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

+is paper is concerned with the feedback flow control of an open-channel hydraulic systemmodeled by a diffusive wave equation with delay. Firstly, we put forward a feedback flow control subject to the action of a constant time delay. +ereafter, 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.


Introduction
e 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. e system consists of two nonlinear coupled hyperbolic partial differential equations: where x is the spatial location (m) and t is the time (s).
Furthermore, E represents the absolute water surface elevation (m), Z(x, E) is the wetted cross-sectional area (m 2 ), q(x, t) is the flow discharge (m 3 /s), S(q, E, x) the friction slope, and L d (x, t) is the lateral discharge (m 2 /s), which can be interpreted as an inflow if L d > 0 and an outflow when L d < 0. Moreover, r � 0 as long as L d > 0, whereas if L d < 0, then r � 1. Finally, V(x, t) is the mean velocity (m/s) in section Z and g the gravitational acceleration (m 2 /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. is has motivated many authors to derive other models. For instance, the following model has been used in [4,5]: 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 (zq/zt) + (z/zx)(q 2 /Z) are negligible with respect to gZ(zE/zx) (for further discussion on the model, the reader is referred to [4,6]). However, for sake of clarity, we shall (8) where ((zf/zu)(u, v)) v denotes the variation of f with respect to u when v is fixed. Combining these identities with (6) gives where α(q, E, x) � 1 W(zS/zq) , 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. e 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][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] (see also [21] for a nonlinear case). Later, using the tools of control theory of partial differential equations [22], the authors in [23][24][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]: together with the boundary conditions and the output observation where U(t) is the control, y(t) is the output, and b(x) is the spatial distribution of the actuator. en, it has been shown that the closed-loop system is exponentially stable under the action of the control U(t) � κq(ℓ, 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| < (2 �� � αβ /(‖b‖ℓ)). 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.
is 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][27][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. e reader can also consult [26][27][28][29][30][31][32][33][34][35], where the effect of time delays in several types of partial differential equations (PDEs) has been studied. e 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 where κ > 0 is the feedback gain and τ > 0 is the delay. en, it is shown that the closed-loop system (11)-(13) is stable even though a delay supervenes in the proposed control. is 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.

Complexity
(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. e rest of this paper is organized as follows. In Section 2, 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.

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 data q 0 and u 0 , the closed-loop system (11)-(13) can be written as follows: For sake of clarity, we recall that is endowed with its standard norm ‖φ‖ � ( is equipped with the usual norm ‖φ‖ H n (0,ℓ) � n i�0 ‖φ (i) ‖ L 2 (0,ℓ) . In view of the well-known change of state variable [33,37], the closed-loop system (14) becomes Complexity 3 whereupon consider the state space equipped with the following inner product: Obviously, the norm induced by such an inner product is equivalent to the usual norm of the space L 2 (0, ℓ) × L 2 (0, 1). For convenience, we shall often denote by f v the derivative of f with respect to the variable v. en, let us assume that the spatial actuator b(·) ∈ L 2 (0, ℓ) (one may suppose that b(·) ∈ L ∞ (0, ℓ)) and consider the system operator: ereby, system (18) can be brought to the following form: where ϕ � (q, u) and ϕ 0 � (q 0 , u 0 ).

Open-Loop
System. First and foremost, we shall investigate the main properties of the uncontrolled system which is given by Taking the Hilbert state space L 2 (0, ℓ) equipped with the usual inner product, system (24) can be written in the ab- where A 0 is an unbounded linear operator defined by e following results will be systematically used.
Theorem 1 (Lumer-Phillips) (see [38]). Given a linear operator P on a Hilbert space Y with D(P) � Y, if P is dissipative and there is a scalar λ 0 such that λ 0 I − P is onto Y, then P generates a C 0 semigroup of contractions on Y.
Theorem 2 (see [39]). Assume that P be a closed densely defined operator on a Hilbert space Y satisfying the following where P * is the adjoint operator of P. en, the operator P e basic properties of the operator A 0 (see (25)) are summarized below.

Proposition 1.
e operator A 0 defined by (25) generates a C 0 -semigroup of contractions S 0 (t) on L 2 (0, ℓ). In addition, the resolvent of A 0 is compact and the spectrum σ(A 0 ) consists entirely of isolated eigenvalues with finite multiplicities.
Proof. Using (25) and integrating by parts, we obtain erefore, the resolvent equation A 0 R � Q has a unique solution R ∈ D(A 0 ) for each Q ∈ L 2 (0, ℓ). Evoking eorem 1, one can deduce that A 0 generates a C 0 -semigroup of contractions S 0 (t) on L 2 (0, ℓ). On the other hand, using the Sobolev embedding theorem [40], we conclude that the resolvent operator of A 0 is compact on L 2 (0, ℓ) and hence the spectrum of A 0 only has isolated eigenvalues with finite multiplicities [41].
For sake of completeness, let us recall the following definition and theorem.

Definition 1
If there exist positive constants C and ω such that ||S(t)|| ≤ Ce − ωt , for all t ≥ 0, then the semigroup is exponentially stable.
Theorem 3 (see [42]). Let S(t) be a C 0 -semigroup in Y whose generator P has a compact resolvent.

en, S(t) 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 operator A 0 (see (25)) is as follows.

Complexity
Proof. Let R(x, t) be the unique solution stemmed from an initial condition R 0 in D(A 0 ). Integrating by parts and using the boundary conditions in (25), we have Moreover, for each φ ∈ f ∈ H 1 (0, ℓ); f(0) � 0 , we have the following Wirtinger's inequality [43]: which together with the last estimate imply that thanks to a classical density argument. □ Now, we turn to the characterization of the spectrum of A 0 .

Proposition 3.
e eigenvalues of A 0 form the following set of negative real numbers: where e n is a nonzero real solution of 2αe n cos e n � − βℓ sin e n .
Proof. It follows from (25) that a complex μ is an eigenvalue of A 0 if and only if the system has a nontrivial solution. For sake of clarity, let Two cases will be treated, namely, G μ � 0 and G μ ≠ 0. i) G μ � 0, which means that μ � − β 2 /(4α) and hence σ 1 � σ 2 � β/(2α) > 0. ereby, it is easy to check that (34) has only the trivial solution and hence μ � − β 2 /(4α) cannot be an eigenvalue of A 0 .
ii) G μ ≠ 0, i.e., σ 1 ≠ σ 2 . In such a case, the general solution of (34) is R(x) � k 1 e σ 1 x + k 2 e σ 2 x , where the constants satisfy the linear system: whereupon μ is an eigenvalue if and only if σ 1 e σ 1 ℓ � σ 2 e σ 2 ℓ , which can be written as follows: where ζ � ( �� � Gμ /(2α))ℓ. e 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) is implies that a and e must obey the following condition: It is a simple task to check that the above equation is valid only in the event that a � 0 and thus (38) becomes βℓ sin e + 2αe cos e � 0. Recalling that ζ � ie � ( �� � Gμ /(2α))ℓ and using (35), we obtain the desired result. □ e asymptotic expansion of the eigenvalues of A 0 is provided in the following proposition. Proposition 4. Let μ n be an eigenvalue of system (24). en, for n large enough, we have Proof. In view of Proposition 3, we know that μ is an eigenvalue of A 0 if and only if ζ � ( �� � Gμ /(2α))ℓ 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. ereby, we just need to find the solutions of (37) in a sector S defined by arg(θ) � 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: for n large enough. Finally, by virtue of (35) and the fact that ζ � ( �� � Gμ /(2α))ℓ, it follows that μ n � − (β/ℓ) − (β 2 /(4α)) − (π 2 /(αℓ 2 ))(n + (1/2)) 2 + O(1/n) + iO(1/n), which is the expression in our proposition.

Semigroup Generation for the Closed-
Loop System e main objective of this section is to show that the closedloop system (23) is well posed in the state space H � L 2 (0, ℓ) × L 2 (0, 1). e proof of such a desirable result is based on the application of semigroup theory of linear operators. Specifically, we shall use eorem 2. We have indeed the following result.

Theorem 4.
e operator A, defined by (21) and (22), generates a C 0 -semigroup S(t) on the Hilbert space H.
Proof. Let ϕ � (q, u) ∈ D(A). Invoking (21) and (22) and integrating by parts, we get Applying Cauchy-Schwartz and Young's inequalities, the latter becomes for any positive constant C. Choosing C � β/|κ|, we deduce that e task ahead is to show that the above inequality also holds for the adjoint operator of A * . 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 Subsequently, let us consider (q, u) ∈ D(A * ). Integrating by parts twice and using (45) and (46), it follows that By virtue of (44) and (47), we conclude, thanks to eorem 2, that the operator D(A) generates a C 0 -semigroup S(t) satisfying ‖S(t)‖ L(H) ≤ exp ((|κ| 2 ‖b‖ 2 / (2β))t).
□ Remark 1. A direct consequence of eorem 4 is that the C 0 -semigroup S(t) 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 S(t) is not dissipative for all values of the feedback gain κ > 0, unlike the uncontrolled system whose operator A 0 is dissipative and hence its semigroup is contractive (see Proposition 1).

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 operator A is indeed a discrete operator.
erewith, A is a discrete operator and the spectrum set σ(A) consists of isolated eigenvalues with finite multiplicities. Moreover, if λ ∈ ρ(A), then the resolvent operator (λI − A) − 1 is compact.
Proof. For arbitrary (f, g) ∈ H, let us solve the resolvent equation with λ > 0: Solving the equation of u in the above system and using the condition u(0) � q(ℓ), we obtain 6 Complexity and hence whereupon resolvent (48) admits a unique solution (q, u) ∈ D(A) once we show the existence of a unique q ∈ H 2 (0, ℓ) satisfying Since λ > 0 and recalling (35), one can readily check that the general solution of (51) is given by In (52), l 1 and l 2 are two constants that satisfy, thanks to the boundary conditions in (51), the following linear system: in which We suppose that λ > 0 is large enough so that σ 2 e σ 2 ℓ − σ 1 e σ 1 ℓ ≠ 0. Accordingly, the above linear system has a unique solution: Combining the last findings with (52) yields is clearly gives which together with (50) implies that In light of (35), the latter becomes Consequently, q(ℓ) is uniquely determined if and only if the following expression 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 ∈ L(H). 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 ∈ ρ(A) [41].

Complexity 7
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: where and G λ is defined by (35).
Proof. Let λ ∈ C. e eigenvalue problem of A consists in seeking a nontrivial element (q, u) ∈ D(A) such that One can check from the second equation of (65) that u is given by Herewith, it suffices to find a nontrivial solution q ∈ H 2 (0, ℓ) to the system We shall treat 3 cases.

Case 1. A straightforward computation shows that λ � 0 is an eigenvalue if and only if
Consequently, ereby, we conclude that λ � 0 is an eigenvalue of the system if and only if the feedback gain satisfies the following condition: (70) (72) e assertion (ii) is then proved.

Stability of the Closed-Loop System
e 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 ℓ 0 (1− e − β/αs )b(s)ds (see eorem 6). Otherwise, the system is unstable by eorem 3. Our first stability result is as follows.

Theorem 7. Suppose that the feedback gain κ fulfills the condition
en, the C 0 -semigroup S(t) generated by the operator A, defined by (21) and (22), is exponentially stable in H. In other words, given any solution q(x, t) of system (14), the following energy-norm exponentially tends to zero as long as condition (78) is fulfilled.
Proof. Let (q(x, t), u(ρ, t)) be the unique solution of system (18) stemmed from an initial condition (q 0 , u 0 ) in D(A). Subsequently, differentiating (79) with respect to t, integrating by parts, and using the boundary conditions in (18), we get Next, evoking Young's and Cauchy-Schwartz inequalities yields (as for (43) and (44)) Furthermore, by virtue of Wirtinger's inequality (31), estimate (81) becomes anks to assumption (78), one can write (82) as follows: in which On the other hand, let us introduce a new functional as where ς is a positive constant to be determined, while c is an arbitrary positive constant. One can check that Moreover, differentiating (85), we have Complexity 9 In view of the well-known inequality one can deduce from (87) that Now, let Using (79), (85), and (86), an elementary computation shows that there exist positive constants L 1 and L 2 such that for any t ≥ 0, we have Furthermore, differentiating (90) and using (83) and (89), we get where ω 1 is defined by (84). In view of (78), one can choose ς < ω 1 /ℓ and hence (92) yields 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.
Proof. We begin by introducing the following functional: where ξ and ] are positive constants to be obtained and c is an arbitrary positive constant. Arguing as before, we have where K is a positive constant related to the application of Young's inequality. Next, we invoke (31) and (88) and choose ] � c 1 |κ|K and ξ � c 2 |κ|K, where c 1 > 0 and c 2 > 1. ereafter, (96) yields In the light of (94), one can choose the positive constant K such that so that the coefficient of (97) is negative. ereby, (97) together with (31) implies that where E 1 (t) � (1/2)‖(q(x, t), q(ℓ, t − τρ))‖ 2 H (see (79)) and 9 is a negative constant depending solely on the physical parameters α, β, and ℓ. Finally, it is easy to check that F(t) 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 gain κ increases, then the decay in eorems 7 and 8 is slower. is 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 eorems 7 and 8 is valid without any restriction on the positive constant c. Whence, c can be tuned in such a way the decay occurs as we desire.

Numerical Results
e discretization of the closed-loop system (14) is given by with in which q N is the discrete solution given by its value at the grid points x j . Indeed, the expression of q N involves the Chebyshev series as follows: where for each j � 0, . . . , N, the function T j (x) is the Chebyshev polynomial of the first kind, which is defined as the eigenfunction of the singular Sturm-Liouville differential equation: In the above equation, T k (x) � cos kθ and θ � arc(cos x).
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 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. Figure 1 describes the behavior of the solution q(x, t) when the distribution function b(x) � (x/π) − 1, the initial condition is q(x, s) � (1 + s)sin(x/2), and the control gain κ � 0.02, α � 0.4, and β � 0.1, for different values of τ. e output q(ℓ, 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 L 2 -norm of q(x, 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 q(ℓ, 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. is validates the analytical results presented in Section 5, where we have used smallness conditions (78) and (94) (see also Remark 2).    Figures 1-3 except that the initial conditions are q(x, s) � (1 + s)(πx 2 − (x 3 /3)). Again the figures show that as the value of τ is increased, the output q(ℓ, t) slowly tends to zero, and this ties with our theoretical results.

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 of N channels (see [46,47] for finite-dimensional systems). is will be the subject of a future work.

Data Availability
No data were used to support this study. 14 Complexity