Stabilization method for the Saint-Venant equations by boundary control

Abstract

In this paper, we are interested in the stabilization of the flow modeled by the Saint-Venant equations. We have solved two problems in this study. The first, we have proved that the operator associated to the Saint-Venant system has a finite number of unstable eigenvalues. Consequently, the system is not exponentially stable on the space $L^{2} (Ω) \times L^{2} (Ω)$ , but is exponentially stable on a subspace of the space $L^{2} (Ω) \times L^{2} (Ω)$ , ( $Ω$ is a given domain). The second problem, if the advection is dominant, the natural stabilization is very slow. To solve these problems, we have used an extension method due to Russel (1974) and Fursikov (2002). Thanks to this method, we have determined a boundary Dirichlet control able to accelerate the stabilization of the flow. Also, the boundary Dirichlet control is able to kill all the unstable eigenvalues to get an exponentially stable solution on the space $L^{2} (Ω) \times L^{2} (Ω)$ . Then, we extend this method to the finite difference equations analog of the continuous Saint-Venant equations. Also, in this case, we obtained similar results of stabilization. A finite difference scheme is used to compute the control and several numerical experiments are performed to illustrate the efficiency of the control.

Keywords

Saint-Venant equations Dirichlet control linear stability stabilization extension method finite difference equations

Introduction

Solutions of certain hyperbolic and parabolic systems can reach the stable state in finite-time. Such a property, called finite-time stability in Alabau et al. (2015); Perrollaz and Rosier (2013, 2014), was first noticed in Komornik (1991); Majda (1974/75) for the (linear) wave equation, and recently in Gugat et al. (2018) for the (quasilinear) hyperbolic equations.

The finite-time stability of a system may be lost when a small, boundary perturbation is added to the system. A famous example is provided by the telegraph equation, see Gugat (2014). See also Gugat et al. (2017) for the exponential stabilization of the isothermal Euler equations and Datko et al. (1986) for the loss of the stability when incorporating an arbitrarily small delay in a transparent boundary condition for the wave equation.

The aim of this paper is the study of the infinite-time stability of a coupled hyperbolic-parabolic linear system by boundary Dirichlet control. In our model, the coupling between the hyperbolic character and the parabolic character leads to some specific difficulties that are interesting to understand. Indeed, the well posedness of the considered problem, in infinite time, requires special treatment. Moreover, we need adequate methods for the numerical simulation of this type of the coupled problems.

In this paper, we are interested in the stabilization of a linear Saint-Venant system, in one-dimensional domain $Ω =] 0, ℓ [$ , given by the equations

\partial_{t} y - \partial_{x} z - α y = 0,

(1)

\partial_{t} z - a \partial_{x} y - ν \partial_{xx} z - α z = 0,

(2)

z (x, 0) = z_{0} (x), y (x, 0) = y_{0} (x),

(3)

z (0, t) = q_{0} (t), z (ℓ, t) = q_{ℓ} (t) .

(4)

In this setting, $ν > 0$ is the viscosity, a is the advection coefficient and $α$ is a strictly positive real number. The boundary data $q_{0} (t)$ and $q_{ℓ} (t)$ , defined on the boundary of the domain $Ω =] 0, ℓ [$ , are controls.

Statement of the stabilization problem: The stabilization problem for the boundary-value problem (1)–(4) is formulated as follows: Given $(y_{0}, z_{0})$ in an appropriate space $X_{0} (Ω)$ , find controls $q_{0} (t)$ and $q_{ℓ} (t)$ such that the solution $(y, z)$ to the boundary-value problem (1)–(4) satisfies the estimation

‖ (y, z) (., t) ‖_{X (Ω)} \leq C e^{- σ t}, t \geq 0,

where $X_{0} (Ω)$ and $X (Ω)$ are Hilbert spaces that will be defined later, and the decay rate $σ$ is determined under some conditions that will be clarified in Section 4.

The paper is organized as follows. In Section 2, we study the Saint-Venant system (1)–(4) with homogeneous boundary conditions. By means of Fourier analysis, we show that the associated semigroup is not exponentially stable. Indeed, it has a finite number of unstable eigenvalues. Consequently, the natural stabilization of the flow takes place with an exponential decay rate on a subspace of the Hilbert space $L^{2} (Ω) \times L^{2} (Ω)$ . In Section 3, we have studied the well posedness of system (1)–(4) in the presence of non-homogeneous boundary conditions. In Section 4, we gave the first main result, (Theorem 3). We show that the stabilization of the flow, with respect to the $H^{1}$ -norm, can be improved by means of boundary controls which are constructed by an extension method introduced by Russel (1974) and Fursikov (2001, 2002), Ervadoza et al. (2012). Let us mention that in the results in Bensoussan et al. (1993), Part III could be used for the stabilizability in $L^{2}$ -norm with a bounded control operator, but does not operate for the stabilizability in $H^{1}$ with boundary controls as in system (31)–(34). In Section 5, we study the stabilization of the discrete problem. We apply the extension method, described previously in Section 4, for the finite difference equations analog of the continuous Saint-Venant equations (1)–(4). In this case, the second main result is given by Theorem 4.

In this paper, we have studied a more general problem than that is studied in the paper Arfaoui et al. (2011). Indeed, the work in Arfaoui et al. (2011) is only a special case of the work in this paper, if we take $α = 0$ in the system (1)-(4). The terms $(- α y)$ and $(- α z)$ , ( $α > 0$ ), in equations (1) and (2), respectively, lead to the appearance of instability in the system of equations (1)–(4), (existence of a finite number of unstable eigenvalues, see Remark 2), which makes the system unstable. Consequently, we have shown that there are subspaces in $L^{2} (Ω) \times L^{2} (Ω)$ and in $H^{1} (Ω) \times H^{1} (Ω)$ in which the solution is exponentially stable with a decay rate depending on the parameters $α$ , the viscosity $ν$ and the advection coefficient a, (see Corollary 1, Theorem 1). Also, using the extension method, we have proved the existence of a boundary Dirichlet control that makes the solution exponentially stable over the entire space $H^{1} (Ω) \times H^{1} (Ω)$ , (Theorem 3). This means, in a sense, that the unstable eigenvalues are killed by the boundary Dirichlet control.

Also, the novelty in this paper, with respect to Arfaoui et al. (2011), is the extension of the study of stabilization of the continuous problem to the discrete problem. we have been able to adapt the extension method to the discrete problem associated with the continuous problem. In this case, we have also obtained some results similar to those obtained in the continuous case.

Some notations:

(i) The Lebesgue space $L_{m}^{2} (Ω)$ is defined as follows

L_{m}^{2} (Ω) = {f \in L^{2} (Ω) : \int_{Ω} f (x) dx = 0} .

(ii) The space $L^{2} (Ω) = L^{2} (Ω) \times L^{2} (Ω)$ is equipped with the scalar product

\begin{matrix} (Y, V)_{L^{2} (Ω)} = a \int_{Ω} y (x) v (x) dx + \\ \int_{Ω} z (x) w (x) dx, \end{matrix}

(5)

where $Y = (y, z)^{T}$ and $V = (v, w)^{T}$ .

(iii) The space $L_{m}^{2} (Ω) = L_{m}^{2} (Ω) \times L^{2} (Ω)$ is equipped with the $L^{2}$ -inner product defined above.

(iv) The space $H_{m}^{1} (Ω) = H^{1} (Ω) \cap L_{m}^{2} (Ω)$ where $H^{1} (Ω)$ is the standard Sobolev space.

(v) The space $H_{m}^{1} (Ω) = H_{m}^{1} (Ω) \times H_{0}^{1} (Ω)$ is equipped with the scalar product

\begin{matrix} ((Y, V))_{H_{m}^{1} (Ω)} = a \int_{Ω} y' (x) v' (x) dx + \\ \int_{Ω} z' (x) w' (x) dx . \end{matrix}

The associated norm, denoted by $‖ \cdot ‖_{H_{m}^{1} (Ω)}$ , is equivalent to the usual norm of $H_{m}^{1} (Ω) \times H_{0}^{1} (Ω)$ , (stems from the Poincaré-Wirtinger and Poincaré inequalities).

Fourier analysis and stability

Let us fix $Ω =] 0, π [$ and $Q_{\infty} = Ω \times (0, \infty)$ . Consider the homogeneous system

\partial_{t} y - \partial_{x} z - α y = 0, in Q_{\infty},

(6)

\partial_{t} z - a \partial_{x} y - ν \partial_{xx} z - α z = 0, in Q_{\infty},

(7)

z (x, 0) = z_{0} (x), y (x, 0) = y_{0} (x), in Ω,

(8)

z (0, t) = 0, z (π, t) = 0, in (0, + \infty) .

(9)

Let us set $Y = (y, z)$ and $Y_{0} = (y_{0}, z_{0})$ . The system (6)–(9) can be written in the following abstract way

Y' (t) + A_{α} Y (t) = 0, \forall t > 0,

(10)

Y (0) = Y_{0},

(11)

where $A_{α}$ is an unbounded operator on $L^{2} (Ω)$ defined by

A_{α} = (\begin{matrix} - α & - \frac{d}{dx} \\ - a \frac{d}{dx} & - ν \frac{d^{2}}{d x^{2}} - α \end{matrix}) .

(12)

Moreover, we have

\begin{matrix} D (A_{α}) = {(y, z) \in L^{2} (Ω) | z \in H_{0}^{1} (Ω), \\ ay + ν z' \in H^{1} (Ω)} . \end{matrix}

The operator $(- A_{α}, D (A_{α}))$ generates a semigroup on $L^{2} (Ω)$ denoted by

S_{α} (t) = e^{- t A_{α}}, \forall t \geq 0 .

In the following, thanks to a Fourier analysis of the differential operator $A_{α}$ , we prove that the semigroup $(S_{α} (t))_{t \geq 0}$ is only exponentially stable on a Hilbert subspace of the space $L^{2} (Ω)$ , (see Remark 2 and Corollary 1), which is specified later on. Furthermore, we give an exponential decay rate of the solution to the system (6)–(9), which depends essentially on the parameters a, $α$ and $ν$ .

Fourier analysis

We prove that the semigroup $(S_{α} (t))_{t \geq 0}$ , generated by $(- A_{α})$ , is not exponentially stable on the Hilbert space $L_{m}^{2} (Ω)$ . Indeed, thanks to a decomposition in Fourier series for the differential operator $A_{α}$ , we show that the spectrum $σ (A_{α})$ contains unstable eigenvalues.

Let us introduce the Fourier bases $(Φ_{n})_{n \in N}$ in $L^{2} (Ω)$ defined by

\begin{matrix} Φ_{0} (x) = \frac{1}{\sqrt{a π}} {(1, 0)}^{T}, \\ Φ_{2 n} (x) = \sqrt{\frac{2}{a π}} {(\cos (\frac{nx}{2}), 0)}^{T}, for n \geq 1, \\ Φ_{2 n - 1} (x) = \sqrt{\frac{2}{π}} {(0, \sin (\frac{nx}{2}))}^{T}, for n \geq 1, \end{matrix}

and consider the one- or two-dimensional subspaces

\begin{matrix} V_{0} = vect (Φ_{0}), \\ V_{n} = vect (Φ_{2 n}, Φ_{2 n - 1}), for n \geq 1, \end{matrix}

where $' vect'$ denotes the vector space generated by the family between parentheses. Notice that $Φ_{n} \in D (A_{α})$ for all $n \in N$ . In this case, the spaces $L^{2} (Ω)$ and $L_{m}^{2} (Ω)$ can be written as direct sums as follows

L^{2} (Ω) = \underset{n \geq 0}{\oplus} V_{n}, L_{m}^{2} (Ω) = \oplus_{n \geq 1} V_{n} .

They are orthogonal sums. The spaces $H^{1} (Ω) \times H_{0}^{1} (Ω)$ and $H_{m}^{1} (Ω)$ are also orthogonal sums of the subspaces $V_{n}$ :

H^{1} (Ω) \times H_{0}^{1} (Ω) = \oplus_{n \geq 0} V_{n}, H_{m}^{1} (Ω) = \oplus_{n \geq 1} V_{n} .

Also, we can express $Y_{0}$ and $Y$ in the Fourier bases $(Φ_{n})_{n \in N}$ , as follows

\begin{matrix} Y_{0} = \sum_{n = 1}^{\infty} (c_{0, n} Φ_{2 n} + d_{0, n} Φ_{2 n - 1}), \\ Y (t) = \sum_{n = 1}^{\infty} (c_{n} (t) Φ_{2 n} + d_{n} (t) Φ_{2 n - 1}) . \end{matrix}

Remark 1: For all $n \geq 1$ , $V_{n}$ is invariant under $A_{α}$ and $A_{α}^{n} = A_{α} |_{V_{n}} \in L (V_{n})$ has the following matrix representation in the basis $(Φ_{2 n}, Φ_{2 n - 1})$ of $V_{n}$

A_{α}^{n} = (\begin{matrix} - α & - \sqrt{a} n \\ \sqrt{a} n & ν n^{2} - α \end{matrix}) .

(i) When $n \in [1, \frac{2 \sqrt{a}}{ν} [$ , $A_{α}^{n}$ has two conjugate complex eigenvalues

λ_{n}^{α} = (δ_{n} - α) - i η_{n}, μ_{n}^{α} = \bar{λ_{n}^{α}},

where

δ_{n} = \frac{ν n^{2}}{2}, η_{n} = \frac{ν n^{2}}{2} \sqrt{\frac{4 a}{ν^{2} n^{2}} - 1} .

It follows that

ℜ (λ_{n}^{α}) = ℜ (μ_{n}^{α}) = \frac{ν n^{2}}{2} - α \geq \frac{ν}{2} - α,

| ℑ (λ_{n}^{α}) | = | ℑ (μ_{n}^{α}) | = η_{n} \leq \frac{2 a}{ν} \sqrt{\frac{4 a}{ν^{2}} - 1} .

(ii) When $n \in [\frac{2 \sqrt{a}}{ν}, + \infty [$ , the eigenvalues of $A_{α}^{n}$ are real and defined by

λ_{n}^{α} = λ_{n} - α, μ_{n}^{α} = μ_{n} - α,

where $λ_{n}$ and $μ_{n}$ are given by

λ_{n} = \frac{ν n^{2}}{2} (1 - \sqrt{1 - \frac{4 a}{ν^{2} n^{2}}}),

μ_{n} = \frac{ν n^{2}}{2} (1 + \sqrt{1 - \frac{4 a}{ν^{2} n^{2}}}) .

We also have

\frac{a}{ν} - α < λ_{n}^{α} \leq \frac{2 a}{ν} - α \leq \frac{2 a}{ν}

and

\frac{2 a}{ν} - α \leq \frac{ν n^{2}}{2} - α \leq μ_{n}^{α} < ν n^{2} - α < ν n^{2} .

Moreover, the limits at infinity are

lim_{n \to \infty} (λ_{n}^{α}, μ_{n}^{α}) = (\frac{a}{ν} - α, \infty) .

As a consequence, we notice that the spectrum of $A_{α}$ except zero, $σ (A_{α}) \ {0}$ , is located in the complex half-plane

σ (A_{α}) \ {0} \subset {λ \in C, ℜ (λ) > τ_{α}} \cap {λ \in C, | ℑ (λ) | \leq γ}

where the constants $τ_{α}$ and $γ$ are given by

τ_{α} = min (\frac{ν}{2} - α, \frac{a}{ν} - α), γ = \frac{2 a}{ν} \sqrt{\frac{4 a}{ν^{2}} - 1},

(13)

and (ℜ,ℑ) stands for the (real, imaginary) part of a complex number.

We denote by $(S_{α}^{n} (t))_{t \geq 0}$ the semigroup $(S_{α} (t) |_{V_{n}})_{t \geq 0}$ . Let $n \geq 1$ , $(S_{α}^{n} (t))_{t \geq 0}$ stands for the matrix representation of $(S_{α}^{n} (t))_{t \geq 0}$ in the basis $(Φ_{2 n}, Φ_{2 n - 1})$ of $V_{n}$ . We have

Proposition 1: If $n = 0$ , then $S_{α}^{0} (t) = I_{V_{0}}$ , where $I_{V_{0}}$ is the identity in $V_{0}$ .

If $n \in [1, \frac{2 \sqrt{a}}{ν} [$ , we have

\begin{matrix} S_{α}^{n} (t) = \frac{e^{(α - δ_{n}) t}}{η_{n}} \times \\ (\begin{matrix} η_{n} \cos (η_{n} t) + δ_{n} \sin (η_{n} t) & \sqrt{a} n \sin (η_{n} t) \\ - \sqrt{a} n \sin (η_{n} t) & η_{n} \cos (η_{n} t) - δ_{n} \sin (η_{n} t) \end{matrix}) \end{matrix}

If $n \in [\frac{2 \sqrt{a}}{ν}, + \infty [$ , then

\begin{matrix} S_{α}^{n} (t) = \frac{e^{α t}}{μ_{n} - λ_{n}} \times \\ (\begin{matrix} μ_{n} e^{- λ_{n} t} - λ_{n} e^{- μ_{n} t} & \sqrt{a} n (e^{- λ_{n} t} - e^{- μ_{n} t}) \\ - \sqrt{a} n (e^{- λ_{n} t} - e^{- μ_{n} t}) & μ_{n} e^{- μ_{n} t} - λ_{n} e^{- λ_{n} t} \end{matrix}), \end{matrix}

where $λ_{n}$ and $μ_{n}$ are defined in Remark 1.

Remark 2: From a detailed analysis of the eigenvalues of $A_{α} | V_{n}$ and using the Remark 1, we can deduce easily the following two cases:

Case 1: If the parameters $α$ , $ν$ and a verify the condition $(α - \frac{a}{ν}) \leq 0,$ then all the real eigenvalues are stable. On the other hand, the complex eigenvalues are unstable if $n \in [1, \sqrt{\frac{2 α}{ν}} [$ , and stable if $n \in [\sqrt{\frac{2 α}{ν}}, \frac{2 \sqrt{a}}{ν}]$ . This enables us to consider a control of the unstable eigenvalues which are in a finished number. Moreover, we have the following estimates on the real eigenvalues

- (α - \frac{a}{ν}) < λ_{n}^{α} \leq μ_{n}^{α}, \forall n \in [\frac{2 \sqrt{a}}{ν}, + \infty [.

Case 2: If $(α - \frac{a}{ν}) > 0,$ $A_{α} | V_{n}$ has an infinite number of negative eigenvalues $λ_{n}^{α}$ that will give unstable solutions.

It follows from the Remark 2, that the semigroup $(S_{α} (t))_{t > 0}$ on $L_{m}^{2} (Ω)$ , generated by the operator $A_{α}$ , is not exponentially stable on $L_{m}^{2} (Ω)$ . Indeed, its spectrum contains a finite number of unstable eigenvalues, (when $(α - \frac{a}{ν}) \leq 0$ ). Consequently, we show, in what follows, that it exists subspace of the space $L^{2} (Ω) \times L^{2} (Ω)$ , (of the space $H^{1} (Ω) \times H^{1} (Ω)$ ), in which the solution is stable.

We introduce the subspace $X_{α, ν}^{0} (Ω)$ of $L_{m}^{2} (Ω)$ as follows

\begin{matrix} X_{α, ν}^{0} (Ω) = \\ {Y_{0} = \sum_{n = 1}^{\infty} (c_{0, n} Φ_{2 n} + d_{0, n} Φ_{2 n - 1}) \in L_{m}^{2} (Ω) | \\ c_{0, n} = d_{0, n} = 0, \forall n \leq \sqrt{\frac{2 α}{ν}}} . \end{matrix}

The subspace $X_{α, ν}^{0} (Ω)$ is associated with the set of all stable eigenvalues. So, using Remark 1 and Remark 2, we can prove that the restruction of the semigroup $(S_{α} (t))_{t > 0}$ on the space $X_{α, ν}^{0} (Ω)$ is exponentially stable. Consequently, we have the following result.

Corollary 1: For all $Y_{0} \in X_{α, ν}^{0} (Ω)$ , the solution $Y (t) = S_{α} (t) Y_{0}$ obeys the following stability

‖ Y (t) ‖_{L^{2} (Ω)} \leq 12 (1 + \frac{a}{ν} t) e^{- τ_{α} t} ‖ Y_{0} ‖_{L^{2} (Ω)},

where $τ_{α}$ is given by (13).

Let us note $X_{α, ν}^{1} (Ω)$ the subspace of $H_{m}^{1} (Ω)$ defined by

\begin{matrix} X_{α, ν}^{1} (Ω) = \\ {Y_{0} = \sum_{n = 1}^{\infty} (c_{0, n} Φ_{2 n} + d_{0, n} Φ_{2 n - 1}) \in H_{m}^{1} (Ω) | \\ c_{0, n} = d_{0, n} = 0, \forall n \leq \sqrt{\frac{2 α}{ν}}} . \end{matrix}

In this case, we have

Theorem 1: For all $Y_{0} \in X_{α, ν}^{1} (Ω)$ , the solution $Y (t) = S_{α} (t) Y_{0}$ obeys the following stability

\begin{matrix} ‖ Y (t) ‖_{H^{1} (Ω)} \leq \\ 12 (1 + \frac{a}{ν} t) e^{- τ_{α} t} ‖ Y_{0} ‖_{H^{1} (Ω)}, \forall t \geq 0 . \end{matrix}

(14)

In addition, we have

\begin{matrix} ‖ y ‖_{H^{1} (0, \infty, H^{1} (Ω))} + ‖ z ‖_{L^{2} (0, \infty, H^{2} (Ω))} + \\ ‖ z ‖_{H^{1} (0, \infty, L^{2} (Ω))} \leq C ‖ Y_{0} ‖_{H^{1} (Ω)} . \end{matrix}

(15)

The proofs of Corollary 1 and Theorem 1, can be deduced from Arfaoui et al. (2011), Corollary 1 and Theorem 2.3.

Corollary 2: For all $Y_{0} \in L_{m}^{2} (Ω)$ , the solution $Y$ to system (6)–(9) satisfies the following energy identity

\begin{matrix} \frac{1}{2} ‖ Y (t) ‖_{L^{2} (Ω)}^{2} + ν \int_{0}^{t} ‖ \partial_{x} z (s) ‖_{L^{2} (Ω)}^{2} ds = \\ \frac{1}{2} ‖ Y_{0} ‖_{L^{2} (Ω)}^{2} + α ‖ Y ‖_{L^{2} (0, t; L^{2} (Ω))}^{2}, \forall t \geq 0 . \end{matrix}

(16)

Proof: If $Y_{0}$ belongs to $H_{m}^{1} (Ω)$ , from Theorem 1, it follows that the solution $Y$ to system (10)–(11) belongs to $H^{1} (0, \infty, H_{m}^{1} (Ω)) \times [L^{2} (0, \infty, H^{2} (Ω)) \cap H^{1} (0, \infty, L^{2} (Ω))]$ . Multiplying equation (6) by ay, and equation (7) by z, and after integration in space, we have

\begin{matrix} \frac{a}{2} \frac{d}{dt} ‖ y (t) ‖_{L^{2} (Ω)}^{2} = a \int_{Ω} (\partial_{x} z) y dx + \\ α a \int_{Ω} y^{2} dx, \end{matrix}

\begin{matrix} \frac{1}{2} \frac{d}{dt} ‖ z (t) ‖_{L^{2} (Ω)}^{2} + ν \int_{Ω} (\partial_{x} z)^{2} dx = \\ - a \int_{Ω} (\partial_{x} z) y dx + α \int_{Ω} z^{2} dx . \end{matrix}

Summing up, we obtain

\begin{matrix} \frac{d}{dt} (a ‖ y (t) ‖_{L^{2} (Ω)}^{2} + ‖ z (t) ‖_{L^{2} (Ω)}^{2}) + \\ 2 ν ‖ \partial_{x} z (t) ‖_{L^{2} (Ω)}^{2} = α ‖ Y (t) ‖_{L^{2} (Ω)}^{2} . \end{matrix}

(17)

Integrating (17) over $(0, t)$ provides identity (16) when $Y_{0} \in H_{m}^{1} (Ω)$ . A density argument leads to (16) when $Y_{0} \in L_{m}^{2} (Ω)$ . □

Nonhomogeneous boundary condition

We are interested in the Saint-Venant system (1)–(4) with nonhomogeneous boundary conditions. We study the stabilization of the system by only one control on the right-hand side of the channel, thus we have

\partial_{t} y - \partial_{x} z - α y = 0, in Q_{\infty},

(18)

\partial_{t} z - a \partial_{x} y - ν \partial_{xx} z - α z = 0, in Q_{\infty},

(19)

z (x, 0) = z_{0} (x), y (x, 0) = y_{0} (x),

(20)

z (0, t) = 0, z (π, t) = q (t) .

(21)

First, we have to prove regularity results for the system

\partial_{t} y - \partial_{x} z - α y = f_{1}, in Q_{\infty},

(22)

\partial_{t} z - a \partial_{x} y - ν \partial_{xx} z - α z = f_{2}, in Q_{\infty},

(23)

z (x, 0) = y (x, 0) = 0,

(24)

z (0, t) = z (π, t) = 0 .

(25)

This system can be written as follows

Y' (t) + A_{α} Y (t) = F (t) in (0, \infty), Y (0) = 0,

(26)

where $F (t) = (f_{1}, f_{2}) (t)$ represents the source term of the system. The weak solution of equation (26) is given by

Y (t) = \int_{0}^{t} S_{α} (t - s) F (s) ds, \forall t \geq 0 .

Proposition 2: Let us assume that $f_{1}, f_{2} \in L^{2} (0, \infty; L^{2} (Ω))$ . The weak solution $Y$ of equation (26) belongs to $H^{1} (0, \infty; H_{m}^{1} (Ω)) \times [H^{1} (0, \infty; L^{2} (Ω)) \cap L^{2} (0, \infty; H^{2} (Ω) \cap H_{0}^{1} (Ω))]$ and we have

\begin{matrix} ‖ y ‖_{H^{1} (0, \infty; H_{m}^{1} (Ω))} + ‖ z ‖_{L^{2} (0, \infty; H^{2} (Ω))} + \\ ‖ z ‖_{H^{1} (0, \infty; L^{2} (Ω))} \leq C (‖ f_{1} ‖_{L^{2} (0, \infty; L^{2} (Ω))} + \\ ‖ f_{2} ‖_{L^{2} (0, \infty; L^{2} (Ω))}) . \end{matrix}

Proof: The proof can be established as the proof of Arfaoui et al. (2011: Proposition 3).

Let us define $H_{{0}}^{1} (Ω)$ as the space of functions in $H^{1} (Ω)$ vanishing at $x = 0$

Theorem 2: For all $Y_{0} = (y_{0}, z_{0})^{T} \in H_{m}^{1} (Ω) \times H_{{0}}^{1} (Ω)$ , and all $q \in H^{1} (0, + \infty)$ satisfying the compatibility condition $z_{0} (π) - q (0) = 0$ , the system (18)–(21) has a unique solution $Y \in C ([0, + \infty [, H^{1} (Ω))$ and the following estimate holds

\begin{matrix} ‖ Y' ‖_{L^{2} (0, + \infty; H^{1} (Ω))} + ‖ z ‖_{L^{2} (0, + \infty; H^{1} (Ω))} \\ \leq C (‖ Y_{0} ‖_{H^{1} (Ω)} + ‖ q ‖_{H^{1} (0, + \infty)} \\ + α ‖ Q ‖_{L^{2} (0, + \infty)} + α ‖ q ‖_{L^{2} (0, + \infty)}), \end{matrix}

where the function Q is given by

Q (t) = \int_{0}^{t} q (s) ds .

Proof: We look for a solution $Y = (y, z)$ to the system (18)–(21) in the form

\begin{matrix} y (x, t) = \tilde{y} (x, t) + \frac{1}{π} Q (t) e^{- α t}, \\ z (x, t) = \tilde{z} (x, t) + \frac{x}{π} q (t) . \end{matrix}

The function $Y = (y, z)$ is solution to system (18)–(21) if, and only if, the function $\tilde{Y} = (\tilde{y}, \tilde{z})$ satisfies

\partial_{t} \tilde{y} - \partial_{x} \tilde{z} - α \tilde{y} = f_{1}, in Q_{\infty},

(27)

\partial_{t} \tilde{z} - a \partial_{x} \tilde{y} - ν \partial_{xx} \tilde{z} - α \tilde{z} = f_{2}, in Q_{\infty},

(28)

\tilde{z} (x, 0) = {\tilde{z}}_{0} (x), \tilde{y} (x, 0) = y_{0} (x), in Ω,

(29)

\tilde{z} (0, t) = \tilde{z} (π, t) = 0, in (0, + \infty),

(30)

where the source terms $f_{1}, f_{2}$ and the initial condition ${\tilde{z}}_{0}$ are defined by

f_{1} (x, t) = \frac{2 α}{π} Q (t) e^{- α t} + \frac{1}{π} q (t) (1 - e^{- α t}),

f_{2} (x, t) = - \frac{x}{π} q' (t) + \frac{α x}{π} q (t),

{\tilde{z}}_{0} (x) = z_{0} (x) - \frac{x}{π} q (0) .

Let us notice that $f_{1}, f_{2} \in L^{2} (0, + \infty; L^{2} (Ω))$ and that ${\tilde{z}}_{0} \in H_{0}^{1} (Ω)$ . Therefore, with Theorem 1 and Proposition 2, we have

\begin{matrix} ‖ \tilde{y} ‖_{H^{1} (0, \infty; H_{m}^{1} (Ω))} + ‖ \tilde{z} ‖_{H^{1} (0, \infty; L^{2} (Ω))} + \\ ‖ \tilde{z} ‖_{L^{2} (0, \infty; H^{2} (Ω))} \leq C (‖ Y_{0} ‖_{H^{1} (Ω)} + ‖ q ‖_{H^{1} (0, + \infty)} + \\ α ‖ Q ‖_{L^{2} (0, + \infty)} + α ‖ q ‖_{L^{2} (0, + \infty)}) . \end{matrix}

It follows that

\begin{matrix} ‖ z ‖_{H^{1} (0, \infty; L^{2} (Ω))} + ‖ z ‖_{L^{2} (0, \infty; H^{2} (Ω))} \\ \leq C (‖ Y_{0} ‖_{H^{1} (Ω)} + ‖ q ‖_{H^{1} (0, + \infty)} + \\ α ‖ Q ‖_{L^{2} (0, + \infty)} + α ‖ q ‖_{L^{2} (0, + \infty)}) . \end{matrix}

In view of equation satisfied by y, we obtain

‖ \partial_{t} y ‖_{L^{2} (0, \infty; H^{1} (Ω))} \leq C (‖ Y_{0} ‖_{H^{1} (Ω)} + ‖ q ‖_{H^{1} (0, \infty)}) .

Boundary stabilizability

We study the stabilizability of the Saint-Venant system (6)–(9) by Dirichlet boundary controls when the advection is dominating, (when $2 a > ν^{2}$ in Remark 1), and when the operator associated to the Saint-Venant system has a finite number of unstable eigenvalues, (see Remark 2). The advective modes, those related to complex eigenvalues with real part $\leq (a / ν - α)$ , are responsible of this slow natural stabilization. Our aim is to choose a Dirichlet control q at the extremity $x = π$ to obtain an exponential decay rate $- σ$ close to $- (a / ν - α)$ . The tool for the construction of such a control is the extension method introduced by Fursikov (2001, 2002), Ervadoza et al. (2012) and Russel (1974).

We consider the Saint-Venant system with nonhomogeneous Dirichlet boundary conditions

\partial_{t} y - \partial_{x} z - α y = 0, in Q_{\infty},

(31)

\partial_{t} z - a \partial_{x} y - ν \partial_{xx} z - α z = 0, in Q_{\infty},

(32)

z (x, 0) = z_{0} (x), y (x, 0) = y_{0} (x), in Ω,

(33)

z (0, t) = 0, z (π, t) = q (t), in (0, + \infty),

(34)

where the initial condition $(y_{0}, z_{0}) \in H_{m}^{1} (Ω) \times H_{{0}}^{1} (Ω)$ (recall that $H_{{0}}^{1} (Ω)$ is the space of functions in $H^{1} (Ω)$ vanishing at $x = 0$ ). We look for q able to stabilize $(y, z)$ in the norm $H^{1} (Ω) \times H^{1} (Ω)$ with a decay rate as close as possible to $(\frac{a}{ν} - α)$ . We use only one control in the Dirichlet condition at the right extreme point of the channel ( $x = π$ ). The stabilization by two controls leads to the same type of result.

Theorem 3: For all $0 < σ < (\frac{a}{ν} - α)$ , there exists a constant $C_{σ}$ (only depending on σ), such that for all initial condition $Y_{0} = (y_{0}, z_{0}) \in H_{m}^{1} (Ω) \times H_{{0}}^{1} (Ω)$ , there exists a control q for which the solution $(y, z)$ to problem (31)–(34) satisfies

‖ (y, z) (\cdot, t) ‖_{H^{1} (Ω) \times H^{1} (Ω)} \leq C_{σ} e^{- σ t}, t > 0 .

The proof is postponed at the end of the section. The extension method consists in looking for an extension operator of the initial condition into a space of functions defined in an extended geometrical domain such that the solution of the extended system in this domain is stable. More precisely, let us set $Ω_{e} =] 0, 2 π [$ and $Q_{\infty}^{e} =] 0, 2 π [\times] 0, \infty [$ . We consider the system: for all $(x, t) \in Q_{\infty}^{e}$

\partial_{t} y_{e} - \partial_{x} z_{e} - α y_{e} = 0,

(35)

\partial_{t} z_{e} - a \partial_{x} y_{e} - ν \partial_{xx} z_{e} - α z_{e} = 0,

(36)

(y_{e}, z_{e}) (x, 0) = E_{α} (Y_{0}) (x),

(37)

z_{e} (0, t) = z_{e} (2 π, t) = 0,

(38)

where $E_{α} \in L (H_{m}^{1} (Ω) \times H_{{0}}^{1} (Ω), H_{m}^{1} (Ω_{e}))$ is a continuous linear operator precisely defined later on, and $H_{m}^{1} (Ω_{e}) = H_{m}^{1} (Ω_{e}) \times H_{0}^{1} (Ω_{e})$ is defined in the same manner as $H_{m}^{1} (Ω)$ .

Let us set $Y_{e} = (y_{e}, z_{e})^{T}$ and $Y_{0}^{e} = E_{α} (y_{0}, z_{0})$ . In this case, the system (35)–(38) can be written in the following form

Y'_{e} (t) + A_{α}^{e} Y_{e} (t) = 0, \forall t > 0,

(39)

Y_{e} (0) = Y_{0}^{e},

(40)

where $A_{α}^{e}$ is an unbounded operator on $L^{2} (Ω_{e})$ defined by

A_{α}^{e} = (\begin{matrix} - α & - \frac{d}{dx} \\ - a \frac{d}{dx} & - ν \frac{d^{2}}{d x^{2}} - α \end{matrix})

Moreover, we have

\begin{matrix} D (A_{α}^{e}) = {(y_{e}, z_{e}) \in L^{2} (Ω_{e}) | z_{e} \in H_{0}^{1} (Ω_{e}), \\ a y_{e} + ν z'_{e} \in H^{1} (Ω_{e})} . \end{matrix}

We introduce the family $(Φ_{n}^{e})_{n \in N}$ of functions in $L^{2} (Ω_{e})$ , defined by

\begin{matrix} Φ_{0}^{e} (x) = \frac{1}{\sqrt{2 π a}} (1, 0)^{T}, \\ Φ_{2 n}^{e} (x) = \sqrt{\frac{1}{π a}} {(\cos (\frac{nx}{2}), 0)}^{T}, for n \geq 1, \\ Φ_{2 n - 1}^{e} (x) = \sqrt{\frac{1}{π}} {(0, \sin (\frac{nx}{2}))}^{T} for n \geq 1 . \end{matrix}

In the Fourier basis $(Φ_{n}^{e})_{n \in}$ , we have

Y_{0}^{e} = \sum_{n = 1}^{\infty} (c_{0, n} Φ_{2 n}^{e} + d_{0, n} Φ_{2 n - 1}^{e}) .

Consequently, we can define the subspace $X_{α}^{m} (Ω_{e})$ of $H_{m}^{1} (Ω_{e})$ as follows

\begin{matrix} X_{α}^{m} (Ω_{e}) = {Y_{0}^{e} \in H_{m}^{1} (Ω_{e}) | c_{0, n} = d_{0, n} = 0, \\ n \leq n_{f}}, \end{matrix}

(41)

where $n_{f} = 2 [\sqrt{(σ + α) / ν}] + 1$ , (the symbol [·] is for the integer part function). Then, we have the following result.

Proposition 3: There exists a continuous linear operator $E_{α} \in L (H_{m}^{1} (Ω) \times H_{{0}}^{1} (Ω), X_{α}^{m} (Ω_{e}))$ such that: for all $Y_{0} \in H_{m}^{1} (Ω) \times H_{{0}}^{1} (Ω)$ , we have

E_{α} (Y_{0}) |_{Ω} \equiv Y_{0} .

Proof: We have proved the existence of this operator in the case $α = 0$ , see Arfaoui et al. (2011: Proposition 5). So, following the same approach, we can prove the existence of the operator $E_{α}$ in this case.□

The following result, Proposition 4, can be established as the same as the Proposition 4 in Arfaoui et al. (2011). We have

Proposition 4: Let $σ$ be given in $] 0, \frac{a}{ν} - α [$ . For all $Y_{0}^{e} \in X_{α}^{m} (Ω_{e})$ , the problem (35)–(38) admits a unique solution $Y_{e} \in C ([0, + \infty); H_{m}^{1} (Ω_{e}))$ , which satisfies

‖ Y_{e} (t) ‖_{H^{1} (Ω_{e})} \leq C_{σ} e^{- σ t} ‖ Y_{0}^{e} ‖_{H^{1} (Ω_{e})}, \forall t > 0 .

(42)

Proof of Theorem 3: The theorem is a direct consequence of Proposition 3 and Proposition 4. Indeed, it is sufficient to set

\begin{matrix} (y (x, t), z (x, t), q (t)) = \\ (y_{e} (x, t) |_{[0, π]}, z_{e} (x, t) |_{[0, π]}, z_{e} (π, t)), \end{matrix}

where $(y_{e}, z_{e})$ is the solution to the problem (35)–(38), with $E_{α}$ is the extension operator defined in Proposition 3.

Remark 3: Let us mention that the subspace $X_{α}^{m} (Ω_{e})$ is associated with the set of the eigenvalues located strictly on the complex half-plane ${ℜ (λ) > σ}$ .

Remark 4: In the case where $α < 0$ , the spectrum of the operator $A_{α}$ associated to the system (1)–(14) contains no unstable eigenvalues. That means that the system is stable. Moreover, it is easy to prove that the system $(1) - (14)$ with homogeneous Dirichlet conditions is exponentially stable on the space $H^{1} (Ω) \times H^{1} (Ω)$ with an exponential decay rate $τ_{α} = \min (\frac{ν}{2} - α, \frac{a}{ν} - α)$ greater than $τ = \min (\frac{ν}{2}, \frac{a}{ν})$ , (as in Arfaoui et al. (2011)). Recall that $τ$ is the exponential decay rate of the homogeneous system $(1) - (14)$ when $α = 0$ , (see Theorem 2.3 in Arfaoui et al. (2011)). This means that the shifted Saint-Venant system $(1) - (14)$ with $α < 0$ stabilizes faster than the same system when $α = 0$ . We can deduce that $α < 0$ is effective in the natural stabilization of the system (1)–(14).

Stabilization problem for finite difference equations

We construct the finite difference equations system associated with the continuous Saint-Venant system (31)–(34) after discretization by implicit Euler scheme and Preissmann scheme. We use a splitting method which consists in separating the hyperbolic operator from the parabolic operator in the system (31)–(34). So, we obtain two systems of equations that will be solved simultaneously.

Let $Ω =] 0, π [$ . We divide the interval $\bar{Ω}$ into N parts by the points

x_{j} = jh, h = \frac{π}{N}, j = 0, \dots, N .

(43)

In the following, $f_{j}$ is defined by

f_{j} = f (x_{j}), \forall j = 0, \dots, N,

for any function f and $x_{j}$ is given by (43). Then, we can introduce the following notations.

Some notations:

(i) The space $L_{h}^{2} (Ω)$ is understood as the grid functions equipped with the norm

‖ u_{\cdot} ‖_{L_{h}^{2} (Ω)}^{2} = h \sum_{j = 1}^{N} | u_{j} |^{2},

where $u_{\cdot} = {u_{j}}_{j}$ .

(ii) The space $L_{m, h}^{2} (Ω)$ is defined by

L_{m, h}^{2} (Ω) = {u_{\cdot} \in L_{h}^{2} (Ω) : \sum_{j = 1}^{N} u_{j} = 0} .

(iii) The space $H_{m, h}^{1} (Ω) = H_{h}^{1} (Ω) \cap L_{m, h}^{2} (Ω)$ , where the space $H_{h}^{1} (Ω)$ is an approximation of the space $H^{1} (Ω)$ :

H_{h}^{1} (Ω) = {v_{\cdot} \in L_{h}^{2} (Ω) : Δ v_{\cdot} \in L_{h}^{2} (Ω)},

where $v_{\cdot} = {v_{j}}_{j}$ and $Δ v_{\cdot} = {(v_{j} - v_{j - 1}) / h}_{j}$ . Consequently, we introduce the space $H_{h}^{1} (Ω)$ defined by

H_{h}^{1} (Ω) = H_{h}^{1} (Ω) \times H_{h}^{1} (Ω) .

(iv) The space $L_{h}^{2} (Ω)$ coincide with $L_{h}^{2} (Ω) \times L_{h}^{2} (Ω)$ and is understood as the grid functions equipped with the norm, analog to the norm (5), as follows

‖ (u_{\cdot}, v_{\cdot}) ‖_{L_{h}^{2} (Ω)}^{2} = h (a \sum_{j = 1}^{N} | u_{j} |^{2} + \sum_{j = 1}^{N} | v_{j} |^{2}) .

(v) The spaces $H_{0, h}^{1} (Ω)$ and $H_{{0}, h}^{1} (Ω)$ are given by

H_{0, h}^{1} (Ω) = {v_{\cdot} \in H_{h}^{1} (Ω) : v_{0} = v_{N} = 0},

H_{{0}, h}^{1} (Ω) = {v_{\cdot} \in H_{h}^{1} (Ω) : v_{0} = 0} .

Now, we can define the space $H_{m, h}^{1} (Ω)$ as

H_{m, h}^{1} (Ω) = H_{m, h}^{1} (Ω) \times H_{0, h}^{1} (Ω),

equipped with the norm

‖ (u_{\cdot}, v_{\cdot}) ‖_{H_{m, h}^{1} (Ω)}^{2} = h (a \sum_{j = 1}^{N} | Δ u_{j} |^{2} + \sum_{j = 1}^{N} | Δ v_{j} |^{2}),

where $Δ f_{\cdot} = {(f_{j} - f_{j - 1}) / h}_{j}$ .

Discretization of the problem (31)–(34): Let $τ$ be the time step and $t^{n} = n τ$ , for $n \geq 0$ . Then, we introduce the following notations for the states $y (x, t)$ and $z (x, t)$ :

y (x_{j}, t^{n}) = y_{j}^{n}, z (x_{j}, t^{n}) = z_{j}^{n},

and for the initial conditions $y_{0} (x)$ and $z_{0} (x)$ , we have

y_{0} (x_{j}) = y_{0, j}, z_{0} (x_{j}) = z_{0, j},

for all $j = 0, \dots, N$ , $n \geq 0$ . Similarly, for the control q given by the Dirichlet boundary condition (34), we have

z_{N}^{n} = q (t^{n}) = q^{n}, n \geq 0 .

Recall that the left boundary condition, (at $x = 0$ ), $z_{0}^{n} = 0$ for all $n \geq 0$ . Numerically, the problem (31)–(34) is solved using a time splitting method, see Huang et al. (2005) and Khan and Liu (1998). This method consists on the two following steps:

(1) We solve the hyperbolic problem

\partial_{t} y - \partial_{x} z - α y = 0,

(44)

\partial_{t} z - a \partial_{x} y - α z = 0,

(45)

by a finite difference Preissmann scheme. The Preissmann scheme we use is unconditionally stable. Details may be found in Abbott and Minns (1998); Cunge et al. (1980); and Lyn and Goodwin (1987). The solution obtained is noted $(\tilde{y}, \tilde{z})$ .

(2) Then, to get the global solution to the problem (31)–(34), we solve the problem

\partial_{t} y = 0,

(46)

\partial_{t} z - ν \partial_{xx} z = 0,

(47)

by an implicit Euler finite difference scheme using the solution $(\tilde{y}, \tilde{z})$ , (obtained from problem (44)–(45)), as an initial condition. Here, the solution of problem (46)–(47) is noted $(\tilde{\tilde{y}}, \tilde{\tilde{z}})$ . Then, we come back again to solve problem (44)–(45) using as initial condition the solution obtained here $(\tilde{\tilde{y}}, \tilde{\tilde{z}})$ , and so on. This process is repeated simultaneously on every time step.

The space-time discretization of the problem (44)–(45), (on the time interval $[t^{n}, t^{n + \frac{1}{2}}]$ ), by Preissmann scheme, is given by the following system: for all $j = 0, \dots, N - 1$ and for all $n \geq 0$

{\begin{matrix} \frac{y_{j + 1}^{n + \frac{1}{2}} - y_{j + 1}^{n}}{2 τ} + \frac{y_{j}^{n + \frac{1}{2}} - y_{j}^{n}}{2 τ} - \frac{θ}{h} (z_{j + 1}^{n + \frac{1}{2}} - z_{j}^{n + \frac{1}{2}}) - \\ \frac{(1 - θ)}{h} (z_{j + 1}^{n} - z_{j}^{n}) - α y_{j}^{n} = 0, \\ \frac{z_{j + 1}^{n + \frac{1}{2}} - z_{j + 1}^{n}}{2 τ} + \frac{z_{j}^{n + \frac{1}{2}} - z_{j}^{n}}{2 τ} - \frac{a θ}{h} (y_{j + 1}^{n + \frac{1}{2}} - y_{j}^{n + \frac{1}{2}}) - \\ \frac{a (1 - θ)}{h} (y_{j + 1}^{n} - y_{j}^{n}) - α z_{j}^{n} = 0, \end{matrix}

(48)

where $θ$ is a parameter belongs to $[0.5, 1]$ and that makes the Preissmann scheme unconditionally stable. On the interval $[t^{n + \frac{1}{2}}, t^{n + 1}]$ , the space-time discretization of the problem (46)-(47), by the implicit Euler scheme, is given by the system: for $j = 0, \dots, N - 1, n \geq 0$

{\begin{matrix} y_{j}^{n + 1} = y_{j}^{n + \frac{1}{2}}, \\ \frac{z_{j}^{n + 1} - z_{j}^{n + \frac{1}{2}}}{τ} - ν \frac{z_{j - 1}^{n + 1} - 2 z_{j}^{n + 1} + z_{j + 1}^{n + 1}}{h^{2}} = 0 . \end{matrix}

(49)

The main result of stabilizability

The main result is given by Theorem 4. We prove the existence of a control $q^{\cdot} = {q^{n}}_{n}$ able to stabilize the solution of system (48)–(49) with a decay rate close to $(\frac{a}{ν} - α)$ . The proof of theorem 4 follows immediately from Proposition 6 and Proposition 7 below.

Theorem 4: For all $σ$ given in $] 0, \frac{a}{ν} - α [$ , there exist a constant $C_{σ}$ (only depending on $σ$ ), such that for all initial condition $(y_{0, \cdot}, z_{0, \cdot}) = {(y_{0, j}, z_{0, j})}_{j} \in H_{m, h}^{1} (Ω) \times H_{{0}, h}^{1} (Ω)$ , there exist a control $q^{\cdot} = {q^{n}}_{n}$ such that the solution $(y_{\cdot}^{\cdot}, z_{\cdot}^{\cdot}) = {(y_{j}^{n}, z_{j}^{n})}$ to the problems (48) and (49) satisfies the estimation: let $Y_{\cdot}^{n} = (y_{\cdot}^{n}, z_{\cdot}^{n})$

‖ Y_{\cdot}^{n} ‖_{H_{h}^{1} (Ω)}^{2} \leq C_{0} e^{- σ n τ}, as n \to + \infty .

(50)

The stabilization construction for finite difference equations

Now, we adapt the stabilization construction in Section 4 to the finite difference equations (48) and (49). We extend the problems (48) and (49) from $[0, π]$ onto the segment $[0, 2 π]$ imposing zero boundary conditions at $x = 2 π$ . If we designate by the state $(u, v)$ the extension of the state $(y, z)$ , then numerically this translates as follows: for all $n \geq 0$

\begin{matrix} (u_{j}^{n}, v_{j}^{n}) j = 0, \dots, 2 N \\ is an extension to \\ (y_{j}^{n}, z_{j}^{n}) j = 0, \dots, N . \end{matrix}

Consequently, for all $j = 0, \dots, 2 N - 1$ , we obtain:

(a) The extension associated to the system (48) is given by

{\begin{matrix} \frac{u_{j + 1}^{n + \frac{1}{2}} - u_{j + 1}^{n}}{2 τ} + \frac{u_{j}^{n + \frac{1}{2}} - u_{j}^{n}}{2 τ} - \frac{θ}{h} (v_{j + 1}^{n + \frac{1}{2}} - v_{j}^{n + \frac{1}{2}}) - \\ \frac{(1 - θ)}{h} (v_{j + 1}^{n} - v_{j}^{n}) - α u_{j}^{n} = 0, \\ \frac{v_{j + 1}^{n + \frac{1}{2}} - v_{j + 1}^{n}}{2 τ} + \frac{v_{j}^{n + \frac{1}{2}} - v_{j}^{n}}{2 τ} - \frac{a θ}{h} (u_{j + 1}^{n + \frac{1}{2}} - u_{j}^{n + \frac{1}{2}}) - \\ \frac{a (1 - θ)}{h} (u_{j + 1}^{n} - u_{j}^{n}) - α v_{j}^{n} = 0, \end{matrix}

(51)

(b) The extension associated to the system (49) is given by

{\begin{matrix} u_{j}^{n + 1} = u_{j}^{n + \frac{1}{2}}, \\ \frac{v_{j}^{n + 1} - v_{j}^{n + \frac{1}{2}}}{τ} - \\ ν \frac{v_{j - 1}^{n + 1} - 2 v_{j}^{n + 1} + v_{j + 1}^{n + 1}}{h^{2}} = 0 . \end{matrix}

(52)

v_{0}^{n} = v_{N}^{n} = 0, \forall n \geq 0 .

(d) The initial condition $(u_{j}^{0}, v_{j}^{0})$ is the extension of the initial condition $(y_{j}^{0}, z_{j}^{0})$ .

(e) We denote by

w_{0, \cdot} = {w_{0, j} = (u_{j}^{0}, v_{j}^{0})} = E_{α}^{h} (r_{0, \cdot}),

(53)

the extended initial data $r_{0, \cdot} = {(y_{j}^{0}, z_{j}^{0})}$ to the extended interval $[0, 2 π]$ . The construction of the extension operator $E_{α}^{h}$ is the main part of the whole construction.

Let $Ω_{e} =] 0, 2 π [$ . In the space $L_{h}^{2} (Ω_{e})$ , equipped with the inner product

〈 F_{\cdot}, G_{\cdot} 〉 = h (a \sum_{k = 1}^{2 N} f_{k}^{1} {\bar{g}}_{k}^{1} + \sum_{k = 1}^{2 N} f_{k}^{2} {\bar{g}}_{k}^{2}),

(54)

where $F_{\cdot} = (f_{\cdot}^{1}, f_{\cdot}^{2})$ and $G_{\cdot} = (g_{\cdot}^{1}, g_{\cdot}^{2})$ , (bar means the complex conjugation). We introduce the family of functions $(Φ_{n}^{e} (j))_{n = 1, \dots, 2 N}$ , $j = 1, \dots, 2 N$ , defined by

{\begin{matrix} Φ_{0}^{e} (j) = \frac{1}{\sqrt{2 a π}} (1, 0), \\ Φ_{2 n}^{e} (j) = \frac{1}{\sqrt{2 a π}} (e^{i \frac{jhn}{2}}, 0), \\ Φ_{2 n - 1}^{e} (j) = \frac{1}{\sqrt{2 π}} (0, e^{i \frac{jhn}{2}}) . \end{matrix}

(55)

The family $(Φ_{n}^{e} (j))_{n = 1, \dots, 2 N}$ , $j = 1, \dots, 2 N$ , forms a basis for $L_{h}^{2} (0, 2 π)$ , and this basis is orthonormal with respect to the inner product (54).

By decomposition of the solution $w_{\cdot}^{n} = (u_{\cdot}^{n}, v_{\cdot}^{n})$ , to the problems (51)–(52), in the basis $(Φ_{n}^{e} (j))_{n = 1, \dots, 2 N}$ , $j = 1, \dots, 2 N$ , we obtain

\begin{matrix} w_{k}^{n} = \sum_{j = 1}^{2 N} [{\hat{u}}^{n} (j) Φ_{2 k}^{e} (j) + {\hat{v}}^{n} (j) Φ_{2 k - 1}^{e} (j)], \\ = \sum_{j = 1}^{2 N} I_{a} \cdot {\hat{w}}^{n} (j) \frac{1}{\sqrt{2 π}} e^{i \frac{jhk}{2}}, \end{matrix}

(56)

where ${\hat{w}}^{n} = ({\hat{u}}^{n}, {\hat{v}}^{n})$ and $I_{a}$ is a $2 \times 2$ matrix given by

I_{a} = (\begin{matrix} \frac{1}{\sqrt{a}} & 0 \\ 0 & 1 \end{matrix})

We can, also, write $u_{k}^{n}$ and $v_{k}^{n}$ separately as follows

u_{k}^{n} = \sum_{j = 1}^{2 N} {\hat{u}}^{n} (j) \frac{1}{\sqrt{2 a π}} e^{i \frac{jhk}{2}},

(57)

v_{k}^{n} = \sum_{j = 1}^{2 N} {\hat{v}}^{n} (j) \frac{1}{\sqrt{2 π}} e^{i \frac{jhk}{2}} .

(58)

Substituting (57) and (58) into (51) and then into (52), we obtain equations depending on Fourier coefficients ${\hat{u}}^{n} (j)$ and ${\hat{v}}^{n} (j)$ as follows:

By substituting (57) and (58) into (51), we obtain

{\begin{matrix} (1 + e^{i \frac{jh}{2}}) {\hat{u}}^{n + \frac{1}{2}} - γ \sqrt{a} (e^{i \frac{jh}{2}} - 1) {\hat{v}}^{n + \frac{1}{2}} = \\ (e^{i \frac{jh}{2}} + 1 + 2 τ α) {\hat{u}}^{n} + β \sqrt{a} (e^{i \frac{jh}{2}} - 1) {\hat{v}}^{n}, \\ - γ \sqrt{a} (e^{i \frac{jh}{2}} - 1) {\hat{u}}^{n + \frac{1}{2}} + (1 + e^{i \frac{jh}{2}}) {\hat{v}}^{n + \frac{1}{2}} = \\ β \sqrt{a} (e^{i \frac{jh}{2}} - 1) {\hat{u}}^{n} + (e^{i \frac{jh}{2}} + 1 + 2 τ α) {\hat{v}}^{n}, \end{matrix}

(59)

where $γ = \frac{2 τ θ}{h}$ , $β = \frac{2 τ (1 - θ)}{h}$ . The system (59) can be written as

A (j) {\hat{w}}^{n + \frac{1}{2}} (j) = B (j) {\hat{w}}^{n} (j), j = 1, \dots, 2 N,

(60)

where the matrix $A (j)$ and $B (j)$ are square matrices of dimension 2, symmetric and are defined by

A (j) = (\begin{matrix} (1 + e^{i \frac{jh}{2}}) & - γ \sqrt{a} (e^{i \frac{jh}{2}} - 1) \\ - γ \sqrt{a} (e^{i \frac{jh}{2}} - 1) & (1 + e^{i \frac{jh}{2}}) \end{matrix}),

B (j) = (\begin{matrix} (1 + 2 τ α + e^{i \frac{jh}{2}}) & β \sqrt{a} (e^{i \frac{jh}{2}} - 1) \\ β \sqrt{a} (e^{i \frac{jh}{2}} - 1) & (1 + 2 τ α + e^{i \frac{jh}{2}}) \end{matrix}) .

The matrix $A (j)$ is reversal. Indeed

det (A (j)) = 4 e^{i \frac{jh}{2}} [\overset{2}{\cos} (\frac{jh}{4}) + a γ^{2} \overset{2}{\sin} (\frac{jh}{4})] \neq 0,

for all $j = 1, \dots, 2 N$ . So, we can write the equation (60) as

{\hat{w}}^{n + \frac{1}{2}} (j) = C (j) {\hat{w}}^{n} (j), j = 1, \dots, 2 N,

(61)

where the matrix

C (j) = (c_{ik})_{1 \leq i, k \leq 2} = A^{- 1} (j) B (j)

is symmetric, and is defined by

\begin{matrix} c_{11} = c_{22} = (1 + e^{i \frac{jh}{2}}) (1 + 2 τ α + e^{i \frac{jh}{2}}) + \\ a γ β (e^{i \frac{jh}{2}} - 1)^{2}, \\ c_{12} = c_{21} = \sqrt{a} γ (e^{i \frac{jh}{2}} - 1) (1 + 2 τ α + e^{i \frac{jh}{2}}) + \\ \sqrt{a} β (e^{ijh} - 1) . \end{matrix}

By substituting (57) and (58) into (52), we obtain: for all $j = 1, \dots, 2 N$

{\hat{w}}^{n + 1} (j) = D (j) {\hat{w}}^{n + \frac{1}{2}} (j),

(62)

where $D (j)$ is $2 \times 2$ matrix

D (j) = (\begin{matrix} 1 & 0 \\ 0 & R (j) \end{matrix})

and the coefficient $R (j)$ is defined as follows

R (j) = {(1 + \frac{4 τ ν}{h^{2}} \overset{2}{\sin} (\frac{jh}{4}))}^{- 1} .

By substituting (61) into equation (62), we obtain ${\hat{w}}^{n + 1} (j)$ at time step $t^{n + 1}$ as follows

{\hat{w}}^{n + 1} (j) = A (j) {\hat{w}}^{n} (j), j = 1, \dots, 2 N,

where $A (j)$ is $2 \times 2$ matrix given by

A (j) = (a_{ik})_{1 \leq i, k \leq 2} = D (j) C (j),

and is defined by

a_{11} = c_{11}, a_{22} = R (j) a_{11},

a_{12} = c_{12}, a_{21} = R (j) a_{12} .

Consequently, we deduce that

{\hat{w}}^{n} (j) = [A (j)]^{n} {\hat{w}}_{0} (j),

(63)

for all $j = 1, \dots, 2 N, n \geq 0$ , where ${\hat{w}}_{0} (j)$ are the Fourier coefficients of the initial data $w_{0, \cdot}$ in (53).

Proposition 5: Assume that $τ = h$ ¹, the matrix $A (j)$ satisfies the estimation: for all $j = 1, \dots, 2 N$

‖ A (j) ‖_{1} \leq C_{τ} (\frac{1 + τ α}{1 + \frac{4 ν}{τ} \overset{2}{\sin} (\frac{j τ}{4})}),

(64)

where the constant $C_{τ}$ is independent of $τ$ .

Proof: By definition, the norm $‖ A (j) ‖_{1} = max {| a_{11} | + | a_{21} |, | a_{22} | + | a_{12} |}$ . Under the condition $τ = h$ and from the definition of $A (j)$ , we can deduce that

| a_{11} | + | a_{21} | \leq C (\frac{1 + τ α + ε (τ)}{1 + \frac{4 ν}{τ} \overset{2}{\sin} (\frac{j τ}{4})}),

where the function $τ \to ε (τ)$ is defined by the following expression

\begin{matrix} ε (τ) = 4 (1 + τ α) \frac{ν}{τ} \overset{2}{\sin} (\frac{j τ}{4}) + \\ 4 a θ (1 - θ) \overset{2}{\sin} (\frac{j τ}{4}) (1 + \frac{4 ν}{τ} \overset{2}{\sin} (\frac{j τ}{4})) + \\ 8 \sqrt{a} | \sin (\frac{j τ}{4}) | (1 + τ θ α), \end{matrix}

and tends to zero as $τ \to 0$ . Then, the function $(ε (τ))_{τ}$ is bounded and we can deduce that there exists a constant $C_{1}^{τ} > 0$ such that

| a_{11} | + | a_{21} | \leq C_{1}^{τ} (\frac{1 + τ α}{1 + \frac{4 ν}{τ} \overset{2}{\sin} (\frac{j τ}{4})}) .

(65)

We have also

| a_{22} | + | a_{12} | \leq C (\frac{1 + τ α + ε (τ)}{1 + \frac{4 ν}{τ} \overset{2}{\sin} (\frac{j τ}{4})}),

and, in this case, the function $τ \to ε (τ)$ is defined by

\begin{matrix} ε (τ) = 4 a θ (1 - θ) \overset{2}{\sin} (\frac{j τ}{4}) + \\ 8 \sqrt{a} | \sin (\frac{j τ}{4}) | (1 + τ θ α) (1 + \frac{4 ν}{τ} \overset{2}{\sin} (\frac{j τ}{4})) r, \end{matrix}

tends to zero as $τ \to 0$ . Then, the function $(ε (τ))_{τ}$ is bounded and we can deduce that there exists a constant $C_{2}^{τ} > 0$ such that

| a_{22} | + | a_{12} | \leq C_{2}^{τ} (\frac{1 + τ α}{1 + \frac{4 ν}{τ} \overset{2}{\sin} (\frac{j τ}{4})}) .

(66)

Finally, from (65) and (66), we obtain (64). □

By substituting (63) in the expression of $w_{\cdot}^{n} = (u_{\cdot}^{n}, v_{\cdot}^{n})$ given by the relation (56), we obtain

w_{k}^{n} = \sum_{j = 1}^{2 N} I_{a} \cdot [A (j)]^{n} {\hat{w}}_{0, k} (j) e^{i \frac{jhk}{2}},

where ${\hat{w}}_{0, k}$ are the Fourier coefficients of the initial data $w_{0, \cdot}$ in (53). By consequence

‖ w_{\cdot}^{n} ‖_{H_{h}^{1} (Ω_{e})}^{2} \leq C \sum_{j = 1}^{2 N} ‖ A (j) ‖_{1}^{2 n} ‖ {\hat{w}}_{0, k} (j) ‖_{H_{h}^{1} (Ω_{e})}^{2} .

(67)

Assume that $τ = h$ . We estimate $‖ A (j) ‖_{1}^{n}$ , for $n = \frac{t}{τ}$ , as The function $E_{2}^{h} (r_{0})$ has the following form $τ \to 0$ . We know that

\begin{matrix} ‖ A (j) ‖_{1}^{n} \leq C {(\frac{1 + τ α}{1 + \frac{4 ν}{τ} \overset{2}{\sin} (\frac{j τ}{4})})}^{n} \\ = C {(1 + τ \frac{α - \frac{4 ν}{τ^{2}} \overset{2}{\sin} (\frac{j τ}{4})}{1 + \frac{4 ν}{τ} \overset{2}{\sin} (\frac{j τ}{4})})}^{\frac{t}{τ}} \\ = C {(1 + τ \frac{α - \frac{ν}{4} j^{2} + o (τ)}{1 + τ (\frac{ν}{4} j^{2} + o (τ))})}^{\frac{t}{τ}} \\ = C {(1 + τ (α - \frac{ν}{4} j^{2} + o (τ)))}^{\frac{t}{τ}} \\ = C {[{(1 + s)}^{\frac{1}{s}}]}^{t (α - \frac{ν}{4} j^{2} + o (τ))} \\ = C e^{t (α - \frac{ν}{4} j^{2})} (1 + o (τ)) . \end{matrix}

It follows that

‖ A (j) ‖_{1}^{n} \leq C e^{t (α - \frac{ν}{4} j^{2})} (1 + o (τ)),

(68)

for sufficiently small $s = τ (α - \frac{ν}{4} j^{2} + o (τ))$ . As usual, $o (τ)$ is a certain function of $τ$ such that $| o (t) | \leq c | τ |$ , as $τ \to 0$ . Consequently, we deduce from inequalities (67) and (68), that

\begin{matrix} ‖ w_{\cdot}^{n} ‖_{H_{h}^{1} (Ω_{e})}^{2} \leq \\ C \sum_{j = 1}^{2 N} e^{2 t (α - \frac{ν}{4} j^{2})} (1 + o (τ)) ‖ {\hat{w}}_{0, k} (j) ‖_{H_{h}^{1} (Ω_{e})}^{2}, \end{matrix}

(69)

where ${\hat{w}}_{0, k}$ are the Fourier coefficients of the initial data $w_{0, \cdot}$ with respect to the basis (55).

In accordance with (41), we introduce the space

\begin{matrix} X_{α}^{h} (Ω_{e}) = \\ {w_{0, \cdot} \in H_{m, h}^{1} (Ω_{e}) : {\hat{w}}_{0, k} = 0, k \leq n_{f}}, \end{matrix}

(70)

where $n_{f}$ is defined in (41). As in the continuous case, we have the following proposition:

Proposition 6: Let $σ$ be given in $] 0, \frac{a}{ν} - α [$ . For all $w_{0, \cdot} \in X_{α}^{h} (Ω_{e})$ and the estimation (69) imply that the problem (51)–(52) admits a solution $w_{\cdot}^{n} \in H_{m, h}^{1} (Ω_{e})$ , such that

‖ w_{\cdot}^{n} ‖_{H_{h}^{1} (Ω_{e})}^{2} \leq C ‖ w_{0, \cdot} ‖_{H_{h}^{1} (Ω_{e})}^{2} e^{- 2 σ n τ},

(71)

as $n \to + \infty$ .

Proposition 7: There exists a continuous linear operator $E_{α}^{h} \in L (H_{m, h}^{1} (Ω) \times H_{{0}, h}^{1} (Ω), X_{α}^{h} (Ω_{e}))$ such that

E_{α}^{h} (r_{0, \cdot}) \equiv w_{0, \cdot} .

Proof: Our purpose is to construct the operator $E_{α}^{h}$ in the form

E_{α}^{h} (r_{0, \cdot}) = E_{1}^{h} (r_{0, \cdot}) + E_{2}^{h} (r_{0, \cdot}),

where $E_{1}^{h} (r_{0, \cdot}) \in H_{m, h}^{1} (Ω_{e})$ is defined by

E_{1}^{h} (r_{0, \cdot}) = {\begin{matrix} r_{0, j} & if & j \leq N, \\ r_{0, 2 N - j} & if & N \leq j \leq 2 N . \end{matrix}

The function $E_{2}^{h} (r_{0, \cdot})$ has the following form

\begin{matrix} E_{2}^{h} (r_{0, \cdot}) = \\ {\begin{matrix} 0, if j \leq N, \\ (2 N - j) (j - N) h \sum_{1 \leq k \leq n_{f}} (c_{k} Φ_{2 k}^{e} + d_{k} Φ_{2 k - 1}^{e}), \\ if n \leq j \leq 2 N, \end{matrix} \end{matrix}

where the coefficients $c_{j}$ and $d_{j}$ are determined to have $E_{α}^{h} (r_{0, \cdot}) \in X_{α}^{h} (Ω_{e})$ . This is fulfilled if and only if : for all $1 \leq j \leq n_{f}$

\begin{matrix} {(E_{2}^{h} (r_{0, \cdot}), Φ_{2 j}^{e})}_{L^{2} (Ω_{e})} = - {(E_{1}^{h} (r_{0, \cdot}), Φ_{2 j}^{e})}_{L^{2} (Ω_{e})}, \\ {(E_{2}^{h} (r_{0, \cdot}), Φ_{2 j - 1}^{e})}_{L^{2} (Ω_{e})} = - {(E_{1}^{h} (r_{0, \cdot}), Φ_{2 j - 1}^{e})}_{L^{2} (Ω_{e})} . \end{matrix}

Both equations may be put under the following form

MC = B_{1} and MD = B_{2},

where $M = (m_{ℓ j})_{1 \leq ℓ, j \leq n_{f}}$ is squared matrix of dimension $n_{f}$ with the entries

\begin{matrix} m_{ℓ j} = h^{2} \sum_{N \leq k \leq 2 N} (2 N - k) (k - N) e^{i \frac{hk}{2} (j - ℓ)}, \end{matrix}

and $C : = (c_{j})_{1 \leq j \leq n_{f}}$ , $D : = (d_{j})_{1 \leq j \leq n_{f}}$ .

The matrix $M$ is reversal. Indeed, let $X = (x_{j})_{1 \leq j \leq n_{f}}$

\begin{matrix} (M X, X) = \sum_{1 \leq j \leq n_{f}} m_{ℓ j} x_{ℓ} {\bar{x}}_{j} = \\ h^{2} \sum_{N \leq k \leq 2 N} (2 N - k) (k - N) | \sum_{1 \leq j \leq n_{f}} x_{j} e^{i \frac{hk}{2} j} |^{2} \geq 0 . \end{matrix}

If $(M X, X) = 0$ , then $\sum_{1 \leq j \leq n_{f}} x_{j} e^{i \frac{hk}{2} j} = 0$ . This imply that $x_{j} = 0$ , for all $j = 1, \dots, n_{f}$ . The proof is complete.

To complete the stabilization construction for the finite difference problems (48) and (49), we determine the solution by the formula

((y_{k}^{n}, z_{k}^{n}), k < N; q^{n}) = (w_{k}^{n} = (u_{k}^{n}, v_{k}^{n}), k < N; v_{N}^{n})

where ${w_{k}^{n}}$ is the solution to the problems (51)–(53) with the initial data $w_{0, \cdot} = E_{α}^{h} (r_{0, \cdot})$ , where $r_{0, \cdot}$ is the initial data in (48). It obvious that the inequality (50) follows from (71).

Numerical simulation

In this section, we show the efficiency of the extension method in stabilizing perturbations that can affect a stable state of the the linearized Saint-Venant system (72) given by

{\begin{matrix} \partial_{t} y - \partial_{x} z - α y = 0, \\ \partial_{t} z - a \partial_{x} y - ν \partial_{xx} z - α z = 0, \\ z (x, 0) = z_{0} (x), y (x, 0) = y_{0} (x), \\ z (0, t) = q_{0} (t), z (1, t) = q_{1} (t), \end{matrix}

(72)

in the domain $Ω = [0, 1]$ , and we choose $a = 1.5$ , $ν = 0.01$ and $α = 0.2$ , consequently, the advection is dominating. This means that the system is closer to the hyperbolic character than that of the parabolic character. In Figure 1, we represent all the eigenvalues of the operator $A_{α}$ . Note that we are in the first case of Remark 2. Indeed, in Case 1 given by the Remark 2 when $(α - \frac{a}{ν}) \leq 0$ , (all the real eigenvalues are stable), we have a finite number of unstable complex eigenvalues $λ_{n}^{α}, μ_{n}^{α}$ for $n \in [1, \sqrt{\frac{2 α}{ν}} [= [1, 2 \sqrt{1} 0 [$ , which implies that $n \in {1, 2, 3, 4, 5, 6}$ , see Figure 2. Only the stable eigenvalues are represented in Figure 3.

The initial perturbations $y_{0} (x)$ , $z_{0} (x)$ are given by

y_{0} (x) = 0, z_{0} (x) = 10 \cos (7 π x) .

Therefore, we do not expect that the perturbation decreases naturally, (as faster as possible). It may be noticed in Figures 4 and 6 that there is no fast natural stabilization when the controls are inactive, ( $q_{0} (t) = q_{1} (t) = 0$ ). Indeed, we obtain natural stabilization at $t = 14.5$ . In order to accelerate the stabilization of the system (72), we extend it in the extended domain $Ω_{e} =] - 1, 2 [$ with the same initial conditions $(y_{0}, z_{0})$ given above. The extension method built a couple of controls $(q_{0}, q_{1})$ at both sides $x = 0$ and $x = 1$ . In Figures 5 and 7 we represent respectively the state y and the state z of the Saint-Venant system (72) at different times with two stabilizing controls. In this case, thanks to the controls $q_{0}, q_{1}$ , we obtain a very fast stabilization at $t = 2.6$ . Both controls $q_{0}$ and $q_{1}$ are plotted in Figure 8 at $t = 2.6$ .

Conclusion

The analysis of the semigroup associated with the Saint-Venant system (1)–(4) made it possible to conclude the following conclusions:

If the parameter $α$ verifies $(α - a / ν) \leq 0$ , this produces an unstable solution. Indeed, the spectrum of Saint-Venant semigroup contains a finite number of unstable eigenvalues, (see Remark 2).

If the advection is dominating, (when $2 a > ν^{2}$ in Remark 1), the natural stabilization is slow. Indeed, some modes, in finite number, are advection modes that decay as $e^{- (ν / 2 - α)}$ . This situation corresponding to a fluid of weak viscosity.

So, we have been able to determine a boundary Dirichlet control allowing to obtain a fast exponential stabilization of the solution with a decay rate $- σ$ close to $- (a / ν - α)$ .

In a future work, we study the extension of the stabilization problem to the case of two-dimensional Saint-Venant system, (theoretical and numerical).

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Hassen Arfaoui

Notes

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