# Time-Delayed Feedback Control of a Hydraulic Model Governed by a Diffusive Wave System

Authors: Boumediène Chentouf; Nejib Smaoui
Journal: Complexity
 (2020)

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## Abstract

This paper is concerned with the feedback flow control of an open-channel hydraulic system modeled by a diffusive wave equation with delay. Firstly, we put forward a feedback flow control subject to the action of a constant time delay. Thereafter, we invoke semigroup theory to substantiate that the closed-loop system has a unique solution in an energy space. Subsequently, we deal with the eigenvalue problem of the system. More importantly, exponential decay of solutions of the closed-loop system is derived provided that the feedback gain of the control is bounded. Finally, the theoretical findings are validated via a set of numerical results.


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## Body

## 1. Introduction

The dynamics of irrigation canals are often described by nonlinear complex partial differential equations derived from the conservation of mass and momentum. Among several models available in the literature, the de Saint-Venant system [1, 2] has been standing at the forefront of the modeling and analysis of irrigation canals for decades. The system consists of two nonlinear coupled hyperbolic partial differential equations:(1)∂Z∂t+∂q∂x=Ld,(2)∂q∂t+∂∂xq2Z+gZ∂E∂x=−gZS+rVLd,where x is the spatial location m and t is the time s. Furthermore, E represents the absolute water surface elevation m, Zx,E is the wetted cross-sectional area m2, qx,t is the flow discharge m3/s, Sq,E,x the friction slope, and Ldx,t is the lateral discharge m2/s, which can be interpreted as an inflow if Ld>0 and an outflow when Ld<0. Moreover, r=0 as long as Ld>0, whereas if Ld<0, then r=1. Finally, Vx,t is the mean velocity m/s in section Z and g the gravitational acceleration m2/s.As mentioned above, the conservation of mass is reflected by equation (1) and the conservation of momentum is translated by (2) (see also [3] for the 2D model).It turned out that, in practice, system (1)-(2) cannot be well studied unless the geometry of the irrigation canal is completely known and also the roughness coefficient. As pointed out in [2, 4], such requirements may not be reachable for long rivers. This has motivated many authors to derive other models. For instance, the following model has been used in [4, 5]:(3)∂q∂tx,t=α∂2q∂x2x,t−β∂q∂xx,t,where the positive constants α and β are the diffusion and the celerity, respectively. Specifically, the above equation has been derived under several assumptions such as assuming that the lateral inflow is minimum, the flow variations as well as the bed slope of the long river are small, and the inertia terms ∂q/∂t+∂/∂xq2/Z are negligible with respect to gZ∂E/∂x (for further discussion on the model, the reader is referred to [4, 6]). However, for sake of clarity, we shall give some details about the derivation of (3). In fact, under the above assumptions, system (1)-(2) have the following form:(4)∂q∂x=−W∂E∂t,(5)S=−∂E∂x,where W is the water surface width. Differentiating (4) and (5), it follows that(6)W2∂S∂t=W∂2q∂x2−∂W∂x∂q∂x.On the other hand, we have(7)∂W∂xt=∂W∂Ex∂E∂x+∂W∂xE=−S∂W∂Ex+∂W∂xE,(8)∂S∂t=∂S∂q∂q∂tx+∂S∂E∂E∂tx=∂S∂q∂q∂tx−1W∂S∂E∂q∂x,where ∂f/∂uu,vv denotes the variation of f with respect to u when v is fixed. Combining these identities with (6) gives(9)∂q∂t=αq,E,x∂2q∂x2−βq,E,x∂q∂x,where(10)αq,E,x=1W∂S/∂q,βq,E,x=1W∂S/∂q∂S∂E−1W∂W∂E−S+∂W∂x=1W2∂S/∂q∂WS∂E−∂W∂x.Finally, it suffices to linearize the latter around a reference flow and come up with the diffusive wave (3).It is worth mentioning that the model under consideration (3) is not only of practical interest but also does not require too many physical parameters of the canal. Indeed, only two parameters are required in order to characterize the water flow, namely, celerity and diffusivity.The control and stabilization problem of (3) has been addressed in several research papers by adopting different approaches and strategies. To name few, using discretization techniques such as the finite difference method or using a finite dimensional approximation in the frequency domain, the authors provided exhaustive studies, where the main concern was controlling (3) and/or estimating its parameters [1, 4–20] (see also [21] for a nonlinear case). Later, using the tools of control theory of partial differential equations [22], the authors in [23–25] tackled the problem of robust regulation of (3) by proposing a proportional and/or integral controller. In a recent article [6], the authors dealt with the output feedback stabilization of model (3). To be more precise, the following system has been put forward in [6]:(11)∂q∂tx,t=α∂2q∂x2x,t−β∂q∂xx,t+bxUt,together with the boundary conditions and the output observation(12)q0,t=∂q∂xℓ,t=0,yt=qℓ,t,where Ut is the control, yt is the output, and bx is the spatial distribution of the actuator. Then, it has been shown that the closed-loop system is exponentially stable under the action of the control Ut=κqℓ,t, where κ is the feedback gain.Up to our knowledge, the stabilization problem of (11) with delay has not been examined in the literature. In fact, as previously mentioned, the time delay has not been taken into account in [6], where the exponential stability result is obtained under the condition κ<2αβ/bℓ. In this article, we go a step further in the sense that we do take into consideration the presence of the time delay in the input control. This is motivated by the fact that a timedelay naturally occurs in practice due to the time factor needed for the communication among the controllers, the sensors, and the actuators of systems. Obviously, when a controller monitors the state of a system, it will make an adjustment (if a need arises) based on its observation. Notwithstanding, such an adjustment surely requires some time to be completed as it can never be conducted in an instantaneous manner. It is also of great interest to point out that the occurrence of a delay in a system could cause a poor performance of the control and could sometime generate instabilities in the system [26–28]. Based on the above discussion, the study of asymptotic behavior of the solutions of system (11) and (12) with input delay is of theoretical as well as practical prominence. The reader can also consult [26–35], where the effect of time delays in several types of partial differential equations (PDEs) has been studied.The main contribution of this paper is twofold:(1)
Extend the mathematical findings of [6, 36] where the presence of time delay has not been considered in the feedback law. In fact, our control is(13)Ut=κqℓ,t−τ,whereκ>0 is the feedback gain and τ>0 is the delay. Then, it is shown that the closed-loop system (11)–(13) is stable even though a delay supervenes in the proposed control. This result is obtained at the expense of a reasonable restriction on the feedback gain, which has been also used in [6] even if there was no delay in the control.(2)
An extensive numerical study is conducted in order to validate our results. More precisely, adopting the Chebyshev collocation that uses the backward Euler method and the Gauss–Lobatto points, for the delayed system (11)–(13), we are able to illustrate, for different values of the delay and the physical parameters of the canal, the stability result.The rest of this paper is organized as follows. In Section2, the closed-loop system is formulated in an evolution equation and preliminary results are put forward for the open-loop system. Next, we prove in Section 3 that our problem is well posed in the sense of semigroup theory of linear operators. In Section 4, we characterize the spectrum of the closed-loop system. Section 5 is devoted to the proof of the main result of this work, namely, the exponential stability of the closed-loop system. In Section 6, we show the relevance of the theoretical results through several numerical simulations. Finally, our conclusions are given in Section 7.


## 2. Problem Set-Up and Preliminaries

In this section, the closed-loop system is formulated as a differential equation in an appropriate functional space. To do so, the notations used in the paper are presented, as well as some definitions and theorems. Finally, a comprehensive study of the open-loop system is provided.Given two initial dataq0 and u0, the closed-loop system (11)–(13) can be written as follows:(14)∂q∂tx,t=α∂2q∂x2x,t−β∂q∂xx,t+κbxqℓ,t−τ,x,t∈0,ℓ×0,∞,q0,t=∂q∂xℓ,t=0,t>0,qx,0=q0x,x∈0,ℓ,qℓ,θ=u0θ,θ∈−τ,0.For sake of clarity, we recall that(15)L20,ℓ=φis measurable and∫0ℓφ2xdx<∞,is endowed with its standard norm φ=∫0ℓφ2xdx1/2. Furthermore, the Sobolev space(16)Hn0,ℓ=φ:0,ℓ⟶ℝ;φn∈L20,ℓ,forn=0,1,2,…,is equipped with the usual norm φHn0,ℓ=∑i=0nφiL20,ℓ.In view of the well-known change of state variable [33, 37],(17)uρ,t=qℓ,t−τρ,ρ∈0,1,the closed-loop system (14) becomes(18)∂q∂tx,t=α∂2q∂x2x,t−β∂q∂xx,t+κbxu1,t,x,t∈0,ℓ×0,∞,τ∂u∂tρ,t+∂u∂ρρ,t=0,ρ,t∈0,1×0,∞,q0,t=∂q∂xℓ,t=0,t>0,qx,0=q0x,x∈0,ℓ,uρ,0=u0−τρ,ρ∈0,1,whereupon consider the state space(19)ℋ=L20,ℓ×L20,1,equipped with the following inner product:(20)q,u,q˜,u˜ℋ=∫0ℓqq˜dx+τβ∫01uu˜dx.Obviously, the norm induced by such an inner product is equivalent to the usual norm of the spaceL20,ℓ×L20,1. For convenience, we shall often denote by fv the derivative of f with respect to the variable v. Then, let us assume that the spatial actuator b⋅∈L20,ℓ (one may suppose that b⋅∈L∞0,ℓ) and consider the system operator:(21)DA≔q,u∈H20,ℓ×H10,ℓ;q0=qxℓ=0,andu0=qℓ,(22)Aq,u≔αqxx−βqx+κb⋅u1,−uρτ.Thereby, system (18) can be brought to the following form:(23)ϕ˙t=Aϕt.ϕ0=ϕ0,where ϕ=q,u and ϕ0=q0,u0.


### 2.1. Open-Loop System

First and foremost, we shall investigate the main properties of the uncontrolled system which is given by(24)∂Rx,t∂t=α∂2Rx,t∂x2−β∂Rx,t∂x,R0,t=Rxℓ,t=0.Taking the Hilbert state spaceL20,ℓ equipped with the usual inner product, system (24) can be written in the abstract form R˙t=A0Rt, where A0 is an unbounded linear operator defined by(25)DA0=R∈H20,ℓ;R0=Rxℓ=0,A0R=αRxx−βRx.The following results will be systematically used.Theorem 1 (Lumer–Phillips) (see [38]).
Given a linear operatorP on a Hilbert space Y with DP¯=Y, if P is dissipative and there is a scalar λ0 such that λ0I−P is onto Y, then P generates a C0 semigroup of contractions on Y.Theorem 2 (see [39]).
Assume thatP be a closed densely defined operator on a Hilbert space Y satisfying the following property: there exists η∈ℝ such that(26)Pv,vY≤ηvY2,∀v∈DP,(27)P∗v,vY≤ηvY2,∀v∈DP∗,where P∗ is the adjoint operator of P. Then, the operator P generates a C0-semigroup St satisfying StℒY≤eηt.The basic properties of the operatorA0 (see (25)) are summarized below.Proposition 1.
The operatorA0 defined by (25) generates a C0-semigroup of contractions S0t on L20,ℓ. In addition, the resolvent of A0 is compact and the spectrum σA0 consists entirely of isolated eigenvalues with finite multiplicities.Proof.
Using (25) and integrating by parts, we obtain(28)A0R,RL20,ℓ=−α∫0ℓRx2xdx−β2R2ℓ≤0,for any R∈DA0. Thereby, A0 is dissipative.Given nowQ∈L20,ℓ, a direct calculation yields(29)Rx≔A0−1Q=αβ∫0xeα/βx−s−1Qsds+1−eα/βx∫0ℓe−α/βxQsds.Therefore, the resolvent equationA0R=Q has a unique solution R∈DA0 for each Q∈L20,ℓ. Evoking Theorem 1, one can deduce that A0 generates a C0-semigroup of contractions S0t on L20,ℓ. On the other hand, using the Sobolev embedding theorem [40], we conclude that the resolvent operator of A0 is compact on L20,ℓ and hence the spectrum of A0 only has isolated eigenvalues with finite multiplicities [41].For sake of completeness, let us recall the following definition and theorem.Definition 1.
(i)
A semigroupSt is stable on a Hilbert space Y if limt⟶∞Stv=0, for any v∈Y.(ii)
If there exist positive constantsC and ω such that St≤Ce−ωt, for all t≥0, then the semigroup is exponentially stable.Theorem 3 (see [42]).
LetSt be a C0-semigroup in Y whose generator P has a compact resolvent. Then, St is stable if and only if it is uniformly bounded and the eigenvalues of P belong to the left halfplane.
The stability result for the operatorA0 (see (25)) is as follows.Proposition 2.
The semigroupS0t generated by the operator A0 defined by (25) is exponentially stable in L20,ℓ.Proof.
LetRx,t be the unique solution stemmed from an initial condition R0 in DA0. Integrating by parts and using the boundary conditions in (25), we have(30)12ddtRx,t2=−α∫0ℓRxx,t2xdx−β2R2ℓ,t.
Moreover, for eachφ∈f∈H10,ℓ;f0=0, we have the following Wirtinger’s inequality [43]:(31)∫0ℓφ2x,tdx≤4ℓ2π2∫0ℓφx2x,tdx,which together with the last estimate imply that d/dtRx,t2≤−απ2/2ℓ2Rx,t2. Solving the above differential inequality, we get Rx,t≤e−π2α/2ℓtR0, which leads to the exponential stability of Rx,t in L20,ℓ, thanks to a classical density argument.Now, we turn to the characterization of the spectrum ofA0.Proposition 3.
The eigenvalues ofA0 form the following set of negative real numbers:(32)μn=−ℓ2β2+4α2en24αℓ2;n∈ℕ,where en is a nonzero real solution of(33)2αencosen=−βℓsinen.
Furthermore, we havesupμn;μnis  an  eigenvalue  ofA0=−ℓ2β2+4α2e2/4αℓ2, where e is the smallest positive solution of (33).Proof.
It follows from (25) that a complex μ is an eigenvalue of A0 if and only if the system(34)αRxx−βRx−μR=0,R0=Rxℓ=0,has a nontrivial solution. For sake of clarity, let(35)Gμ=4αμ+β2,σ1=β−Gμ2α,σ2=β+Gμ2α.
Two cases will be treated, namely,Gμ=0 and Gμ≠0.
i) Gμ=0, which means that μ=−β2/4α and hence σ1=σ2=β/2α>0. Thereby, it is easy to check that (34) has only the trivial solution and hence μ=−β2/4α cannot be an eigenvalue of A0.
ii) Gμ≠0, i.e., σ1≠σ2. In such a case, the general solution of (34) is Rx=k1eσ1x+k2eσ2x, where the constants satisfy the linear system:(36)k1+k2=0,σ1eσ1ℓk1+σ2eσ2ℓk2=0,whereupon μ is an eigenvalue if and only if σ1eσ1ℓ=σ2eσ2ℓ, which can be written as follows:(37)βℓsinhζ+2αζcoshζ=0,where ζ=Gμ/2αℓ. The immediate task is to characterize the solutions of (37). To do so, write ζ=a+ie, where a and e are two real numbers, and using (37) yields(38)e−a2αa−βℓcose+2αesine+ea2αa+βℓcose−2αesine=0,e−aβℓ−2αasine+2αecose+ea2αa+βℓsine+2αecose=0.
This implies thata and e must obey the following condition:(39)e4a2αa+βℓ2+4α2e2=2αa−βℓ2+4α2e2.
It is a simple task to check that the above equation is valid only in the event thata=0 and thus (38) becomes βℓsine+2αecose=0. Recalling that ζ=ie=Gμ/2αℓ and using (35), we obtain the desired result.The asymptotic expansion of the eigenvalues ofA0 is provided in the following proposition.Proposition 4.
Letμn be an eigenvalue of system (24). Then, for n large enough, we have(40)μn=−β1ℓ+β4α−π2αℓ22n+122+O1n+iO1n.Proof.
In view of Proposition3, we know that μ is an eigenvalue of A0 if and only if ζ=Gμ/2αℓ is a solution of (37). In turn, the solution set of (37) admits the two bisectors of the complex plane as symmetrical axes and the origin as a symmetrical center. Thereby, we just need to find the solutions of (37) in a sector S defined by argθ=υ with ϵ<υ<π/2+ϵ for ϵ small. Next, applying Rouché theorem in S, the solutions of (37) have the following asymptotic representation:(41)ζn=O1n2+iβL2απn+π2n+12+O1n2,for n large enough. Finally, by virtue of (35) and the fact that ζ=Gμ/2αℓ, it follows that μn=−β/ℓ−β2/4α−π2/αℓ2n+1/22+O1/n+iO1/n, which is the expression in our proposition.


## 2.1. Open-Loop System

First and foremost, we shall investigate the main properties of the uncontrolled system which is given by(24)∂Rx,t∂t=α∂2Rx,t∂x2−β∂Rx,t∂x,R0,t=Rxℓ,t=0.Taking the Hilbert state spaceL20,ℓ equipped with the usual inner product, system (24) can be written in the abstract form R˙t=A0Rt, where A0 is an unbounded linear operator defined by(25)DA0=R∈H20,ℓ;R0=Rxℓ=0,A0R=αRxx−βRx.The following results will be systematically used.Theorem 1 (Lumer–Phillips) (see [38]).
Given a linear operatorP on a Hilbert space Y with DP¯=Y, if P is dissipative and there is a scalar λ0 such that λ0I−P is onto Y, then P generates a C0 semigroup of contractions on Y.Theorem 2 (see [39]).
Assume thatP be a closed densely defined operator on a Hilbert space Y satisfying the following property: there exists η∈ℝ such that(26)Pv,vY≤ηvY2,∀v∈DP,(27)P∗v,vY≤ηvY2,∀v∈DP∗,where P∗ is the adjoint operator of P. Then, the operator P generates a C0-semigroup St satisfying StℒY≤eηt.The basic properties of the operatorA0 (see (25)) are summarized below.Proposition 1.
The operatorA0 defined by (25) generates a C0-semigroup of contractions S0t on L20,ℓ. In addition, the resolvent of A0 is compact and the spectrum σA0 consists entirely of isolated eigenvalues with finite multiplicities.Proof.
Using (25) and integrating by parts, we obtain(28)A0R,RL20,ℓ=−α∫0ℓRx2xdx−β2R2ℓ≤0,for any R∈DA0. Thereby, A0 is dissipative.Given nowQ∈L20,ℓ, a direct calculation yields(29)Rx≔A0−1Q=αβ∫0xeα/βx−s−1Qsds+1−eα/βx∫0ℓe−α/βxQsds.Therefore, the resolvent equationA0R=Q has a unique solution R∈DA0 for each Q∈L20,ℓ. Evoking Theorem 1, one can deduce that A0 generates a C0-semigroup of contractions S0t on L20,ℓ. On the other hand, using the Sobolev embedding theorem [40], we conclude that the resolvent operator of A0 is compact on L20,ℓ and hence the spectrum of A0 only has isolated eigenvalues with finite multiplicities [41].For sake of completeness, let us recall the following definition and theorem.Definition 1.
(i)
A semigroupSt is stable on a Hilbert space Y if limt⟶∞Stv=0, for any v∈Y.(ii)
If there exist positive constantsC and ω such that St≤Ce−ωt, for all t≥0, then the semigroup is exponentially stable.Theorem 3 (see [42]).
LetSt be a C0-semigroup in Y whose generator P has a compact resolvent. Then, St is stable if and only if it is uniformly bounded and the eigenvalues of P belong to the left halfplane.
The stability result for the operatorA0 (see (25)) is as follows.Proposition 2.
The semigroupS0t generated by the operator A0 defined by (25) is exponentially stable in L20,ℓ.Proof.
LetRx,t be the unique solution stemmed from an initial condition R0 in DA0. Integrating by parts and using the boundary conditions in (25), we have(30)12ddtRx,t2=−α∫0ℓRxx,t2xdx−β2R2ℓ,t.
Moreover, for eachφ∈f∈H10,ℓ;f0=0, we have the following Wirtinger’s inequality [43]:(31)∫0ℓφ2x,tdx≤4ℓ2π2∫0ℓφx2x,tdx,which together with the last estimate imply that d/dtRx,t2≤−απ2/2ℓ2Rx,t2. Solving the above differential inequality, we get Rx,t≤e−π2α/2ℓtR0, which leads to the exponential stability of Rx,t in L20,ℓ, thanks to a classical density argument.Now, we turn to the characterization of the spectrum ofA0.Proposition 3.
The eigenvalues ofA0 form the following set of negative real numbers:(32)μn=−ℓ2β2+4α2en24αℓ2;n∈ℕ,where en is a nonzero real solution of(33)2αencosen=−βℓsinen.
Furthermore, we havesupμn;μnis  an  eigenvalue  ofA0=−ℓ2β2+4α2e2/4αℓ2, where e is the smallest positive solution of (33).Proof.
It follows from (25) that a complex μ is an eigenvalue of A0 if and only if the system(34)αRxx−βRx−μR=0,R0=Rxℓ=0,has a nontrivial solution. For sake of clarity, let(35)Gμ=4αμ+β2,σ1=β−Gμ2α,σ2=β+Gμ2α.
Two cases will be treated, namely,Gμ=0 and Gμ≠0.
i) Gμ=0, which means that μ=−β2/4α and hence σ1=σ2=β/2α>0. Thereby, it is easy to check that (34) has only the trivial solution and hence μ=−β2/4α cannot be an eigenvalue of A0.
ii) Gμ≠0, i.e., σ1≠σ2. In such a case, the general solution of (34) is Rx=k1eσ1x+k2eσ2x, where the constants satisfy the linear system:(36)k1+k2=0,σ1eσ1ℓk1+σ2eσ2ℓk2=0,whereupon μ is an eigenvalue if and only if σ1eσ1ℓ=σ2eσ2ℓ, which can be written as follows:(37)βℓsinhζ+2αζcoshζ=0,where ζ=Gμ/2αℓ. The immediate task is to characterize the solutions of (37). To do so, write ζ=a+ie, where a and e are two real numbers, and using (37) yields(38)e−a2αa−βℓcose+2αesine+ea2αa+βℓcose−2αesine=0,e−aβℓ−2αasine+2αecose+ea2αa+βℓsine+2αecose=0.
This implies thata and e must obey the following condition:(39)e4a2αa+βℓ2+4α2e2=2αa−βℓ2+4α2e2.
It is a simple task to check that the above equation is valid only in the event thata=0 and thus (38) becomes βℓsine+2αecose=0. Recalling that ζ=ie=Gμ/2αℓ and using (35), we obtain the desired result.The asymptotic expansion of the eigenvalues ofA0 is provided in the following proposition.Proposition 4.
Letμn be an eigenvalue of system (24). Then, for n large enough, we have(40)μn=−β1ℓ+β4α−π2αℓ22n+122+O1n+iO1n.Proof.
In view of Proposition3, we know that μ is an eigenvalue of A0 if and only if ζ=Gμ/2αℓ is a solution of (37). In turn, the solution set of (37) admits the two bisectors of the complex plane as symmetrical axes and the origin as a symmetrical center. Thereby, we just need to find the solutions of (37) in a sector S defined by argθ=υ with ϵ<υ<π/2+ϵ for ϵ small. Next, applying Rouché theorem in S, the solutions of (37) have the following asymptotic representation:(41)ζn=O1n2+iβL2απn+π2n+12+O1n2,for n large enough. Finally, by virtue of (35) and the fact that ζ=Gμ/2αℓ, it follows that μn=−β/ℓ−β2/4α−π2/αℓ2n+1/22+O1/n+iO1/n, which is the expression in our proposition.


## 3. Semigroup Generation for the Closed-Loop System

The main objective of this section is to show that the closed-loop system (23) is well posed in the state space ℋ=L20,ℓ×L20,1. The proof of such a desirable result is based on the application of semigroup theory of linear operators. Specifically, we shall use Theorem 2. We have indeed the following result.Theorem 4.
The operatorA, defined by (21) and (22), generates a C0-semigroup St on the Hilbert space ℋ.Proof.
Letϕ=q,u∈DA. Invoking (21) and (22) and integrating by parts, we get(42)Aϕ,ϕℋ=−α∫0ℓqx2xdx−β2q2ℓ+κu1∫0ℓbxqxdx−β2u21−u20=−α∫0ℓqx2xdx+κu1∫0ℓbxqxdx−β2u21.
Applying Cauchy–Schwartz and Young’s inequalities, the latter becomes(43)Aϕ,ϕℋ≤−α∫0ℓqx2xdx+κC−β2u21+κb22C∫0ℓq2xdx,for any positive constant C. Choosing C=β/κ, we deduce that(44)ReAϕ,ϕℋ≤κ22βb2∫0ℓq2xdx.
The task ahead is to show that the above inequality also holds for the adjoint operator ofA∗. First, one can readily verify that the operator A is closed and its domain is dense in ℋ. Furthermore, the adjoint operator of A∗ is given by(45)DA∗≔f,g∈H20,ℓ×H10,1;f0=0,βg0=αfxℓ+βfℓ,βg1=κ∫0ℓbxfxdx,(46)A∗f,g≔αfxx+βfx,−gρτ,∀q∈DA∗.
Subsequently, let us considerq,u∈DA∗. Integrating by parts twice and using (45) and (46), it follows that(47)A∗q,u,q,uℋ=αqxℓqℓ−α∫0ℓqx2xdx+β2q2ℓ+β2u21−u20≤−α∫0ℓqx2xdx−α22βqx2ℓ+κ22βb2∫0ℓq2xdx≤κ22βb2∫0ℓq2xdx.
By virtue of (44) and (47), we conclude, thanks to Theorem 2, that the operator DA generates a C0-semigroup St satisfying Stℒℋ≤expκ2b2/2βt.Remark 1.
A direct consequence of Theorem4 is that the C0-semigroup St of the closed-loop system is not necessarily uniformly bounded for all values of the feedback gain κ>0. One can explain this property by recalling that the generator A of St is not dissipative for all values of the feedback gain κ>0, unlike the uncontrolled system whose operator A0 is dissipative and hence its semigroup is contractive (see Proposition 1).


## 4. Eigenvalue Problem

It is the intention of this section to describe the spectrum and the resolvent set of the closed-loop system. To this end, we first prove that the system operatorA is indeed a discrete operator.Theorem 5.
The operatorλI−A−1 is compact on ℋ for some λ>0. Therewith, A is a discrete operator and the spectrum set σA consists of isolated eigenvalues with finite multiplicities. Moreover, if λ∈ρA, then the resolvent operator λI−A−1 is compact.Proof.
For arbitraryf,g∈ℋ, let us solve the resolvent equation with λ>0:(48)αqxxx−βqxx−λqx+κbxu1=−fx,uρρ+λτuρ=τgρ,u0=qℓ,q0=qxℓ=0.
Solving the equation ofu in the above system and using the condition u0=qℓ, we obtain(49)uρ=qℓe−λτρ+τ∫0ρgre−λτρ−rdr,and hence(50)ϖ≔u1=qℓe−λτ+τ∫01gre−λτ1−rdr,whereupon resolvent (48) admits a unique solution q,u∈DA once we show the existence of a unique q∈H20,ℓ satisfying(51)αqxxx−βqxx−λqx=−fx−κϖbx,q0=qxℓ=0.
Sinceλ>0 and recalling (35), one can readily check that the general solution of (51) is given by(52)qx=l1eσ1x+l2eσ2x+1σ2−σ1∫0xeσ1x−s−eσ2x−sfs+κϖbsds.
In (52), l1 and l2 are two constants that satisfy, thanks to the boundary conditions in (51), the following linear system:(53)l1+l2=0,σ1eσ1ℓl1+σ2eσ2ℓl2=M,in which(54)M=1σ1−σ2∫0ℓσ1eσ1ℓ−s−σ2eσ2ℓ−sfs+κϖbsds.
We suppose thatλ>0 is large enough so that σ2eσ2ℓ−σ1eσ1ℓ≠0. Accordingly, the above linear system has a unique solution:(55)l1=−Mσ2eσ2ℓ−σ1eσ1ℓ,l2=Mσ2eσ2ℓ−σ1eσ1ℓ.
Combining the last findings with (52) yields(56)qx=1σ2−σ1∫0xeσ1x−s−eσ2x−sfs+κϖbsds−Meσ1x−eσ2xσ2eσ2ℓ−σ1eσ1ℓ.
This clearly gives(57)qℓ=1σ2−σ1∫0ℓeσ1ℓ−s−eσ2ℓ−sfs+κϖbsds−Meσ1ℓ−eσ2ℓσ2eσ2ℓ−σ1eσ1ℓ,which together with (50) implies that(58)qℓ1+κe−λτeσ1+σ2ℓσ2eσ2ℓ−σ1eσ1ℓ∫0ℓe−σ2s−e−σ1sbsds=eσ1+σ2ℓσ2eσ2ℓ−σ1eσ1ℓ∫0ℓe−σ1s−e−σ2sfs+κτbs∫01gre−λτ1−rdrds.
In light of (35), the latter becomes(59)qℓ1+2ακeβℓ/α−λτβ+Gλeβ+Gλ/2αℓ+Gλ−βeβ−Gλ/2αℓ∫0ℓe−β+Gλ/2αs−eGλ−β/2αℓsbsds=2αeβℓ/αβ+Gλeβ+Gλ/2αℓ+Gλ−βeβ−Gλ/2αℓ×∫0ℓeGλ−β/2αℓs−e−β+Gλ/2αsfs+κτbs∫01gre−λτ1−rdrds.
Consequently,qℓ is uniquely determined if and only if the following expression(60)1+2ακeβℓ/α−λτβ+Gλeβ+Gλ/2αℓ+Gλ−βeβ−Gλ/2αℓ∫0ℓe−β+Gλ/2αs−eGλ−β/2αℓsbsds,is not zero, which leads us to claim that resolvent (48) has a unique solution q,u given by (49) and (56). Whereupon, the operator λI−A−1 exists for λ>0 and λI−A−1∈ℒℋ. Furthermore, by virtue of the Sobolev embedding theorem [40], the operator λI−A−1 is compact on ℋ, for λ>0. Finally, the spectrum of A consists of isolated eigenvalues with finite multiplicities and λI−A−1 is compact for any λ∈ρA [41].We now turn to the eigenvalue problem. We have the following result.Theorem 6.
(i)
λ=0 is an eigenvalue of the operator A if and only if the feedback gain obeys the following condition:(61)β=κ∫0ℓ1−e−β/αsbsds.(ii)
λ=−β/4α2 is an eigenvalue of the operator A if and only if the feedback gain satisfies(62)κeβ2τ/4α∫0ℓse−βs/2αbsds=e−βℓ/2α1+βℓ2α.(iii)
Ifβ≠κ∫0ℓ1−e−β/αsbsds and 1+β/αℓ≠κeβ/αℓ∫0ℓse−β/αsbsds, then the spectrum σA consists of eigenvalues with finite multiplicity and(63)σA=λ∈ℂ;Δλ=0,where(64)Δλ≔1+2ακeβℓ/α−λτβ+Gλeβ+Gλ/2αℓ+Gλ−βeβ−Gλ/2αℓ∫0ℓe−β+Gλ/2αs−eGλ−β/2αℓsbsds,and Gλ is defined by (35).Proof.
Letλ∈ℂ. The eigenvalue problem of A consists in seeking a nontrivial element q,u∈DA such that(65)αqxxx−βqxx−λqx+κbxu1=0,uρρ+λτuρ=0,u0=qℓ,q0=qxℓ=0.
One can check from the second equation of (65) that u is given by(66)uρ=qℓe−λτρ.
Herewith, it suffices to find a nontrivial solutionq∈H20,ℓ to the system(67)αqxxx−βqxx−λqx+κbxqℓe−λτ=0,q0=qxℓ=0.
We shall treat 3 cases.Case 1.
A straightforward computation shows thatλ=0 is an eigenvalue if and only if(68)qx=κβqℓeβx/α−1∫0ℓe−βs/αbsds+κβqℓ∫0xeβ/αx−s−1bsds.
Consequently,(69)qℓ1−κβ∫0ℓ1−e−βs/αbsds=0.
Thereby, we conclude thatλ=0 is an eigenvalue of the system if and only if the feedback gain satisfies the following condition:(70)κ=β∫0ℓ1−e−βs/αbsds−1.Case 2.
It is not difficult to verify from (67) that λ=−β2/4α is an eigenvalue if and only if(71)qx=−κeβ2τ/4α2α+βℓqℓ∫0ℓ2α+βℓ+se−βs/2αbsdsxeβx/2α+κqℓeβ2τ/4αeβx/2α∫0xs−xe−βs/2αbsds.
Accordingly,λ=−β/4α2 is an eigenvalue if and only if the feedback gain verifies(72)κeβ2τ/4α∫0ℓse−βs/2αbsds=e−βℓ/2α1+βℓ2α.
The assertion (ii) is then proved.Case 3.
Assume thatλ∉0,−β/4α2. Thereby, (67) gives(73)qx=a1eσ1x+a2eσ2x+αGλκqℓe−λτ∫0xeσ1x−s−eσ2x−sbsds,where Gλ, σ1, and σ2 are given by (35). Moreover, a1 and a2 are constants to be determined by the boundary conditions q0=qxℓ=0. In fact, we have(74)a1+a2=0,σ1eσ1ℓa1+σ2eσ2ℓa2=ϒ,with(75)ϒ=αGλκqℓe−λτ∫0ℓσ1eσ1ℓ−s−σ2eσ2ℓ−sbsds.
Ifσ1eσ1ℓ=σ2eσ2ℓ, then necessarily ∫0ℓe−σ1s−e−σ2sbsds=0. However, if σ1eσ1ℓ≠σ2eσ2ℓ, then the solution qx obtained in (73) takes the following form:(76)qx=1σ2−σ1κqℓe−λτ∫0xeσ1x−s−eσ2x−sbsds+ϒeσ1x−eσ2xσ2eσ2ℓ−σ1eσ1ℓ.
Substitutingℓ for x in (76) and recalling (35), we finally obtain that λ obeys the following condition:(77)1+2ακeβℓ/α−λτ∫0ℓe−β+Gλ/2αs−eGλ−β/2αℓsbsdsβ+Gλeβ+Gλ/2αℓ+Gλ−βeβ−Gλ/2αℓ=0.
The third assertion is then proved.


## 5. Stability of the Closed-Loop System

The aim of this section is to investigate the stability of the closed-loop system (14) according to the values of κ. To proceed, we assume throughout this section that λ≠0 is not an eigenvalue of our system, that is, β≠κ∫0ℓ1−e−β/αsbsds (see Theorem 6). Otherwise, the system is unstable by Theorem 3.Our first stability result is as follows.Theorem 7.
Suppose that the feedback gainκ fulfills the condition(78)κ<πbℓαβ2.Then, theC0-semigroup St generated by the operator A, defined by (21) and (22), is exponentially stable in ℋ. In other words, given any solution qx,t of system (14), the following energy-norm(79)E1t≔12qx,t,qℓ,t−τρℋ2=12∫01q2x,tdx+τβ∫01q2ℓ,t−τρdρ,exponentially tends to zero as long as condition (78) is fulfilled.Proof.
Letqx,t,uρ,t be the unique solution of system (18) stemmed from an initial condition q0,u0 in DA. Subsequently, differentiating (79) with respect to t, integrating by parts, and using the boundary conditions in (18), we get(80)E1′t=−α∫0ℓqx2x,tdx−β2q2ℓ,t−τ+κqℓ,t−τ∫0ℓqx,tbxdx.
Next, evoking Young’s and Cauchy–Schwartz inequalities yields (as for (43) and (44))(81)E1′t≤−α∫0ℓqx2x,tdx+κ22βb2∫0ℓq2x,tdx.
Furthermore, by virtue of Wirtinger’s inequality (31), estimate (81) becomes(82)E1′t≤2κ2b2ℓ2−αβπ24βℓ2∫0ℓq2x,tdx.
Thanks to assumption (78), one can write (82) as follows:(83)E1′t≤−ω1∫0ℓq2x,tdx,in which(84)ω1=αβπ2−2κ2b2ℓ24βℓ2>0.
On the other hand, let us introduce a new functional as(85)E2t=ςτ∫01e−2γτρq2ℓ,t−τρdρ,where ς is a positive constant to be determined, while γ is an arbitrary positive constant. One can check that(86)ςτ∫01e−2γτq2ℓ,t−τρdρ≤E2t≤ςτ∫01q2ℓ,t−τρdρ.
Moreover, differentiating (85), we have(87)E2′t=2ςτ∫01e−2γτρqℓ,t−τρqtℓ,t−τρdρ=2ςτ∫01e−2γτρqℓ,t−τρ1τqρℓ,t−τρdρ=−ςe−2γτq2ℓ,t−τ+ςq2ℓ,t−2ςτγ∫01e−2γτρq2ℓ,t−τρdρ.
In view of the well-known inequality(88)q2ℓ,t≤ℓ∫0ℓqx2x,tdx,one can deduce from (87) that(89)E2′t≤−ςe−2γτq2ℓ,t−τ+ςℓ∫0ℓqx2x,tdx−2ςτγ∫01e−2γτρq2ℓ,t−τρdρ.
Now, let(90)ℰt=E1t+E2t.
Using (79), (85), and (86), an elementary computation shows that there exist positive constants L1 and L2 such that for any t≥0, we have(91)L1E1t≤ℰt≤L2E1t.
Furthermore, differentiating (90) and using (83) and (89), we get(92)ℰ′t≤−ςe−2γτq2ℓ,t−τ−2ςτγ∫01e−2γτρq2ℓ,t−τρdρ+ςℓ−ω1∫0ℓqx2x,tdx,where ω1 is defined by (84). In view of (78), one can choose ς<ω1/ℓ and hence (92) yields(93)ℰ′t≤−θE1t,for some positive constant θ depending solely on the physical parameters α,β, and ℓ. Amalgamating (90), (91), and (93), we reach our desired result.Our second stability result is as follows.Theorem 8.
Assume that the feedback gainκ satisfies the condition(94)κ<πα2bℓ3/22c1+c2,for any constant c1>0 and c2>1. Then, the closed-loop system (23) is exponentially stable.Proof.
We begin by introducing the following functional:(95)ℱt=12∫01q2x,tdx+τξ2∫01q2ℓ,t−τρdρ+ντ∫01e−2γτρq2ℓ,t−τρdρ,where ξ and ν are positive constants to be obtained and γ is an arbitrary positive constant. Arguing as before, we have(96)ℱ′t≤−α∫0ℓqx2x,tdx+12ξ−β+νq2ℓ,t+κ2Kb2∫0ℓq2x,tdx+12κK−ξ−νe−2γτq2ℓ,t−τ−2ντγ∫01e−2γτρq2ℓ,t−τρdρ,where K is a positive constant related to the application of Young’s inequality. Next, we invoke (31) and (88) and choose ν=c1κK and ξ=c2κK, where c1>0 and c2>1. Thereafter, (96) yields(97)ℱ′t=2c1+c2κℓπ2K2−2π2αK+4κb2ℓ22π2K∫0ℓqx2x,tdx−β2q2ℓ,t+κ2K1−c2−νe−2γτq2ℓ,t−τ−2ντγ∫01e−2γτρq2ℓ,t−τρdρ.
In the light of (94), one can choose the positive constant K such that(98)K∈πα−π2α2−42c1+c2κ2b2ℓ3π2c1+c2κℓ,πα−π2α2−42c1+c2κ2b2ℓ3π2c1+c2κℓ,so that the coefficient of ∫01qx2x,tdx in (97) is negative. Thereby, (97) together with (31) implies that(99)ℱ′t≤−ϱE1t,where E1t=1/2qx,t,qℓ,t−τρℋ2 (see (79)) and ϱ is a negative constant depending solely on the physical parameters α,β, and ℓ. Finally, it is easy to check that ℱt satisfies similar inequalities to (91) and hence the exponential stability of the system follows.Remark 2.
(a)
We emphasize on the fact that if the value of the feedback gainκ increases, then the decay in Theorems 7 and 8 is slower. This is as a direct consequence of (78) (see also (94)). In turn, such conditions often occur for systems with time delay (see [27, 28, 33, 34, 44]) and can be viewed as a “smallness” of the delay term.(b)
It is also worth mentioning that the stability result obtained in Theorems7 and 8 is valid without any restriction on the positive constant γ. Whence, γ can be tuned in such a way the decay occurs as we desire.


## 6. Numerical Results

The discretization of the closed-loop system (14) is given by(100)∂qN∂t−α∂2qN∂x2+β∂qN∂x+κqNℓ,t−τbxx=xj=0;j=1,…,N−1,with(101)qN0,t=0,∂qN∂xℓ,t=0,t>0,(102)qNxj,s=fxj,s,j=0,…,N,−τ≤s≤0,in which qN is the discrete solution given by its value at the grid points xj. Indeed, the expression of qN involves the Chebyshev series as follows:(103)qNx,t=∑k=0Nq^ktTkx,where for each j=0,…,N, the function Tjx is the Chebyshev polynomial of the first kind, which is defined as the eigenfunction of the singular Sturm–Liouville differential equation:(104)ddx1−x2dTkdxx+k21−x2Tkx=0.In the above equation,(105)Tkx=coskθandθ=arccosx.The transformationx˜=2x/ℓ−1 will be applied on (100) and (102) so that the spatial domain 0,ℓ becomes −1,1.In the numerical simulations presented in this section, the Chebyshev collocation method (see [45] for more details) combined with backward Euler method and the Gauss–Lobatto points given by(106)xj=cosjπN,j=0,…,N,are used to deal with the feedback time delay models (100)–(102) for different values of the delay and the physical parameters of the canal.Figure1 describes the behavior of the solution qx,t when the distribution function bx=x/π−1, the initial condition is qx,s=1+ssinx/2, and the control gain κ=0.02,α=0.4,andβ=0.1, for different values of τ. The output qℓ,t versus time, for different values of τ, is shown in Figure 2(a) and also in Figure 2(b) but for different values of κ. Figure 3 displays the L2-norm of qx,t versus time for different values of τ (Figure 3(a)) and for different values of κ (Figure 3(b)). Note that when κ=0.02 and as the value of τ increases from τ=0.02 to τ=0.1, the output qℓ,t slowly converges to zero. Also, when τ=0.02 and as the value of κ is increased from κ=0.02 to κ=0.12, the solution slowly converges to the zero solution. This validates the analytical results presented in Section 5, where we have used smallness conditions (78) and (94) (see also Remark 2).Figure 1
Time evolution ofqx,t in (14) with the spatial distribution function bx=x/π−1, qx,s=1+ssinx/2, α=0.4,β=0.1,κ=0.02, and for different values of τ: (a) τ=0.02; (b) τ=0.04; (c) τ=0.06; (d) τ=0.1.
(a)
(b)
(c)
(d)Figure 2
(a)qℓ,t vs. time when bx=x/π−1, qx,s=1+ssinx/2, α=0.4,β=0.1,κ=0.02, and for different values of τ. (b) qℓ,t vs. time when α=0.4,β=0.1,τ=0.02, and for different values of κ.
(a)
(b)Figure 3
(a) TheL2-norm of qx,t vs. time with bx=x/π−1, qx,s=1+ssinx/2, α=0.4,β=0.1,κ=0.02 and for different values of τ. (b) The L2-norm of qx,t vs. time when α=0.4,β=0.1,τ=0.02, and for different values of κ.
(a)
(b)Figures4–6 are similar to Figures 1–3 except that the initial conditions are qx,s=1+sπx2−x3/3. Again the figures show that as the value of τ is increased, the output qℓ,t slowly tends to zero, and this ties with our theoretical results.Figure 4
Time evolution ofqx,t in (14) when bx=x/π−1, qx,s=1+sπx2−x3/3, α=0.4,β=0.1,κ=0.02, and for different values of τ: (a) τ=0.02; (b) τ=0.04; (c) τ=0.06; (d) τ=0.1.
(a)
(b)
(c)
(d)Figure 5
(a)qℓ,t vs. time with the spatial distribution function bx=x/π−1, qx,s=1+sπx2−x3/3, α=0.4,β=0.1,andκ=0.02 and for different values of τ. (b) qℓ,t vs. time when α=0.4,β=0.1,τ=0.02, and for different values of κ.
(a)
(b)Figure 6
(a) TheL2-norm of qx,t vs. time when bx=x/π−1, qx,s=1+sπx2−x3/3, α=0.4,β=0.1,κ=0.02, and for different values of τ. (b) The L2-norm of qx,t vs. time when α=0.4,β=0.1,τ=0.02, and for different values of κ.
(a)
(b)


## 7. Conclusion

In this article, an irrigation system has been considered when a constant time delay occurs in the flow control. Despite the presence of such a delay which could be a source of instability of the system or poor performance of the control, it has been shown that the impact of the delay can be eliminated provided the feedback gain satisfies some smallness conditions. Numerical results are also provided to demonstrate the correctness of the theoretical outcomes.We would like to point out that it would be worthwhile to consider a time-dependent delay in the control and investigate the stability of the system. One may also investigate the stability of a networked system composed ofN channels (see [46, 47] for finite-dimensional systems). This will be the subject of a future work.



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