_{1}

This paper is concerned with the optimal distributed control problem governed by b-equation. We firstly investigate the existence and uniqueness of weak solution for the controlled system with appropriate initial value and boundary condition. By contrasting with our previous result, the proof without considering viscous coefficient is a big improvement. Secondly, based on the well-posedness result, we find a unique optimal control for the controlled system with the quadratic cost functional. Moreover, by means of the optimal control theory, we obtain the sufficient and necessary optimality condition of an optimal control, which is another major novelty of this paper. Finally, we also present the optimality conditions corresponding to two physical meaningful distributive observation cases.

Recently, Escher and Yin [

{ u t − α 2 u x x t + c 0 u x + ( b + 1 ) u u x + Γ u x x x = α 2 ( b u x u x x + u u x x x ) , t > 0 , x ∈ R , u ( 0 , x ) = u 0 ( x ) , x ∈ R , (1.1)

where c 0 , b , Γ and α are arbitrary real constants. Denoting y = u − α 2 u x x , we can rewrite b-equation in the following form:

{ y t + c 0 u x + u y x + b u x y + Γ u x x x = 0 , t > 0 , x ∈ R , u ( 0 , x ) = u 0 ( x ) , x ∈ R . (1.2)

Equation (1.2) can be derived as a family of asymptotically equivalent shallow water wave equations that emerge at quadratic order accuracy for ∀ b ≠ − 1 by an appropriate Kodama transformation [

If α = 0 and b = 2 , then b-Equation (1.1) becomes the well-known KdV equation

u t + c 0 u x + 3 u u x + Γ u x x x = 0 , t > 0 , x ∈ R , (1.3)

which describes the unidirectional propagation of waves at the free surface of shallow water under the influence of gravity [

For Γ = 0 and b = 2 , b-Equation (1.1) becomes the CH equation

u t − α 2 u x x t + c 0 u x + 3 u u x = 2 α 2 u x u x x + α 2 u u x x x , t > 0 , x ∈ R , (1.4)

modelling the unidirectional propagation of shallow water waves over a flat bottom. Again u ( t , x ) stands for the fluid velocity at time t in the spatial x direction and c 0 is a nonnegative parameter related to the critical shallow water speed [

Since the CH equation is structurally very rich, many physicists and mathematicians pay great attention to it. Local well-posedness for the initial datum u 0 ∈ H s ( I ) with s > 3 / 2 was proved by several authors, as in [

For c 0 = Γ = 0 and b = 3 in b-Equation (1.1), then we find the DP equation of the form [

u t − α 2 u x x t + 4 u u x = 3 α 2 u x u x x + α 2 u u x x x , t > 0 , x ∈ R . (1.5)

Degasperis, Holm and Hone [

u ( t , x ) = − 1 t + k sgn ( x ) e − | x | , k > 0.

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 [

The Cauchy problem for the DP equation has been studied widely. Local well-posedness of this equation is established in [

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 [

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 [

Recently Gui, Liu, and Tian [

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 [

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

{ u t − u x x t + c 0 u x + ( b + 1 ) u u x + Γ u x x x − b u x u x x − u u x x x = B v , u ( t , x + L ) = u ( t , x ) , ∀ x ∈ R , ∀ t ∈ [ 0 , T ] , y ( 0 , x ) = y 0 ( x ) = u ( 0 , x ) − u ( 0 , x ) ∈ V , (1.6)

where B v is the external control term which is L-periodic in spatial x,

We mainly consider the two following problems:

・ for the nonlinear control system governed by the b-equation with quadratic cost functional I ( v ) = ‖ C u ( v ; t , x ) − z d ‖ M 2 + ( N v , v ) U , can one find v ∗ ∈ U a d

such that I ( v ∗ ) = inf ∀ v ∈ U a d I ( v ) and whether this v ∗ is unique?

・ if one finds the unique optimal control v ∗ ∈ U a d 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 S ( 0 , T ) . 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 b ∈ R , the proof without relying on viscous coefficient is a major improvement in comparison with our results in [

Without loss of generality, we assume Ω = [ 0 , L ] . Denote the usual Hilbert

space H = L 2 ( Ω ) equipped with the norm ‖ u ‖ H = ( ∫ Ω | u | 2 d x ) 1 2 , and the inner

product in H is denoted by ( u , u ) H = ‖ u ‖ H 2 . Let H s ( Ω ) = W s , 2 ( Ω ) , s ∈ N be the integral exponent Sobolev spaces. By using the Poincare’s inequality in

H s ( Ω ) , we can define norm ‖ ξ ‖ H s ( Ω ) = ( ∑ 0 ≤ | α | ≤ s ‖ ∂ x α ξ ‖ H 2 ) 1 2 ≅ ‖ ∂ x s ξ ‖ H , where

∂ x s ξ ( 0 ) = ∂ x s ξ ( L ) and s = 0 , 1 , 2 , 3 , ⋅ ⋅ ⋅ . Especially, taking m = 1 , we get the Hilbert space V = H 1 ( Ω ) supplied with the inner product ( φ , ψ ) V = ( φ x , ψ x ) H , where ∀ φ , ψ ∈ V . Let us denote that V ∗ = H − 1 ( Ω ) and H ∗ = L 2 ( Ω ) are the dual spaces of V and H respectively. Then we can find that V embeds into H and H ∗ embeds into V ∗ , 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)

{ u t − u x x t + c 0 u x + ( b + 1 ) u u x + Γ u x x x − b u x u x x − u u x x x = f ( t , x ) , u ( t , x + L ) = u ( t , x ) , ∀ x ∈ R , ∀ t ∈ [ 0 , T ] , u ( 0 , x ) = u 0 ( x ) , ∀ x ∈ R , (2.1)

where f ( t , x ) is forcing item which is L-periodic in spatial x.

With y ( t , x ) = u ( t , x ) − u x x ( t , x ) and y 0 ( x ) = u ( 0 , x ) − u x x ( 0 , x ) , Equation (2.1) takes the form:

{ y t ( t , x ) + c 0 u x ( t , x ) + u ( t , x ) y x ( t , x ) + b u x ( t , x ) y ( t , x ) + Γ u x x x ( t , x ) = f ( t , x ) , u ( t , x + L ) = u ( t , x ) , ∀ x ∈ R , ∀ t ∈ [ 0 , T ] , y ( 0 , x ) = y 0 ( x ) , ∀ x ∈ R . (2.2)

In order to study the weak solution of Equation (2.2), we introduce the following two special spaces firstly.

W ( 0 , T ) is defined by W ( 0 , T ) = { ξ | ξ ∈ L 2 ( 0 , T ; V ) , ξ t ∈ L 2 ( 0 , T ; V ∗ ) } , which is equipped with the norm ‖ ξ ‖ W ( 0 , T ) = ( ‖ ξ ‖ L 2 ( 0 , T ; V ) 2 + ‖ ξ t ‖ L 2 ( 0 , T ; V ∗ ) 2 ) 1 2 .

S ( 0 , T ) is defined by S ( 0 , T ) = { ξ | ξ ∈ L 2 ( 0 , T ; H 3 ( Ω ) ) , ξ t ∈ L 2 ( 0 , T ; V ) } endowed with the norm ‖ ξ ‖ S ( 0 , T ) = ( ‖ ξ ‖ L 2 ( 0 , T ; H 3 ( Ω ) ) 2 + ‖ ξ t ‖ L 2 ( 0 , T ; V ) 2 ) 1 2 .

It is easy to verify that the spaces W ( 0 , T ) and S ( 0 , T ) are both Hilbert spaces.

Definition 2.1. A function u ( t , x ) ∈ S ( 0 , T ) is said to be a weak solution of Equation (2.2), if y ( t , x ) = u ( t , x ) − u x x ( t , x ) ∈ W ( 0 , T ) satisfies

{ ( y t ( t , ⋅ ) , φ ( ⋅ ) ) H + ( c 0 u x ( t , ⋅ ) , φ ( ⋅ ) ) H + ( u ( t , ⋅ ) y x ( t , ⋅ ) , φ ( ⋅ ) ) H + ( b u x ( t , ⋅ ) y ( t , ⋅ ) , φ ( ⋅ ) ) H + ( Γ u x x x ( t , ⋅ ) , φ ( ⋅ ) ) H = ( f ( t , ⋅ ) , φ ( ⋅ ) ) H , u ( t , x + L ) = u ( t , x ) , ∀ x ∈ R , ∀ t ∈ [ 0 , T ] , y ( 0 , x ) = y 0 ( x ) ∈ V , (2.3)

for ∀ φ ( ⋅ ) ∈ H in the sense of D ′ ( 0 , T ) .

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 A u = − u x x , where ∀ u ∈ Math_106#. Then the set of all linearly independent

eigenvectors { ω j } j ∈ N + of A with the eigenvalues { λ j ∗ } j ∈ N + , i.e., A ω j = λ j ∗ ω j ,

0 < λ 1 ∗ ≤ λ 2 ∗ ≤ ⋅ ⋅ ⋅ ≤ λ j ∗ → ∞ as j → ∞ ,

is an orthonormal basis of H.

Furthermore, we can define the powers A s of A for s ∈ N + , where the space D ( A s ) is a Hilbert space which is endowed with the norm ‖ A s ⋅ ‖ H . It can be found that the following expression holds

A s ω j = ( − 1 ) s ∂ x 2 s ω j = λ j s ω j ,

where { ω j } j ∈ N + are eigenvectors of A s and { λ j s } j ∈ N + are eigenvalues.

Definition 2.2. A function u m ( t , x ) = ∑ j = 1 m a j m ( t ) ω j ( x ) ∈ C 1 ( [ 0 , T ] ; S m ) is called

an approximate solution to Equation (2.2), if it satisfies

{ ( y m , t ( t , x ) , ω j ) H + ( c 0 u m , x ( t , x ) , ω j ) H + ( u m ( t , x ) y m , x ( t , x ) , ω j ) H + ( b u m , x ( t , x ) y m ( t , x ) , ω j ) H + ( Γ u m , x x x ( t , x ) , ω j ) H = ( f ( t , x ) , ω j ) H , u m ( t , x + L ) = u m ( t , x ) , ∀ x ∈ R , ∀ t ∈ [ 0 , T ] , y m ( 0 , x ) = ∑ j = 1 m χ j m ω j → y 0 ( x ) ∈ V , as m → ∞ , (2.4)

where y m ( t , x ) = u m ( t , x ) − u m , x x ( t , x ) , S m = s p a n { ω 1 ( x ) , ω 2 ( x ) , ⋯ , ω m ( x ) } and

a j m ( t ) ∈ C 1 ( [ 0 , T ] ; R ) .

Lemma 2.1. Let y ( t , x ) = u ( t , x ) − u x x ( t , x ) ∈ W ( 0 , T ) and u ( t , x ) satisfies the boundary conditions of Equation (2.1). Then, we get

‖ u ( t , x ) ‖ S ( 0 , T ) ≤ C ‖ y ( t , x ) ‖ W ( 0 , T ) ,

where C > 0 is a constant.

The proof of Lemma 2.1 can be referred to our article [

Theorem 2.1. Assume that f ( t , x ) ∈ L 2 ( 0 , T ; V ) and y 0 ( x ) ∈ V . Then, Equation (2.2) exhibits a unique weak solution u ( t , x ) ∈ S ( 0 , T ) .

Proof: Multiplying both sides of the first equation in Equation (2.4) by a j m ( t ) and summing up over j from 1 to m, we have

( y m , t , u m ) H + ( c 0 u m , x , u m ) H + ( u m y m , x , u m ) H + ( b u m , x y m , u m ) H + ( Γ u m , x x x , u m ) H = ( f , u m ) H .

This gives

d d t ( ‖ u m ‖ H 2 + ‖ u m ‖ V 2 ) = ( 2 − b ) ∫ Ω u m , x 3 d x + 2 ( f , u m ) H . (2.5)

Because f ( t , x ) ∈ L 2 ( 0 , T ; V ) is a forcing function, we can assume that ‖ f ‖ V ≤ M 1 , where M 1 > 0 is constant.

It then derives from Equation (2.5) that

d d t ( ‖ u m ‖ H 2 + ‖ u m ‖ V 2 ) ≤ | 2 − b | λ 2 ‖ u m ‖ H 2 ( Ω ) ‖ u m ‖ V 2 + λ 1 2 M 1 2 + ‖ u m ‖ H 2 , (2.6)

where λ i > 0 , i = 1 , 2 are embedding constants. In order to estimate the term ‖ u m ‖ H 2 + ‖ u m ‖ V 2 , we should estimate the term { u m } m ∈ N + in H 2 ( Ω ) .

Multiplying both sides of the first equation in Equation (2.4) by λ j ∗ a j m ( t ) and summing up over j from 1 to m, we get

( y m , t , − u m , x x ) H + ( c 0 u m , x , − u m , x x ) H + ( u m y m , x , − u m , x x ) H + ( b u m , x y m , − u m , x x ) H + ( Γ u m , x x x , − u m , x x ) H = ( f , − u m , x x ) H .

The above equation implies that

d d t ( ‖ u m ‖ V 2 + ‖ u m ‖ H 2 ( Ω ) 2 ) + ( b + 1 ) ∫ Ω u m , x 3 d x + ( 2 b − 1 ) ∫ Ω u m , x u m , x x 2 d x = 2 ( f , − u m , x x ) H . (2.7)

By the use of the Sobolev embedding theorem, we can estimate the following items as

− ( b + 1 ) ∫ Ω u m , x 3 d x ≤ | b + 1 | ‖ u m , x ‖ L ∞ ‖ u m ‖ V 2 ≤ | b + 1 | λ 2 ‖ u m ‖ H 2 ( Ω ) ‖ u m ‖ V 2 ;

− ( 2 b − 1 ) ∫ Ω u m , x u m , x x 2 d x ≤ | 2 b − 1 | ‖ u m , x ‖ L ∞ ‖ u m ‖ H 2 ( Ω ) 2 ≤ | 2 b − 1 | λ 2 ‖ u m ‖ H 2 ( Ω ) 3

and

2 ( f , − u m , x x ) H ≤ 2 ‖ f ‖ H ‖ u m ‖ H 2 ( Ω ) ≤ 2 λ 1 M 1 ‖ u m ‖ H 2 ( Ω ) ,

where λ i > 0 , i = 1 , 2 are embedding constants.

Therefore, we can deduce from Equation (2.7) that

d d t ( ‖ u m ‖ V 2 + ‖ u m ‖ H 2 ( Ω ) 2 ) ≤ | b + 1 | λ 2 ‖ u m ‖ H 2 ( Ω ) ‖ u m ‖ V 2 + | 2 b − 1 | λ 2 ‖ u m ‖ H 2 ( Ω ) 3 + 2 λ 1 M 1 ‖ u m ‖ H 2 ( Ω ) ≤ β 1 λ 2 ( ‖ u m ‖ V 2 + ‖ u m ‖ H 2 ( Ω ) 2 + 2 λ 1 M 1 β 1 λ 2 ) 3 2 ,

where

β 1 = max { | b + 1 | , | 2 b − 1 | } . (2.8)

From inequality (2.8), we can obtain that

‖ u m ‖ V 2 + ‖ u m ‖ H 2 ( Ω ) 2 ≤ ‖ u m ( 0 , x ) ‖ V 2 + ‖ u m ( 0 , x ) ‖ H 2 2 + 2 λ 1 M 1 β 1 λ 2 [ 1 − β 1 λ 2 2 t ‖ u m ( 0 , x ) ‖ V 2 + ‖ u m ( 0 , x ) ‖ H 2 2 + 2 λ 1 M 1 β 1 λ 2 ] 2 − 2 λ 1 M 1 β 1 λ 2 ≜ M 2 2 , (2.9)

where ∀ t ∈ [ 0 , T ] , T < 2 β 1 2 λ 2 2 ( ‖ u m ( 0 , x ) ‖ V 2 + ‖ u m ( 0 , x ) ‖ H 2 2 ) + 2 β 1 λ 1 λ 2 M 1 and M 2 > 0 is a constant.

Therefore, combining the boundedness of the sequence { u m } m ∈ N + in H 2 ( Ω ) with the inequality (2.6), we can derive that

‖ u m ‖ H 2 + ‖ u m ‖ V 2 ≤ ( ‖ u m ( 0 , x ) ‖ H 2 + ‖ u m ( 0 , x ) ‖ V 2 + λ 1 2 M 1 2 ) exp ( β 2 t ) − λ 1 2 M 1 2 ≜ M 3 2 , (2.10)

where ∀ t ∈ [ 0 , T ] , β 2 = max { | 2 − b | λ 2 M 2 , 1 } and M 3 is some positive constant.

Similarly, multiplying both sides of the first equation in Equation (2.4) by ( λ j ∗ ) 2 a j m ( t ) and summing up over j from 1 to m, we can get

( y m , t , u m , x x x x ) H + ( c 0 u m , x , u m , x x x x ) H + ( u m y m , x , u m , x x x x ) H + ( b u m , x y m , u m , x x x x ) H + ( Γ u m , x x x , u m , x x x x ) = ( f , u m , x x x x ) H .

By integration by parts in the above equation, we can deduce that

d d t ( ‖ u m ‖ H 2 ( Ω ) 2 + ‖ u m ‖ H 3 ( Ω ) 2 ) + 5 ( b + 1 ) ∫ Ω u m , x u m , x x 2 d x + ( 2 b + 1 ) ∫ Ω u m , x u m , x x x 2 d x = 2 ( f , u m , x x x x ) H . (2.11)

Using the Sobolev embedding theorem, inequality (2.9) and boundary conditions of Equation (2.4), we can estimate the following each item

− 5 ( b + 1 ) ∫ Ω u m , x u m , x x 2 d x ≤ 5 | b + 1 | ‖ u m , x ‖ L ∞ ‖ u m ‖ H 2 ( Ω ) 2 ≤ 5 | b + 1 | λ 2 ‖ u m ‖ H 2 ( Ω ) 3 ≤ 5 | b + 1 | λ 2 M 2 3 ;

− ( 2 b + 1 ) ∫ Ω u m , x u m , x x x 2 d x ≤ | 2 b + 1 | ‖ u m , x ‖ L ∞ ‖ u m ‖ H 3 ( Ω ) 2 ≤ | 2 b + 1 | λ 2 M 2 ‖ u m ‖ H 3 ( Ω ) 2

and

2 ( f , u m , x x x x ) H ≤ 2 | ( f x , − u m , x x x ) H | ≤ 2 ‖ f ‖ V ‖ u m ‖ H 3 ( Ω ) ≤ M 1 2 + ‖ u m ‖ H 3 ( Ω ) 2 .

Combining above estimates, Equation (2.11) can be deduced into the following inequality

d d t ( ‖ u m ‖ H 2 ( Ω ) 2 + ‖ u m ‖ H 3 ( Ω ) 2 ) ≤ ( | 2 b + 1 | λ 2 M 2 + 1 ) ‖ u m ‖ H 3 ( Ω ) 2 + ( 5 | b + 1 | λ 2 M 2 3 + M 1 2 ) ≤ ( | 2 b + 1 | λ 2 M 2 + 1 ) ( ‖ u m ‖ H 2 ( Ω ) 2 + ‖ u m ‖ H 3 ( Ω ) 2 ) + ( 5 | b + 1 | λ 2 M 2 3 + M 1 2 ) . (2.12)

From inequality (2.12), we can obtain that

‖ u m ‖ H 2 ( Ω ) 2 + ‖ u m ‖ H 3 ( Ω ) 2 ≤ [ ( | 2 b + 1 | λ 2 M 2 + 1 ) ( ‖ u m ( 0 , x ) ‖ H 2 2 + ‖ u m ( 0 , x ) ‖ H 3 2 ) + ( 5 | b + 1 | λ 2 M 2 3 + M 1 2 ) ] exp [ ( | 2 b + 1 | λ 2 M 2 + 1 ) t ] | 2 b + 1 | λ 2 M 2 + 1 − 5 | b + 1 | λ 2 M 2 3 + M 1 2 | 2 b + 1 | λ 2 M 2 + 1 ≜ M 4 2 (2.13)

where ∀ t ∈ [ 0 , T ] , T < 2 β 1 2 λ 2 2 ( ‖ u m ( 0 , x ) ‖ V 2 + ‖ u m ( 0 , x ) ‖ H 2 2 ) + 2 β 1 λ 1 λ 2 M 1 and M 4 > 0 is a constant.

Hence, combining estimate inequality (2.9) and (2.13), we can find that

‖ y m ‖ V 2 = ‖ u m , x − u m , x x x ‖ H 2 = ‖ u m ‖ V 2 + 2 ‖ u m ‖ H 2 ( Ω ) 2 + ‖ u m ‖ H 3 ( Ω ) 2 ≤ M 2 2 + M 4 2 , (2.14)

which indicate y m ∈ V . We also can have y m ∈ H from the fact of V embeds into H.

Combining estimate inequality (2.9) and (2.10), we also can know that

‖ y m ‖ H 2 = ‖ u m − u m , x x ‖ H 2 = ‖ u m ‖ H 2 + 2 ‖ u m ‖ V 2 + ‖ u m ‖ H 2 ( Ω ) 2 ≤ M 2 2 + M 3 2 . (2.15)

Therefore, we deduce from inequality (2.14) that

‖ y m ‖ L 2 ( 0 , T ; V ) 2 ≤ ( M 2 2 + M 4 2 ) T , (2.16)

which indicates { y m } m ∈ N + is uniformly bounded in L 2 ( 0 , T ; V ) .

Afterward, we will prove uniform boundedness of sequence { y m , t } m ∈ N + in L 2 ( 0 , T ; V ∗ ) . Indeed, from the first equation of Equation (2.2) and the Sobolev embedding theorem, we have

‖ y m , t ‖ V ∗ ≤ ‖ f ‖ V ∗ + | c 0 | ‖ u m ‖ H + λ 2 ‖ u m ‖ V ‖ y m ‖ H + | b | λ 2 ‖ u m ‖ H ‖ y m ‖ + V | Γ | ‖ u m ‖ H 2 ( Ω ) ≤ λ 3 M 1 + | c 0 | M 3 + λ 2 M 3 M 2 2 + M 3 2 + | b | λ 2 M 2 M 2 2 + M 4 2 + | Γ | M 2 , (2.17)

where λ i > 0 , i = 2 , 3 are embedding constants as before.

It derives from inequality (2.17) that

‖ y m , t ‖ L 2 ( 0 , T ; V ∗ ) 2 ≤ [ λ 3 M 1 + | c 0 | M 3 + λ 2 M 3 M 2 2 + M 3 2 + | b | λ 2 M 2 M 2 2 + M 4 2 + | Γ | M 2 ] 2 T .

Collecting the analysis above, one has:

(I) For ∀ t ∈ [ 0 , T ] , where T < 2 β 1 2 λ 2 2 ( ‖ u m ( 0 , x ) ‖ V 2 + ‖ u m ( 0 , x ) ‖ H 2 2 ) + 2 β 1 λ 1 λ 2 M 1 ,

the sequence { y m } m ∈ N + is bounded in L 2 ( 0 , T ; H ) as well as in L 2 ( 0 , T ; V ) , which is independent of the dimension of ansatz space S m .

(II) For ∀ t ∈ [ 0 , T ] , where

T < 2 β 1 2 λ 2 2 ( ‖ u m ( 0 , x ) ‖ V 2 + ‖ u m ( 0 , x ) ‖ H 2 2 ) + 2 β 1 λ 1 λ 2 M 1 ,

the sequence { y m , t } m ∈ N + is bounded in L 2 ( 0 , T ; V ∗ ) , which is also independent of the dimension of ansatz space S m .

So, we obtain the boundedness of { y m } m ∈ N + in W ( 0 , T ) from (I) and (II) mentioned above. By the extraction theorem of Rellich’s, there may extract a subsequence { y m k } of { y m } m ∈ N + and find a y ∈ W ( 0 , T ) such that

y m k → weakly y in W ( 0 , T ) , as k → ∞ . (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 { y m k } is pre-compact in L 2 ( 0 , T ; H ) . Hence we can choose a subsequence (denoted again by { y m k } ) of { y m k } such that

y m k → strongly y in L 2 ( 0 , T ; H ) , as k → ∞ . (2.19)

Because W ( 0 , T ) embeds into C ( 0 , T ; H ) , we can obtain that u m ∈ Math_214#. Then, by virtue of (2.19), we can find a subsequence (denoted again by { u m k } ) of { u m k } such that

u m k → strongly u in H 2 ( Ω ) , as k → ∞ , for ∀ t ∈ [ 0 , T ] a.e.. (2.20)

Combining (2.18)-(2.20) and the Lebesgue dominated convergence theorem, we have

u m k y m k , x → weakly u y x in L 2 ( 0 , T ; H ) , as k → ∞ ; (2.21)

u m k , x y m k → strongly u x y in L 2 ( 0 , T ; H ) , as k → ∞ ; (2.22)

u m k , x x x → weakly u x x x in L 2 ( 0 , T ; H ) , as k → ∞ . (2.23)

We replace y m and u m by y m k and u m k respectively in the first equation of Equation (2.4), which yields

( y m k , t , ω j ) H + ( c 0 u m k , x , ω j ) H + ( u m k y m k , x , ω j ) H + ( b u m k , x y m k , ω j ) H + ( Γ u m k , x x x , ω j ) H = ( f , ω j ) H . (2.24)

Multiplying both sides of Equation (2.24) by α ( t ) , where α ( t ) ∈ C 1 [ 0 , T ] , α ( T ) = 0 and integrating the result equation over [ 0 , T ] , we have

∫ 0 T [ − ( y m k , , α t ω j ) H + ( c 0 u m k , x , α ω j ) H + ( u m k y m k , x , α ω j ) H + ( b u m k , x y m k , α ω j ) H + ( Γ u m k , x x x , α ω j ) H ] d t = ∫ 0 T ( f , α ω j ) H d t + ( y m k ( 0 , x ) , α ( 0 ) ω j ) H (2.25)

Utilizing (2.19), (2.21)-(2.23), we may pass to the limit in Equation (2.25). Then, we get

∫ 0 T [ − ( y , α t ω j ) H + ( c 0 u x , α ω j ) H + ( u y x , α ω j ) H + ( b u x y , α ω j ) H + ( Γ u x x x , α ω j ) H ] d t = ∫ 0 T ( f , α ω j ) H d t + ( y 0 , α ( 0 ) ω j ) H . (2.26)

We can find Equation (2.26) is true for any α ( t ) . Therefore, we may take

d d t ( y ( t , x ) , ω j ) H + ( c 0 u x ( t , x ) , α ω j ) H + ( u ( t , x ) y x ( t , x ) , ω j ) H + ( b u x ( t , x ) y ( t , x ) , ω j ) H + ( Γ u x x x ( t , x ) , ω j ) H = ( f ( t , x ) , ω j ) H

in the sense of D ′ ( 0 , T ) .

Since j is arbitrary and finite linear combinations of ω j is dense in H, we can find that y ( t , x ) ∈ W ( 0 , T ) satisfies Definition 2.1. Hence, from complex analysis above and Lemma 2.1, we obtain the existence of weak solution u ( t , x ) ∈ Math_248# to Equation (2.2).

Next we will discuss the uniqueness of this weak solution.

Let u 1 and u 2 be any two weak solutions of Equation (2.1) and set η ( t , x ) = u 1 ( t , x ) − u 2 ( t , x ) . Then η satisfies

{ η t − η x x t + c 0 η x + ( b + 1 ) u 1 η x + ( b + 1 ) u 2 , x η + Γ η x x x − b u 1 , x η x x − b u 2 , x x η x − u 1 η x x x − u 2 , x x x η = 0 , η ( t , x + L ) = η ( t , x ) , ∀ x ∈ R , ∀ t ∈ [ 0 , T ] , η ( 0 , x ) = η x ( 0 , x ) = η x x ( 0 , x ) = 0 , ∀ x ∈ R . (2.27)

Taking the inner product of both sides of the first equation in Equation (2.27) with η , we obtain

1 2 d d t ( ‖ η ‖ H 2 + ‖ η ‖ V 2 ) = − ( ( b + 1 ) u 1 η x , η ) − ( ( b + 1 ) u 2 , x η , η ) + ( b u 1 , x η x x , η ) + ( b u 2 , x x η x , η ) + ( u 1 η x x x , η ) + ( u 2 , x x x η , η ) . (2.28)

The right hand side of Equation (2.28) can be estimated as follows:

− ( ( b + 1 ) u 1 η x , η ) ≤ | b + 1 | ‖ u 1 ‖ L ∞ ∫ Ω | η x η | d x ≤ | b + 1 | λ 2 C 1 2 ( ‖ η ‖ H 2 + ‖ η ‖ V 2 ) ;

− ( ( b + 1 ) u 2 , x η , η ) ≤ | b + 1 | ‖ u 2 , x ‖ L ∞ ∫ Ω | η | 2 d x ≤ | b + 1 | λ 2 ‖ u 2 ‖ H 2 ( Ω ) ‖ η ‖ H 2 ≤ | b + 1 | λ 2 C 2 ‖ η ‖ H 2 ;

( b u 1 , x η x x , η ) = − b ∫ Ω u 1 , x x η η x d x − b ∫ Ω u 1 , x η x 2 d x ≤ | b | ‖ u 1 , x x ‖ L ∞ ∫ Ω | η η x | d x + | b | ‖ u 1 , x ‖ L ∞ ∫ Ω | η x | 2 d x ≤ | b | λ 2 2 ‖ u 1 ‖ H 3 ( Ω ) ( ‖ η ‖ H 2 + ‖ η ‖ V 2 ) + | b | λ 2 ‖ u 1 ‖ H 2 ( Ω ) ‖ η ‖ V 2 ≤ | b | λ 2 2 C 3 ( ‖ η ‖ H 2 + ‖ η ‖ V 2 ) + | b | λ 2 C 4 ‖ η ‖ V 2 ;

( b u 2 , x x η x , η ) ≤ | b | λ 2 2 ‖ u 2 ‖ H 3 ( Ω ) ( ‖ η ‖ H 2 + ‖ η ‖ V 2 ) ≤ | b | λ 2 2 C 5 ( ‖ η ‖ H 2 + ‖ η ‖ V 2 ) ;

( u 1 η x x x , η ) = ∫ Ω u 1 , x x η η x d x + 3 2 ∫ Ω u 1 , x η x 2 d x ≤ ‖ u 1 , x x ‖ L ∞ ∫ Ω | η x η | d x + 3 2 ‖ u 1 , x ‖ L ∞ ∫ Ω | η x | 2 d x ≤ λ 2 2 ‖ u 1 ‖ H 3 ( Ω ) ( ‖ η ‖ H 2 + ‖ η ‖ V 2 ) + 3 λ 2 2 ‖ u 1 ‖ H 2 ( Ω ) ‖ η ‖ V 2 ≤ λ 2 C 3 2 ( ‖ η ‖ H 2 + ‖ η ‖ V 2 ) + 3 λ 2 C 4 2 ‖ η ‖ V 2 ;

( u 2 , x x x η , η ) = − 2 ∫ Ω u 2 , x x η η x d x ≤ λ 2 ‖ u 2 ‖ H 3 ( Ω ) ( ‖ η ‖ H 2 + ‖ η ‖ V 2 ) ≤ λ 2 C 5 ( ‖ η ‖ H 2 + ‖ η ‖ V 2 ) ,

where λ 2 > 0 is an embedding constant and C i > 0 , i = 1 , 2 , ⋅ ⋅ ⋅ , 5 are some con- stants.

Combining all complex estimates above and Equation (2.28), we can deduce that

d d t ( ‖ η ‖ H 2 + ‖ η ‖ V 2 ) ≤ β ( ‖ η ‖ H 2 + ‖ η ‖ V 2 ) , (2.29)

where

β = max { | b + 1 | λ 2 C 1 + 2 | b + 1 | λ 2 C 2 + ( | b | + 1 ) λ 2 C 3 + ( | b | + 2 ) λ 2 C 5 , | b + 1 | λ 2 C 1 + ( | b | + 1 ) λ 2 C 3 + ( 2 | b | + 3 ) λ 2 C 4 + ( | b | + 2 ) λ 2 C 5 } .

Integrating inequality (2.29) with respect to t over [ 0 , t ) , we have

( ‖ η ( t , x ) ‖ H 2 + ‖ η ( t , x ) ‖ V 2 ) ≤ ( ‖ η ( 0 , x ) ‖ H 2 + ‖ η ( 0 , x ) ‖ V 2 ) exp ( β t ) , (2.30)

where ∀ t ∈ [ 0 , T ] . It follows from η ( 0 , x ) = 0 that ‖ η ( t , x ) ‖ H 2 + ‖ η ( t , x ) ‖ V 2 = 0 , which implies u 1 ( t , x ) = u 2 ( t , x ) .

This completes the proof of uniqueness.

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 U be a Hilbert space of control variables, and B ∈ L ( U , L 2 ( 0 , T ; V ) ) be an operator called a controller. We assume that the admissible set U a d be a bounded closed convex set, which has the non-empty interior with respect to U topology, i.e. int L 2 ( 0 , T ) U a d ≠ ∅ .

We study the following nonlinear control system:

{ y t ( v ; t , x ) + c 0 u x ( v ; t , x ) + u ( v ; t , x ) y x ( v ; t , x ) + b u x ( v ; t , x ) y ( v ; t , x ) + Γ u x x x ( v ; t , x ) = B v , u ( v ; t , x + L ) = u ( v ; t , x ) , ∀ x ∈ R , ∀ t ∈ [ 0 , T ] , y ( v ; 0 , x ) = y 0 ( x ) ∈ V , (3.1)

where v ∈ U a d is a control. By virtue of Theorem 2.1 and Equation (3.1), we can uniquely define the solution mapping v → u ( v ; t , x ) of U a d into S ( 0 , T ) . The weak solution u ( v ; t , x ) is called the state variable of the nonlinear control system (3.1).

The observation of the state is assumed to be given by

z ( v ; t , x ) = C u ( v ; t , x ) , (3.2)

where C ∈ L ( S ( 0 , T ) , M ) is an operator called the observer and M is a Hilbert space of the observation variables.

We shall consider the following quadratic cost functional associated with the nonlinear control system (3.1):

I ( v ) = ‖ C u ( v ; t , x ) − z d ‖ M 2 + ( N v , v ) U , (3.3)

where z d ∈ M is a desired value of u ( v ; t , x ) . N ∈ L ( U , U ) is symmetric and positive definite, i.e., ( N v , v ) U = ( v , N v ) U ≥ λ ‖ v ‖ U 2 , where λ > 0 is some constant.

Hence, the discussed optimal control problem is to find an element v ∗ ∈ U a d such that

I ( v ∗ ) = inf { I ( v ) | ∀ v ∈ U a d } ,

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 v ∗ 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 v ∗ ∈ U a d for the nonlinear control

system (3.1) with the cost functional (3.3), such that I ( v ∗ ) = inf ∀ v ∈ U a d I ( v ) .

Proof. Because U a d ≠ ∅ is a closed convex set, there exists a minimizing sequence { v n } n ∈ N + in U a d such that

inf ∀ v ∈ U a d I ( v ) = lim n → ∞ I ( v n ) .

We set

π ( v 1 , v 2 ) = ( C ( u ( v 1 ; t , x ) − u ( 0 ; t , x ) ) , C ( u ( v 2 ; t , x ) − u ( 0 ; t , x ) ) ) M + ( N v 1 , v 2 ) U

and

L ( v ) = ( z d − C u ( 0 ; t , x ) , C ( u ( v ; t , x ) − u ( 0 ; t , x ) ) ) M .

Then cost functional (3.3) can be rewritten as

I ( v ) = π ( v , v ) − 2 L ( v ) + ‖ z d − C u ( 0 ; t , x ) ‖ M 2 , (3.4)

where π ( v 1 , v 2 ) is a continuous symmetric bilinear form on U and L ( v ) is a continuous linear form on U .

Obviously, { I ( v n ) } is bounded in R + . So, the quadratic cost functional (3.3) implies that there exists a constant M 0 > 0 such that

λ ‖ v n ‖ U 2 ≤ ( N v n , v n ) U ≤ I ( v n ) ≤ M 0 , (3.5)

which indicates that { v n } n ∈ N + is bounded in U . Because U a d is closed and convex set, we can extract a subsequence { v n k } ⊂ { v n } n ∈ N + and find a v ∗ ∈ U a d such that

v n k → weakly v ∗ in U , as k → ∞ . (3.6)

From now on, each state variable u n ( t , x ) = u ( v n ; t , x ) ∈ S ( 0 , T ) corresponding to v n is the solution of

{ y n , t + c 0 u n , x + u n y n , x + b u n , x y n + Γ u n , x x x = B v n , u n ( t , x + L ) = u n ( t , x ) , ∀ x ∈ R , ∀ t ∈ [ 0 , T ] , y n ( 0 , x ) → y 0 ( x ) , (3.7)

where y n = u n − u n , x x .

From inequality (3.5), the right hand side of the first equation in Equation (3.7) can be estimated as

‖ B v n ‖ L 2 ( 0 , T ; V ) ≤ ‖ B ‖ L ( U , L 2 ( 0 , T ; V ) ) ‖ v n ‖ U ≤ ‖ B ‖ L ( U , L 2 ( 0 , T ; V ) ) λ − 1 M 0 ≤ M , (3.8)

where M > 0 is some constant.

Utilizing inequality (3.8), we can apply the same method used in Theorem 2.1 to deduce that { y n } n ∈ N + is bounded in W ( 0 , T ) . Hence, by the extraction theorem of Rellich’s, we can extract a subsequence { y n k } of { y n } n ∈ N + and find a y = u − u x x ∈ W ( 0 , T ) such that

y n k → weakly y in W ( 0 , T ) , as k → ∞ . (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 { y n k } is pre-compact in L 2 ( 0 , T ; H ) . So we can also choose a subsequence (denoted again by { y n k } ) of { y n k } such that

y n k → strongly y , in L 2 ( 0 , T ; H ) as k → ∞ . (3.10)

On the other hand, because W ( 0 , T ) embeds into C ( 0 , T ; H ) , we can infer that u n ∈ C ( 0 , T ; H 2 ( Ω ) ) . And from (3.10), we can get a subsequence (denoted again by { u n k } ) of { u n k } such that

u n k → strongly u in H 2 ( Ω ) , as k → ∞ , for t ∈ [ 0 , T ] a.e.. (3.11)

Combining (3.9)-(3.11) and the Lebesgue dominated convergence theorem, it is not difficult to obtain that

u n k , x → strongly u x in L 2 ( 0 , T ; H ) , as k → ∞ ; (3.12)

u n k y n k , x → weakly u y x in L 2 ( 0 , T ; H ) , as k → ∞ ; (3.13)

u n k , x y n k → strongly u x y in L 2 ( 0 , T ; H ) , as k → ∞ ; (3.14)

u n k , x x x → weakly u x x x in L 2 ( 0 , T ; H ) , as k → ∞ . (3.15)

We replace u n and v n by u n k and v n k in Equation (3.7) respectively, and take k → ∞ . Then, by the standard arguments as in [

{ y t + c 0 u + u y x + b u x y + Γ u x x x = B v ∗ , u ( t , x + L ) = u ( t , x ) , ∀ x ∈ R , ∀ t ∈ [ 0 , T ] , y ( 0 , x ) = y 0 ( x ) , (3.16)

in weak sense, where y = u − u x x . Moreover, by the uniqueness of weak solution of Equation (3.16) via Theorem 2.1 and Lemma 2.1, we can conclude that u = Math_371#, which implies u ( v n ; t , x ) → weakly u ( v ∗ ; t , x ) in S ( 0 , T ) .

Because the mapping v → π ( v , v ) is lower semi-continuous in the weak topology of U and ‖ ⋅ ‖ M is also lower semi-continuous. The mapping v → L ( v ) is continuous in the weak topology of U . Thus the mapping v → I ( v ) is weakly lower semi-continuous.

So, we can deduce from cost functional (3.4) that

lim inf k → ∞ I ( v n ) ≥ I ( v ∗ ) . (3.17)

At the same time, from inequality (3.17), we have

inf ∀ v ∈ U a d I ( v ) = lim inf n → ∞ I ( v n ) ≥ I ( v ∗ ) .

Moreover, combining I ( v ∗ ) ≥ inf ∀ v ∈ U a d I ( v ) by definition, we can obtain that

I ( v ∗ ) = inf ∀ v ∈ U a d I ( v ) . (3.18)

Next, we will prove the uniqueness of v ∗ ∈ U a d in (3.18).

Because the mapping v → π ( v , v ) is strictly convex and the mapping v → L ( v ) is continuous. Hence the mapping v → I ( v ) is also strictly convex.

Let v 1 ∗ ∈ U a d and v 2 ∗ ∈ U a d be two optimal controls, which satisfy I ( v 1 ∗ ) =

inf ∀ v ∈ U a d I ( v ) and I ( v 2 ∗ ) = inf ∀ v ∈ U a d I ( v ) respectively. Because U a d is a bounded closed convex set, we can get that 1 2 ( v 1 ∗ + v 2 ∗ ) ∈ U a d . We thus can deduce that I ( 1 2 ( v 1 ∗ + v 2 ∗ ) ) < 1 2 I ( v 1 ∗ ) + 1 2 I ( v 2 ∗ ) = inf ∀ v ∈ U a d I ( v ) , which is a contradiction unless v 1 ∗ = v 2 ∗ . This completes the proof.

From the above analysis, we can conclude that ( u ( v ∗ ; t , x ) , v ∗ ) of S ( 0 , T ) × U a d is a unique optimal solution to the optimal control problem investigated.

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 v → I ( v ) is differentiable, strictly convex and U a d is bounded. Then the unique element (optimal control) v ∗ in

U a d satisfying I ( v ∗ ) = inf v ∈ U a d I ( v ) can be characterized by

I ′ ( v ∗ ) ( v − v ∗ ) ≥ 0 , (4.1)

where ∀ v ∈ U a d and I ′ ( v ∗ ) denote the derivative of I ( v ) at v = v ∗ .

Proof. Let v ∗ be the optimal control subject to Theorem 3.1. Then for ∀ v ∈ U a d and θ ∈ ( 0 , 1 ) , we have

I ( v ∗ ) = I ( ( 1 − θ ) v ∗ + θ v ∗ ) ≤ I ( ( 1 − θ ) v ∗ + θ v ) . (4.2)

From inequality (4.2), we can derive that

θ − 1 [ I ( v ∗ + θ ( v − v ∗ ) ) − I ( v ∗ ) ] ≥ 0 . (4.3)

Therefore, if we pass to the limit in inequality (4.3), we obtain that

I ′ ( v ∗ ) ( v − v ∗ ) ≥ 0 , where ∀ v ∈ U a d .

Alternatively, suppose inequality (4.1) remains true. Because the mapping v → I ( v ) is strictly convex, we can get

I ( ( 1 − θ ) v ∗ + θ v ) < ( 1 − θ ) I ( v ∗ ) + θ I ( v ) , for ∀ θ ∈ ( 0 , 1 ) . (4.4)

From inequality (4.4), we deduce that

θ − 1 [ I ( v ∗ + θ ( v − v ∗ ) ) − I ( v ∗ ) ] < I ( v ) − I ( v ∗ ) . (4.5)

If we pass the limit in inequality (4.5), we can get

0 ≤ I ′ ( v ∗ ) ( v − v ∗ ) = lim θ → 0 I ( v ∗ + θ ( v − v ∗ ) ) − I ( v ∗ ) θ < I ( v ) − I ( v ∗ ) ,

for ∀ v ∈ U a d , 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 v → u ( v ; t , x ) of U a d → S ( 0 , T ) is differentiable at v = v ∗ .

Definition 4.1. The solution mapping v → u ( v ; t , x ) of U into S ( 0 , T ) is said to be differentiable at v = v ∗ in any direction w, if for ∀ w ∈ U and θ ∈ ( 0 , 1 ) , there exists a u ′ ( v ∗ ; t , x ) ∈ L ( U , S ( 0 , T ) ) such that

θ − 1 [ u ( v ∗ + θ w ; t , x ) − u ( v ∗ ; t , x ) ] → u ′ ( v ∗ ; t , x ) w in S ( 0 , T ) , as θ → 0 .

The function u ′ ( v ∗ ; t , x ) w ∈ S ( 0 , T ) is called the directional derivative of u ( v ; t , x ) , which plays crucial in the following discussion.

Theorem 4.1. The mapping v → u ( v ; t , x ) of U a d into S ( 0 , T ) is derivative at v = v ∗ and such the derivative of u ( v ; t , x ) at v = v ∗ in the direction w = v − v ∗ ∈ U a d , say g = u ′ ( v ∗ ; t , x ) w , is a weak solution of the following equation:

{ G t + c 0 g x + g y x + u ( v ∗ ; t , x ) G x + b g x y + b u x ( v ∗ ; t , x ) G + Γ g x x x = B w , g ( t , x + L ) = g ( t , x ) , ∀ x ∈ R , ∀ t ∈ [ 0 , T ] , G ( 0 , x ) = 0 , (4.6)

where y = u ( v ∗ ; t , x ) − u x x ( v ∗ ; t , x ) and G = g − g x x .

Proof. Let θ ∈ ( − 1 , 0 ) ∪ ( 0 , 1 ) . We set g θ = θ − 1 ( u ( v ∗ + θ w ; t , x ) − u ( v ∗ ; t , x ) ) and G θ = g θ − g θ , x x . Then g θ satisfies

{ G θ , t + c 0 g θ , x + g θ y θ , x + u ( v ∗ ; t , x ) G θ , x + b g θ , x y θ + b u x ( v ∗ ; t , x ) G θ + Γ g θ , x x x = B w , g θ ( t , x + L ) = g θ ( t , x ) , ∀ x ∈ R , ∀ t ∈ [ 0 , T ] , G θ ( 0 , x ) = 0 , (4.7)

where y θ = u ( v ∗ + θ w ; t , x ) − u x x ( v ∗ + θ w ; t , x ) .

In order to estimate G θ , we multiply both sides of the first equation in Equation (4.7) by 2 g θ and integrate it over Ω . Then we get

d d t ( ‖ g θ ‖ H 2 + ‖ g θ ‖ V 2 ) = ( 4 − 2 b ) ∫ Ω y θ g θ g θ , x d x + ( 2 − 2 b ) ∫ Ω u x x ( v ∗ ; t , x ) g θ g θ , x d x + ( 3 − 2 b ) ∫ Ω u x ( v ∗ ; t , x ) g θ , x 2 d x + ( 1 − 2 b ) ∫ Ω u x ( v ∗ ; t , x ) g θ 2 d x + 2 ∫ Ω ( B w ) g θ d x . (4.8)

Each item on the right hand of Equation (4.8) can be estimated as follows:

( 4 − 2 b ) ∫ Ω y θ g θ g θ , x d x ≤ | 4 − 2 b | ‖ y θ ‖ L ∞ ∫ Ω | g θ g θ , x | d x ≤ | 4 − 2 b | m 1 2 ( ‖ g θ ‖ H 2 + ‖ g θ ‖ V 2 ) ;

( 2 − 2 b ) ∫ Ω u x x ( v ∗ ; t , x ) g θ g θ , x d x ≤ | 2 − 2 b | ‖ u x x ( v ∗ ; t , x ) ‖ L ∞ ∫ Ω | g θ g θ , x | d x ≤ | 2 − 2 b | m 2 2 ( ‖ g θ ‖ H 2 + ‖ g θ ‖ V 2 ) ;

( 3 − 2 b ) ∫ Ω u x ( v ∗ ; t , x ) g θ , x 2 d x ≤ | 3 − 2 b | ‖ u x ( v ∗ ; t , x ) ‖ L ∞ ‖ g θ ‖ V 2 ≤ | 3 − 2 b | m 3 ‖ g θ ‖ V 2 ;

( 1 − 2 b ) ∫ Ω u x ( v ∗ ; t , x ) g θ 2 d x ≤ | 1 − 2 b | ‖ u x ( v ∗ ; t , x ) ‖ L ∞ ‖ g θ ‖ H 2 ≤ | 1 − 2 b | m 3 ‖ g θ ‖ H 2

and

2 ∫ Ω ( B w ) g θ d x ≤ ‖ B w ‖ H 2 + ‖ g θ ‖ H 2 ≤ λ 1 2 ‖ B w ‖ V 2 + ‖ g θ ‖ H 2 ,

where λ 1 > 0 is an embedding constant and m i > 0 , i = 1 , 2 , 3 are some constants.

Hence, Equation (4.8) can be changed into

d d t ( ‖ g θ ‖ H 2 + ‖ g θ ‖ V 2 ) ≤ β 3 ( ‖ g θ ‖ H 2 + ‖ g θ ‖ V 2 ) + λ 1 2 ‖ B w ‖ V 2 , (4.9)

where

β 3 = { | 4 − 2 b | m 1 2 + | 2 − 2 b | m 2 2 + | 1 − 2 b | m 3 + 1 , | 4 − 2 b | m 1 2 + | 2 − 2 b | m 2 2 + | 3 − 2 b | m 3 } .

It follows from inequality (4.9) and the Gronwall’s lemma that

‖ g θ ‖ H 2 + ‖ g θ ‖ V 2 ≤ exp ( β 3 t ) [ ( ‖ g θ ( 0 , x ) ‖ H 2 + ‖ g θ ( 0 , x ) ‖ V 2 ) + λ 1 2 ∫ 0 t ‖ B w ‖ V 2 exp ( − β 3 s ) d s ] ≜ Ζ 1 , (4.10)

where ∀ t ∈ [ 0 , T ] .

Next, multiplying both sides of the first equation in Equation (4.7) by − 2 g θ , x x and integrating it over Ω , which gives

d d t ( ‖ g θ ‖ V 2 + ‖ g θ ‖ H 2 2 ) = ( 2 b − 2 ) ∫ Ω g θ , x g θ , x x y θ d x − 2 ∫ Ω g θ g θ , x x x y θ d x + 2 ∫ Ω u ( v ∗ ; t , x ) g θ , x g θ , x x d x + 2 b ∫ Ω u x ( v ∗ ; t , x ) g θ g θ , x x d x + ( 1 − 2 b ) ∫ Ω u x ( v ∗ ; t , x ) g θ , x x 2 d x − 2 ∫ Ω ( B w ) g θ , x x d x . (4.11)

Then, we estimate the each item of the right hand of Equation (4.11) as follows:

( 2 b − 2 ) ∫ Ω g θ , x g θ , x x y θ d x ≤ | 2 b − 2 | m 1 2 ( ‖ g θ ‖ V 2 + ‖ g θ ‖ H 2 ( Ω ) 2 ) ;

− 2 ∫ Ω g θ g θ , x x x y θ d x ≤ ‖ y θ ‖ L ∞ | ∫ Ω 2 g θ g θ , x x x d x | = 0 ;

2 ∫ Ω u ( v ∗ ; t , x ) g θ , x g θ , x x d x ≤ ‖ u ( v ∗ ; t , x ) ‖ L ∞ ∫ Ω | 2 g θ , x g θ , x x | d x ≤ m 4 ( ‖ g θ ‖ V 2 + ‖ g θ ‖ H 2 ( Ω ) 2 ) ;

2 b ∫ Ω u x ( v ∗ ; t , x ) g θ g θ , x x d x ≤ | b | ‖ u x ( v ∗ ; t , x ) ‖ L ∞ ( ‖ g θ ‖ H 2 + ‖ g θ , x x ‖ H 2 ) ≤ | b | m 3 ( λ 1 2 ‖ g θ ‖ V 2 + ‖ g θ ‖ H 2 ( Ω ) 2 )

( 1 − 2 b ) ∫ Ω u x ( v ∗ ; t , x ) g θ , x x 2 d x ≤ | 1 − 2 b | ‖ u x ( v ∗ ; t , x ) ‖ L ∞ ‖ g θ ‖ H 2 ( Ω ) 2 ≤ | 1 − 2 b | m 3 ‖ g θ ‖ H 2 ( Ω ) 2

and

− 2 ∫ Ω ( B w ) g θ , x x d x ≤ 2 ‖ B w ‖ H ‖ g θ , x x ‖ H ≤ ‖ B w ‖ H 2 + ‖ g θ ‖ H 2 ( Ω ) 2 ≤ λ 1 2 ‖ B w ‖ V 2 + ‖ g θ ‖ H 2 ( Ω ) 2 ,

where λ 1 > 0 is an embedding constant and m i > 0 , i = 1 , 3 , 4 are some constants.

By the above estimates, we can deduce from Equation (4.11) that

d d t ( ‖ g θ ‖ V 2 + ‖ g θ ‖ H 2 ( Ω ) 2 ) ≤ β 4 ( ‖ g θ ‖ V 2 + ‖ g θ ‖ H 2 ( Ω ) 2 ) + λ 1 2 ‖ B w ‖ V 2 , (4.12)

where

β 4 = max { | 2 b − 2 | m 1 2 + m 4 + | b | m 3 λ 1 2 , | 2 b − 2 | m 1 2 + m 4 + | b | m 3 + | 1 − 2 b | m 3 + 1 } .

Applying Gronwall’s lemma to inequality (4.12), which yields

‖ g θ ‖ V 2 + ‖ g θ ‖ H 2 ( Ω ) 2 ≤ exp ( β 4 t ) [ ( ‖ g θ ( 0 , x ) ‖ V 2 + ‖ g θ ( 0 , x ) ‖ H 2 ( Ω ) 2 ) + λ 1 2 ∫ 0 t ‖ B w ‖ V 2 exp ( − β 4 s ) d s ] ≜ Ζ 2 , (4.13)

where ∀ t ∈ [ 0 , T ] .

Similarly, multiplying both sides of the first equation in Equation (4.7) by 2 g θ , x x x x and integrating it over Ω , which gives

d d t ( ‖ g θ ‖ H 2 ( Ω ) 2 + ‖ g θ ‖ H 3 ( Ω ) 2 ) = ( 2 − 2 b ) ∫ Ω g θ , x g θ , x x x x y θ d x + 2 ∫ Ω g θ g θ , x x x x x y θ d x − ( 2 b + 3 ) ∫ Ω u x ( v ∗ ; t , x ) g θ , x x 2 d x − ( 2 b + 1 ) ∫ Ω u x ( v ∗ ; t , x ) g θ , x x x 2 d x − ( 2 b + 2 ) ∫ Ω u x x ( v ∗ ; t , x ) g θ , x g θ , x x d x + 2 b ∫ Ω u x x ( v ∗ ; t , x ) g θ g θ , x x x d x − 2 b ∫ Ω u x x ( v ∗ ; t , x ) g θ , x x g θ , x x x d x + 2 ∫ Ω ( B w ) g θ , x x x x d x . (4.14)

We can also estimate each item of the right hand of Equation (4.14) as follows:

( 2 − 2 b ) ∫ Ω g θ , x g θ , x x x x y θ d x ≤ | 2 − 2 b | ‖ y θ ‖ L ∞ | ∫ Ω g θ , x g θ , x x x x d x | = 0 ;

2 ∫ Ω g θ g θ , x x x x x y θ d x ≤ ‖ y θ ‖ L ∞ | ∫ Ω 2 g θ g θ , x x x x x d x | = 0 ;

− ( 2 b + 3 ) ∫ Ω u x ( v ∗ ; t , x ) g θ , x x 2 d x ≤ | 2 b + 3 | ‖ u x ( v ∗ ; t , x ) ‖ L ∞ ∫ Ω | g θ , x x 2 | d x ≤ | 2 b + 3 | m 3 ‖ g θ ‖ H 2 ( Ω ) 2 ;

− ( 2 b + 1 ) ∫ Ω u x ( v ∗ ; t , x ) g θ , x x x 2 d x ≤ | 2 b + 1 | ‖ u x ( v ∗ ; t , x ) ‖ L ∞ ∫ Ω | g θ , x x x 2 | d x ≤ | 2 b + 1 | m 3 ‖ g θ ‖ H 3 ( Ω ) 2 ;

− ( 2 b + 2 ) ∫ Ω u x x ( v ∗ ; t , x ) g θ , x g θ , x x d x ≤ | 2 b + 2 | m 2 2 ( λ 4 2 ‖ g θ ‖ H 2 ( Ω ) 2 + ‖ g θ ‖ H 2 ( Ω ) 2 ) ;

2 b ∫ Ω u x x ( v ∗ ; t , x ) g θ g θ , x x x d x ≤ | b | m 2 ( λ 5 2 ‖ g θ ‖ H 2 ( Ω ) 2 + ‖ g θ ‖ H 3 ( Ω ) 2 ) ;

− 2 b ∫ Ω u x x ( v ∗ ; t , x ) g θ , x x g θ , x x x d x ≤ | b | m 2 ( ‖ g θ ‖ H 2 ( Ω ) 2 + ‖ g θ ‖ H 3 ( Ω ) 2 )

and

2 ∫ Ω ( B w ) g θ , x x x x d x ≤ 2 ‖ ( B w ) x ‖ H ‖ g θ , x x x ‖ H ≤ ‖ B w ‖ V 2 + ‖ g θ ‖ H 3 ( Ω ) 2 ,

where m i > 0 , i = 2 , 3 are some constants and λ i > 0 , i = 4 , 5 are some embedding constants.

Combining a series of complex estimates above and Equation (4.14), we can obtain that

d d t ( ‖ g θ ‖ H 2 ( Ω ) 2 + ‖ g θ ‖ H 3 ( Ω ) 2 ) ≤ β 5 ( ‖ g θ ‖ H 2 ( Ω ) 2 + ‖ g θ ‖ H 3 ( Ω ) 2 ) + ‖ B w ‖ V 2 , (4.15)

where

β 5 = max { [ | 2 b + 2 | ( λ 4 2 + 1 ) 2 + | b | + | b | λ 5 2 ] m 2 + | 2 b + 3 | m 3 , 2 | b | m 2 + | 2 b + 1 | m 3 + 1 } .

By applying the Gronwall’s lemma to inequality (4.15), we can get

‖ g θ ‖ H 2 ( Ω ) 2 + ‖ g θ ‖ H 3 ( Ω ) 2 ≤ exp ( β 5 t ) [ ( ‖ g θ ( 0 , x ) ‖ H 2 ( Ω ) 2 + ‖ g θ ( 0 , x ) ‖ H 3 ( Ω ) 2 ) + ∫ 0 t ‖ B w ‖ V 2 exp ( − β 5 s ) d s ] ≜ Ζ 3 , (4.16)

where ∀ t ∈ [ 0 , T ] .

Combining estimate inequality (4.13) and (4.16), we can deduce that

‖ G θ ‖ V 2 = ‖ g θ , x − g θ , x x x ‖ H 2 = ‖ g θ ‖ V 2 + 2 ‖ g θ ‖ H 2 ( Ω ) 2 + ‖ g θ ‖ H 3 ( Ω ) 2 ≤ Ζ 2 + Ζ 3 . (4.17)

Similarly, combining estimate inequality (4.10) and (4.13), we can obtain that

‖ G θ ‖ H 2 = ‖ g θ − g θ , x x ‖ H 2 = ‖ g θ ‖ H 2 + 2 ‖ g θ ‖ V 2 + ‖ g θ ‖ H 2 ( Ω ) 2 ≤ Ζ 1 + Ζ 2 . (4.18)

From inequality (4.17), we derive that

‖ G θ ‖ L 2 ( 0 , T ; V ) 2 ≤ ( Ζ 2 + Ζ 3 ) T , (4.19)

which indicates a uniformly L 2 ( 0 , T ; V ) bounded of G θ .

Afterward, we will prove a uniformly L 2 ( 0 , T ; V ∗ ) bounded of G θ , t .

From the first equation in Equation (4.7) and the Sobolev embedding theorem, we have

‖ G θ , t ‖ V ∗ ≤ ‖ B w ‖ V ∗ + c 0 ‖ g θ ‖ H + λ 2 ‖ g θ ‖ ‖ y θ ‖ V H + ‖ u ( v ∗ ; t , x ) ‖ L ∞ ‖ G θ ‖ H + | b | ‖ y θ ‖ L ∞ ‖ g θ ‖ + H | b | λ 2 ‖ G θ ‖ ‖ u ( v ∗ ; t , x ) ‖ V H + | Γ | ‖ g θ ‖ H 2 ( Ω ) ≤ λ 3 ‖ B w ‖ V + c 0 Ζ 1 1 2 + λ 2 m 5 Ζ 1 1 2 + m 4 ( Ζ 1 + Ζ 2 ) 1 2 + | b | m 1 Ζ 1 1 2 + | b | λ 2 m 6 ( Ζ 2 + Ζ 3 ) 1 2 + | Γ | Ζ 2 1 2 , (4.20)

where λ i > 0 , i = 2 , 3 are some embedding constants and m i > 0 , i = 1 , 4 , 5 , 6 are some constants.

Analogously, from inequality (4.20), we can get

‖ G θ , t ‖ L 2 ( 0 , T ; V ∗ ) 2 ≤ [ λ 3 ‖ B w ‖ V + c 0 Ζ 1 1 2 + λ 2 m 5 Ζ 1 1 2 + m 4 ( Ζ 1 + Ζ 2 ) 1 2 + | b | m 1 Ζ 1 1 2 + | b | λ 2 m 6 ( Ζ 2 + Ζ 3 ) 1 2 + | Γ | Ζ 2 1 2 ] 2 T . (4.21)

Combining inequality (4.19) and (4.21), we can establish the boundedness of G θ in W ( 0 , T ) . Hence, from Lemma 2.1, we can deduce that

‖ g θ ‖ S ( 0 , T ) ≤ C ‖ G θ ‖ W ( 0 , T ) < + ∞ .

From now on, we can infer that there exists a g ∈ S ( 0 , T ) and a sequence { θ k } ⊂ ( − 1 , 1 ) tending to 0 such that

g θ k → weakly g in S ( 0 , T ) , as k → ∞ . (4.22)

Because the imbedding S ( 0 , T ) into L 2 ( 0 , T ; H 2 ( Ω ) ) is compact, then it can deduce from (4.22) that

g θ k → strongly g in H 2 ( Ω ) a.e. t ∈ [ 0 , T ] , (4.23)

for some { θ k } ⊂ ( − 1 , 1 ) tending to 0 as k → ∞ . Whence by (4.22) - (4.23), Theorem 2.1 and the Lebesgue dominated convergence theorem, we can easily obtain that

g θ k y θ k , x → weakly g y x in L 2 ( 0 , T ; H ) ; (4.24)

g θ k , x y θ k → strongly g x y in L 2 ( 0 , T ; H ) ; (4.25)

G θ k → weakly G in L 2 ( 0 , T ; V ) ; (4.26)

g θ k , x x x → weakly g x x x in L 2 ( 0 , T ; H ) ; (4.27)

as k → ∞ , where G = g − g x x . And also we can derive from Equation (4.7) and inequality (4.21) that

G θ k , t → weakly G t in L 2 ( 0 , T ; V ∗ ) , as k → ∞ . (4.28)

Therefore, we can infer from (4.24) to (4.28) that

in S ( 0 , T ) as θ → 0 in which g is a solution of Equation (4.6).

Consequently, the solution mapping v → u ( v ; t , x ) of U a d into S ( 0 , T ) is differentiable in the weak topology of S ( 0 , T ) . This completes the proof.

The conclusion of Theorem 4.1 means that the cost I ( v ) is derivative at v ∗ in the direction v − v ∗ . So, we can get that

I ′ ( v ∗ ) ( v − v ∗ ) = lim θ → 0 I ( v ∗ + θ ( v − v ∗ ) ) − I ( v ∗ ) θ = lim θ → 0 θ − 1 [ ( C u ( v ∗ + θ ( v − v ∗ ) ) − z d , C u ( v ∗ + θ ( v − v ∗ ) ) − z d ) M − ( C u ( v ∗ ) − z d , C u ( v ∗ ) − z d ) M ] + lim θ → 0 θ − 1 [ ( N ( v ∗ + θ ( v − v ∗ ) ) , v ∗ + θ ( v − v ∗ ) ) U − ( N v ∗ , v ∗ ) U ] = 2 ( C u ( v ∗ ) − z d , C u ′ ( v ∗ ) ( v − v ∗ ) ) M + 2 ( N v ∗ , v − v ∗ ) U

Then the sufficient and necessary optimality condition (4.1) can be rewritten as

( C u ( v ∗ ; t , x ) − z d , C u ′ ( v ∗ ; t , x ) ( v − v ∗ ) ) M + ( N v ∗ , v − v ∗ ) U = 〈 C ∗ Λ M ( C u ( v ∗ ; t , x ) − z d ) , u ′ ( v ∗ ; t , x ) ( v − v ∗ ) 〉 S ( 0 , T ) ′ , S ( 0 , T ) + ( N v ∗ , v − v ∗ ) U ≥ 0 , (4.29)

for

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 M = L 2 ( 0 , T ; H ) and C ∈ L ( S ( 0 , T ) , M ) , then observe that z ( v ; t , x ) = C u ( v ; t , x ) = u ( v ; t , x ) ∈ L 2 ( 0 , T ; H ) .

(II) We set M = L 2 ( 0 , T ; H ) and C ∈ L ( S ( 0 , T ) , M ) , then observe that z ( v ; t , x ) = C u ( v ; t , x ) = ( I − ∂ x 2 ) u ( v ; t , x ) = y ( v ; t , x ) ∈ L 2 ( 0 , T ; H ) .

Firstly, we discuss the cost functional expressed by

where z d ( t , x ) ∈ M is a desired value. Let v ∗ 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 g = u ′ ( v ∗ ; t , x ) ( v − v ∗ ) is the weak solution of Equation (4.6). Now we will introduce the adjoint system to describe the optimality condition (5.2):

{ − Ψ t ( v ∗ ; t , x ) − c 0 ψ x ( v ∗ ; t , x ) − u ( v ∗ ; t , x ) Ψ x ( v ∗ ; t , x ) + ( 3 − 2 b ) u x x ( v ∗ ; t , x ) ψ x ( v ∗ ; t , x ) + ( 3 − b ) u x ( v ∗ ; t , x ) ψ x x ( v ∗ ; t , x ) − b y ( v ∗ ; t , x ) ψ x ( v ∗ ; t , x ) − Γ ψ x x x ( v ∗ ; t , x ) = u ( v ∗ ; t , x ) − z d ( t , x ) , ψ ( v ∗ ; t , x + L ) = ψ ( v ∗ ; t , x ) , ∀ x ∈ R , ∀ t ∈ [ 0 , T ] , Ψ ( v ∗ ; T , x ) = 0 , (5.3)

where

Ψ ( v ∗ ; t , x ) = ψ ( v ∗ ; t , x ) − ψ x x ( v ∗ ; t , x )

and

y ( v ∗ ; t , x ) = u ( v ∗ ; t , x ) − u x x ( v ∗ ; t , x ) .

Therefore, we can provide the characterization for the optimal control v ∗ of the quadratic cost functional (5.1) as follows:

Theorem 5.1. The optimal control v ∗ of the quadratic cost functional (5.1) is characterized by the following control system, adjoint system and inequality:

{ y t ( v ∗ ; t , x ) + c 0 u x ( v ∗ ; t , x ) + u ( v ∗ ; t , x ) y x ( v ∗ ; t , x ) + b u x ( v ∗ ; t , x ) y ( v ∗ ; t , x ) + Γ u x x x ( v ∗ ; t , x ) = B v ∗ , u ( v ∗ ; t , x + L ) = u ( v ∗ ; t , x ) , ∀ x ∈ R , ∀ t ∈ [ 0 , T ] , y ( v ∗ ; 0 , x ) = y 0 ( x ) ∈ V ,

{ − Ψ t ( v ∗ ; t , x ) − c 0 ψ x ( v ∗ ; t , x ) − u ( v ∗ ; t , x ) Ψ x ( v ∗ ; t , x ) + ( 3 − 2 b ) u x x ( v ∗ ; t , x ) ψ x ( v ∗ ; t , x ) + ( 3 − b ) u x ( v ∗ ; t , x ) ψ x x ( v ∗ ; t , x ) − b y ( v ∗ ; t , x ) ψ x ( v ∗ ; t , x ) − Γ ψ x x x ( v ∗ ; t , x ) = u ( v ∗ ; t , x ) − z d ( t , x ) , ψ ( v ∗ ; t , x + L ) = ψ ( v ∗ ; t , x ) , ∀ x ∈ R , ∀ t ∈ [ 0 , T ] , Ψ ( v ∗ ; T , x ) = 0 ,

where

y ( v ∗ ; t , x ) = u ( v ∗ ; t , x ) − u x x ( v ∗ ; t , x )

and

Ψ ( v ∗ ; t , x ) = ψ ( v ∗ ; t , x ) − ψ x x ( v ∗ ; t , x ) .

Proof. Taking inner product of the first equation in Equation (5.3) by g over Ω , then integrating the result equation with respect to t on [ 0 , T ] , we get

∫ 0 T ∫ Ω − Ψ t g d x d t − c 0 ∫ 0 T ∫ Ω ψ x g d x d t − ∫ 0 T ∫ Ω u Ψ x g d x d t + ( 3 − 2 b ) ∫ 0 T ∫ Ω u x x ψ x g d x d t + ( 3 − b ) ∫ 0 T ∫ Ω u x ψ x x g d x d t − b ∫ 0 T ∫ Ω y ψ x g d x d t − Γ ∫ 0 T ∫ Ω ψ x x x x g d x d t = ∫ 0 T ∫ Ω ( u − z d ) g d x d t . (5.4)

Combining Equation (4.6) and Equation (5.3) and taking integration by parts, the left hand side of Equation (5.4) yields

∫ 0 T ∫ Ω ψ ( G t + c 0 g x + g y x + u G x + b g x y + b u x G + Γ g x x x ) d x d t = ∫ 0 T ∫ Ω ψ B ( v − v ∗ ) d x d t , (5.5)

where G = g − g x x . 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 z d ( t , x ) ∈ M is a desired value. Let v ∗ 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 G = g − g x x and g = u ′ ( v ∗ ; t , x ) w is the weak solution of Equation (4.6). Similarly, we formulate the adjoint system to describe the optimality condition (5.7):

{ − Ψ t ( v ∗ ; t , x ) − c 0 ψ x ( v ∗ ; t , x ) − u ( v ∗ ; t , x ) Ψ x ( v ∗ ; t , x ) + ( 3 − 2 b ) u x x ( v ∗ ; t , x ) ψ x ( v ∗ ; t , x ) + ( 3 − b ) u x ( v ∗ ; t , x ) ψ x x ( v ∗ ; t , x ) − b y ( v ∗ ; t , x ) ψ x ( v ∗ ; t , x ) − Γ ψ x x x ( v ∗ ; t , x ) = ( I − ∂ x 2 ) ( y ( v ∗ ; t , x ) − z d ( t , x ) ) , ψ ( v ∗ ; t , x + L ) = ψ ( v ∗ ; t , x ) , ∀ x ∈ R , ∀ t ∈ [ 0 , T ] , Ψ ( v ∗ ; T , x ) = 0 , (5.8)

where

y ( v ∗ ; t , x ) = u ( v ∗ ; t , x ) − u x x ( v ∗ ; t , x )

and

Ψ ( v ∗ ; t , x ) = ψ ( v ∗ ; t , x ) − ψ x x ( v ∗ ; t , x ) .

Hence, we can give the following theorem.

Theorem 5.2. The optimal control v ∗ of the quadratic cost functional (5.7) is characterized by the following control system, adjoint system and inequality:

{ y t ( v ∗ ; t , x ) + c 0 u x ( v ∗ ; t , x ) + u ( v ∗ ; t , x ) y x ( v ∗ ; t , x ) + b u x ( v ∗ ; t , x ) y ( v ∗ ; t , x ) + Γ u x x x ( v ∗ ; t , x ) = B v ∗ , u ( v ∗ ; t , x + L ) = u ( v ∗ ; t , x ) , ∀ x ∈ R , ∀ t ∈ [ 0 , T ] , y ( v ∗ ; 0 , x ) = y 0 ( x ) ∈ V ,

{ − Ψ t ( v ∗ ; t , x ) − c 0 ψ x ( v ∗ ; t , x ) − u ( v ∗ ; t , x ) Ψ x ( v ∗ ; t , x ) + ( 3 − 2 b ) u x x ( v ∗ ; t , x ) ψ x ( v ∗ ; t , x ) + ( 3 − b ) u x ( v ∗ ; t , x ) ψ x x ( v ∗ ; t , x ) − b y ( v ∗ ; t , x ) ψ x ( v ∗ ; t , x ) − Γ ψ x x x ( v ∗ ; t , x ) = ( I − ∂ x 2 ) ( y ( v ∗ ; t , x ) − z d ( t , x ) ) , ψ ( v ∗ ; t , x + L ) = ψ ( v ∗ ; t , x ) , ∀ x ∈ R , ∀ t ∈ [ 0 , T ] , Ψ ( v ∗ ; T , x ) = 0 ,

where

y ( v ∗ ; t , x ) = u ( v ∗ ; t , x ) − u x x ( v ∗ ; t , x )

and

Ψ ( v ∗ ; t , x ) = ψ ( v ∗ ; t , x ) − ψ x x ( v ∗ ; t , x ) .

Proof. As we did before, we multiply both sides of the first equation of Equation (5.8) by g and integrate it over [ 0 , T ] × Ω . Then we have

∫ 0 T ∫ Ω [ − Ψ t − c 0 ψ x − u Ψ x + ( 3 − 2 b ) u x x ψ x + ( 3 − b ) u x ψ x x − b y ψ x − Γ ψ x x x ] g d x d t = ∫ 0 T ∫ Ω [ ( I − ∂ x 2 ) ( y − z d ) ] g d x d t = ∫ 0 T ∫ Ω ( y − z d ) G d x d t , (5.9)

where G = g − g x x .

Utilizing Equation (4.6), the integration by parts on the left hand side of Equation (5.9) yields

∫ 0 T ∫ Ω ψ ( G t + c 0 g x + g y x + u G x + b g x y + b u x G + Γ g x x x ) d x d t = ∫ 0 T ∫ Ω ψ B ( v − v ∗ ) d x d t , (5.10)

where G = g − g x x . Therefore, combining Equation (5.9) and Equation (5.10), the sufficient and necessary optimality condition (5.7) is equivalent to

∫ 0 T ∫ Ω ψ ( v ∗ ; t , x ) B ( v − v ∗ ) d x d t + ( N v ∗ , v − v ∗ ) U ≥ 0 ,

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).

Shen, C.Y. (2017) Optimal Distributed Control Problem for the b-Equation. Journal of Applied Mathe- matics and Physics, 5, 1269-1300. https://doi.org/10.4236/jamp.2017.56108