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Advances in Mathematical Physics
Volume 2013 (2013), Article ID 715959, 8 pages
http://dx.doi.org/10.1155/2013/715959
Research Article

Optimal Control of a Viscous Shallow Water Equation

1Taizhou Institute of Science and Technology (NUST) Taizhou, Jiangsu 225300, China
2Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, China

Received 22 June 2013; Accepted 23 August 2013

Academic Editor: Shao-Ming Fei

Copyright © 2013 Wenbin Zhang and Jiangbo Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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: which describes the balance between convection and stretching for small viscosity in the dynamics of one-dimensional nonlinear waves in fluids. Here , , and is chosen to be the Green’s function for the Helmholtz operator on the line. In a recent study of soliton equations, it is found that (1) for and any is included in the family of shallow water equations at quadratic order accuracy that are asymptotically equivalent under Kodama transformations [2].

When , (1) becomes the -family of equations: 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 or ; compare [2, 4, 5].

For in (2), it becomes the Camassa-Holm (CH) equation: 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 with . 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 .

For in (2), it becomes the Degasperis-Procesi (DP) equation: 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 [2123]. After (4) appeared, it has attracted many researchers to discover its dynamics (see [2431]). Yin [24, 25] proved the local well-posedness of (4) with initial data () 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, 2224], (4) has not only global strong solutions [27, 28] but also blow-up solutions [26, 2830]. Apart from these, Coclite and Karlsen [31] proved that it has global entropy weak solutions in and .

Recently, Lai and Wu [32] found that, in both models (1) and (2), the coefficient of is equal to the coefficient of plus the one of . That is, Then, they studied the global solutions and blow-up phenomena to the following generalized equation:

In this paper, we study the optimal control problem for the following equation: where 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. [4648] 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 , the optimal control problem for (7) we intend to investigate is subject to where . Clearly, our control target is to match the given desired state by adjusting the body force in a control volume in the -sense.

Notation. In this paper, we set that , , , and are dual space, respectively. Then, , with each embedding being dense. The extension operator is given by We supply the Lebesgue space with the norm and the Sobolev space , with the norm . For a fixed , we also defined a space as 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: with the boundary conditions and the initial value where , , , and a control . Now, we give the definition of the weak solution in the space .

Definition 1. A function is called a weak solution to (12), if
is valid for all , a.e. and .

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 and then the problem (12)–(14) admits a local weak solution which satisfies Definition 1.

Proof. Let be an orthonormal basis in the space consisting of eigenfunctions of the operator . For , we define the discrete ansatz space by . Let with in .
By analyzing the limiting behavior of the sequences of smooth functions and , we can prove the existence of a weak solution to (12).
Performing the Galerkin procedure for the problem (12)–(14), we obtain with the boundary conditions and the initial value
Clearly, (16) is an ordinary differential equation, and, according to standard ODE’s theory, there is a unique solution to (16) in the interval . What we should do is to show that the solution is uniformly bounded as . We will prove the existence of weak solution in the following steps.
Step  1. Taking the inner product of (16) with in , we have Let us estimate the first term of the right hand side of (19) as follows: By the Sobolev embedding theorem and the Poincaré inequality, we obtain where are constants.
It follows from (20) and (21) that
Next, we estimate the second term of the right hand side of (19). Noting that is a control item, we can assume that , where is positive constant. Then Combining inequalities (22) and (23) with (19) and using the Young inequality, we have Therefore, we obtain It, thus, transpires that where , and .
From the above discussion, we know that and . By the Sobolev embedding theorem, we obtain
Step  2. We prove a uniform bound on a sequence . Taking the inner product of (16) with in , we have Let us estimate the first term of the right hand side of (28). By the Poincaré inequality and the Hölder inequality, we have where .
Furthermore, we estimate the second term of the right hand side of (28) in the following way: Combining inequalities (29) and (30) with (28), we have where . It follows from (31) with the Young inequality that Integrating the pervious inequality with respect to on , we obtain Furthermore, we have From (32), we also have Integrating the pervious inequality with respect to on , we obtain
Step  3. We prove a uniform bound on a sequence . By (16) and the Sobolev embedding theorem, we have Furthermore, we have Integrating (38) with respect to on , we deduce that From the pervious discussion one has the following.(a)For every , and are two constants and the sequence is bounded in as well as in , which is independent of the dimension of ansatz space .(b)For every and is constant, the sequence is bounded in , which is independent of the dimension of ansatz space .
Note that, (a) and (b) the above mentioned are equivalent to bounded, and is compactly embedded into . Then, we conclude the convergence of a subsequence, again denoted by weak in , weakstar in , and strong in to a function .
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 and ; then there exists a positive constant , such that

Proof. Taking the inner product of (12) with in , by using the same argument as in the proof of Theorem 2, we have where and are positive constants as in Theorem 2. Taking the inner product of (12) with in , we have Let us estimate the first term of the right hand side of (42) as follows: Furthermore, we estimate the second term of the right hand side of (42) in the following way: Combining inequalities (43) and (44) with (42), we have where .
Integrating (45) with respect to on , we derive that According to the Young inequality, we have Substituting (47) into (46) yields It follows from (48) that where .
Combining inequality (45) with , we also have Integrating the pervious inequality with respect to yields where .
Combining (12) with the Sobolev embedding theorem, we obtain Furthermore, we have Integrating (53) with respect to on , we deduce that where .
Taking into account (49) and (54), we arrive at where . 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 , we study the following control system: where and . Let be an observation operator, and let be a real Hilbert space of observations. Then, we choose performance index of tracking type where is a desired state and is fixed. Optimal control problem of (7) is where satisfies (56).

Let , ; we define an operator by where is an operator from to . Then, we rewrite (58) in the following form:

Now, we give the main result.

Theorem 4. There exists an optimal control solution to the problem (56).

Proof. Let satisfying the equation . According to (57) and Theorem 3, we have Then, Note that the norm is weakly lower semicontinuous [51]; we have that is weakly lower semicontinuous. Since , for all , holds, there exists defined by which means the existence of a minimizing sequence in such that
According to (62), there exists an element such that, when ,
From (65), it is easy to check that
Thanks to the fact that is compactly embedded into [52] and is continuously embedded into [53], we obtain that strongly in and strongly in . Furthermore, we have that strongly in .
Since the sequence converges weakly and is bounded [54], based on embedding theorem, we obtain that is bounded. In fact, is also bounded, because strongly in .
Thus, it follows from the Hölder inequality that Similarly, Then, we have By view of (66), we can obtain that According to the above discussion, we have that
Thanks to the fact that and weakly in , we have and weakly as . Furthermore, we obtain which means that .
Therefore, we have that
Thus, there exists an optimal solution to the problem (56). In the meantime, we can infer that there exists an optimal solution to the viscous shallow water equation due to the relation . 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.

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