Optimal Control of a Viscous Shallow Water Equation

Abstract

This paper is concerned with a viscous shallow water equation, which includes both the viscous Camassa-Holm equation and the viscous Degasperis-Procesi equation as its special cases. The optimal control under boundary conditions is given, and the existence of optimal solution to the equation is proved.

1. Introduction

Holm and Staley [1] studied the following family of evolutionary 1+1 PDEs: 𝑚_𝑡+𝑢_𝑥𝑚_convection+𝑏𝑢𝑚_{𝑥stretching}=𝜀𝑚_{𝑥𝑥viscosity}, (1) which describes the balance between convection and stretching for small viscosity in the dynamics of one-dimensional nonlinear waves in fluids. Here 𝑢=𝑝∗𝑚=∫^∞_−∞𝑝(𝑥−𝑦)𝑚(𝑦)𝑑𝑦, 𝑚=𝑢−𝑢_𝑥𝑥, and 𝑝 is chosen to be the Green’s function for the Helmholtz operator 1−𝜕²_𝑥 on the line. In a recent study of soliton equations, it is found that (1) for 𝜀=0 and any 𝑏≠−1 is included in the family of shallow water equations at quadratic order accuracy that are asymptotically equivalent under Kodama transformations [2].

When 𝜀=0, (1) becomes the 𝑏-family of equations: 𝑚_𝑡+𝑢_𝑥𝑚+𝑏𝑢𝑚_𝑥=0, (2) which describes a one-dimensional version of active fluid transport. It was shown by Degasperis and Procesi [3] that (2) cannot satisfy the asymptotic integrability condition unless 𝑏=2 or 𝑏=3; compare [2, 4, 5].

For 𝑏=2 in (2), it becomes the Camassa-Holm (CH) equation: 𝑢_𝑡−𝑢_𝑥𝑥𝑡+3𝑢𝑢_𝑥=2𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥, (3) which is a model describing the unidirectional propagation of shallow water waves over a flat bottom [4]. Equation (3) has a bi-Hamiltonian structure [6] and is completely integrable [7, 8]. It admits, in addition to smooth waves, a multitude of traveling wave solutions with singularities: peakons, cuspons, stumpons, and composite waves [4, 9]. Its solitary waves are stable solitons [10, 11], retaining their shape and form after interactions [10]. The Cauchy problem of (3) has been studied extensively. Constantin [12] and Rodríguez-Blanco [13] investigated the locally well-posed for initial data 𝑢₀∈𝐻^𝑠(𝑅) with 𝑠>3/2. More interestingly, it has strong solutions that are global in time [11, 14] as well as solutions that blow up in finite time [11, 15, 16]. On the other hand, Bressan and Constantin [17] and Xin and Zhang [18] showed that (3) has global weak solutions with initial data 𝑢₀∈𝐻¹.

For 𝑏=3 in (2), it becomes the Degasperis-Procesi (DP) equation: 𝑢_𝑡−𝑢_𝑥𝑥𝑡+4𝑢𝑢_𝑥=3𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥, (4) which can be used as a model for nonlinear shallow water dynamics, and its asymptotic accuracy is the same as (3). Degasperis et al. [5] presented that (4) has a bi-Hamiltonian structure with an infinite sequence of conserved quantities and admits exact peakon solutions which are analogous to (3) peakons [4, 10, 19]. Dullin et al. [20] showed that (4) can be obtained from the shallow water elevation equation by an appropriate Kodama transformation. The numerical stability of solitons and peakons, the multisoliton solutions and their peakon limits, and an inverse scattering method to compute N-peakon solutions to (4) have been investigated, respectively, in [21–23]. After (4) appeared, it has attracted many researchers to discover its dynamics (see [24–31]). Yin [24, 25] proved the local well-posedness of (4) with initial data 𝑢₀∈𝐻^𝑠(𝑅) (𝑠>3/2) on the line and on the circle and derived the precise blow-up scenario and a blow-up result. The global existence of strong solutions and global weak solutions to (4) were studied in [29, 30]. Similar to (3) [11, 15, 18, 22–24], (4) has not only global strong solutions [27, 28] but also blow-up solutions [26, 28–30]. Apart from these, Coclite and Karlsen [31] proved that it has global entropy weak solutions in 𝐿²(𝑅)∩𝐵𝑉(𝑅) and 𝐿²(𝑅)∩𝐿⁴(𝑅).

Recently, Lai and Wu [32] found that, in both models (1) and (2), the coefficient of 𝑢𝑢_𝑥 is equal to the coefficient of 𝑢_𝑥𝑢_𝑥𝑥 plus the one of 𝑢𝑢_𝑥𝑥𝑥. That is, 3=2+1,4=3+1. (5) Then, they studied the global solutions and blow-up phenomena to the following generalized equation: 𝑢_𝑡−𝑢_𝑥𝑥𝑡+(𝑎+𝑏)𝑢𝑢_𝑥=𝑎𝑢_𝑥𝑢_𝑥𝑥+𝑏𝑢𝑢_𝑥𝑥𝑥. (6)

In this paper, we study the optimal control problem for the following equation: 𝑢_𝑡−𝑢_𝑥𝑥𝑡−𝜀𝑢_𝑥𝑥−𝑢_{𝑥𝑥𝑥𝑥}+(𝑎+𝑏)𝑢𝑢_𝑥=𝑎𝑢_𝑥𝑢_𝑥𝑥+𝑏𝑢𝑢_𝑥𝑥𝑥, (7) where 𝑎,𝑏 are positive constants. The optimal control is an important component of modern control theories and has a wider application in modern engineering. Two methods are introduced to study the control problems in PDE: one is using a low model method and then changing to an ODE model [33]; the other is using a quasi-optimal control method [34]. No matter which one is chosen, it is necessary to prove the existence of optimal solution according to the basic theory [35]. The control problems of nonlinear PDE have been studied extensively. Kunisch and Volkwein solved open-loop and closed-loop optimal control problems for the Burgers equation [36] and discussed the instantaneous control of the equation [37]. Vedantham [38] developed a technique to utilize the Cole-Hopf transformation to solve an optimal control problem for the Burgers equation. Øksendal [39] proved a sufficient maximum principle for the optimal control systems described by a quasi-linear stochastic heat equation. In [40], Ghattas and Bark studied the optimal control of two- and three-dimensional incompressible Navier-Stokes flows. Lagnese and Leugering [41] considered the problem of boundary optimal control of a wave equation with boundary dissipation. In [42], Yong established a unified existence theory of optimal controls for general semilinear evolutionary distributed parameter systems under the framework of mild (or weak) solutions for evolution equations. Yong and Zheng [43] considered the Cahn-Hilliard equation in a bounded domain with smooth boundary. Based on the energy estimates and the compact method, Ryu and Yagi [44, 45] studied Keller-Segel equations and adsorbate-induced phase transition model. Tian et al. [46–48] studied the optimal control problems for parabolic equations, such as viscous CH equation, viscous DP equation, and viscous DGH equation. Under boundary condition, Zhao and Liu [49] studied viscous Cahn-Hilliard equation.

With 𝑚=𝑢−𝑢_𝑥𝑥, the optimal control problem for (7) we intend to investigate is ()=1min𝐽𝑚,𝜛2‖𝐶𝑚−𝑧‖²_𝑆+𝛿2‖𝜛‖²_𝐿2𝑄0, (8) subject to 𝑚_𝑡−𝜀𝑚_𝑥𝑥+𝑎𝑢_𝑥𝑚+𝑏𝑢𝑚_𝑥=𝐵^∗𝜛in(0,𝑇)×Ω,𝑢=𝑢_𝑥=𝑢_𝑥𝑥()𝑚()=0on0,𝑇×𝜕Ω,0,𝑥=𝑚₀(𝑥)inΩ, (9) where Ω=(0,1). Clearly, our control target is to match the given desired state 𝑧 by adjusting the body force 𝜛 in a control volume 𝑄₀⊆𝑄=(0,𝑇)×Ω in the 𝐿²-sense.

Notation. In this paper, we set that 𝑉=𝐻¹₀(0,1), 𝐻=𝐿²(0,1), 𝑉^∗=𝐻⁻¹(0,1), and 𝐻^∗=𝐿²(0,1) are dual space, respectively. Then, 𝑉↪𝐻=𝐻^∗↪𝑉^∗, with each embedding being dense. The extension operator 𝐵^∗∈ℒ(𝐿²(𝑄₀),𝐿²(𝑉^∗)) is given by 𝐵^∗𝑞=𝑞,𝑞∈𝑄₀,𝑄0,𝑞∈𝑄₀. (10) We supply the Lebesgue space 𝐿² with the norm ‖⋅‖ and the Sobolev space 𝐻^𝑠, 𝑠∈𝑅 with the norm ‖⋅‖_𝐻^𝑠. For a fixed 𝑇>0, we also defined a space 𝑊(0,𝑇;𝑉) as 𝑊(0,𝑇;𝑉)=𝑚∶𝑚∈𝐿²(0,𝑇;𝑉),𝑚_𝑡∈𝐿²0,𝑇;𝑉^∗, (11) which is a Hilbert space endowed with common inner product.

2. Existence of Unique Weak Solution

In this section, we prove the existence of a weak solution for the following equation: 𝑚_𝑡−𝜀𝑚_𝑥𝑥+𝑎𝑢_𝑥𝑚+𝑏𝑢𝑚_𝑥=𝐵^∗𝜛in(0,𝑇)×Ω, (12) with the boundary conditions 𝑢=𝑢_𝑥=𝑢_𝑥𝑥=0on(0,𝑇)×𝜕Ω, (13) and the initial value 𝑚(𝑥,0)=𝑚₀(𝑥)inΩ, (14) where 𝑚=𝑢−𝑢_𝑥𝑥, Ω=(0,1), 𝐵^∗𝜛∈𝐿²(𝑉^∗), and a control 𝜛∈𝐿²(𝑄₀). Now, we give the definition of the weak solution in the space 𝑊(0,𝑇;𝑉).

Definition 1. A function 𝑚(𝑥,𝑡)∈𝑊(0,𝑇;𝑉) is called a weak solution to (12), if 𝑑𝑚𝑑𝑡𝑚,𝜙+𝜀_𝑥,𝜙_𝑥+𝑎𝑢_𝑥𝑚,𝜙+𝑏𝑢𝑚_𝑥=𝐵,𝜙_∗𝜛,𝜙_𝑉∗,𝑉, (15)
is valid for all 𝜙∈𝑉, a.e. 𝑡∈[0,𝑇] and 𝑚₀(𝑥)∈𝐻.

By using the standard Galerkin method and some a priori estimates, one can obtain the following theorem, which ensures the existence of a unique weak solution to the viscous shallow water equation.

Theorem 2. Let 𝑚₀(𝑥)∈𝐻 and 𝐵^∗𝜛∈𝐿²(𝑉^∗) then the problem (12)–(14) admits a local weak solution 𝑚(𝑥,𝑡) which satisfies Definition 1.

Proof. Let {𝜑_𝑖}^∞_𝑖=1 be an orthonormal basis in the space 𝐻 consisting of eigenfunctions of the operator 𝐴=−𝜕²_𝑥. For 𝑘∈𝒩, we define the discrete ansatz space by 𝑉_𝑘=span{𝜑₁,𝜑₂,…,𝜑_𝑘}⊂𝑉. Let 𝑚_𝑘(𝑡)=𝑚_𝑘(𝑥,𝑡)=∑^𝑘_𝑖=1𝑚^𝑘_𝑖(𝑡)𝜑_𝑖(𝑥) with 𝑚_𝑘(⋅,0)→𝑚₀ in 𝐻.
By analyzing the limiting behavior of the sequences of smooth functions {𝑢_𝑘} and {𝑚_𝑘}, we can prove the existence of a weak solution to (12).
Performing the Galerkin procedure for the problem (12)–(14), we obtain 𝑚_𝑘,𝑡−𝜀𝑚_{𝑘,𝑥𝑥}+𝑎𝑢_𝑘,𝑥𝑚_𝑘+𝑏𝑢_𝑘𝑚_𝑘,𝑥=𝐵^∗𝜛in(0,𝑇)×Ω, (16) with the boundary conditions 𝑢_𝑘=𝑢_𝑘,𝑥=𝑢_{𝑘,𝑥𝑥}=0on(0,𝑇)×𝜕Ω, (17) and the initial value 𝑚_𝑘(𝑥,0)=𝑚_𝑘,0(𝑥)∈𝐻inΩ. (18)
Clearly, (16) is an ordinary differential equation, and, according to standard ODE’s theory, there is a unique solution to (16) in the interval [0,𝑡_𝑘). What we should do is to show that the solution is uniformly bounded as 𝑡_𝑘→𝑇. We will prove the existence of weak solution in the following steps.
Step  1. Taking the inner product of (16) with 𝑢_𝑘 in Ω, we have 12𝑑𝑢𝑑𝑡^𝑘2+𝑢_𝑘,𝑥²+𝜀𝑢_𝑘,𝑥²+𝑢_{𝑘,𝑥𝑥}²=−𝑎−2𝑏2𝑢_𝑘,𝑥,𝑢²_𝑘,𝑥+𝐵^∗𝜛,𝑢_{𝑘𝑉∗,𝑉}. (19) Let us estimate the first term of the right hand side of (19) as follows: −𝑎−2𝑏2𝑢_𝑘,𝑥,𝑢²_𝑘,𝑥≤|𝑎−2𝑏|2𝑢_{𝑘,𝑥𝐿∞}𝑢_𝑘,𝑥². (20) By the Sobolev embedding theorem and the Poincaré inequality, we obtain 𝑢_{𝑘,𝑥𝐿∞}≤𝑐₁𝑢_{𝑘,𝑥𝐻1}≤𝑐₁𝑢_𝑘𝐻2≤𝑐₂𝑢_{𝑘,𝑥𝑥}, (21) where 𝑐₁,𝑐₂ are constants.
It follows from (20) and (21) that −𝑎−2𝑏2𝑢_𝑘,𝑥,𝑢²_𝑘,𝑥≤|𝑎−2𝑏|2𝑐₂𝑢_{𝑘,𝑥𝑥}⋅𝑢_𝑘,𝑥². (22)
Next, we estimate the second term of the right hand side of (19). Noting that 𝐵^∗𝜛∈𝐿²(𝑉^∗) is a control item, we can assume that ‖𝐵^∗𝜛‖_𝑉^∗≤𝑀, where 𝑀 is positive constant. Then 𝐵^∗𝜛,𝑢_{𝑘𝑉∗,𝑉}≤𝐵^∗𝜛_𝑉∗𝑢_𝑘𝑉≤𝑐₃𝑀𝑢_𝑘,𝑥. (23) Combining inequalities (22) and (23) with (19) and using the Young inequality, we have 12𝑑𝑢𝑑𝑡^𝑘2+𝑢_𝑘,𝑥²+𝜀𝑢_𝑘,𝑥²+𝑢_{𝑘,𝑥𝑥}²≤|𝑎−2𝑏|2𝑐₂𝑢_{𝑘,𝑥𝑥}⋅𝑢_𝑘,𝑥²+𝑐₃𝑀𝑢_𝑘,𝑥≤𝜀𝑢_{𝑘,𝑥𝑥}²+(𝑎−2𝑏)²𝑐₂²𝑢4𝜀^𝑘,𝑥4+𝜀𝑢_𝑘²+𝑐₃²𝑀²𝜀≤𝜀𝑢_{𝑘,𝑥𝑥}²+𝑢_𝑘²+(𝑎−2𝑏)²𝑐₂²𝑢4𝜀^𝑘2+𝑢_𝑘,𝑥²²+𝑐₃²𝑀²𝜀. (24) Therefore, we obtain 𝑑𝑢𝑑𝑡^𝑘2+𝑢_𝑘,𝑥²≤(𝑎−2𝑏)²𝑐₂²𝑢2𝜀^𝑘2+𝑢_𝑘,𝑥²²+2𝑐₃²𝑀²𝜀. (25) It, thus, transpires that 𝑢_𝑘²+𝑢_𝑘,𝑥²≤2𝑐₂𝑀|𝑎−2𝑏|𝑐₂𝑐tan²𝑐₃𝑀|𝑎−2𝑏|𝜀𝑡+𝑐₄≜𝜌²₁, (26) where 𝑡∈[0,𝑇], 𝑇<𝜋𝜀/2𝑐₂𝑐₃𝑀|𝑎−2𝑏| and 𝑐₄=arctan(|𝑎−2𝑏|𝑐₂(‖𝑢_𝑘,0‖²+‖𝑢_𝑘,𝑥(0)‖²)/2𝑐₃𝑀).
From the above discussion, we know that ‖𝑢_𝑘‖≤𝜌₁ and ‖𝑢_𝑘,𝑥‖≤𝜌₁. By the Sobolev embedding theorem, we obtain 𝑢_𝑘𝐿∞≤𝑢_𝑘𝐻1=𝑢_𝑘𝑉≤𝑢_𝑘+𝑢_𝑘,𝑥≤𝑐₅. (27)
Step  2. We prove a uniform 𝐿²(0,𝑇;𝑉) bound on a sequence {𝑚_𝑘}. Taking the inner product of (16) with 𝑚_𝑘 in Ω, we have 12𝑚_𝑘²+𝜀𝑚_𝑘,𝑥²=(2𝑎−𝑏)𝑢_𝑘,𝑚_𝑘𝑚_𝑘,𝑥+𝐵^∗𝜛,𝑚_{𝑘𝑉∗,𝑉}. (28) Let us estimate the first term of the right hand side of (28). By the Poincaré inequality and the Hölder inequality, we have (2𝑎−𝑏)𝑢_𝑘,𝑚_𝑘𝑚_𝑘,𝑥≤|2𝑎−𝑏|𝑢_𝑘𝐿∞𝑚_𝑘⋅𝑚_𝑘,𝑥≤𝑐₅|2𝑎−𝑏|2𝑚_𝑘²+𝑚_𝑘,𝑥²≤𝑐₅|2𝑎−𝑏|2𝑐₆𝑚_𝑘,𝑥²+𝑚_𝑘,𝑥²=𝑐₇𝑚_𝑘,𝑥², (29) where 𝑐₇=(𝑐₅|2𝑎−𝑏|/2)(𝑐₆+1).
Furthermore, we estimate the second term of the right hand side of (28) in the following way: 𝐵^∗𝜛,𝑚_{𝑘𝑉∗,𝑉}≤𝑀𝑚_𝑘𝑉≤𝑐₈𝑀𝑚_𝑘,𝑥. (30) Combining inequalities (29) and (30) with (28), we have 12𝑑𝑚𝑑𝑡^𝑘2+𝜀𝑚_𝑘,𝑥²≤𝑐₇𝑚_𝑘,𝑥²+𝑐₈𝑀𝑚_𝑘,𝑥, (31) where 𝜀>𝑐₇. It follows from (31) with the Young inequality that 𝑑𝑚𝑑𝑡^𝑘2+𝜀−𝑐₇𝑚_𝑘,𝑥²≤𝑐₈²𝑀²𝜀−𝑐₇. (32) Integrating the pervious inequality with respect to 𝑡 on [0,𝑇], we obtain ^𝑇₀𝑚_𝑘,𝑥²1𝑑𝑡≤𝜀−𝑐₇𝑐₈²𝑀²𝑇𝜀−𝑐₇+𝑚_𝑘,0²≜𝜌²₂. (33) Furthermore, we have 𝑚_𝑘²_{𝐿2()0,𝑇;𝑉}=^𝑇₀𝑚_𝑘²_𝐻1𝑑𝑡≤𝑐₈^𝑇₀𝑚_𝑘,𝑥²𝑑𝑡≜𝜌²₃. (34) From (32), we also have 𝑑𝑚𝑑𝑡^𝑘2≤𝑐₈²𝑀²𝜀−𝑐₇. (35) Integrating the pervious inequality with respect to 𝑡<𝑇 on [0,𝑡], we obtain 𝑚_𝑘²≤𝑐₈²𝑀²𝑡𝜀−𝑐₇+𝑚_𝑘,0²≜𝜌²₄,∀𝑡∈[0,𝑇]. (36)
Step  3. We prove a uniform 𝐿²(0,𝑇;𝑉^∗) bound on a sequence {𝑚_𝑘,𝑡}. By (16) and the Sobolev embedding theorem, we have 𝑚_{𝑘,𝑡𝑉∗}≤𝜀𝑚_𝑘𝑉+𝑎𝑢_𝑘𝑚_𝑘𝑉+𝑏𝑢_𝑘𝑉𝑚_𝑘+𝐵^∗𝜛_𝑉∗≤𝜀𝑚_𝑘𝑉+𝑎𝜌₁𝑚_𝑘𝑉+𝑏𝑐₅𝑚_𝑘+𝑀. (37) Furthermore, we have 𝑚_𝑘,𝑡²_𝑉∗≤3𝜀+𝑎𝜌₁²𝑚_𝑘²_𝑉+3𝑏²𝑐₅²𝑚_𝑘²+3𝑀². (38) Integrating (38) with respect to 𝑡 on [0,𝑇], we deduce that 𝑚_𝑘,𝑡²_{𝐿2(0,𝑇;𝑉∗)}≤3𝜀+𝑎𝜌₁²𝜌²₃+3𝑏²𝑐₅²𝜌²₄+𝑀²𝑇≜𝜌²₅. (39) From the pervious discussion one has the following.(a)For every 𝑇<𝜋𝜀/2𝑐₂𝑐₃𝑀|𝑎−2𝑏|, 𝜌₃ and 𝜌₄ are two constants and the sequence {𝑚_𝑘} is bounded in 𝐿²(0,𝑇;𝐻) as well as in 𝐿²(0,𝑇;𝑉), which is independent of the dimension of ansatz space 𝑘.(b)For every 𝑇<𝜋𝜀/2𝑐₂𝑐₃𝑀|𝑎−2𝑏| and 𝜌₅ is constant, the sequence {𝑚_𝑘,𝑡} is bounded in 𝐿²(0,𝑇;𝑉^∗), which is independent of the dimension of ansatz space 𝑘.
Note that, (a) and (b) the above mentioned are equivalent to {𝑚_𝑘}⊂𝑊(0,𝑇;𝑉) bounded, and 𝑊(0,𝑇;𝑉) is compactly embedded into 𝐶(0,𝑇;𝐻). Then, we conclude the convergence of a subsequence, again denoted by {𝑚_𝑘} weak in 𝑊(0,𝑇;𝑉), weakstar in 𝐿^∞(0,𝑇;𝐻), and strong in 𝐿²(0,𝑇;𝐻) to a function 𝑚(𝑥,𝑡)∈𝑊(0,𝑇;𝑉).
The proof of the uniqueness of the solution is similar to Theorem  1 in [37], so we omit it here. This completes the proof of the theorem.

Our next result describes that the norm of weak solution can be controlled by initial value and control item.

Theorem 3. Let 𝑚₀∈𝐻 and 𝐵^∗𝜛∈𝐿²(𝑉^∗); then there exists a positive constant 𝐶, such that ‖𝑚‖²_{𝑊()0,𝑇;𝑉}≤𝐶𝑚₀²+‖𝜛‖²_𝐿2𝑄0. (40)

Proof. Taking the inner product of (12) with 𝑢 in Ω, by using the same argument as in the proof of Theorem 2, we have ‖𝑢‖≤𝜌₁,𝑢_𝑥≤𝜌₁,‖𝑢‖_𝐿^∞≤𝑐₅, (41) where 𝜌₁ and 𝑐₅ are positive constants as in Theorem 2. Taking the inner product of (12) with 𝑚 in Ω, we have 12‖𝑚‖²+𝜀𝑚_𝑥²=(2𝑎−𝑏)𝑢,𝑚𝑚_𝑥+𝐵^∗𝜛,𝑚_𝑉∗,𝑉. (42) Let us estimate the first term of the right hand side of (42) as follows: (2𝑎−𝑏)𝑢,𝑚𝑚_𝑥≤|2𝑎−𝑏|‖𝑢‖_𝐿^∞‖𝑚‖⋅𝑚_𝑥≤𝑐₅|2𝑎−𝑏|2‖𝑚‖²+𝑚_𝑥²≤𝑐₅|2𝑎−𝑏|2𝑐₆𝑚_𝑥²+𝑚_𝑥²=𝑐₇𝑚_𝑥². (43) Furthermore, we estimate the second term of the right hand side of (42) in the following way: 𝐵^∗𝜛,𝑚_𝑉∗,𝑉≤𝐵^∗𝜛_𝑉∗‖𝑚‖_𝑉≤𝑐₈𝐵^∗𝜛_𝑉∗𝑚_𝑥. (44) Combining inequalities (43) and (44) with (42), we have 12𝑑‖𝑚‖𝑑𝑡²+𝜀−𝑐₇𝑚_𝑥²≤𝑐₈𝐵^∗𝜛_𝑉∗𝑚_𝑥, (45) where 𝜀>𝑐₇.
Integrating (45) with respect to 𝑡 on [0,𝑇], we derive that 12‖𝑚(𝑇)‖²−12𝑚₀²+𝜀−𝑐₇𝑚_𝑥²_{𝐿2()0,𝑇;𝐻}≤𝑐₈𝐵^∗𝜛_{𝐿2(0,𝑇;𝑉∗)}𝑚_{𝑥𝐿2(0,𝑇;𝐻)}. (46) According to the Young inequality, we have 𝑐₈𝐵^∗𝜛_{𝐿2(0,𝑇;𝑉∗)}𝑚_{𝑥𝐿2(0,𝑇;𝐻)}≤𝜀−𝑐₇2𝑚_𝑥²_{𝐿2()0,𝑇;𝐻}+𝑐₈²2𝜀−𝑐₇𝐵^∗𝜛²_{𝐿2(0,𝑇;𝑉∗)}. (47) Substituting (47) into (46) yields 𝜀−𝑐₇𝑚_𝑥²_{𝐿2()0,𝑇;𝐻}≤𝑚₀²+𝑐₈²𝜀−𝑐₇𝐵^∗𝜛²_{𝐿2(0,𝑇;𝑉∗)}. (48) It follows from (48) that ‖𝑚‖²_{𝐿2()0,𝑇;𝑉}≤𝑐₈^𝑇₀𝑚_𝑥²≤𝑐𝑑𝑡₈𝜀−𝑐₇𝑚₀²+𝑐₈³𝜀−𝑐₇²𝐵^∗𝜛²_{𝐿2(0,𝑇;𝑉∗)}≤𝑐₉𝑚₀+𝐵^∗𝜛_{𝐿2(0,𝑇;𝑉∗)}², (49) where 𝑐₉=max{𝑐₈/(𝜀−𝑐₇),𝑐₈³/(𝜀−𝑐₇)²}.
Combining inequality (45) with 𝜀>𝑐₇, we also have 12𝑑‖𝑚‖𝑑𝑡²≤𝑐₈𝐵^∗𝜛_𝑉∗‖𝑚‖_𝑉. (50) Integrating the pervious inequality with respect to 𝑡 yields ‖𝑚‖²≤𝑚₀²+2𝑐₈𝐵^∗𝜛_{𝐿2(0,𝑇;𝑉∗)}‖𝑚‖_𝐿²_{(0,𝑇;𝑉)}≤𝑚₀²+2𝑐₈𝐵^∗𝜛_{𝐿2(0,𝑇;𝑉∗)}√𝑐₉×𝑚₀+𝐵^∗𝜛_{𝐿2(0,𝑇;𝑉∗)}≤𝑐₁₀𝑚₀+𝐵^∗𝜛_{𝐿2(0,𝑇;𝑉∗)}², (51) where 𝑐₁₀=2max{1,𝑐₈√𝑐₉}.
Combining (12) with the Sobolev embedding theorem, we obtain 𝑚_𝑡𝑉∗≤𝜀‖𝑚‖_𝑉+𝑎‖𝑢‖‖𝑚‖_𝑉+𝑏‖𝑢‖_𝑉‖𝑚‖+𝐵^∗𝜛_𝑉∗≤𝜀+𝑎𝜌₁‖𝑚‖_𝑉+𝑏𝑐₅‖𝑚‖+𝐵^∗𝜛_𝑉∗. (52) Furthermore, we have 𝑚_𝑡²_𝑉∗≤3𝜀+𝑎𝜌₁²‖𝑚‖²_𝑉+3𝑏²𝑐₅²‖𝑚‖²+3𝐵^∗𝜛²_𝑉∗. (53) Integrating (53) with respect to 𝑡 on [0,𝑇], we deduce that 𝑚_𝑡²_{𝐿2(0,𝑇;𝑉∗)}≤3𝜀+𝑎𝜌₁²‖𝑚‖²_{𝐿2()0,𝑇;𝑉}+3𝑏²𝑐₅^2𝑇₀‖𝑚‖²𝑑𝑡+3𝐵^∗𝜛²_{𝐿2(0,𝑇;𝑉∗)}≤3𝜀+𝑎𝜌₁²𝑐₉𝑚₀+𝐵^∗𝜛_{𝐿2(0,𝑇;𝑉∗)}²+3𝑏²𝑐₅²𝑐₁₀𝑇𝑚₀+𝐵^∗𝜛_{𝐿2(0,𝑇;𝑉∗)}²+3𝐵^∗𝜛²_{𝐿2(0,𝑇;𝑉∗)}≤𝑐₁₁𝑚₀+𝐵^∗𝜛_{𝐿2(0,𝑇;𝑉∗)}², (54) where 𝑐₁₁=3[(𝜀+𝑎𝜌₁)²𝑐₉+𝑏²𝑐₅²𝑐₁₀𝑇+1].
Taking into account (49) and (54), we arrive at ‖𝑚‖²_{𝑊()0,𝑇;𝑉}=‖𝑚‖²_{𝐿2()0,𝑇;𝑉}+𝑚_𝑡²_{𝐿2(0,𝑇;𝑉∗)}≤𝑐₉+𝑐₁₁𝑚₀+𝐵^∗𝜛_{𝐿2(0,𝑇;𝑉∗)}²≤𝐶𝑚₀+‖𝜛‖_𝐿²_(𝑄0)², (55) where 𝐶=𝑐₉+𝑐₁₁. This completes the proof of the theorem.

3. Existence of Optimal Solution to the Problem of (7)

In this section, we consider the optimal control problem of (7) and prove the existence of optimal solution based on Lions’ theory (see [50]).

Allowing a control 𝜛∈𝐿²(𝑄₀), we study the following control system: 𝑚_𝑡−𝜀𝑚_𝑥𝑥+𝑎𝑢_𝑥𝑚+𝑏𝑢𝑚_𝑥=𝐵^∗𝜛in(0,𝑇)×Ω,𝑢=𝑢_𝑥=𝑢_𝑥𝑥()𝑚()=0on0,𝑇×𝜕Ω,𝑥,0=𝑚₀(𝑥)∈𝐻inΩ, (56) where 𝑚=𝑢−𝑢_𝑥𝑥 and Ω=(0,1). Let 𝐶∈𝐿(𝑊(0,𝑇;𝑉),𝑆) be an observation operator, and let 𝑆 be a real Hilbert space of observations. Then, we choose performance index of tracking type 𝐽()=1𝑚,𝜛2‖𝐶𝑚−𝑧‖²_𝑆+𝛿2‖𝜛‖²_𝐿2𝑄0, (57) where 𝑧∈𝑆 is a desired state and 𝛿>0 is fixed. Optimal control problem of (7) is min𝐽(𝑚,𝜛), (58) where (𝑚,𝜛) satisfies (56).

Let 𝑋=𝑊(0,𝑇;𝑉)×𝐿²(𝑄₀), 𝑌=𝐿²(0,𝑇;𝑉)×𝐻; we define an operator 𝑒=𝑒(𝑒₁,𝑒₂)∶𝑋→𝑌 by 𝑒₁=(−Δ)⁻¹𝑚_𝑡−𝜀𝑚_𝑥𝑥+𝑎𝑢_𝑥𝑚+𝑏𝑢𝑚_𝑥−𝐵^∗𝜛,𝑒₂=𝑚(𝑥,0)−𝑚₀, (59) where Δ is an operator from 𝑉 to 𝑉^∗. Then, we rewrite (58) in the following form: ()()min𝐽𝑚,𝜛s.t.𝑒𝑚,𝜛=0. (60)

Now, we give the main result.

Theorem 4. There exists an optimal control solution (𝑚^∗,𝜛^∗) to the problem (56).

Proof. Let (𝑚,𝜛)∈𝑋 satisfying the equation 𝑒(𝑚,𝜛)=0. According to (57) and Theorem 3, we have 𝐽()≥𝛿𝑚,𝜛2‖𝜛‖²_𝐿2𝑄0,‖𝜛‖_𝐿²_(𝑄0)⟶∞,as‖𝑚‖_{𝑊(0,𝑇;𝑉)}⟶∞. (61) Then, 𝐽(𝑚,𝜛)⟶∞,as‖(𝑚,𝜛)‖_𝑋⟶∞. (62) Note that the norm is weakly lower semicontinuous [51]; we have that 𝐽 is weakly lower semicontinuous. Since 𝐽(𝑚,𝜛)≥0, for all (𝑚,𝜛)∈𝑋, holds, there exists 𝜅≥0 defined by 𝜅=inf{𝐽(𝑚,𝜛)∣(𝑚,𝜛)∈𝑋,𝑒(𝑚,𝜛)=0}, (63) which means the existence of a minimizing sequence {(𝑚^𝑛,𝜛^𝑛)}_𝑛∈𝒩 in 𝑋 such that 𝜅=lim_𝑛→∞𝐽𝑚^𝑛,𝜛^𝑛,𝑒𝑚^𝑛,𝜛^𝑛=0,∀𝑛∈𝒩. (64)
According to (62), there exists an element (𝑚^∗,𝜛^∗)∈𝑋 such that, when 𝑛→∞, 𝑚^𝑛⟶𝑚^∗(),𝜛,weakly,𝑚∈𝑊0,𝑇;𝑉_𝑛⟶𝜛^∗,weakly,𝜛∈𝐿²𝑄₀. (65)(66)
From (65), it is easy to check that lim_𝑛→∞^𝑇₀𝑚^𝑛_𝑡(𝑡)−𝑚^∗_𝑡(𝑡),𝜙(𝑡)_𝑉∗,𝑉𝑑𝑡=0,∀𝜙∈𝐿²(0,𝑇;𝑉). (67)
Thanks to the fact that 𝑊(0,𝑇;𝑉) is compactly embedded into 𝐿²(0,𝑇;𝐿^∞) [52] and 𝑊(0,𝑇;𝑉) is continuously embedded into 𝐶(0,𝑇;𝐻) [53], we obtain that 𝑚^𝑛→𝑚^∗ strongly in 𝐿²(0,𝑇;𝐿^∞) and 𝑚^𝑛→𝑚^∗ strongly in 𝐶(0,𝑇;𝐻). Furthermore, we have that 𝑢^𝑛→𝑢^∗ strongly in 𝐶(0,𝑇;𝐻).
Since the sequence {𝑚^𝑛}_𝑛∈𝒩 converges weakly and ‖𝑚^𝑛‖_{𝑊(0,𝑇;𝑉)} is bounded [54], based on embedding theorem, we obtain that ‖𝑚^𝑛‖_𝐿²_(0,𝑇;𝐿^∞₎ is bounded. In fact, ‖𝑚^∗‖_𝐿²_(0,𝑇;𝐿^∞₎ is also bounded, because 𝑚^𝑛→𝑚^∗ strongly in 𝐿²(0,𝑇;𝐿^∞).
Thus, it follows from the Hölder inequality that ^𝑇₀¹₀𝑢^𝑛_𝑥𝑚^𝑛−𝑢^∗_𝑥𝑚^∗=𝜙𝑑𝑥𝑑𝑡_𝑇010𝑢^𝑛_𝑥𝑚^𝑛−𝑢^𝑛_𝑥𝑚^∗+𝑢^𝑛_𝑥𝑚^∗−𝑢^∗_𝑥𝑚^∗≤𝜙𝑑𝑥𝑑𝑡_𝑇010𝑢^𝑛_𝑥𝑚^𝑛−𝑚^∗+𝜙𝑑𝑥𝑑𝑡_𝑇010𝑢^𝑛_𝑥−𝑢^∗_𝑥𝑚^∗≤𝜙𝑑𝑥𝑑𝑡_𝑇0𝑚^𝑛−𝑚^∗_𝐿∞𝑢^𝑛_𝐻𝜙_𝑉+𝑑𝑡_𝑇0𝑚^∗_𝐿∞𝑢^𝑛−𝑢^∗_𝐻𝜙_𝑉≤𝑚𝑑𝑡_𝑛−𝑚^∗_{𝐿2(0,𝑇;𝐿∞)}𝑢^𝑛_{𝐶(0,𝑇;𝐻)}𝜙_{𝐿2(0,𝑇;𝑉)}+𝑚^∗_{𝐿2(0,𝑇;𝐿∞)}𝑢^𝑛−𝑢^∗_{𝐶(0,𝑇;𝐻)}𝜙_{𝐿2(0,𝑇;𝑉)}⟶0,∀𝜙∈𝐿²(0,𝑇;𝑉). (68) Similarly, ^𝑇₀¹₀𝑢^∗𝑚^∗−𝑢^𝑛𝑚^𝑛𝜙_𝑥=𝑑𝑥𝑑𝑡_𝑇010𝑢^∗𝑚^∗−𝑢^𝑛𝑚^∗+𝑢^𝑛𝑚^∗−𝑢^𝑛𝑚^𝑛𝜙_𝑥≤𝑑𝑥𝑑𝑡_𝑇010𝑢^∗𝑚^∗−𝑢^𝑛𝑚^∗𝜙_𝑥+𝑑𝑥𝑑𝑡_𝑇010𝑢^𝑛𝑚^∗−𝑢^𝑛𝑚^𝑛𝜙_𝑥≤𝑑𝑥𝑑𝑡_𝑇0𝑚^∗_𝐿∞𝑢^∗−𝑢^𝑛_𝐻𝜙_𝑉+𝑑𝑡_𝑇0𝑢^𝑛_𝐻𝑚^∗−𝑚^𝑛_𝐿∞𝜙_𝑉≤𝑚𝑑𝑡_{∗𝐿2(0,𝑇;𝐿∞)}𝑢^∗−𝑢^𝑛_{𝐶(0,𝑇;𝐻)}𝜙_{𝐿2(0,𝑇;𝑉)}+𝑚^∗−𝑢^𝑛_{𝐿2(0,𝑇;𝐿∞)}𝑢^𝑛_{𝐶(0,𝑇;𝐻)}𝜙_{𝐿2(0,𝑇;𝑉)}⟶0,∀𝜙∈𝐿²(0,𝑇;𝑉). (69) Then, we have ^𝑇₀¹₀𝑢^𝑛𝑚^𝑛_𝑥−𝑢^∗𝑚^∗_𝑥=𝜙𝑑𝑥𝑑𝑡_𝑇010−𝑢^𝑛_𝑥𝑚^𝑛𝜙−𝑢^𝑛𝑚^𝑛𝜙_𝑥+𝑢^∗_𝑥𝑚^∗𝜙+𝑢^∗𝑚^∗𝜙_𝑥≤𝑑𝑥𝑑𝑡_𝑇010𝑢^∗_𝑥𝑚^∗−𝑢^𝑛_𝑥𝑚^𝑛+𝜙𝑑𝑥𝑑𝑡_𝑇010𝑢^∗𝑚^∗−𝑢^𝑛𝑚^𝑛𝜙_𝑥𝑑𝑥𝑑𝑡⟶0,∀𝜙∈𝐿²(0,𝑇;𝑉). (70) By view of (66), we can obtain that ^𝑇₀¹₀𝐵^∗𝜛^𝑛−𝐵^∗𝜛^∗𝜙𝑑𝑥𝑑𝑡⟶0,∀𝜙∈𝐿²(0,𝑇;𝑉). (71) According to the above discussion, we have that 𝑒₁𝑚^∗,𝜛^∗=0,∀𝑛∈𝒩. (72)
Thanks to the fact that 𝑚^∗∈𝑊(0,𝑇;𝑉) and 𝑚^𝑛→𝑚^∗ weakly in 𝑊(0,𝑇;𝑉), we have 𝑚^∗(0)∈𝐻 and 𝑚^𝑛(0)→𝑚^∗(0) weakly as 𝑛→∞. Furthermore, we obtain 𝑚^𝑛(0)−𝑚^∗(0),𝜙⟶0,𝑛⟶∞,∀𝜙∈𝐻, (73) which means that 𝑒₂(𝑚^∗,𝜛^∗)=0.
Therefore, we have that 𝑒𝑚^∗,𝜛^∗=0,in𝑌. (74)
Thus, there exists an optimal solution (𝑚^∗,𝜛^∗) to the problem (56). In the meantime, we can infer that there exists an optimal solution (𝑢^∗,𝜛^∗) to the viscous shallow water equation due to the relation 𝑢=(1−𝜕²_𝑥)⁻¹𝑚. The proof of this theorem is completed.

Acknowledgments

The authors would like to thank all the anonymous referees for their careful reading and constructive comments. This work was supported by the Natural Science Foundation of China (nos. 11171135, 10420130638) and the Taizhou Social Development Project.