogk-(ντ+1)x-a1-(x+a4)2.
As a result, introducing two sets
(37)S1={x+iy:y=(2n+1)νπ(x+a4)2(ντ+1)x+a1+a4(ντ+1)-νlogk,x,y∈ℝ,x≠a4,n∈ℤ(2n+1)νπ(x+a4)2(ντ+1)x+a1+a4(ντ+1)-νlogk},S2={a2a3(x+a4)νlogk-(ντ+1)x-a1-(x+a4)2x+iy:y=±a2a3(x+a4)νlogk-(ντ+1)x-a1-(x+a4)2,x,y∈ℝ,x≠a4a2a3(x+a4)νlogk-(ντ+1)x-a1-(x+a4)2},
we see that the point spectrum σp(A) of A is given by σp(A)=S1∩S2. But we remark that the resolvent set ρ(A) of A is (S1c∪S2c)∖{-a4} (see the Appendix). This means that
(38)σ(A)=(S1∩S2)∪{-a4}.
Whenλ=x+iy∈σp(A), from the definition of the set S2, x must satisfy the inequality
(39)a2a3(x+a4)νlogk-(ντ+1)x-a1-(x+a4)2≥0,
which is equivalent to
(40)(x+a4)(x-νlogk-a1ντ+1)×(a2a3+a4(a1-νlogk)ντ+1x2+a1-νlogk+a4ντ+1x+a2a3+a4(a1-νlogk)ντ+1)≤0.
From the inequality and the definition of the μ(k,τ), it is obvious that
(41)sup{Re(λ):λ∈σ(A)}≤μ(k,τ).Lemma 3.
Suppose that the assumption of Theorem1 is satisfied. Then, for any ε>0, the following holds:
(42)sup{∥(λI-A)-1∥ℒ(ℋ):λ∈ℂandReλ≥μ(k,τ)+ε}<∞,
in which μ(k,τ) is the number defined in Lemma 2.Proof.
In Theorem1, it is shown that the operator A generates a uniformly bounded C0-semigroup T(t) on ℋ when the feedback gain k is chosen such that k2<1/(ντ)<1. Then it follows from Theorem 5.3 and Remark 5.4 of [17] that, for any ε>0, there exists some constant M such that
(43)∥(λI-A)-1∥ℒ(ℋ)≤MRe(λ)≤Mε
holds for all λ∈ℂ with Re(λ)≥ε.
Now, let the subsetE1 of the complex domain ℂ be given by
(44)E1={λ∈ℂ:μ(k,τ)+ε≤Re(λ)≤ε}.
In order to apply Theorem 1.1 of [14], it must be shown that
(45)sup{∥(λI-A)-1∥ℒ(ℋ):λ∈E1}<∞.
First, for each λ∈E1 and each g=(g1,g2,g3)T∈ℋ, we consider the resolvent equation (λI-A)f=g, which is equivalent to
(46)νf1′(x)+(λ+a1)f1(x)+a2f2(x)=g1(x),(47)-a3f1(x)(x)+(λ+a4)f2(x)=g2(x),(48)f3′(x)+τλf3(x)=τg3(x).
Solving (47) and (48), we have
(49)f2(x)=a3λ+a4f1(x)+1λ+a4g2(x),(50)f3(x)=f3(0)e-τλx+∫0x‍τe-τλ(x-s)g3(s)ds.
Substituting (49) with (46) and solving it, we have
(51)f1(x)=f1(0)eα(λ)x+∫0x‍τeα(λ)(x-s)×[1νg1(s)-a2ν(λ+a4)g2(s)]ds.
Since f=(f1,f2,f3)T belongs to the domain D(A) of A, f1(x) and f3(x) should satisfy the relations
(52)f1(0)=-kf3(1),f1(1)=f3(0).
Putting x=1 in (50) and (51), respectively, we obtain
(53)f3(1)=e-τλ1+ke-β(λ)×[∫01‍eα(λ)(1-s)(1νg1(s)-a2ν(λ+a4)g2(s))ds+∫01‍τeτλsg3(s)ds],f3(0)=eτλf3(1)-∫01‍τeτλsg3(s)ds.
If substitute them into (50) and (51), then we have
(54)f1(x)=-ke-τλeα(λ)x1+ke-β(λ)×[∫01‍eα(λ)(1-s)(1νg1(s)-a2ν(λ+a4)g2(s))ds+∫01‍τeτλsg3(s)ds]+∫0x‍τeα(λ)(x-s)[1νg1(s)-a2ν(λ+a4)g2(s)]ds,f3(x)=eτλx1+ke-β(λ)×[∫01‍eα(λ)(1-s)(1νg1(s)-a2ν(λ+a4)g2(s))ds+∫01‍τe-τλ(1-s)g3(s)ds]-∫x1‍τe-τλ(x-s)g3(s)ds.
Also, f2(x) can be obtained from the previous equations and (49).
Next, we will estimate a bound ofℋ norm of f=(f1,f2,f3)T. It follows from
(55)ε≤μ(k,τ)+ε+a4ε=|μ(k,τ)+ε+a4|≤Reλ+a4=|λ+a4|
that
(56)|e-(1/ν)(a2a3/(λ+a4))x|=e-(1/ν)a2a3(Reλ+a4)x/((Reλ+a4)2+(Imλ)2)≤e-(1/ν)a2a3εx/((Reλ+a4)2+(Imλ)2)≤1.
Noting that
(57)νlogk+(1+ντ)ε≤a1+(1+ντ)[μ(k,τ)+ε]νlogk+(1+ντ)ε≤a1+(1+ντ)Reλ,νlogk+ε≤νlogk+ε+ντ(ε-Reλ)≤a1+Reλ,
we have
(58)|e-β(λ)x|=e-(1/ν)[(1+ντ)Reλ+a1]|e-(1/ν)(a2a3/(λ+a4))x|≤e-(1/ν)[νlogk+(1+ντ)ε]x=k-1e-(1/ν)(1+ντ)εx<k-1,(59)|eα(λ)x|=e-(1/ν)[Reλ+a1]|e-(1/ν)(a2a3/(λ+a4))x|≤e-(1/ν)[νlogk+ε]x=k-1e-(1/ν)εx<k-1,
for all x∈[0,1]. Putting x=1 in (58), we have
(60)k|e-β(λ)|≤e-(1/ν)(1+ντ)ε<1.
Moreover, the continuous function h(x,y):=e-τxy defined on the compact set [0,1]×[μ(k,τ)+ε,ε] has absolute maximum and absolute minimum, which are denoted by L and l, respectively. Thus, we have
(61)|f1(x)|≤k|e-τλ||eα(λ)x||1+ke-β(λ)||f1(x)|≤×[∫01‍|eα(λ)(1-s)|(1ν|g1(s)|+a2ν|λ+a4||g2(s)|)ds|f1(x)|≤+∫01‍τ|eτλs||g3(s)|ds]|f1(x)|≤+∫0x‍τ|eα(λ)(x-s)||f1(x)|≤×[1ν|g1(s)|+a2ν|λ+a4||g2(s)|]ds|f1(x)|≤e-τReλ1-|ke-β(λ)||f1(x)|≤×[∫01‍1kν|g1(s)|+a2kνε|g2(s)|ds+∫01‍τeτsReλ|g3(s)|ds]|f1(x)|≤+∫01‍τkν|g1(s)|+τa2kνε|g2(s)|ds|f1(x)|≤L1-e-(1/ν)(1+ντ)ε|f1(x)|≤×[∫01‍1kν|g1(s)|+a2kνε|g2(s)|ds+∫01‍τl|g3(s)|ds]|f1(x)|≤+∫01‍τkν|g1(s)|+τa2kνε|g2(s)|ds|f1(x)|≤(Lkν[1-e-(1/ν)(1+ντ)ε]+τkν)∫01‍|g1(s)|2dx|f1(x)|≤+a2Lkνε[1-e-(1/ν)(1+ντ)ε]∫01‍|g2(s)|2dx|f1(x)|≤+(τLl[1-e-(1/ν)(1+ντ)ε]+τa2kνε)∫01‍|g3(s)|2dx.
The Cauchy-Schwarz inequality is applied in the last step. This means that
(62)|f1(x)|≤M1∥g∥ℋor∥f1(x)∥L2[0,1]≤M1∥g∥ℋ,
in which
(63)M1=max{τLl[1-e-(1/ν)(1+ντ)ε]Lkν[1-e-(1/ν)(1+ντ)ε]+τkν,a2Lkνε[1-e-(1/ν)(1+ντ)ε],τLl[1-e-(1/ν)(1+ντ)ε]+τa2kνετLl[1-e-(1/ν)(1+ντ)ε]}.
Similarly, we have
(64)|f3(x)|≤|eτλx||1+ke-β(λ)||f3(x)|≤×[∫01‍|eα(λ)(1-s)|×(1ν|g1(s)|+a2ν|λ+a4||g2(s)|)ds+∫01‍τ|e-τλ(1-s)||g3(s)|ds]|f3(x)|≤+∫x1‍τ|e-τλ(x-s)||g3(s)|ds|f3(x)|≤eτxReλ1-e-(1/ν)(1+ντ)ε|f3(x)|≤×[∫01‍(1ν|g1(s)|+a2νε|g2(s)|)ds+∫01‍τe-τReλ(1-s)|g3(s)|ds]|f3(x)|≤+∫x1‍τe-τReλ(x-s)|g3(s)|ds|f3(x)|≤1l[1-e-(1/ν)(1+ντ)ε]|f3(x)|≤×[∫01‍(1ν|g1(s)|+a2νε|g2(s)|)ds+∫01‍τLl|g3(s)|ds]|f3(x)|≤+∫01‍τLl|g3(s)|ds|f3(x)|≤1lν[1-e-(1/ν)(1+ντ)ε]∫01‍|g1(s)|2dx|f3(x)|≤+a2lνε[1-e-(1/ν)(1+ντ)ε]∫01‍|g2(s)|2dx|f3(x)|≤+(τLl2[1-e-(1/ν)(1+ντ)ε]+τLl)∫01‍|g3(s)|2dx.
This means that
(65)|f3(x)|≤M3∥g∥ℋor∥f3(x)∥L2[0,1]≤M3∥g∥ℋ,
in which
(66)M3=max{1lν[1-e-(1/ν)(1+ντ)ε],a2lνε[1-e-(1/ν)(1+ντ)ε],τLl2[1-e-(1/ν)(1+ντ)ε]+τLl}.
It follows from (49) and (62) that
(67)|f2(x)|≤a3ε|f1(x)|+1ε|g2(x)|≤a3M1ε∥g∥ℋ+1ε|g2(x)|.
This implies that
(68)∥f2(x)∥L2[0,1]≤M2∥g∥ℋ
in which
(69)M2=a3M1ε+1ε.
Therefore, by using (62), (65), and (68), we can estimate a bound of ℋ norm of f=(f1,f2,f3)T as follows:
(70)∥f∥ℋ2=∥f1(x)∥L2[0,1]2+∥f2(x)∥L2[0,1]2+∥f3(x)∥L2[0,1]2∥f∥ℋ2≤(M12+M22+M32)∥g∥ℋ,
which is equivalent to
(71)∥(λI-A)-1∥L(ℋ)≤M12+M22+M32.
Since the inequality holds for all g∈ℋ and for all λ∈E1, we have
(72)supλ∈E1∥(λI-A)-1∥L(ℋ)≤M12+M22+M32<∞.
This shows that (45) holds. In this way, we finally obtain
(73)sup{∥(λI-A)-1∥ℒ(ℋ):λ∈ℂandReλ≥μ(k,τ)+ε}<∞.Theorem 4.
Suppose that the assumption of Theorem1 is satisfied. Then, for any ε>0, there exists a constant Mε,τ such that
(74)∥T(t)∥ℒ(ℋ)≤Mε,τe[μ(k,τ)+ε]t
in which μ(k,τ) 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 boundedC0-semigroup of linear operators if the feedback gain k satisfies 0<k<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.



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*Source: 274857-2013-08-06.xml*