Recently, Escher and Yin  studied the following nonlinear dispersive equation (b-equation):
where , , and are arbitrary real constants. Denoting , we can rewrite b-equation in the following form:
Equation (1.2) can be derived as a family of asymptotically equivalent shallow water wave equations that emerge at quadratic order accuracy for by an appropriate Kodama transformation   . For the case , the corresponding Kodama transformation is singular and the asymptotic ordering is violated   . The solutions of the b-Equation (1.2) with were studied numerically for various values of b in   , where b was taken as a bifurcation parameter. The symmetry condition necessary for integrability of the b-Equation (1.2) was investigated in  . The Korteweg-de Vries (KdV) equation, the Camassa-Holm (CH) equation and the Degasperis-Procesi (DP) equation are the only three integrable equations in the b-Equation (1.2), which was shown in   by using Painleve analysis. The b-equation with admits peaked solutions for    . The peaked solutions replicate a feature that is characteristic for the waves of great height: waves of the largest amplitude that are exact solutions of the governing equations for water waves    .
If and , then b-Equation (1.1) becomes the well-known KdV equation
which describes the unidirectional propagation of waves at the free surface of shallow water under the influence of gravity  . In this model, re- presents the wave’s height above a flat bottom; x is proportional to distance in the direction of propagation and t is proportional to the elapsed time. The KdV equation is completely integrable, and its solitary waves are solitons  . The Cauchy problem of the KdV equation has been studied by many authors    and a satisfactory local or global (in time) existence theory is now available (for example, in   ). The solution of the KdV equation is global for   . It is also observed that the KdV equation does not accommodate wave breaking (by wave breaking we mean the phenomenon that a wave remains bounded but its slope becomes unbounded in finite time)  .
For and , b-Equation (1.1) becomes the CH equation
modelling the unidirectional propagation of shallow water waves over a flat bottom. Again stands for the fluid velocity at time t in the spatial x direction and is a nonnegative parameter related to the critical shallow water speed  . The CH equation is derived physically by approximating directly the Hamiltonian for Euler’s equations in the shallow water regime (it also appears in the context of hereditary symmetries studied by Fuchssteiner and Fokas  ). Recently, the alternative derivations of the CH equation as a model for water waves, respectively, as the equation for geodesic flow on the diffeomorphism group of the circle were presented in  and in  . For the physical derivation, we refer to the work in  . The geometric interpretation is important because it can be used to prove that the least action principle holds for the CH equation  . It is worth pointing out that the fundamental aspect of the CH equation, the fact that it is a completely integrable system, was shown in   for the periodic case and in   for the non-periodic case. Its solitary waves are smooth if and peaked in the limiting case  . They are orbitally stable and interact like solitons   and the explicit interaction of the peaked solitons is given in  .
Since the CH equation is structurally very rich, many physicists and mathematicians pay great attention to it. Local well-posedness for the initial datum with was proved by several authors, as in     . For the initial data with lower regularity, we refer to Molinet’s paper  and also the paper  . Moreover, wave breaking for a large class of initial data has been established in     . However, in  , global existence of weak solutions was proved but uniqueness was obtained only under a prior assumption that is known to hold only for the initial data such that is a sign-definite Radon measure (under this condition, global existence and uniqueness was shown in  ). Also it is worth noting that CH equation has global conservative solutions in    and global dissipative solutions (with energy being lost when wave breaking occurs) in   . In  , the authors showed the infinite propagation speed for the CH equation in the sense that a strong solution of the Cauchy problem with compact initial profile cannot be compactly supported at any later time unless it is the zero solution, which is an improvement of the previous results in this direction obtained in  .
For and in b-Equation (1.1), then we find the DP equation of the form 
Degasperis, Holm and Hone  proved the formal integrability of the DP Equation (1.5) by constructing a Lax pair. They also showed that DP equation has a bi-Hamiltonian structure and an infinite sequence of conserved quantities, and that it admits exact peakon solutions which are analogous to the CH peakons. Peakons for either b = 2 or b = 3 are true solitons that interact via elastic collisions under CH dynamics, or DP dynamics, respectively. Recently, Lundmark  showed that the DP equation has not only peaked solitons, but also shock peakons of the form
The DP equation can be regarded as a model for nonlinear shallow water dynamics and its asymptotic accuracy is the same as for the CH shallow water equation    . An inverse scattering approach for computing n-peakon solutions to the DP equation was presented in  . Its traveling wave solutions were investigated in  .
The Cauchy problem for the DP equation has been studied widely. Local well-posedness of this equation is established in   for the initial data with . Similar to the CH equation, the DP equation has also global strong solutions     as well as finite time blow-up solutions      . On the other hand, it has global weak solutions in    . Analogous to the case of the CH equation, Henry  and Mustafa  showed that smooth solutions to the DP equation have infinite speed of propagation. Coclite and Karlsen  also obtained global existence results for entropy weak solutions belonging to the class of and the class of .
Although the DP equation is similar to the CH equation in several aspects, these two equations are truly different. One of the novel features of the DP equation different from the CH equation is that it has not only peakon solutions  and periodic peakon solutions  , but also shock peakons   and the periodic shock waves  .
Despite the abundant literature on the above three special cases of the b-equ- ation, there are few results on the b-equation. Recently, some authors devoted to studying the Cauchy problem of the b-equation. Since the conservation laws of the b-equation are much weaker, there are only a few kinds of global or blow-up results.
In  , Escher and Yin studied b-equation on the line for and . They established the local well-posedness, described the precise blow-up scenario, and proved that the equation has strong solutions which exist globally in time and blow up in finite time. Moreover, the authors showed the uniqueness and existence of global weak solutions to b-equation, provided the initial data satisfy certain sign conditions. The similar discussions for b-equation on the circle can be found in  . The author expanded the result of corresponding solutions blow-up in finite time where conditions on the initial data and the bifurcation parameter in  to the case  . In  , the authors established the local well-posedness for the nonuniform weakly dissipative b-equation which includes both the weakly dissipative CH equation and the weakly dissipative DP equation as its special cases. They studied the blow-up phenomena and the long time behavior of the solutions.
Recently Gui, Liu, and Tian  considered b-Equation (1.1) with on the real line. They proved that the equation is locally well-posed in the Sobolev space for . Moreover, they give the precise blow-up scenario of strong solution of the equation with certain initial data. In  , Zhou established blow-up results for b-equation with under various classes of initial data. He also proved that the solutions with compact support initial data do not have compact support. In the periodic case of b-equation with , sufficient conditions on the initial data were obtained in  to guarantee the finite time blow-up and global existence. The local well-posedness of b-equation with in the critical Besov space was studied in  . They showed that if a weaker -topology is used, the solution map becomes Hölder continuous. Moreover, they showed that the dependence on initial data is optimal in in the sense that the solution map is continuous but not uniformly continuous. They also obtained the periodic peaked solutions and applied them to obtain the ill-posedness in . There are some other papers concerned with b-equation of and we will not attempt to mention all here.
In the past decades, the optimal control of distributed parameter systems has become much more active in academic field. Especially, the optimal control of nonlinear solitary wave equation lies in the front of the intersection of mathematics, engineering and computer science and so on. Recently, people have taken a considerable interest in realizing the operation mechanism of prototype tsunami in the laboratory and in looking for a really efficient control mechanism to generate exact long water waves in the man-made pool. The CH equation attracted much more attention also in the context of the relevance of integrable equations to the modelling of tsunami waves    . Naturally, an optimization problem needs to be considered in this shallow water wave equation. It seems to the author that the study of nonlinear shallow water equation from the point of view of control theory was an open field. There are only some research results reported. For instance, Zhang studied the control problems for two nonlinear dispersive wave equations--the KdV equation and the Benjamin-Bona- Mahony (BBM) equation. Moreover, for the BBM equation, he showed that the wave-maker, by choosing a proper boundary value, can make a wave to approach a given state as closely as desired as long as the given state is small in some sense  . Glass investigated the problem of exact controllability and asymptotic stabilization of the CH equation on the circle, by means of a distributed control. The results are global, and in particular the control prevents the solution from blowing up  . The distributed optimal control problems for the viscous CH equation, the viscous DP equation, the viscous Dullin-Gottwald- Holm (DGH) equation were considered by our research team respectively. We proved the existence and uniqueness of weak solution in short interval. Further, we employed the quadratic cost objective functional to be minimized within an admissible control set with the distributive observation and discussed the existence of optimal control which minimizes the quadratic cost functional    . Subsequently, by the Dubovitskii-Milyutin functional analytical approach, Sun considered the optimal distributed control problem of the viscous generalized CH equation and viscous DGH equation respectively and obtained the Pontryagin maximum principle of the systems studied. The necessary optimality condition is established for an optimal control problem in fixed final horizon case   . In   , recently, our research team studied optimal distributed control of the Fornberg-Whitham equation and the θ-equation which involve complex nonlinear items respectively. We clarified the well-posedness of weak solution without relying on viscous coefficient, which is major improvement in comparison with our previous results. Utilizing the Dubovitskii- Milyutin functional analytical approach, we also proved the necessary optimality condition for the control systems in fixed final horizon case. Hwang studied the quadratic cost optimal control problems for the viscous DGH equation. He derived the necessary optimality conditions of optimal controls, corresponding to physically meaningful distributive observations. By making use of the second order Gateaux differentiability of solution mapping on control variables, he also proved the local uniqueness of optimal control  .
Inspired by the papers mentioned above, in present work, we investigate the b-equation from the point of view of distributed control. More precisely, we consider the following governing equation
where is the external control term which is L-periodic in spatial x, is a control and B be an operator called a controller. The explicit formulation of the control problem will be provided after the investigation of well-posedness of the state equation.
We mainly consider the two following problems:
・ for the nonlinear control system governed by the b-equation with quadratic cost functional , can one find
such that and whether this is unique?
・ if one finds the unique optimal control for the above control problem, how can we characterize this optimal control?
The plan of the remaining sections can be summarized as follows. In Section 2, we study the initial-boundary problem of the b-equation with forcing function in a special space . Adopting the Faedo-Galerkin method and utilizing a uniformly prior estimate of the approximate solution, we prove the existence and uniqueness of weak solution under the definition introduced in the paper. For general , the proof without relying on viscous coefficient is a major improvement in comparison with our results in    and other discussions in    . In Section 3, based on the well-posedness result, we give the formulation of the quadratic cost optimal control problem for the b-equation and investigate the existence and uniqueness of the optimal solution. In Section 4, by the method of control theory (for more detailed discussion, we refer readers to book  ), we establish the sufficient and necessary optimality condition of an optimal control in fixed final horizon case. In order to obtain this result, we also prove the Gateaux differentiability of the state variable which is used to define the associate adjoint systems. Comparing with the research in our previous works    and the related works      , the sufficient and necessary optimality condition of an optimal control which is not limited to the necessary condition is another novelty in this paper. At last, in Section 5, we give the specific sufficient and necessary optimality condition of optimal control for two physical meaningful distributed observation cases employing the associate adjoint systems.
2. The Existence and Uniqueness of Weak Solution
Without loss of generality, we assume . Denote the usual Hilbert
space equipped with the norm , and the inner
product in H is denoted by . Let , be the integral exponent Sobolev spaces. By using the Poincare’s inequality in
, we can define norm , where
and . Especially, taking , we get the Hilbert space supplied with the inner product , where . Let us denote that and are the dual spaces of V and H respectively. Then we can find that V embeds into H and embeds into , where each embedding is dense and corresponding injections are continuous.
For convenience, we shall consider the following initial-boundary value problem for Equation (1.1)
where is forcing item which is L-periodic in spatial x.
With and , Equation (2.1) takes the form:
In order to study the weak solution of Equation (2.2), we introduce the following two special spaces firstly.
is defined by , which is equipped with the norm .
is defined by endowed with the norm .
It is easy to verify that the spaces and are both Hilbert spaces.
Definition 2.1. A function is said to be a weak solution of Equation (2.2), if satisfies
for in the sense of .
From now on, when we speak of a solution of Equation (2.2), we shall always mean the weak solution in the sense of Definition 2.1 unless noted otherwise.
We set an unbounded linear self-adjoint operator , where Math_106#. Then the set of all linearly independent
eigenvectors of A with the eigenvalues , i.e., ,
is an orthonormal basis of H.
Furthermore, we can define the powers of A for , where the space is a Hilbert space which is endowed with the norm . It can be found that the following expression holds
where are eigenvectors of and are eigenvalues.
Definition 2.2. A function is called
an approximate solution to Equation (2.2), if it satisfies
where , and
Lemma 2.1. Let and satisfies the boundary conditions of Equation (2.1). Then, we get
where is a constant.
The proof of Lemma 2.1 can be referred to our article   .
Theorem 2.1. Assume that and . Then, Equation (2.2) exhibits a unique weak solution .
Proof: Multiplying both sides of the first equation in Equation (2.4) by and summing up over j from 1 to m, we have
Because is a forcing function, we can assume that , where is constant.
It then derives from Equation (2.5) that
where are embedding constants. In order to estimate the term , we should estimate the term in .
Multiplying both sides of the first equation in Equation (2.4) by and summing up over j from 1 to m, we get
The above equation implies that
By the use of the Sobolev embedding theorem, we can estimate the following items as
where are embedding constants.
Therefore, we can deduce from Equation (2.7) that
From inequality (2.8), we can obtain that
where , and is a constant.
Therefore, combining the boundedness of the sequence in with the inequality (2.6), we can derive that
where , and is some positive constant.
Similarly, multiplying both sides of the first equation in Equation (2.4) by and summing up over j from 1 to m, we can get
By integration by parts in the above equation, we can deduce that
Using the Sobolev embedding theorem, inequality (2.9) and boundary conditions of Equation (2.4), we can estimate the following each item
Combining above estimates, Equation (2.11) can be deduced into the following inequality
From inequality (2.12), we can obtain that
where , and is a constant.
Hence, combining estimate inequality (2.9) and (2.13), we can find that
which indicate . We also can have from the fact of V embeds into H.
Combining estimate inequality (2.9) and (2.10), we also can know that
Therefore, we deduce from inequality (2.14) that
which indicates is uniformly bounded in .
Afterward, we will prove uniform boundedness of sequence in . Indeed, from the first equation of Equation (2.2) and the Sobolev embedding theorem, we have
where are embedding constants as before.
It derives from inequality (2.17) that
Collecting the analysis above, one has:
(I) For , where ,
the sequence is bounded in as well as in , which is independent of the dimension of ansatz space .
(II) For , where
the sequence is bounded in , which is also independent of the dimension of ansatz space .
So, we obtain the boundedness of in from (I) and (II) mentioned above. By the extraction theorem of Rellich’s, there may extract a subsequence of and find a such that
in , as . (2.18)
Utilizing the fact that V embeds H compactly and (2.18), we can refer to the conclusion of Aubin-Lions-Teman’s compact embedding theorem to verify that is pre-compact in . Hence we can choose a subsequence (denoted again by ) of such that
in , as . (2.19)
Because embeds into , we can obtain that Math_214#. Then, by virtue of (2.19), we can find a subsequence (denoted again by ) of such that
in , as , for a.e.. (2.20)
Combining (2.18)-(2.20) and the Lebesgue dominated convergence theorem, we have
in , as ; (2.21)
in , as ; (2.22)
in , as . (2.23)
We replace and by and respectively in the first equation of Equation (2.4), which yields
Multiplying both sides of Equation (2.24) by , where , and integrating the result equation over , we have
Utilizing (2.19), (2.21)-(2.23), we may pass to the limit in Equation (2.25). Then, we get
We can find Equation (2.26) is true for any . Therefore, we may take, then Equation (2.26) gives
in the sense of .
Since j is arbitrary and finite linear combinations of is dense in H, we can find that satisfies Definition 2.1. Hence, from complex analysis above and Lemma 2.1, we obtain the existence of weak solution Math_248# to Equation (2.2).
Next we will discuss the uniqueness of this weak solution.
Let and be any two weak solutions of Equation (2.1) and set . Then satisfies
Taking the inner product of both sides of the first equation in Equation (2.27) with , we obtain
The right hand side of Equation (2.28) can be estimated as follows:
where is an embedding constant and are some con- stants.
Combining all complex estimates above and Equation (2.28), we can deduce that
Integrating inequality (2.29) with respect to t over , we have
where . It follows from that , which implies .
This completes the proof of uniqueness.
3. The Existence and Uniqueness of an Optimal Control
In this section, we will give the formulation of the quadratic cost optimal control problem for b-equation and investigate the existence and uniqueness of an optimal solution.
Let be a Hilbert space of control variables, and be an operator called a controller. We assume that the admissible set be a bounded closed convex set, which has the non-empty interior with respect to topology, i.e. .
We study the following nonlinear control system:
where is a control. By virtue of Theorem 2.1 and Equation (3.1), we can uniquely define the solution mapping of into . The weak solution is called the state variable of the nonlinear control system (3.1).
The observation of the state is assumed to be given by
where is an operator called the observer and is a Hilbert space of the observation variables.
We shall consider the following quadratic cost functional associated with the nonlinear control system (3.1):
where is a desired value of . is symmetric and positive definite, i.e., , where is some constant.
Hence, the discussed optimal control problem is to find an element such that
which subject to the controlled system (3.1) together with the control constraints.
Now, we shall discuss the existence and uniqueness of an optimal control for the cost functional (3.3), which is the content of the following theorem.
Theorem 3.1. Let us suppose that the hypotheses of Theorem 2.1 are satisfied. Then there exists a unique optimal control for the nonlinear control
system (3.1) with the cost functional (3.3), such that .
Proof. Because is a closed convex set, there exists a minimizing sequence in such that
Then cost functional (3.3) can be rewritten as
where is a continuous symmetric bilinear form on and is a continuous linear form on .
Obviously, is bounded in . So, the quadratic cost functional (3.3) implies that there exists a constant such that
which indicates that is bounded in . Because is closed and convex set, we can extract a subsequence and find a such that
in , as . (3.6)
From now on, each state variable corresponding to is the solution of
From inequality (3.5), the right hand side of the first equation in Equation (3.7) can be estimated as
where is some constant.
Utilizing inequality (3.8), we can apply the same method used in Theorem 2.1 to deduce that is bounded in . Hence, by the extraction theorem of Rellich’s, we can extract a subsequence of and find a such that
in , as . (3.9)
Using the fact that V embeds H compactly and the result of (3.9), we can refer to the conclusion of Aubin-Lions-Teman’s compact embedding theorem to verify that is pre-compact in . So we can also choose a subsequence (denoted again by ) of such that
, in as . (3.10)
On the other hand, because embeds into , we can infer that . And from (3.10), we can get a subsequence (denoted again by ) of such that
in , as , for a.e.. (3.11)
Combining (3.9)-(3.11) and the Lebesgue dominated convergence theorem, it is not difficult to obtain that
in , as ; (3.12)
in , as ; (3.13)
in , as ; (3.14)
in , as . (3.15)
We replace and by and in Equation (3.7) respectively, and take . Then, by the standard arguments as in  , we find that the limit u satisfies the following equations:
in weak sense, where . Moreover, by the uniqueness of weak solution of Equation (3.16) via Theorem 2.1 and Lemma 2.1, we can conclude that Math_371#, which implies in .
Because the mapping is lower semi-continuous in the weak topology of and is also lower semi-continuous. The mapping is continuous in the weak topology of . Thus the mapping is weakly lower semi-continuous.
So, we can deduce from cost functional (3.4) that
At the same time, from inequality (3.17), we have
Moreover, combining by definition, we can obtain that
Next, we will prove the uniqueness of in (3.18).
Because the mapping is strictly convex and the mapping is continuous. Hence the mapping is also strictly convex.
Let and be two optimal controls, which satisfy
and respectively. Because is a bounded closed convex set, we can get that . We thus can deduce that , which is a contradiction unless . This completes the proof.
From the above analysis, we can conclude that of is a unique optimal solution to the optimal control problem investigated.
4. The Sufficient and Necessary Optimality Condition
In this section, we shall characterize the optimal control by giving the sufficient and necessary condition for optimality. We firstly give the following lemma according to optimal control theory.
Lemma 4.1. Assume that the mapping is differentiable, strictly convex and is bounded. Then the unique element (optimal control) in
satisfying can be characterized by
where and denote the derivative of at .
Proof. Let be the optimal control subject to Theorem 3.1. Then for and , we have
From inequality (4.2), we can derive that
Therefore, if we pass to the limit in inequality (4.3), we obtain that
, where .
Alternatively, suppose inequality (4.1) remains true. Because the mapping is strictly convex, we can get
, for . (4.4)
From inequality (4.4), we deduce that
If we pass the limit in inequality (4.5), we can get
for , which completes the proof.
Conditions of the type (4.1) are usually termed as “first order sufficient and necessary condition”, in terminology of calculus of variations. In order to analyze inequality (4.1), we need to prove that the mapping of is differentiable at .
Definition 4.1. The solution mapping of into is said to be differentiable at in any direction w, if for and , there exists a such that
in , as .
The function is called the directional derivative of , which plays crucial in the following discussion.
Theorem 4.1. The mapping of into is derivative at and such the derivative of at in the direction , say , is a weak solution of the following equation:
where and .
Proof. Let . We set and . Then satisfies
In order to estimate , we multiply both sides of the first equation in Equation (4.7) by and integrate it over . Then we get
Each item on the right hand of Equation (4.8) can be estimated as follows:
where is an embedding constant and are some constants.
Hence, Equation (4.8) can be changed into
It follows from inequality (4.9) and the Gronwall’s lemma that
Next, multiplying both sides of the first equation in Equation (4.7) by and integrating it over , which gives
Then, we estimate the each item of the right hand of Equation (4.11) as follows:
where is an embedding constant and are some constants.
By the above estimates, we can deduce from Equation (4.11) that
Applying Gronwall’s lemma to inequality (4.12), which yields
Similarly, multiplying both sides of the first equation in Equation (4.7) by and integrating it over , which gives
We can also estimate each item of the right hand of Equation (4.14) as follows:
where are some constants and are some embedding constants.
Combining a series of complex estimates above and Equation (4.14), we can obtain that
By applying the Gronwall’s lemma to inequality (4.15), we can get
Combining estimate inequality (4.13) and (4.16), we can deduce that
Similarly, combining estimate inequality (4.10) and (4.13), we can obtain that
From inequality (4.17), we derive that
which indicates a uniformly bounded of .
Afterward, we will prove a uniformly bounded of .
From the first equation in Equation (4.7) and the Sobolev embedding theorem, we have
where are some embedding constants and are some constants.
Analogously, from inequality (4.20), we can get
Combining inequality (4.19) and (4.21), we can establish the boundedness of in . Hence, from Lemma 2.1, we can deduce that
From now on, we can infer that there exists a and a sequence tending to 0 such that
in , as . (4.22)
Because the imbedding into is compact, then it can deduce from (4.22) that
in a.e. , (4.23)
for some tending to 0 as . Whence by (4.22) - (4.23), Theorem 2.1 and the Lebesgue dominated convergence theorem, we can easily obtain that
in ; (4.24)
in ; (4.25)
in ; (4.26)
in ; (4.27)
as , where . And also we can derive from Equation (4.7) and inequality (4.21) that
in , as . (4.28)
Therefore, we can infer from (4.24) to (4.28) that
in as in which g is a solution of Equation (4.6).
Consequently, the solution mapping of into is differentiable in the weak topology of . This completes the proof.
The conclusion of Theorem 4.1 means that the cost is derivative at in the direction . So, we can get that
Then the sufficient and necessary optimality condition (4.1) can be rewritten as
for, where is the canonical isomorphism onto and is desired value.
5. The Two Cases of Distributive Observations
In this section, we will characterize the optimal control by giving the sufficient and necessary optimality condition (4.29) for the following two cases of physical meaningful observations:
(I) We set and , then observe that .
(II) We set and , then observe that .
Firstly, we discuss the cost functional expressed by
where is a desired value. Let be the optimal control subject to Equation (3.1) and cost functional (5.1). Then the sufficient and necessary optimality condition (4.29) can be represented by
where is the weak solution of Equation (4.6). Now we will introduce the adjoint system to describe the optimality condition (5.2):
Therefore, we can provide the characterization for the optimal control of the quadratic cost functional (5.1) as follows:
Theorem 5.1. The optimal control of the quadratic cost functional (5.1) is characterized by the following control system, adjoint system and inequality:
Proof. Taking inner product of the first equation in Equation (5.3) by g over , then integrating the result equation with respect to t on , we get
Combining Equation (4.6) and Equation (5.3) and taking integration by parts, the left hand side of Equation (5.4) yields
where . Therefore, utilizing Equation (5.4) and Equation (5.5), the sufficient and necessary optimality condition (5.2) is equivalent to
Hence, the theorem is proved.
Secondly, we discuss the cost functional expressed by
where is a desired value. Let be the optimal control subject to Equation (3.1) and cost functional (5.6). Then the sufficient and necessary optimality condition (4.29) is represented by
where and is the weak solution of Equation (4.6). Similarly, we formulate the adjoint system to describe the optimality condition (5.7):
Hence, we can give the following theorem.
Theorem 5.2. The optimal control of the quadratic cost functional (5.7) is characterized by the following control system, adjoint system and inequality:
Proof. As we did before, we multiply both sides of the first equation of Equation (5.8) by g and integrate it over . Then we have
Utilizing Equation (4.6), the integration by parts on the left hand side of Equation (5.9) yields
where . Therefore, combining Equation (5.9) and Equation (5.10), the sufficient and necessary optimality condition (5.7) is equivalent to
which completes the proof.
b-equation is an important shallow water wave equation which has many practical meanings. In this paper, we aim at pursuing an in-depth study of the optimal control issue of the classical b-equation. So, we investigate firstly the local existence and uniqueness of solution to the initial-boundary problem of the b-equation with source term, and then discuss the formulation of the quadratic cost optimal control problem for the b-equation, obtain the existence and uniqueness of an optimal control, establish the sufficient and necessary optimality condition of an optimal control in fixed final horizon case. Moreover, we give the specific sufficient and necessary optimality condition for two physical meaningful distributive observation cases by employing associate adjoint systems. Compared with other papers in similar directions, the weak solution analysis of b-equation without relying on viscous item is one technical innovation, and the sufficient and necessary optimality condition of an optimal control which is not limited to the necessary condition is another novelty. However, much work remains to be done in this direction. For example, it is an optimal control problem of the distributed parameter system governed by the nonlinear partial differential equation, to obtain the numerical solutions for the optimal control-trajectory pair is not an easy job due to the tremendous calculation and possible model difficulties. We try to finish this non-trivial work in the follow-up research by optimizing numerical algorithm and carrying out numerical simulation, which can provide a basis for application in the engineering field.
Research was supported in part by the National Natural Science Foundation of China (No: 11371175, 11501253, 11571140).
 Escher, J. and Yin, Z. (2008) Well-Posedness, Blow-Up Phenomena, and Global Solutions for the b-Equation. Journal für die reine und angewandte Mathematik, 624, 51-80.
 Dullin, H.R., Gottwald, G.A. and Holm, D.D. (2003) Camassa-Holm, Korteweg-de Vries-5 and Other Asymptotically Equivalent Equations for Shallow Water Waves. Fluid Dynamics Research, 33, 73-95.
 Holm, D.D. and Staley, M.F. (2003) Wave Structure and Nonlinear Balances in a Family of Evolutionary PDEs. SIAM Journal on Applied Dynamical Systems, 2, 323-380.
 Holm, D.D. and Staley, M.F. (2003) Nonlinear Balance and Exchange of Stability in Dynamics of Solitons, Peakons, Ramps/Cliffs and Leftons in a 1-1 Nonlinear Evolutionary PDE. Physics Letters A, 308, 437-444.
 Degasperis, A., Holm, D.D. and Hone, A.N.W. (2003) Integrable and Non-Integrable Equations with Peakons. In: Ablowitz, M.J., Boiti, M., Pempinelli, F. and Prinari, B., Eds., Nonlinear Physics: Theory and Experiment II, World Scientific Publication, River Edge, NJ, 37-43.
 Dullin, H.R., Gottwald, G.A. and Holm, D.D. (2001) An Integrable Shallow Water Equation with Linear and Nonlinear Dispersion. Physical Review Letters, 87, 4501-4504.
 Mckean, H.P. (1979) Integrable Systems and Algebraic Curves. In: Grmela, M. and Marsden, J.E., Eds., Global Analysis, Lecture Notes in Mathematics Vol. 755, Springer, Berlin, 83-200.
 Bona, J.L. and Scott, R. (1976) Solutions of the Korteweg-de Vries Equation in Fractional Order Sobolev Spaces. Duke Mathematical Journal, 43, 87-99.
 Kenig, C., Ponce, G. and Vega, L. (1993) Well-Posedness and Scattering Results for the Generalized Korteweg-de Vries Equation via the Contraction Principle. Communications on Pure and Applied Mathematics, 46, 527-620.
 Tao, T. (2002) Low-Regularity Global Solutions to Nonlinear Dispersive Equations. In: Hassell, A., Ed., Surveys in Analysis and Operator Theory, Centre for Mathematics and Its Applications, Mathematical Sciences Institute, The Australian National University, Canberra, 19-48.
 Fuchssteiner, B. and Fokas, A.S. (1981) Symplectic Structures, Their Bäcklund Transformation and Hereditary Symmetries. Journal of Physics D, 4, 47-66.
 Constantin, A. and Lannes, D. (2009) The Hydrodynamical Relevance of the Camassa-Holm and Degasperis-Procesi Equations. Archive for Rational Mechanics and Analysis, 192, 165-186.
 Constantin, A. and McKean, H.P. (1999) A Shallow Water Equation on the Circle. Communications on Pure and Applied Mathematics, 52, 949-982.
 Constantin, A. and Strauss, W.A. (2000) Stability of Peakons. Communications on Pure and Applied Mathematics, 53, 603-610.
 Constantin, A. and Escher, J. (1998) Global Existence and Blow-Up for a Shallow Water Equation. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, 26, 303-328.
 Danchin, R. (2001) A Few Remarks on the Camassa-Holm Equation. Differential and Integral Equations, 14, 953-988.
 Li, Y.A. and Olver, P.J. (2000) Well-Posedness and Blow-Up Solutions for an Integrable Nonlinearly Dispersive Model Wave Equation. Journal of Differential Equations, 162, 27-63.
 Constantin, A. and Escher, J. (1998) Well-Posedness, Global Existence, and Blowup Phenomena for a Periodic Quasilinear Hyperbolic Equation. Communications on Pure and Applied Mathematics, 51, 475-504.
 Bressan, A. and Constantin, A. (2007) Global Conservative Solutions of the Camassa-Holm Equation. Archive for Rational Mechanics and Analysis, 183, 215-239.
 Constantin, A. (2000) Existence of Permanent and Breaking Waves for a Shallow Water Equation: A Geometric Approach. Annales de l'Institut Fourier (Grenoble), 50, 321-362.
 Xin, Z. and Zhang, P. (2000) On the Weak Solutions to a Shallow Water Equation. Communications on Pure and Applied Mathematics, 53, 1411-1433.
 Holden, H. and Raynaud, X. (2007) Global Conservative Multipeakon Solutions of the Camassa-Holm Equation. Journal of Hyperbolic Differential Equations, 4, 39-63. https://doi.org/10.1142/S0219891607001045
 Holden, H. and Raynaud, X. (2007) Global Conservative Solutions of the Camassa-Holm Equation—A Lagrangian Point of View. Communications in Partial Differential Equations, 32, 1511-1549.
 Holden, H. and Raynaud, X. (2008) Global Dissipative Multipeakon Solutions of the Camassa-Holm Equation. Communications in Partial Differential Equations, 33, 2040-2063.
 Himonas, A., Misiolek, G., Ponce, G. and Zhou, Y. (2007) Persistence Properties and Unique Continuation of Solutions of the Camassa-Holm Equation. Communications in Mathematical Physics, 271, 511-522.
 Yin, Z. (2003) On the Cauchy Problem for an Integrable Equation with Peakon Solutions. Illinois Journal of Mathematics, 47, 649-666.
 Escher, J., Liu, Y. and Yin, Z. (2006) Global Weak Solutions and Blow-Up Structure for the Degasperis-Procesi Equation. Journal of Functional Analysis, 241, 457-485.
 Liu, Y. and Yin, Z. (2006) Global Existence and Blow-Up Phenomena for the Degasperis-Procesi Equation. Communications in Mathematical Physics, 267, 801-820.
 Escher, J., Liu, Y. and Yin, Z. (2007) Shock Waves and Blow-Up Phenomena for the Periodic Degasperis-Procesi Equation. Indiana University Mathematics Journal, 56, 87-117.
 Escher, J. (2007) Wave Breaking and Shock Waves for a Periodic Shallow Water Equation. Philosophical Transactions of the Royal Society Series A, 365, 2281-2289.
 Kodzha, M. (2012) On the Blow-Up of Solutions for the b-Equation. Acta Mathematica Universitatis Comenianae, 1, 117-126.
 Niu, W.S. and Zhang, S.H. (2011) Blow-Up Phenomena and Global Existence for the Non-Uniform Weakly Dissipative b-Equation. Journal of Mathematical Analysis and Applications, 374, 166-177.
 Gui, G., Liu, Y. and Tian, L. (2008) Global Existence and Blow-Up Phenomena for the Peakon b-Family of Equations. Indiana University Mathematics Journal, 57, 1209-1234.
 Christov, O. and Hakkaev, S. (2009) On the Cauchy Problem for the Periodic b-Family of Equations and of the Non-Uniform Continuity of Degasperis-Procesi Equation. Journal of Mathematical Analysis and Applications, 360, 47-56.
 Tang, H., Shi, S. and Liu, Z. (2015) The Dependences on Initial Data for the b-Family Equation in Critical Besov Space. Monatshefte für Mathematik, 177, 471-492. https://doi.org/10.1007/s00605-014-0673-8
 Lakshmanan, M. (2007) Integrable Nonlinear Wave Equations and Possible Connections to Tsunami Dynamics. In: Kundu, A., Ed., Tsunami and Nonlinear Waves, Springer-Verlag, Berlin Heidelberg, 31-49.
 Constantin, A. and Johnson, R.S. (2008) Propagation of Very Long Water Waves with Vorticity, over Variable Depth, with Applications to Tsunamis. Fluid Dynamics Research, 40, 175-211.
 Tian, L., Shen, C. and Ding, D. (2009) Optimal Control of the Viscous Camassa-Holm Equation. Nonlinear Analysis: Real World Applications, 10, 519-530.
 Shen, C., Tian, L. and Gao, A. (2010) Optimal Control of the Viscous Dullin-Gottwald-Holm Equation. Nonlinear Analysis: Real World Applications, 11, 480-491.
 Sun, B. (2013) An Optimal Distributed Control Problem of the Viscous Generalized Camassa-Holm Equation. Transactions of the Institute of Measurement and Control, 35, 409-416.
 Sun, B. (2012) Maximum Principle for Optimal Distributed Control of the Viscous Dullin-Gottwald-Holm Equation. Nonlinear Analysis: Real World Applications, 13, 325-332.