# Optimal Control Problem with Necessary Optimality Conditions for the Viscous Dullin-Gottwald-Holm Equation

Authors: Jinsoo Hwang
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We study the quadratic cost optimal control problems for the viscous Dullin-Gottwald-Holm equation. The main novelty of this paper is to derive the necessary optimality conditions of optimal controls, corresponding to physically meaningful distributive observations. For this we prove the Gâteaux differentiability of nonlinear solution mapping on control variables. Moreover by making use of the second order Gâteaux differentiability of solution mapping on control variables, we prove the local uniqueness of optimal control. This is another novelty of the paper.


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## Body

## 1. Introduction

Recently, in the study of shallow water wave, Dullin et al. [1] derived a new integrable shallow water wave equation with linear and nonlinear dispersion as follows:
(1)&yt+2ωyx+3yyx-α2(yxxt+2yxyxx+yyxxx)+γyxxx=0,
where y is fluid velocity, α2 and γ/2ω are squares of length scales, and 2ω=gh is the linear wave speed for undisturbed water at rest at spatial infinity, where y and its spatial derivatives are taken to vanish. Letting α2→0, (1) reduces to the well-known Korteweg-de Vries (KDV) equation (linear dispersion case). And when letting γ→0, (1) reduces to the Camassa-Holm equation of [2] (nonlinear dispersion case). Usually we can rewrite (1) into
(2)mt+2ωyx+ymx+2yxm+γyxxx=0,
where m=y-α2yxx is a momentum variable. Physically, the third and fourth terms of the left side of (2) represent convection and stretching effects of unidirectional propagation of shallow water waves over a flat bottom, respectively (see [2, 3]).Many researchers studied the well-posedness of Cauchy problem for the DGH equation including various properties of the solution of it (see [4–6]).Recently, Shen et al. [7] studied the optimal control problem of the following viscous DGH equation (cf. [3]):
(3)mt+2ωyx+ymx+2yxm+γyxxx=νmxx,
where m=y-yxx and ν>0 stands for the viscosity constant of shallow water wave. As explained in Holm and Staley [8], the small viscosity makes sense to take into account the balance or relaxation between convection and stretching dynamics of shallow water wave.In [7] Shen et al. studied the distributive optimal control problems of (3) (cf. [3]). For this purpose, they modified (3) to Dirichlet boundary value problem in short interval and proved the existence and uniqueness of m in (3) by weak formulation. However the well-posedness of (3) with respect to y is unclear and the proof contained in [7] relies on the size of ν which is an extra condition. Further, in [7] they employed the quadratic cost objective functional to be minimized within an admissible control set with the distributive observation of m in (3) and only discussed the existence of optimal controls which minimize the quadratic cost. But the necessary optimality conditions of optimal controls have not been studied there.As for the necessary optimality condition of optimal controls, we can find a recent paper Sun [9]. By employing the Dubovitskii and Milyutin functional analytic approach, Sun has established in [9, Theorem 3] the Pontryagin maximum principle of the optimal control for the viscous DGH equation with the quite general cost which depends on m and not on y. Meanwhile, in this paper, we propose the quadratic cost functional for y, which is actually more reasonable than that for m, and establish the necessary optimality conditions of optimal controls due to Lions [10] in Theorems 5 and 7 for the physically meaningful observations z=y(u) and z=m(u), respectively. To this end, we successfully characterize the Gâteaux derivative Dy(u)(v-u) of y(v) in the direction v-u∈𝒰, where 𝒰 is a Hilbert space of control variables and u is an optimal control for quadratic cost.Actually, the extension of optimal control theory to quasilinear equations is not easy. Some researches have been devoted to the study of optimal control or identification problems in specific quasilinear equations. For instance, we can refer to Hwang and Nakagiri [11, 12] and Hwang [13, 14].The aim of this paper can be summarized as follows. Firstly, we clarify the well-posedness of (3) with respect to y in the Hadamard sense with appropriate initial value condition in short interval as posed in [7]. Secondly, based on the well-posedness result, we expand the optimal control theory due to Lions [10] with emphasis on deriving necessary optimality conditions of optimal controls in the following distributive control system:
(4)mt(v)-νmxx(v)+2ωyx(v)+2yx(v)m(v)+y(v)mx(v)+γyxxx(v)=f+Bvin(0,T)×(0,1),y(v;0,t)=y(v;1,t)=yx(v;0,t)=yx(v;1,t)=yxx(v;0,t)=yxx(v;1,t)=0,t∈[0,T],m(v;x,0)=m0=y0-y0,xxin(0,1),
where m(v)=y(v)-yxx(v), f is a forcing term, B is a controller, v is a control, and y(v) denotes the state for a given v∈𝒰.In order to apply the variational approach due to Lions [10] to our problem, we propose the quadratic cost functional J(v) as studied in Lions [10] which is to be minimized within 𝒰ad;  𝒰ad is an admissible set of control variables in 𝒰. We show the existence of u∈𝒰ad which minimizes the quadratic cost functional J(v). Then, we establish the necessary conditions of optimality of the optimal control u for some physically meaningful observation cases employing the associate adjoint systems. For this we successfully prove the Gâteaux differentiability of the nonlinear solution mapping v→y(v), which is used to define the associate adjoint systems.Moreover, in this paper we discuss the local uniqueness of optimal control. As widely known, it is unclear and difficult to verify the uniqueness of optimal control in nonlinear control problems. By employing the idea given in Ryu [15], we show the strict convexity of the quadratic cost J(v) in local time interval by utilizing the second order Gâteaux differentiability of the nonlinear solution mapping v→y(v). Whence by proving strict convexity of the quadratic cost with respect to the control variable, we prove the local uniqueness of optimal control. This is another novelty of the paper.


## 2. Preliminaries

For fixedT>0, we set Ω=(0,1) and Q=Ω×(0,T). The scalar products and norms on L2(Ω) and H0k(Ω), k=1,2,3, are denoted by (·,·)2, |·|2 and ((·,·))k, ∥·∥k, k=1,2,3, respectively. Then, by virtue of Poincare inequality, we can replace these scalar products and norms by ((·,·))k=(∂xk·,∂xk·)2 and ∥·∥k=|∂xk·|2, k=1,2,3, respectively. Let us denote the topological dual spaces of H0k(Ω), k=1,2,3, by H-k(Ω), k=1,2,3. We denote their duality pairing between H0k(Ω) and H-k(Ω) by 〈·,·〉k,-k, k=1,2,3.We consider the following Dirichlet boundary value problem for the viscous Dullin-Gottwald-Holm (DGH) equation:(5)mt-νmxx+2ωyx+2yxm+ymx+γyxxx=finQ,y(0,t)=y(1,t)=yx(0,t)=yx(1,t)=yxx(0,t)=yxx(1,t)=0,t∈[0,T],m(x,0)=m0=y0-y0,xxinΩ,
where m=y-yxx,  f is a forcing function, and m0 is an initial value.In order to define weak solutions of (5), we define some Hilbert spaces. At first, 𝒮(0,T) is defined by
(6)𝒮(0,T)={ϕ′∈L2(0,T;H01(Ω))}ϕ∣ϕ∈L2(0,T;H03(Ω)),ϕ′∈L2(0,T;H01(Ω))}
endowed with the norm
(7)∥ϕ∥𝒮(0,T)=(∥ϕ∥L2(0,T;H03(Ω))2+∥ϕ′∥L2(0,T;H01(Ω))2)1/2,
where ϕ′ denotes the first order distributional derivatives of ϕ. Further, 𝒲(0,T) is defined by
(8)𝒲(0,T)={ψ′∈L2(0,T;H-1(Ω))H01}ψ∣ψ∈L2(0,T;H01(Ω)),ψ′∈L2(0,T;H-1(Ω))H01}
endowed with the norm
(9)∥ψ∥𝒲(0,T)=(∥ψ∥L2(0,T;H01(Ω))2+∥ψ′∥L2(0,T;H-1(Ω))2)1/2,
where ψ′ denotes the first order distributional derivatives of ψ. We note here that 𝒮(0,T) and 𝒲(0,T) are continuously imbedded in C([0,T];H02(Ω)) and C([0,T];L2(Ω)), respectively (cf. Dautray and Lions [16, page 555]).From now on, we will omit writing the integral variables in the definite integral without any confusion.Lemma 1.
Letϕ satisfy the boundary conditions of (5) and ϕ-ϕxx∈𝒲(0,T). Then one has
(10)∥ϕ∥𝒮(0,T)≤C∥ϕ-ϕxx∥𝒲(0,T),
where C>0 is a constant.Proof.
According to the boundary conditions ofϕ, we have
(11)∥ϕ-ϕxx∥𝒲(0,T)2=∫0T|ϕx-ϕxxx|22dt∥ϕ-ϕxx∥𝒲(0,T)2+∫0T∥ϕt-ϕxxt∥H-1(Ω)2dt‍∥ϕ-ϕxx∥𝒲(0,T)2=∫0T(|ϕx|22+2|ϕxx|22+|ϕxxx|22)dt‍∥ϕ-ϕxx∥𝒲(0,T)2+∫0T∥ϕt-ϕxxt∥H-1(Ω)2dt.‍SinceI-∂x2:H01(Ω)→H-1(Ω) is an isomorphism, we can deduce that
(12)R.H.S.of(11)≥∫0T‍|ϕxxx|22dt+c0∫0T‍|ϕtx|22dt≥min⁡⁡{1,c0}(∥ϕ∥L2(0,T;H03(Ω))2+∥ϕ′∥L2(0,T;H01(Ω))2)=c1∥ϕ∥𝒮(0,T)2,
where c0,c1>0 are constants. Thus we prove this lemma.The following variational formulation is used to define theweak solution of (5).Definition 2.
A functiony∈𝒮(0,T) is said to be a weak solution of (5) if m=y-yxx∈𝒲(0,T) and m=y-yxx satisfies
(13)〈m′(·),ϕ〉-1,1+ν(mx(·),ϕx)2+2ω(yx(·),ϕ)2+2(yxm,ϕ)2+(y(·)mx(·),ϕ)2+γ(yxxx(·),ϕ)2=〈f(·),ϕ〉-1,1forallϕ∈H01(Ω)inthesenseof𝒟′(0,T),m(0)=m0=y0-y0,xx.In order to verify the well-posedness of (5), we partially refer to the result by Shen et al. [7]. The well-posedness of (5) in the sense of Hadamard can be given as follows.Theorem 3.
Assume thatν>0, f∈L2(0,T;H-1(Ω)), and y0∈H02(Ω). Then the problem (5) has a unique solution y in 𝒮(0,T). And the solution mapping p=(y0,f)→y(p) of P≡H02(Ω)×L2(0,T;H-1(Ω)) into 𝒮(0,T) is a local Lipschitz continuous; that is, for each p1=(y01,f1)∈P and p2=(y02,f2)∈P, one has the inequality
(14)∥y1(p1)-y2(p2)∥𝒮(0,T)≤C(∥y01-y02∥H02(Ω)+∥f1-f2∥L2(0,T;H-1(Ω))),
where C is a constant which depends on y1 and y2.Remark 4.
In [7], the well-posedness of (5) is partially verified, which is indeed the case that the viscosity constant ν>0 is large enough. However, as we will see in the Appendix, such an extra assumption can be removed.Proof of Theorem3.
By utilizing the result of [7], combined with Lemma 1, we can know that (5) possesses a unique solution y∈𝒮(0,T) under the data condition p=(y0,f)∈H02(Ω)×L2(0,T;H-1(Ω)).
Based on the above result, we prove the inequality (14). For that purpose, we denote y1-y2≡y(p1)-y(p2) by ψ and yi-yi,xx by mi,i=1,2. Then, we can observe from (5) that
(15)Ψt-νΨxx+2ωψx+2ψxm1+2y2,xΨ+ψm1,x+y2Ψx+γψxxx=f1-f2inQ,ψ(0,t)=ψ(1,t)=ψx(0,t)=ψx(1,t)=ψxx(0,t)ψ(0,t)=ψxx(1,t)=0,t∈[0,T],Ψ(x,0)=Ψ0=m01-m02inΩ,
where Ψ=ψ-ψxx and m0i=y0i-y0,xxi, i=1,2. Multiplying Ψ in both sides of (15), we have
(16)12ddt|Ψ|22+ν|Ψx|22=-2ω(ψx,Ψ)212ddt|Ψ|22+ν|Ψx|22-(2ψxm1,Ψ)2-(2y2,xΨ,Ψ)212ddt|Ψ|22+ν|Ψx|22-(ψm1,x,Ψ)2-(y2Ψx,Ψ)212ddt|Ψ|22+ν|Ψx|22-γ(ψxxx,Ψ)2+〈f1-f2,Ψ〉-1,1.
And we integrate (16) over [0,t] to have
(17)12|Ψ(t)|22+ν∫0t‍|Ψx|22ds=12|Ψ0|22-2ω∫0t‍(ψx,Ψ)2ds-∫0t‍(2ψxm1,Ψ)2ds-∫0t‍(2y2,xΨ,Ψ)2ds-∫0t‍(ψm1,x,Ψ)2ds-∫0t‍(y2Ψx,Ψ)2ds-γ∫0t‍(ψxxx,Ψ)2ds+∫0t‍〈f1-f2,Ψ〉-1,1ds.
By Sobolev embedding theorem, mi∈𝒲(0,T)↪C([0,T];L2(Ω)), i=1,2, and |Ψ|22=|ψ|22+2|ψx|22+|ψxx|22, the right members of (17) can be estimated as follows:
(18)|-2ω∫0t‍(ψx,Ψ)2ds|≤2ω∫0t‍|ψx|2|Ψ|2ds≤2ω∫0t‍|Ψ|22ds;|-∫0t‍(2ψxm1,Ψ)2ds|≤2∫0t‍∥ψx∥L∞(Ω)|m1|2|Ψ|2ds≤c0∥m1∥C([0,T];L2(Ω))∫0t‍|ψxx|2|Ψ|2ds≤c1∫0t‍|Ψ|22ds;|-∫0t‍(2y2,xΨ,Ψ)2ds|≤2∥y2,x∥L∞(Q)∫0t‍|Ψ|22ds≤c2∫0t‍|Ψ|22ds;|-∫0t‍(ψm1,x,Ψ)2ds|≤∫0t‍∥ψ∥L∞(Ω)|m1,x|2|Ψ|2ds≤c3∫0t‍|ψx|2|m1,x|2|Ψ|2ds≤c4∫0t‍|m1,x|2|Ψ|22ds;|-∫0t‍(y2Ψx,Ψ)2ds|≤∥y2∥L∞(Q)∫0t‍|Ψx|2|Ψ|2ds≤c5∫0t‍|Ψ|22ds+ν6∫0t‍|Ψx|22ds;|-γ∫0t‍(ψxxx,Ψ)2ds|=|γ∫0t‍(ψxx,Ψx)2ds|≤γ∫0t‍|ψxx|2|Ψx|2ds≤γ∫0t‍|Ψ|2|Ψx|2ds≤c6∫0t‍|Ψ|22ds+ν6∫0t‍|Ψx|22ds|∫0t‍〈f1-f2,Ψ〉-1,1ds|≤∫0t‍∥f1-f2∥H-1(Ω)|Ψx|2ds≤c7∥f1-f2∥L2(0,T;H-1(Ω))2+ν6∫0t‍|Ψx|22ds,
where c0,…,c7 are constants. We replace the right hand side of (17) by the right members of (18). Then we have
(19)|Ψ(t)|22+∫0t‍|Ψx|22ds≤c8(|Ψ0|22+∥f1-f2∥L2(0,T;H-1(Ω))2+∫0t‍|m1,x|2|Ψ|22ds),
where c8 is a constant, depending on y1 and y2. And we apply the Gronwall inequality to (19) to obtain
(20)|Ψ(t)|22+∫0t‍|Ψx|22ds≤c8(|Ψ0|22+∥f1-f2∥L2(0,T;H-1(Ω))2)×exp⁡⁡(c8∥m1∥L1(0,T;H01(Ω)))≤c9(|Ψ0|22+∥f1-f2∥L2(0,T;H-1(Ω))2),
where c9 is a constant, depending on y1 and y2. Next we estimate Ψt in (15) as follows:
(21)∥Ψt∥H-1(Ω)≤ν|Ψx|2+2ω|ψx|2+2∥m1∥L∞(Ω)|ψx|2∥Ψt∥H-1(Ω)+2∥y2,x∥L∞(Q)|Ψ|2∥Ψt∥H-1(Ω)+∥ψ∥L∞(Ω)|m1,x|2+∥y2∥L∞(Q)|Ψx|2∥Ψt∥H-1(Ω)+γ|ψxxx|2+∥f1-f2∥H-1(Ω).
By Sobolev embedding theorem, we have the following inequality:
(22)R.H.S.of(21)≤c10|Ψx|2+c11(|m1,x|2+1)|Ψ|2R.H.S.of(21)+∥f1-f2∥H-1(Ω),
where c10, c11 are constants, depending on y1 and y2. By (21) and (22) we can obtain
(23)∥Ψt∥L2(0,T;H-1(Ω))2≤c12(∥Ψ∥L2(0,T;H01(Ω))2+(∥m1∥L2(0,T;H01(Ω))2+1)×∥Ψ∥L∞(0,T;L2(Ω))2+∥f1-f2∥L2(0,T;H-1(Ω))2)≤c13(|Ψ0|22+∥f1-f2∥L2(0,T;H-1(Ω))2),
where c12, c13 are constants, depending on y1 and y2. We can deduce that (20) and (23) imply
(24)∥Ψ∥𝒲(0,T)≤c14(|Ψ0|2+∥f1-f2∥L2(0,T;H-1(Ω))),
where c14 is a constant, depending on y1 and y2. Finally from Lemma 1 and (24) we have
(25)∥ψ∥𝒮(0,T)≡∥y1(p1)-y2(p2)∥𝒮(0,T)≤c15(|Ψ0|2+∥f1-f2∥L2(0,T;H-1(Ω)))≤C(∥y01-y02∥H02(Ω)+∥f1-f2∥L2(0,T;H-1(Ω))),
where c15, C are constants, depending on y1 and y2. Thus we complete the proof.


## 3. Quadratic Cost Optimal Control Problems

In this section we study the quadratic cost optimal control problems for the viscous DGH equation due to the theory of Lions [10]. Let 𝒰 be a Hilbert space of control variables, and let B be an operator,
(26)B∈ℒ(𝒰,L2(0,T;L2(Ω))),
called a controller. We consider the following nonlinear control system:
(27)mt(v)-νmxx(v)+2ωyx(v)+2yx(v)m(v)+y(v)mx(v)+γyxxx(v)=f+BvinQ,y(v;0,t)=y(v;1,t)=yx(v;0,t)y(v;0,t)=yx(v;1,t)=yxx(v;0,t)y(v;0,t)=yxx(v;1,t)=0,t∈[0,T],m(v;x,0)=m0=y0-y0,xx,inΩ,
where m(v)=y(v)-yxx(v), m0∈L2(Ω), f∈L2(0,T;L2(Ω)), and v∈𝒰 is a control. By virtue of Theorem 3 and (26), we can define uniquely the solution map v→y(v) of 𝒰 into 𝒮(0,T). We will call the solution y(v) of (27) the state of the control system (27). The observation of the state is assumed to be given by
(28)z(v)=Cy(v),C∈ℒ(𝒮(0,T),M),
where C is an operator called the observer and M is a Hilbert space of observation variables. The quadratic cost function associated with the control system (27) is given by
(29)J(v)=∥Cy(v)-Yd∥M2+(Rv,v)𝒰forv∈𝒰,
where Yd∈M is a desired value of y(v) and R∈ℒ(𝒰,𝒰) is symmetric and positive; that is,
(30)(Rv,v)𝒰=(v,Rv)𝒰≥d∥v∥𝒰2
for some d>0. Let 𝒰ad be a closed convex subset of 𝒰, which is called the admissible set. An element u∈𝒰ad which attains the minimum of J(v) over 𝒰ad is called an optimal control for the cost (29).In this section we will characterize the optimal controls by giving necessary conditions for optimality. For this it is necessary to write down the necessary optimality condition,(31)DJ(u)(v-u)≥0forallv∈𝒰ad,
and to analyze (31) in view of the proper adjoint state system, where DJ(u) denote the Gâteaux derivative of J(v) at v=u. And we study local uniqueness of the optimal control.As indicated in Section1, we show the existence of an optimal control and give the characterizations of them.


### 3.1. Existence of the Optimal Control

Now we show the existence of an optimal controlu for the cost (29).Theorem 5.
Assume that the hypotheses of Theorem3 are satisfied. Then there exists at least one optimal control u for the control problem (27) with the cost (29).Proof.
SetJ=inf⁡v∈𝒰adJ(v). Since 𝒰ad is nonempty, there is a sequence {vn} in 𝒰 such that
(32)inf⁡v∈𝒰adJ(v)=lim⁡n→∞J(vn)=J.
Obviously {J(vn)} is bounded in R+. Then by (30) there exists a constant K0>0 such that
(33)d∥vn∥𝒰2≤(Rvn,vn)𝒰≤J(vn)≤K0.
This shows that {vn} is bounded in 𝒰. Since 𝒰ad is closed and convex, we can choose a subsequence (denoted again by {vn}) of {vn} and find a u∈𝒰ad such that
(34)vn⟶uweaklyin𝒰
as n→∞. From now on, each state yn=y(vn)∈𝒮(0,T) corresponding to vn is the solution of
(35)mn,t-νmn,xx+2ωyn,x+2yn,xmn+ynmn,x+γyn,xxx=f+BvninQ,yn(0,t)=yn(1,t)=yn,x(0,t)=yn,x(1,t)=yn,xx(0,t)=yn,xx(1,t)=0,t∈[0,T],mn(x,0)=m0=y0-y0,xxinΩ,
where mn=yn-yn,xx. By (33) the term Bvn is estimated as
(36)∥Bvn∥L2(0,T;L2(Ω))≤∥B∥ℒ(𝒰,L2(0,T;L2(Ω)))∥vn∥𝒰∥Bvn∥L2(0,T;L2(Ω))≤∥B∥ℒ(𝒰,L2(0,T;L2(Ω)))K0d-1≡K1.
Hence we can deduce from Theorem 3 that
(37)∥mn∥𝒲(0,T)≤C0(∥y0∥H02(Ω)+∥f∥L2(0,T;L2(Ω))+K1)
for some C0>0. And also from Lemma 1 we can know that
(38)∥yn∥𝒮(0,T)≤C1(∥y0∥H02(Ω)+∥f∥L2(0,T;L2(Ω))+K1)
for some C1>0. Therefore, by the extraction theorem of Rellich’s, we can find a subsequence of {mn}, say again {mn}, and find a m=y-yxx∈𝒲(0,T) such that
(39)mn⟶mweaklyin𝒲(0,T).
By using the fact that H01(Ω)↪L2(Ω) is compact and by virtue of (39), we can refer to the result of the Aubin-Lions-Temam’s compact imbedding theorem (cf. Temam [17, page 271]) to verify that {mn} is precompact in L2(0,T;L2(Ω)). Hence there exists a subsequence {mnk}⊂{mn} such that
(40)mnk⟶mstronglyinL2(0,T;L2(Ω))ask⟶∞.
Since mn=yn-yn,xx∈𝒲(0,T)↪C([0,T];L2(Ω)), we know that yn∈C([0,T];H02(Ω)). And from (40) we can choose a subsequence of {ynk}, denoted again by {ynk} such that
(41)ynk(t)⟶y(t)stronglyinH02(Ω)fora.e.t∈[0,T].
We use (39)–(41) and apply the Lebesgue dominated convergence theorem to have
(42)2ynk,xmnk⟶2yxmstronglyinL2(0,T;L2(Ω)),ynkmnkx⟶ymxweaklyinL2(0,T;L2(Ω))
as k→∞. We replace yn and mn by ynk and mnk, respectively, and take k→∞ in (35). Then by the standard argument in Dautray and Lions [16, pages 561–565], we conclude that the limit m satisfies
(43)mt-νmxx+2ωyx+2yxm+ymx+γyxxx=f+BuinQ,y(0,t)=y(1,t)=yx(0,t)=yx(1,t)=yxx(0,t)y(0,t)=yxx(1,t)=0,t∈[0,T],m(x,0)=m0=y0-y0,xx,inΩ
in weak sense, where m=y-yxx. Moreover the uniqueness of weak solutions in (43) via Theorem 3 enables us to conclude that y=y(u) in 𝒮(0,T), which implies y(vn)→y(u) weakly in 𝒮(0,T). Since C is continuous on 𝒮(0,T) and ∥·∥M is lower semicontinuous, it follows that
(44)∥Cy(u)-zd∥M≤liminf⁡n→∞∥Cy(vn)-zd∥M.
It is also clear from liminf⁡k→∞∥R1/2vn∥𝒰≥∥R1/2v∥𝒰 that liminf⁡k→∞(Rvn,vn)𝒰≥(Ru,u)𝒰. Hence
(45)J=liminf⁡n→∞J(vn)≥J(u).
But since J(u)≥J by definition, we conclude that J(u)=inf⁡v∈𝒰adJ(v). This completes the proof.


### 3.2.  Gâteaux Differentiability of Solution Mapping

In order to characterize the optimal control which satisfies the necessary optimality condition (31), we need to prove the Gâteaux differentiability of the mapping v→y(v) of 𝒰→𝒮(0,T).Definition 6.
The solution mapv→y(v) of 𝒰 into 𝒮(0,T) is said to be Gâteaux differentiable at v=u if for any w∈𝒰 there exists a Dy(u)∈ℒ(𝒰,𝒮(0,T)) such that
(46)∥1λ(y(u+λw)-y(u))-Dy(u)w∥𝒮(0,T)⟶0asλ⟶0.
The operator Dy(u) denotes the Gâteaux derivative of y(u) at v=u and the function Dy(u)w∈𝒮(0,T) is called the Gâteaux derivative in the direction w∈𝒰, which plays an important role in the nonlinear optimal control problem.Theorem 7.
The mapv→y(v) of 𝒰 into 𝒮(0,T) is Gâteaux differentiable at v=u and such the Gâteaux derivative of y(v) at v=u in the direction v-u∈𝒰, say z=Dy(u)(v-u), is a unique solution of the following problem:
(47)𝒵t-ν𝒵xx+2ωzx+2zxm+2yx(u)𝒵+zmx+y(u)𝒵x+γzxxx=B(v-u)inQ,z(0,t)=z(1,t)=zx(0,t)=zx(1,t)=zxx(0,t)=zxx(1,t)=0,t∈[0,T],𝒵(x,0)=0inΩ,
where m=y(u)-yxx(u) and 𝒵=z-zxx.Proof.
Letλ∈(-1,1), λ≠0. We set w=v-u and
(48)zλ=λ-1(y(u+λw)-y(u)).
Then zλ satisfies
(49)𝒵λ,t-ν𝒵λ,xx+2ωzλ,x+2zλ,xmλ+2yx(u)𝒵λ+zλmλ,x+y(u)𝒵λ,x+γzλ,xxx=BwinQ,zλ(0,t)=zλ(1,t)=zλ,x(0,t)=zλ,x(1,t)=zλ,xx(0,t)zλ(0,t)=zλ,xx(1,t)=0,t∈[0,T],𝒵λ(x,0)=0inΩ,
where mλ=y(u+λw)-yxx(u+λw) and 𝒵λ=zλ-zλ,xx.
By the continuity of (14), we have
(50)∥y(u+λw)-y(u)∥𝒮(0,T)≤|λ|C∥Bw∥L2(0,T;L2(Ω)),
where C is a constant, depending on y(u) and y(u+λw). Hence we have
(51)∥zλ∥𝒮(0,T)≤C∥Bw∥L2(0,T;L2(Ω))<∞.
Therefore, we can infer that there exists a z∈𝒮(0,T) and a sequence {λk}⊂(-1,1) tending to 0 such that
(52)zλk⟶zweaklyin𝒮(0,T)
as k→∞. Since the imbedding 𝒮(0,T)↪L2(0,T;H02(Ω)) is compact ([17, page 271]), it is implied from (52) that
(53)zλk(t)⟶z(t)stronglyinH02(Ω)a.e.t∈[0,T]
for some {λk}⊂(-1,1) tending to 0 as k→∞. Whence by (50)–(53) and Lebesgue dominated convergence theorem we can easily show that
(54)2zλk,xmλk⟶2zxmstronglyinL2(0,T;L2(Ω)),(55)zλkmλk,x⟶zmxstronglyinL2(0,T;L2(Ω)),(56)𝒵λk⟶𝒵weaklyinL2(0,T;H01(Ω))
as k→∞, where m=y(u)-yxx(u) and 𝒵=z-zxx. And also we can deduce from (49), (52), and (56) that
(57)𝒵λk,t⟶𝒵tweaklyinL2(0,T;H-1(Ω))
as k→∞.
Hence we can see from (52) to (57) that zλ→z=Dy(u)w weakly in 𝒮(0,T) as λ→0 in which z is a solution of (47). This convergence can be improved by showing the strong convergence of {zλ} also in the topology of 𝒮(0,T).
Subtracting (47) from (49) and denoting zλ-z by ϕλ, we obtain that
(58)Φλ,t-νΦλ,xx+2ωϕλ,x+2yx(u)Φλ+y(u)Φλ,x+γϕλ,xxx=ϵ(λ)inQ,ϕλ(0,t)=ϕλ(1,t)=ϕλ,x(0,t)=ϕλ,x(1,t)=ϕλ,xx(0,t)ϕλ(0,t)=ϕλ,xx(1,t)=0,t∈[0,T],Φλ(x,0)=0inΩ,
where Φλ=ϕλ-ϕλ,xx and ϵ(λ)=-2zλ,xmλ+2zxm-zλmλ,x+zmx.
From (54) and (55) we know that
(59)ϵ(λ)⟶0stronglyinL2(0,T;L2(Ω))asλ⟶0.
In order to estimate ϕλ we multiply Φλ in both sides of (58) and integrate it over [0,t] to have
(60)|Φλ(t)|22+2ν∫0t‍|Φλ,x|22ds=-2∫0t‍(2ωϕλ,x,Φλ)2ds-2∫0t‍(2yx(u)Φλ,Φλ)2ds-2∫0t‍(y(u)Φλ,x,Φλ)2ds-2∫0t‍(γϕλ,xxx,Φλ)2ds+2∫0t‍(ϵ(λ),Φλ)2ds.
The integral parts of the right member of (60) can be estimated as follows:
(61)|-2∫0t‍(2ωϕλ,x,Φλ)2ds|≤4ω∫0t‍|ϕλ,x|2|Φλ|2ds≤4ω∫0t‍|Φλ|22ds;(62)|-2∫0t‍(2yx(u)Φλ,Φλ)2ds|≤4∥yx(u)∥L∞(Q)∫0t‍|Φλ|22ds≤c0∫0t‍|Φλ|22ds;(63)|-2∫0t‍(y(u)Φλ,x,Φλ)2ds|≤2∥y(u)∥L∞(Q)×∫0t‍|Φλ,x|2|Φλ|2ds≤ν2∫0t‍|Φλ,x|22ds+c1∫0t‍|Φλ|22ds;(64)|-2∫0t‍(γϕλ,xxx,Φλ)2ds|=|2∫0t‍(γϕλ,xx,Φλ,x)2ds|≤2γ∫0t‍|Φλ|2|Φλ,x|2ds≤ν2∫0t‍|Φλ,x|22ds+c2∫0t‍|Φλ|22ds;(65)|2∫0t‍(ϵ(λ),Φλ)2ds|≤2∫0t‍|ϵ(λ)|2|Φλ|2ds≤∫0t‍|ϵ(λ)|22ds+∫0t‍|Φλ|22ds,
where c0, c1, and c2 are constants. We replace the right hand side of (62) by the right members of (61)–(65). And we apply the Gronwall inequality to the replaced inequality; then we arrive at
(66)|Φλ(t)|22+∫0t‍|Φλ,x|22ds≤C∥ϵ(λ)∥L2(0,T;L2(Ω))2,
where C is a constant. By virtue of (59) and (66) we deduce that
(67)Φλ⟶0inC([0,T];L2(Ω))∩L2(0,T;H01(Ω))00000000000000000000000000000000asλ⟶0.
As in (21)–(23), we have from (58), (66), and (67) that
(68)Φλ,t⟶0stronglyinL2(0,T;H-1(Ω))asλ⟶0.
Therefore (67) and (68) mean
(69)Φλ⟶0stronglyin𝒲(0,T)asλ⟶0.
Whence from Lemma 1(70)zλ(·)⟶z(·)stronglyin𝒮(0,T)asλ⟶0.
This completes the proof.Theorem7 means that the cost J(v) is Gâteaux differentiable at u in the direction v-u and the optimality condition (31) is rewritten by
(71)(Cy(u)-Yd,C(Dy(u)(v-u)))M+(Ru,v-u)𝒰=〈C*ΛM(Cy(u)-Yd),Dy(u)(v-u)〉𝒮(0,T)′,𝒮(0,T)+(Ru,v-u)𝒰≥0,∀v∈𝒰ad,
where ΛM is the canonical isomorphism M onto M′ and Yd∈M is a desired value.


### 3.3.  Necessary Condition of Optimal Control

In this section we will characterize the optimal controls by giving necessary condition (71) for optimality for the following physically meaningful observations.(1)
We takeM=L2(Q) and C1∈ℒ(𝒮(0,T),M) and observe
(72)z(v)=C1y(v)=y(v;·)∈L2(Q).(2)
We takeM=L2(Q) and C2∈ℒ(𝒮(0,T),M) and observe
(73)z(v)=C2y(v)=(I-∂x2)y(v;·)≡m(v;·)∈L2(Q). Since y∈𝒮(0,T)⊂C([0,T];H02(Ω)) by Theorem 3, the above observations are meaningful.Due to Lions [10], we construct the necessary condition of optimal control via appropriate adjoint equation. In order to follow the idea we need to introduce and analyze the following adjoint equation for distributive observations:
(74)-𝒫t-ν𝒫xx-2ωpx(u)-2(m(u)p(u)),x+(I-∂x2)(2yx(u)p(u))+mx(u)p(u)-(I-∂x2)(y(u)p(u)),x-γpxxx(u)=C*ΛM(Cy(u)-Yd)inQ,p(u;0,t)=p(u;1,t)=px(u;0,t)=px(u;1,t)=0,t∈[0,T],𝒫(x,T)=0inΩ,
where C=C1 or C2, 𝒫=p(u)-pxx(u), and m(u)=y(u)-yxx(u). In order to show the well-posedness of (74), we introduce the solution Hilbert space W(H02(Ω),L2(Ω)) defined by
(75)W(H02(Ω),L2(Ω))≜{ψ∣ψ∈L2(0,T;H02(Ω)),ψ′∈L2(0,T;L2(Ω))}
equipped with the norm
(76)∥ψ∥W(H02(Ω),L2(Ω))=(∥ψ∥L2(0,T;H02(Ω))2+∥ψ′∥L2(0,T;L2(Ω))2)1/2.
We remark that W(H02(Ω),L2(Ω)) is continuously embedded in C([0,T];H01(Ω)) (cf. Dautray and Lions [16, page 555]).In the following proposition we show the well-posedness of (74).Proposition 8.
Assume thatC*ΛM(Cy(u)-Yd)∈L2(0,T;H-2(Ω)); then by reversing the direction of time t→T-t, (74) admits a unique solution p(u) satisfying
(77)(i)p(u)∈W(H02(Ω),L2(Ω)),(ii)(pt(u)-νpxx(u)-y(u)px(u)+yx(u)p(u),Φ)2-(2ωpx(u)-mx(u)p(u),ϕ)2+(2m(u)p(u)+γpxx(u),ϕx)2=〈g,ϕ〉-2,2,∀ϕ∈H02(Ω)inthesenseof𝒟′(0,T),(iii)𝒫(x,0)=0inΩ,
where g=C*ΛM(Cy(u)-Yd) and Φ=ϕ-ϕxx.The proof of Proposition8 is given in the Appendix.Remark 9.
As we will see, there are some merits in takingW(H02(Ω),L2(Ω)) as the solution space for adjoint equations. For the observation (72), even though we can take the adjoint system in the space 𝒮(0,T) with additional boundary conditions, we can derive the same necessary optimality condition of optimal controls through the less regular solution p(u)∈W(H02(Ω),L2(Ω)). Therefore, W(H02(Ω),L2(Ω)) is preferred solution space of adjoint equation for the observation (72). And also, for the observation (73), we need to solve adjoint equation in W(H02(Ω),L2(Ω)) because of the less regular data condition than that of the observation (72).


#### 3.3.1. Case of the Observation (72)

In this subsection we consider the cost functional expressed by(78)J(v)=∫Q|y(v;x,t)-Yd(x,t)|2‍dxdt+(Rv,v)𝒰,000000000000000000000000000∀v∈𝒰ad⊂𝒰,
where Yd∈L2(Q) is a desired value. Let u be the optimal control subject to (27) and (78). Then the optimality condition (71) is represented by
(79)∫Q‍(y(u;x,t)-Yd(x,t))z(x,t)dxdt+(Ru,v-u)𝒰≥0,000000000000000000000000000000000000∀v∈𝒰ad,
where z is the solution of (47).Now we will formulate the following adjoint system to describe the optimality condition:(80)-𝒫t-ν𝒫xx-2ωpx(u)-2(m(u)p(u)),x+(I-∂x2)(2yx(u)p(u))+mx(u)p(u)-(I-∂x2)(y(u)p(u)),x-γpxxx(u)=y(u)-YdinQ,p(u;0,t)=p(u;1,t)=px(u;0,t)p(u;0,t)=px(u;1,t)=0,t∈[0,T],𝒫(x,T)=0inΩ,
where 𝒫=p(u)-pxx(u) and m(u)=y(u)-yxx(u).Remark 10.
Taking into account the observation conditionsy(u)-Yd∈L2(Q)=L2(0,T;L2(Ω))⊂L2(0,T;H-2(Ω)), we can assert that (80), reversing the direction of time t→T-t, admits a unique solution p(u)∈W(H02(Ω),L2(Ω)) by Proposition 8.Now we proceed the calculations. We multiply both sides of the weak form of (80) by z(t) and integrate it over [0,T]. Then we have
(81)∫0T〈-𝒫t-ν𝒫xx-2ωpx(u)-2(m(u)p(u)),x‍00000+(I-∂x2)(2yx(u)p(u)),z〉-2,2dt+∫0T〈mx(u)p(u)-(I-∂x2)(y(u)p(u)),x‍000000-γpxxx(u),z∂x2〉-1,1dt=∫0T(y(u)-Yd,z)2dt,‍
where 𝒫=p(u)-pxx(u). By (47) for z, we can verify by integration by parts that the left hand side of (81) yields
(82)∫0T(p(u),𝒵t-ν𝒵xx+2ωzx+2zxm(u)‍+2yx(u)𝒵+mx(u)z+y(u)𝒵x+γzxxx)2dt=∫0T(p(u),B(v-u))2dt,‍
where 𝒵=z-zxx. Therefore, by (81) and (82) we can deduce that the optimality condition (79) is equivalent to
(83)∫0T‍(p(u),B(v-u))2dt+(Ru,v-u)𝒰≥0,∀v∈𝒰ad.
Hence, we give the following theorem.Theorem 11.
The optimal controlu for (78) is characterized by the following system of equations and inequality:
(84)mt(u)-νmxx(u)+2ωyx(u)+2yx(u)m(u)+y(u)mx(u)+γyxxx(u)=f+BuinQ,y(u;0,t)=y(u;1,t)=yx(u;0,t)y(u;0,t)=yx(u;1,t)=yxx(u;0,t)y(u;0,t)=yxx(u;1,t)=0,t∈[0,T],m(u;x,0)=m0=y0-y0,xx,inΩ,(85)-𝒫t-ν𝒫xx-2ωpx(u)-2(m(u)p(u)),x+(I-∂x2)(2yx(u)p(u))+mx(u)p(u)-(I-∂x2)(y(u)p(u)),x-γpxxx(u)=y(u)-YdinQ,p(u;0,t)=p(u;1,t)=px(u;0,t)p(u;0,t)=px(u;1,t)=0,t∈[0,T],𝒫(x,T)=0inΩ,(86)∫0T‍(p(u),B(v-u))2dt+(Ru,v-u)𝒰≥0,000000000000000000000000000∀v∈𝒰ad,
where m(u)=y(u)-yxx(u) and 𝒫=p(u)-pxx(u).


#### 3.3.2. Case of the Observation (73)

We consider the following momentum’s distributive cost functional expressed by(87)J(v)=∫Q‍|m(v)-Yd|2dxdt+(Rv,v)𝒰,∀v∈𝒰ad,
where m(v)=y(v)-yxx(v) and Yd∈L2(Q). Let u be the optimal control subject to (27) and (87). Then the optimality condition (71) is rewritten as
(88)∫0T‍(m(u)-Yd,𝒵)2dxdt+(Ru,v-u)𝒰≥0,∀v∈𝒰ad,
where 𝒵=z-zxx and z is the solution of (47). As before we formulate the following adjoint system to describe the optimality condition:
(89)-𝒫t-ν𝒫xx-2ωpx(u)-2(m(u)p(u)),x+(I-∂x2)(2yx(u)p(u))+mx(u)p(u)-(I-∂x2)(y(u)p(u)),x-γpxxx(u)=(I-∂x2)(m(u)-Yd)inQ,p(u;0,t)=p(u;1,t)=px(u;0,t)p(u;0,t)=px(u;1,t)=0,t∈[0,T],𝒫(x,T)=0inΩ,
where 𝒫=p(u)-pxx(u) and m(u)=y(u)-yxx(u).Remark 12.
Since the observation conditionsm(u)-Yd∈L2(Q)=L2(0,T;L2(Ω)), we know that (I-∂x2)(m(u)-Yd)∈L2(0,T;H-2(Ω)). Hence by reversing the direction of time t→T-t and applying Proposition 8, we deduce that (89) admits a unique solution p(u)∈W(H02(Ω),L2(Ω)).As we did before, we multiply both sides of the weak form of (89) by z(t) and integrate it over [0,T]. Then we have
(90)∫0T〈-𝒫t-ν𝒫xx-2ωpx(u)-2(m(u)p(u)),x‍00000+(I-∂x2)(2yx(u)p(u)),z〉-2,2dt+∫0T〈mx(u)p(u)-(I-∂x2)(y(u)p(u)),x‍0000000000-γpxxx(u),z∂x2〉-1,1dt=∫0T‍〈(I-∂x2)(m(u)-Yd),z〉-2,2dt=∫0T‍(m(u)-Yd,𝒵)2dt,
where 𝒫=p(u)-pxx(u), m(u)=y(u)-yxx(u), and 𝒵=z-zxx. By (47) for z the integration by parts of the left hand side of (90) yields
(91)∫0T(p(u),𝒵t-ν𝒵xx+2ωzx+2zxm(u)+2yx(u)𝒵‍+mx(u)z+y(u)𝒵x+γzxxx)2dt=∫0T‍(p(u),B(v-u))2dt,
where 𝒵=z-zxx. Therefore, combining (90) and (91), we can deduce that the optimality condition (88) is equivalent to
(92)∫0T‍(p(u),B(v-u))2dt+(Ru,v-u)𝒰≥0,∀v∈𝒰ad.
Hence, we give the following theorem.Theorem 13.
The optimal controlu for (87) is characterized by the following system of equations and inequality:
(93)mt(u)-νmxx(u)+2ωyx(u)+2yx(u)m(u)+y(u)mx(u)+γyxxx(u)=f+BuinQ,y(u;0,t)=y(u;1,t)=yx(u;0,t)y(u;0,t)=yx(u;1,t)=yxx(u;0,t)y(u;0,t)=yxx(u;1,t)=0,t∈[0,T],m(u;x,0)=m0=y0-y0,xxinΩ,(94)-𝒫t-ν𝒫xx-2ωpx(u)-2(m(u)p(u)),x+(I-∂x2)(2yx(u)p(u))+mx(u)p(u)-(I-∂x2)(y(u)p(u)),x-γpxxx(u)=(I-∂x2)(m(u)-Yd)inQ,p(u;0,t)=p(u;1,t)=px(u;0,t)p(u;0,t)=px(u;1,t)=0,t∈[0,T],𝒫(x,T)=0inΩ,(95)∫0T‍(p(u),B(v-u))2dt+(Ru,v-u)𝒰≥0,∀v∈𝒰ad,
where m(u)=y(u)-yxx(u) and 𝒫=p(u)-pxx(u).


### 3.4. Local Uniqueness of an Optimal Control

We note that the uniqueness of an optimal control in nonlinear problem is not assured. However, referring to the result in [15], we can show the local uniqueness of the optimal control for our problem. In order to show the uniqueness of optimal control by using strict convexity of quadratic cost (cf. [18]) we consider the following proposition.Proposition 14.
The mapv→y(v) of 𝒰 into 𝒮(0,T) is second order Gâteaux differentiable at v=u and such the second order Gâteaux derivative of y(v) at v=u in the direction v-u∈𝒰, say g=D2y(u)(v-u,v-u), is a unique solution of the following problem:
(96)𝒢t-ν𝒢xx+2ωgx+y(u)𝒢x+2yx(u)𝒢+2gxm+gmx+4zx𝒵+2z𝒵x+γgxxx=0inQ,g(0,t)=g(1,t)=gx(0,t)=gx(1,t)=gxx(0,t)g(0,t)=gxx(1,t)=0,t∈[0,T],𝒢(x,0)=0inΩ,
where 𝒢=g-gxx, 𝒵=z-zxx, and z is the solution of (47).Proof.
The proof is similar to that of Theorem7.Lemma 15.
Letg be the solution of (96). Then we can show that
(97)∥g∥𝒮(0,T)≤C∥v-u∥𝒰2,
where C>0 is a constant.Proof.
Letz be the solution of (47). Then, using the same arguments as in (5), we can deduce that
(98)∥𝒵∥𝒲(0,T)≤c0∥B(v-u)∥L2(0,T;L2(Ω))∥𝒵∥𝒲(0,T)≤c0∥B∥ℒ(𝒰;L2(0,T;L2(Ω)))∥v-u∥𝒰∥𝒵∥𝒲(0,T)≤c1∥v-u∥𝒰,
where 𝒵=z-zxx and c0, c1 are constants which does not depend on g. And also for the solution g of (96), we can show that
(99)∥g∥𝒮(0,T)≤c2∥-4zx𝒵-2z𝒵x∥L2(0,T;L2(Ω))∥g∥𝒮(0,T)≤c3(∥zx∥L∞(Q)+∥z∥L∞(Q))∥𝒵∥L2(0,T;H01(Ω))∥g∥𝒮(0,T)≤c4∥z∥𝒮(0,T)∥𝒵∥𝒲(0,T)∥g∥𝒮(0,T)≤c5∥𝒵∥𝒲(0,T)2,
where c2,…,c5 are constants. Combining (98) with (99), we have (97).We prove the local uniqueness of the optimal control.Theorem 16.
WhenT is small enough, then there is a unique optimal control for the problem (29) for observations (72) and (73).Proof.
We prove the case (73). Then the same result will be followed for the case (72).We show the local uniqueness by proving the strict convexity of the mapv∈𝒰ad→J(v). Therefore as in [18], we need to show for all u,v∈𝒰ad(u≠v)(100)D2J(u+ξ(v-u))(v-u,v-u)>0(0<ξ<1).For simplicity, we denotey(u+ξ(v-u)), z(u+ξ(v-u)), and g(u+ξ(v-u)) by y(ξ), z(ξ), and g(ξ), respectively.We calculate(101)DJ(u+ξ(v-u))(v-u)=lim⁡l→0⁡J(u+(ξ+l)(v-u))-J(u+ξ(v-u))l=2∫0T‍(m(ξ)-Yd,𝒵(ξ))2ds+2(R(u+ξ(v-u)),v-u)𝒰,
where m(ξ)=y(ξ)-yxx(ξ) and 𝒵(ξ)=z(ξ)-zxx(ξ). From (101) we obtain the second order Gâteaux derivative of J as follows:
(102)D2J(u+ξ(v-u))(v-u,v-u)=lim⁡k→0⁡(k-1(DJ(u+(ξ+k)(v-u))(v-u)-DJ(u+ξ(v-u))(v-u))×k-1)=2∫0T‍(m(ξ)-Yd,𝒢(ξ))2ds+2∫0T‍|𝒵(ξ)|22ds+2(R(v-u),v-u)𝒰=2∫0T‍〈(I-∂x2)(m(ξ)-Yd),g(ξ)〉-2,2ds+2∫0T‍|𝒵(ξ)|22ds+2(R(v-u),v-u)𝒰,
where 𝒢(ξ)=g(ξ)-gxx(ξ).By Lemma15 and (102) we deduce that
(103)D2J(u+ξ(v-u))(v-u,v-u)≥-2∥g(ξ)∥L∞(0,T;H02(Ω))×∫0T‍∥(I-∂x2)(m(ξ)-Yd)∥H-2(Ω)ds+2∫0T‍|𝒵(ξ)|22ds+2d∥v-u∥𝒰2≥-2c0T∥g(ξ)∥𝒮(0,T)∥m(ξ)-Yd∥L2(0,T;L2(Ω))+2∫0T‍|𝒵(ξ)|22ds+2d∥v-u∥𝒰2≥2(d-c1T∥m(ξ)-Yd∥L2(0,T;L2(Ω)))∥v-u∥𝒰2+2∫0T‍|𝒵(ξ)|22ds,
where c0 and c1 are constants. Here we can take T>0 small enough so that the right hand side of (103) is strictly greater than 0. Therefore we obtain the strict convexity of the quadratic cost J(v), v∈𝒰ad, which prove this theorem.Remark 17.
If we assumed is large enough then we can obtain the strict convexity of the quadratic cost (29) in global sense. Therefore we can obtain the desired result of Theorem 16 in global sense for the cost (29).


## 3.1. Existence of the Optimal Control

Now we show the existence of an optimal controlu for the cost (29).Theorem 5.
Assume that the hypotheses of Theorem3 are satisfied. Then there exists at least one optimal control u for the control problem (27) with the cost (29).Proof.
SetJ=inf⁡v∈𝒰adJ(v). Since 𝒰ad is nonempty, there is a sequence {vn} in 𝒰 such that
(32)inf⁡v∈𝒰adJ(v)=lim⁡n→∞J(vn)=J.
Obviously {J(vn)} is bounded in R+. Then by (30) there exists a constant K0>0 such that
(33)d∥vn∥𝒰2≤(Rvn,vn)𝒰≤J(vn)≤K0.
This shows that {vn} is bounded in 𝒰. Since 𝒰ad is closed and convex, we can choose a subsequence (denoted again by {vn}) of {vn} and find a u∈𝒰ad such that
(34)vn⟶uweaklyin𝒰
as n→∞. From now on, each state yn=y(vn)∈𝒮(0,T) corresponding to vn is the solution of
(35)mn,t-νmn,xx+2ωyn,x+2yn,xmn+ynmn,x+γyn,xxx=f+BvninQ,yn(0,t)=yn(1,t)=yn,x(0,t)=yn,x(1,t)=yn,xx(0,t)=yn,xx(1,t)=0,t∈[0,T],mn(x,0)=m0=y0-y0,xxinΩ,
where mn=yn-yn,xx. By (33) the term Bvn is estimated as
(36)∥Bvn∥L2(0,T;L2(Ω))≤∥B∥ℒ(𝒰,L2(0,T;L2(Ω)))∥vn∥𝒰∥Bvn∥L2(0,T;L2(Ω))≤∥B∥ℒ(𝒰,L2(0,T;L2(Ω)))K0d-1≡K1.
Hence we can deduce from Theorem 3 that
(37)∥mn∥𝒲(0,T)≤C0(∥y0∥H02(Ω)+∥f∥L2(0,T;L2(Ω))+K1)
for some C0>0. And also from Lemma 1 we can know that
(38)∥yn∥𝒮(0,T)≤C1(∥y0∥H02(Ω)+∥f∥L2(0,T;L2(Ω))+K1)
for some C1>0. Therefore, by the extraction theorem of Rellich’s, we can find a subsequence of {mn}, say again {mn}, and find a m=y-yxx∈𝒲(0,T) such that
(39)mn⟶mweaklyin𝒲(0,T).
By using the fact that H01(Ω)↪L2(Ω) is compact and by virtue of (39), we can refer to the result of the Aubin-Lions-Temam’s compact imbedding theorem (cf. Temam [17, page 271]) to verify that {mn} is precompact in L2(0,T;L2(Ω)). Hence there exists a subsequence {mnk}⊂{mn} such that
(40)mnk⟶mstronglyinL2(0,T;L2(Ω))ask⟶∞.
Since mn=yn-yn,xx∈𝒲(0,T)↪C([0,T];L2(Ω)), we know that yn∈C([0,T];H02(Ω)). And from (40) we can choose a subsequence of {ynk}, denoted again by {ynk} such that
(41)ynk(t)⟶y(t)stronglyinH02(Ω)fora.e.t∈[0,T].
We use (39)–(41) and apply the Lebesgue dominated convergence theorem to have
(42)2ynk,xmnk⟶2yxmstronglyinL2(0,T;L2(Ω)),ynkmnkx⟶ymxweaklyinL2(0,T;L2(Ω))
as k→∞. We replace yn and mn by ynk and mnk, respectively, and take k→∞ in (35). Then by the standard argument in Dautray and Lions [16, pages 561–565], we conclude that the limit m satisfies
(43)mt-νmxx+2ωyx+2yxm+ymx+γyxxx=f+BuinQ,y(0,t)=y(1,t)=yx(0,t)=yx(1,t)=yxx(0,t)y(0,t)=yxx(1,t)=0,t∈[0,T],m(x,0)=m0=y0-y0,xx,inΩ
in weak sense, where m=y-yxx. Moreover the uniqueness of weak solutions in (43) via Theorem 3 enables us to conclude that y=y(u) in 𝒮(0,T), which implies y(vn)→y(u) weakly in 𝒮(0,T). Since C is continuous on 𝒮(0,T) and ∥·∥M is lower semicontinuous, it follows that
(44)∥Cy(u)-zd∥M≤liminf⁡n→∞∥Cy(vn)-zd∥M.
It is also clear from liminf⁡k→∞∥R1/2vn∥𝒰≥∥R1/2v∥𝒰 that liminf⁡k→∞(Rvn,vn)𝒰≥(Ru,u)𝒰. Hence
(45)J=liminf⁡n→∞J(vn)≥J(u).
But since J(u)≥J by definition, we conclude that J(u)=inf⁡v∈𝒰adJ(v). This completes the proof.


## 3.2.  Gâteaux Differentiability of Solution Mapping

In order to characterize the optimal control which satisfies the necessary optimality condition (31), we need to prove the Gâteaux differentiability of the mapping v→y(v) of 𝒰→𝒮(0,T).Definition 6.
The solution mapv→y(v) of 𝒰 into 𝒮(0,T) is said to be Gâteaux differentiable at v=u if for any w∈𝒰 there exists a Dy(u)∈ℒ(𝒰,𝒮(0,T)) such that
(46)∥1λ(y(u+λw)-y(u))-Dy(u)w∥𝒮(0,T)⟶0asλ⟶0.
The operator Dy(u) denotes the Gâteaux derivative of y(u) at v=u and the function Dy(u)w∈𝒮(0,T) is called the Gâteaux derivative in the direction w∈𝒰, which plays an important role in the nonlinear optimal control problem.Theorem 7.
The mapv→y(v) of 𝒰 into 𝒮(0,T) is Gâteaux differentiable at v=u and such the Gâteaux derivative of y(v) at v=u in the direction v-u∈𝒰, say z=Dy(u)(v-u), is a unique solution of the following problem:
(47)𝒵t-ν𝒵xx+2ωzx+2zxm+2yx(u)𝒵+zmx+y(u)𝒵x+γzxxx=B(v-u)inQ,z(0,t)=z(1,t)=zx(0,t)=zx(1,t)=zxx(0,t)=zxx(1,t)=0,t∈[0,T],𝒵(x,0)=0inΩ,
where m=y(u)-yxx(u) and 𝒵=z-zxx.Proof.
Letλ∈(-1,1), λ≠0. We set w=v-u and
(48)zλ=λ-1(y(u+λw)-y(u)).
Then zλ satisfies
(49)𝒵λ,t-ν𝒵λ,xx+2ωzλ,x+2zλ,xmλ+2yx(u)𝒵λ+zλmλ,x+y(u)𝒵λ,x+γzλ,xxx=BwinQ,zλ(0,t)=zλ(1,t)=zλ,x(0,t)=zλ,x(1,t)=zλ,xx(0,t)zλ(0,t)=zλ,xx(1,t)=0,t∈[0,T],𝒵λ(x,0)=0inΩ,
where mλ=y(u+λw)-yxx(u+λw) and 𝒵λ=zλ-zλ,xx.
By the continuity of (14), we have
(50)∥y(u+λw)-y(u)∥𝒮(0,T)≤|λ|C∥Bw∥L2(0,T;L2(Ω)),
where C is a constant, depending on y(u) and y(u+λw). Hence we have
(51)∥zλ∥𝒮(0,T)≤C∥Bw∥L2(0,T;L2(Ω))<∞.
Therefore, we can infer that there exists a z∈𝒮(0,T) and a sequence {λk}⊂(-1,1) tending to 0 such that
(52)zλk⟶zweaklyin𝒮(0,T)
as k→∞. Since the imbedding 𝒮(0,T)↪L2(0,T;H02(Ω)) is compact ([17, page 271]), it is implied from (52) that
(53)zλk(t)⟶z(t)stronglyinH02(Ω)a.e.t∈[0,T]
for some {λk}⊂(-1,1) tending to 0 as k→∞. Whence by (50)–(53) and Lebesgue dominated convergence theorem we can easily show that
(54)2zλk,xmλk⟶2zxmstronglyinL2(0,T;L2(Ω)),(55)zλkmλk,x⟶zmxstronglyinL2(0,T;L2(Ω)),(56)𝒵λk⟶𝒵weaklyinL2(0,T;H01(Ω))
as k→∞, where m=y(u)-yxx(u) and 𝒵=z-zxx. And also we can deduce from (49), (52), and (56) that
(57)𝒵λk,t⟶𝒵tweaklyinL2(0,T;H-1(Ω))
as k→∞.
Hence we can see from (52) to (57) that zλ→z=Dy(u)w weakly in 𝒮(0,T) as λ→0 in which z is a solution of (47). This convergence can be improved by showing the strong convergence of {zλ} also in the topology of 𝒮(0,T).
Subtracting (47) from (49) and denoting zλ-z by ϕλ, we obtain that
(58)Φλ,t-νΦλ,xx+2ωϕλ,x+2yx(u)Φλ+y(u)Φλ,x+γϕλ,xxx=ϵ(λ)inQ,ϕλ(0,t)=ϕλ(1,t)=ϕλ,x(0,t)=ϕλ,x(1,t)=ϕλ,xx(0,t)ϕλ(0,t)=ϕλ,xx(1,t)=0,t∈[0,T],Φλ(x,0)=0inΩ,
where Φλ=ϕλ-ϕλ,xx and ϵ(λ)=-2zλ,xmλ+2zxm-zλmλ,x+zmx.
From (54) and (55) we know that
(59)ϵ(λ)⟶0stronglyinL2(0,T;L2(Ω))asλ⟶0.
In order to estimate ϕλ we multiply Φλ in both sides of (58) and integrate it over [0,t] to have
(60)|Φλ(t)|22+2ν∫0t‍|Φλ,x|22ds=-2∫0t‍(2ωϕλ,x,Φλ)2ds-2∫0t‍(2yx(u)Φλ,Φλ)2ds-2∫0t‍(y(u)Φλ,x,Φλ)2ds-2∫0t‍(γϕλ,xxx,Φλ)2ds+2∫0t‍(ϵ(λ),Φλ)2ds.
The integral parts of the right member of (60) can be estimated as follows:
(61)|-2∫0t‍(2ωϕλ,x,Φλ)2ds|≤4ω∫0t‍|ϕλ,x|2|Φλ|2ds≤4ω∫0t‍|Φλ|22ds;(62)|-2∫0t‍(2yx(u)Φλ,Φλ)2ds|≤4∥yx(u)∥L∞(Q)∫0t‍|Φλ|22ds≤c0∫0t‍|Φλ|22ds;(63)|-2∫0t‍(y(u)Φλ,x,Φλ)2ds|≤2∥y(u)∥L∞(Q)×∫0t‍|Φλ,x|2|Φλ|2ds≤ν2∫0t‍|Φλ,x|22ds+c1∫0t‍|Φλ|22ds;(64)|-2∫0t‍(γϕλ,xxx,Φλ)2ds|=|2∫0t‍(γϕλ,xx,Φλ,x)2ds|≤2γ∫0t‍|Φλ|2|Φλ,x|2ds≤ν2∫0t‍|Φλ,x|22ds+c2∫0t‍|Φλ|22ds;(65)|2∫0t‍(ϵ(λ),Φλ)2ds|≤2∫0t‍|ϵ(λ)|2|Φλ|2ds≤∫0t‍|ϵ(λ)|22ds+∫0t‍|Φλ|22ds,
where c0, c1, and c2 are constants. We replace the right hand side of (62) by the right members of (61)–(65). And we apply the Gronwall inequality to the replaced inequality; then we arrive at
(66)|Φλ(t)|22+∫0t‍|Φλ,x|22ds≤C∥ϵ(λ)∥L2(0,T;L2(Ω))2,
where C is a constant. By virtue of (59) and (66) we deduce that
(67)Φλ⟶0inC([0,T];L2(Ω))∩L2(0,T;H01(Ω))00000000000000000000000000000000asλ⟶0.
As in (21)–(23), we have from (58), (66), and (67) that
(68)Φλ,t⟶0stronglyinL2(0,T;H-1(Ω))asλ⟶0.
Therefore (67) and (68) mean
(69)Φλ⟶0stronglyin𝒲(0,T)asλ⟶0.
Whence from Lemma 1(70)zλ(·)⟶z(·)stronglyin𝒮(0,T)asλ⟶0.
This completes the proof.Theorem7 means that the cost J(v) is Gâteaux differentiable at u in the direction v-u and the optimality condition (31) is rewritten by
(71)(Cy(u)-Yd,C(Dy(u)(v-u)))M+(Ru,v-u)𝒰=〈C*ΛM(Cy(u)-Yd),Dy(u)(v-u)〉𝒮(0,T)′,𝒮(0,T)+(Ru,v-u)𝒰≥0,∀v∈𝒰ad,
where ΛM is the canonical isomorphism M onto M′ and Yd∈M is a desired value.


## 3.3.  Necessary Condition of Optimal Control

In this section we will characterize the optimal controls by giving necessary condition (71) for optimality for the following physically meaningful observations.(1)
We takeM=L2(Q) and C1∈ℒ(𝒮(0,T),M) and observe
(72)z(v)=C1y(v)=y(v;·)∈L2(Q).(2)
We takeM=L2(Q) and C2∈ℒ(𝒮(0,T),M) and observe
(73)z(v)=C2y(v)=(I-∂x2)y(v;·)≡m(v;·)∈L2(Q). Since y∈𝒮(0,T)⊂C([0,T];H02(Ω)) by Theorem 3, the above observations are meaningful.Due to Lions [10], we construct the necessary condition of optimal control via appropriate adjoint equation. In order to follow the idea we need to introduce and analyze the following adjoint equation for distributive observations:
(74)-𝒫t-ν𝒫xx-2ωpx(u)-2(m(u)p(u)),x+(I-∂x2)(2yx(u)p(u))+mx(u)p(u)-(I-∂x2)(y(u)p(u)),x-γpxxx(u)=C*ΛM(Cy(u)-Yd)inQ,p(u;0,t)=p(u;1,t)=px(u;0,t)=px(u;1,t)=0,t∈[0,T],𝒫(x,T)=0inΩ,
where C=C1 or C2, 𝒫=p(u)-pxx(u), and m(u)=y(u)-yxx(u). In order to show the well-posedness of (74), we introduce the solution Hilbert space W(H02(Ω),L2(Ω)) defined by
(75)W(H02(Ω),L2(Ω))≜{ψ∣ψ∈L2(0,T;H02(Ω)),ψ′∈L2(0,T;L2(Ω))}
equipped with the norm
(76)∥ψ∥W(H02(Ω),L2(Ω))=(∥ψ∥L2(0,T;H02(Ω))2+∥ψ′∥L2(0,T;L2(Ω))2)1/2.
We remark that W(H02(Ω),L2(Ω)) is continuously embedded in C([0,T];H01(Ω)) (cf. Dautray and Lions [16, page 555]).In the following proposition we show the well-posedness of (74).Proposition 8.
Assume thatC*ΛM(Cy(u)-Yd)∈L2(0,T;H-2(Ω)); then by reversing the direction of time t→T-t, (74) admits a unique solution p(u) satisfying
(77)(i)p(u)∈W(H02(Ω),L2(Ω)),(ii)(pt(u)-νpxx(u)-y(u)px(u)+yx(u)p(u),Φ)2-(2ωpx(u)-mx(u)p(u),ϕ)2+(2m(u)p(u)+γpxx(u),ϕx)2=〈g,ϕ〉-2,2,∀ϕ∈H02(Ω)inthesenseof𝒟′(0,T),(iii)𝒫(x,0)=0inΩ,
where g=C*ΛM(Cy(u)-Yd) and Φ=ϕ-ϕxx.The proof of Proposition8 is given in the Appendix.Remark 9.
As we will see, there are some merits in takingW(H02(Ω),L2(Ω)) as the solution space for adjoint equations. For the observation (72), even though we can take the adjoint system in the space 𝒮(0,T) with additional boundary conditions, we can derive the same necessary optimality condition of optimal controls through the less regular solution p(u)∈W(H02(Ω),L2(Ω)). Therefore, W(H02(Ω),L2(Ω)) is preferred solution space of adjoint equation for the observation (72). And also, for the observation (73), we need to solve adjoint equation in W(H02(Ω),L2(Ω)) because of the less regular data condition than that of the observation (72).


### 3.3.1. Case of the Observation (72)

In this subsection we consider the cost functional expressed by(78)J(v)=∫Q|y(v;x,t)-Yd(x,t)|2‍dxdt+(Rv,v)𝒰,000000000000000000000000000∀v∈𝒰ad⊂𝒰,
where Yd∈L2(Q) is a desired value. Let u be the optimal control subject to (27) and (78). Then the optimality condition (71) is represented by
(79)∫Q‍(y(u;x,t)-Yd(x,t))z(x,t)dxdt+(Ru,v-u)𝒰≥0,000000000000000000000000000000000000∀v∈𝒰ad,
where z is the solution of (47).Now we will formulate the following adjoint system to describe the optimality condition:(80)-𝒫t-ν𝒫xx-2ωpx(u)-2(m(u)p(u)),x+(I-∂x2)(2yx(u)p(u))+mx(u)p(u)-(I-∂x2)(y(u)p(u)),x-γpxxx(u)=y(u)-YdinQ,p(u;0,t)=p(u;1,t)=px(u;0,t)p(u;0,t)=px(u;1,t)=0,t∈[0,T],𝒫(x,T)=0inΩ,
where 𝒫=p(u)-pxx(u) and m(u)=y(u)-yxx(u).Remark 10.
Taking into account the observation conditionsy(u)-Yd∈L2(Q)=L2(0,T;L2(Ω))⊂L2(0,T;H-2(Ω)), we can assert that (80), reversing the direction of time t→T-t, admits a unique solution p(u)∈W(H02(Ω),L2(Ω)) by Proposition 8.Now we proceed the calculations. We multiply both sides of the weak form of (80) by z(t) and integrate it over [0,T]. Then we have
(81)∫0T〈-𝒫t-ν𝒫xx-2ωpx(u)-2(m(u)p(u)),x‍00000+(I-∂x2)(2yx(u)p(u)),z〉-2,2dt+∫0T〈mx(u)p(u)-(I-∂x2)(y(u)p(u)),x‍000000-γpxxx(u),z∂x2〉-1,1dt=∫0T(y(u)-Yd,z)2dt,‍
where 𝒫=p(u)-pxx(u). By (47) for z, we can verify by integration by parts that the left hand side of (81) yields
(82)∫0T(p(u),𝒵t-ν𝒵xx+2ωzx+2zxm(u)‍+2yx(u)𝒵+mx(u)z+y(u)𝒵x+γzxxx)2dt=∫0T(p(u),B(v-u))2dt,‍
where 𝒵=z-zxx. Therefore, by (81) and (82) we can deduce that the optimality condition (79) is equivalent to
(83)∫0T‍(p(u),B(v-u))2dt+(Ru,v-u)𝒰≥0,∀v∈𝒰ad.
Hence, we give the following theorem.Theorem 11.
The optimal controlu for (78) is characterized by the following system of equations and inequality:
(84)mt(u)-νmxx(u)+2ωyx(u)+2yx(u)m(u)+y(u)mx(u)+γyxxx(u)=f+BuinQ,y(u;0,t)=y(u;1,t)=yx(u;0,t)y(u;0,t)=yx(u;1,t)=yxx(u;0,t)y(u;0,t)=yxx(u;1,t)=0,t∈[0,T],m(u;x,0)=m0=y0-y0,xx,inΩ,(85)-𝒫t-ν𝒫xx-2ωpx(u)-2(m(u)p(u)),x+(I-∂x2)(2yx(u)p(u))+mx(u)p(u)-(I-∂x2)(y(u)p(u)),x-γpxxx(u)=y(u)-YdinQ,p(u;0,t)=p(u;1,t)=px(u;0,t)p(u;0,t)=px(u;1,t)=0,t∈[0,T],𝒫(x,T)=0inΩ,(86)∫0T‍(p(u),B(v-u))2dt+(Ru,v-u)𝒰≥0,000000000000000000000000000∀v∈𝒰ad,
where m(u)=y(u)-yxx(u) and 𝒫=p(u)-pxx(u).


### 3.3.2. Case of the Observation (73)

We consider the following momentum’s distributive cost functional expressed by(87)J(v)=∫Q‍|m(v)-Yd|2dxdt+(Rv,v)𝒰,∀v∈𝒰ad,
where m(v)=y(v)-yxx(v) and Yd∈L2(Q). Let u be the optimal control subject to (27) and (87). Then the optimality condition (71) is rewritten as
(88)∫0T‍(m(u)-Yd,𝒵)2dxdt+(Ru,v-u)𝒰≥0,∀v∈𝒰ad,
where 𝒵=z-zxx and z is the solution of (47). As before we formulate the following adjoint system to describe the optimality condition:
(89)-𝒫t-ν𝒫xx-2ωpx(u)-2(m(u)p(u)),x+(I-∂x2)(2yx(u)p(u))+mx(u)p(u)-(I-∂x2)(y(u)p(u)),x-γpxxx(u)=(I-∂x2)(m(u)-Yd)inQ,p(u;0,t)=p(u;1,t)=px(u;0,t)p(u;0,t)=px(u;1,t)=0,t∈[0,T],𝒫(x,T)=0inΩ,
where 𝒫=p(u)-pxx(u) and m(u)=y(u)-yxx(u).Remark 12.
Since the observation conditionsm(u)-Yd∈L2(Q)=L2(0,T;L2(Ω)), we know that (I-∂x2)(m(u)-Yd)∈L2(0,T;H-2(Ω)). Hence by reversing the direction of time t→T-t and applying Proposition 8, we deduce that (89) admits a unique solution p(u)∈W(H02(Ω),L2(Ω)).As we did before, we multiply both sides of the weak form of (89) by z(t) and integrate it over [0,T]. Then we have
(90)∫0T〈-𝒫t-ν𝒫xx-2ωpx(u)-2(m(u)p(u)),x‍00000+(I-∂x2)(2yx(u)p(u)),z〉-2,2dt+∫0T〈mx(u)p(u)-(I-∂x2)(y(u)p(u)),x‍0000000000-γpxxx(u),z∂x2〉-1,1dt=∫0T‍〈(I-∂x2)(m(u)-Yd),z〉-2,2dt=∫0T‍(m(u)-Yd,𝒵)2dt,
where 𝒫=p(u)-pxx(u), m(u)=y(u)-yxx(u), and 𝒵=z-zxx. By (47) for z the integration by parts of the left hand side of (90) yields
(91)∫0T(p(u),𝒵t-ν𝒵xx+2ωzx+2zxm(u)+2yx(u)𝒵‍+mx(u)z+y(u)𝒵x+γzxxx)2dt=∫0T‍(p(u),B(v-u))2dt,
where 𝒵=z-zxx. Therefore, combining (90) and (91), we can deduce that the optimality condition (88) is equivalent to
(92)∫0T‍(p(u),B(v-u))2dt+(Ru,v-u)𝒰≥0,∀v∈𝒰ad.
Hence, we give the following theorem.Theorem 13.
The optimal controlu for (87) is characterized by the following system of equations and inequality:
(93)mt(u)-νmxx(u)+2ωyx(u)+2yx(u)m(u)+y(u)mx(u)+γyxxx(u)=f+BuinQ,y(u;0,t)=y(u;1,t)=yx(u;0,t)y(u;0,t)=yx(u;1,t)=yxx(u;0,t)y(u;0,t)=yxx(u;1,t)=0,t∈[0,T],m(u;x,0)=m0=y0-y0,xxinΩ,(94)-𝒫t-ν𝒫xx-2ωpx(u)-2(m(u)p(u)),x+(I-∂x2)(2yx(u)p(u))+mx(u)p(u)-(I-∂x2)(y(u)p(u)),x-γpxxx(u)=(I-∂x2)(m(u)-Yd)inQ,p(u;0,t)=p(u;1,t)=px(u;0,t)p(u;0,t)=px(u;1,t)=0,t∈[0,T],𝒫(x,T)=0inΩ,(95)∫0T‍(p(u),B(v-u))2dt+(Ru,v-u)𝒰≥0,∀v∈𝒰ad,
where m(u)=y(u)-yxx(u) and 𝒫=p(u)-pxx(u).


## 3.3.1. Case of the Observation (72)

In this subsection we consider the cost functional expressed by(78)J(v)=∫Q|y(v;x,t)-Yd(x,t)|2‍dxdt+(Rv,v)𝒰,000000000000000000000000000∀v∈𝒰ad⊂𝒰,
where Yd∈L2(Q) is a desired value. Let u be the optimal control subject to (27) and (78). Then the optimality condition (71) is represented by
(79)∫Q‍(y(u;x,t)-Yd(x,t))z(x,t)dxdt+(Ru,v-u)𝒰≥0,000000000000000000000000000000000000∀v∈𝒰ad,
where z is the solution of (47).Now we will formulate the following adjoint system to describe the optimality condition:(80)-𝒫t-ν𝒫xx-2ωpx(u)-2(m(u)p(u)),x+(I-∂x2)(2yx(u)p(u))+mx(u)p(u)-(I-∂x2)(y(u)p(u)),x-γpxxx(u)=y(u)-YdinQ,p(u;0,t)=p(u;1,t)=px(u;0,t)p(u;0,t)=px(u;1,t)=0,t∈[0,T],𝒫(x,T)=0inΩ,
where 𝒫=p(u)-pxx(u) and m(u)=y(u)-yxx(u).Remark 10.
Taking into account the observation conditionsy(u)-Yd∈L2(Q)=L2(0,T;L2(Ω))⊂L2(0,T;H-2(Ω)), we can assert that (80), reversing the direction of time t→T-t, admits a unique solution p(u)∈W(H02(Ω),L2(Ω)) by Proposition 8.Now we proceed the calculations. We multiply both sides of the weak form of (80) by z(t) and integrate it over [0,T]. Then we have
(81)∫0T〈-𝒫t-ν𝒫xx-2ωpx(u)-2(m(u)p(u)),x‍00000+(I-∂x2)(2yx(u)p(u)),z〉-2,2dt+∫0T〈mx(u)p(u)-(I-∂x2)(y(u)p(u)),x‍000000-γpxxx(u),z∂x2〉-1,1dt=∫0T(y(u)-Yd,z)2dt,‍
where 𝒫=p(u)-pxx(u). By (47) for z, we can verify by integration by parts that the left hand side of (81) yields
(82)∫0T(p(u),𝒵t-ν𝒵xx+2ωzx+2zxm(u)‍+2yx(u)𝒵+mx(u)z+y(u)𝒵x+γzxxx)2dt=∫0T(p(u),B(v-u))2dt,‍
where 𝒵=z-zxx. Therefore, by (81) and (82) we can deduce that the optimality condition (79) is equivalent to
(83)∫0T‍(p(u),B(v-u))2dt+(Ru,v-u)𝒰≥0,∀v∈𝒰ad.
Hence, we give the following theorem.Theorem 11.
The optimal controlu for (78) is characterized by the following system of equations and inequality:
(84)mt(u)-νmxx(u)+2ωyx(u)+2yx(u)m(u)+y(u)mx(u)+γyxxx(u)=f+BuinQ,y(u;0,t)=y(u;1,t)=yx(u;0,t)y(u;0,t)=yx(u;1,t)=yxx(u;0,t)y(u;0,t)=yxx(u;1,t)=0,t∈[0,T],m(u;x,0)=m0=y0-y0,xx,inΩ,(85)-𝒫t-ν𝒫xx-2ωpx(u)-2(m(u)p(u)),x+(I-∂x2)(2yx(u)p(u))+mx(u)p(u)-(I-∂x2)(y(u)p(u)),x-γpxxx(u)=y(u)-YdinQ,p(u;0,t)=p(u;1,t)=px(u;0,t)p(u;0,t)=px(u;1,t)=0,t∈[0,T],𝒫(x,T)=0inΩ,(86)∫0T‍(p(u),B(v-u))2dt+(Ru,v-u)𝒰≥0,000000000000000000000000000∀v∈𝒰ad,
where m(u)=y(u)-yxx(u) and 𝒫=p(u)-pxx(u).


## 3.3.2. Case of the Observation (73)

We consider the following momentum’s distributive cost functional expressed by(87)J(v)=∫Q‍|m(v)-Yd|2dxdt+(Rv,v)𝒰,∀v∈𝒰ad,
where m(v)=y(v)-yxx(v) and Yd∈L2(Q). Let u be the optimal control subject to (27) and (87). Then the optimality condition (71) is rewritten as
(88)∫0T‍(m(u)-Yd,𝒵)2dxdt+(Ru,v-u)𝒰≥0,∀v∈𝒰ad,
where 𝒵=z-zxx and z is the solution of (47). As before we formulate the following adjoint system to describe the optimality condition:
(89)-𝒫t-ν𝒫xx-2ωpx(u)-2(m(u)p(u)),x+(I-∂x2)(2yx(u)p(u))+mx(u)p(u)-(I-∂x2)(y(u)p(u)),x-γpxxx(u)=(I-∂x2)(m(u)-Yd)inQ,p(u;0,t)=p(u;1,t)=px(u;0,t)p(u;0,t)=px(u;1,t)=0,t∈[0,T],𝒫(x,T)=0inΩ,
where 𝒫=p(u)-pxx(u) and m(u)=y(u)-yxx(u).Remark 12.
Since the observation conditionsm(u)-Yd∈L2(Q)=L2(0,T;L2(Ω)), we know that (I-∂x2)(m(u)-Yd)∈L2(0,T;H-2(Ω)). Hence by reversing the direction of time t→T-t and applying Proposition 8, we deduce that (89) admits a unique solution p(u)∈W(H02(Ω),L2(Ω)).As we did before, we multiply both sides of the weak form of (89) by z(t) and integrate it over [0,T]. Then we have
(90)∫0T〈-𝒫t-ν𝒫xx-2ωpx(u)-2(m(u)p(u)),x‍00000+(I-∂x2)(2yx(u)p(u)),z〉-2,2dt+∫0T〈mx(u)p(u)-(I-∂x2)(y(u)p(u)),x‍0000000000-γpxxx(u),z∂x2〉-1,1dt=∫0T‍〈(I-∂x2)(m(u)-Yd),z〉-2,2dt=∫0T‍(m(u)-Yd,𝒵)2dt,
where 𝒫=p(u)-pxx(u), m(u)=y(u)-yxx(u), and 𝒵=z-zxx. By (47) for z the integration by parts of the left hand side of (90) yields
(91)∫0T(p(u),𝒵t-ν𝒵xx+2ωzx+2zxm(u)+2yx(u)𝒵‍+mx(u)z+y(u)𝒵x+γzxxx)2dt=∫0T‍(p(u),B(v-u))2dt,
where 𝒵=z-zxx. Therefore, combining (90) and (91), we can deduce that the optimality condition (88) is equivalent to
(92)∫0T‍(p(u),B(v-u))2dt+(Ru,v-u)𝒰≥0,∀v∈𝒰ad.
Hence, we give the following theorem.Theorem 13.
The optimal controlu for (87) is characterized by the following system of equations and inequality:
(93)mt(u)-νmxx(u)+2ωyx(u)+2yx(u)m(u)+y(u)mx(u)+γyxxx(u)=f+BuinQ,y(u;0,t)=y(u;1,t)=yx(u;0,t)y(u;0,t)=yx(u;1,t)=yxx(u;0,t)y(u;0,t)=yxx(u;1,t)=0,t∈[0,T],m(u;x,0)=m0=y0-y0,xxinΩ,(94)-𝒫t-ν𝒫xx-2ωpx(u)-2(m(u)p(u)),x+(I-∂x2)(2yx(u)p(u))+mx(u)p(u)-(I-∂x2)(y(u)p(u)),x-γpxxx(u)=(I-∂x2)(m(u)-Yd)inQ,p(u;0,t)=p(u;1,t)=px(u;0,t)p(u;0,t)=px(u;1,t)=0,t∈[0,T],𝒫(x,T)=0inΩ,(95)∫0T‍(p(u),B(v-u))2dt+(Ru,v-u)𝒰≥0,∀v∈𝒰ad,
where m(u)=y(u)-yxx(u) and 𝒫=p(u)-pxx(u).


## 3.4. Local Uniqueness of an Optimal Control

We note that the uniqueness of an optimal control in nonlinear problem is not assured. However, referring to the result in [15], we can show the local uniqueness of the optimal control for our problem. In order to show the uniqueness of optimal control by using strict convexity of quadratic cost (cf. [18]) we consider the following proposition.Proposition 14.
The mapv→y(v) of 𝒰 into 𝒮(0,T) is second order Gâteaux differentiable at v=u and such the second order Gâteaux derivative of y(v) at v=u in the direction v-u∈𝒰, say g=D2y(u)(v-u,v-u), is a unique solution of the following problem:
(96)𝒢t-ν𝒢xx+2ωgx+y(u)𝒢x+2yx(u)𝒢+2gxm+gmx+4zx𝒵+2z𝒵x+γgxxx=0inQ,g(0,t)=g(1,t)=gx(0,t)=gx(1,t)=gxx(0,t)g(0,t)=gxx(1,t)=0,t∈[0,T],𝒢(x,0)=0inΩ,
where 𝒢=g-gxx, 𝒵=z-zxx, and z is the solution of (47).Proof.
The proof is similar to that of Theorem7.Lemma 15.
Letg be the solution of (96). Then we can show that
(97)∥g∥𝒮(0,T)≤C∥v-u∥𝒰2,
where C>0 is a constant.Proof.
Letz be the solution of (47). Then, using the same arguments as in (5), we can deduce that
(98)∥𝒵∥𝒲(0,T)≤c0∥B(v-u)∥L2(0,T;L2(Ω))∥𝒵∥𝒲(0,T)≤c0∥B∥ℒ(𝒰;L2(0,T;L2(Ω)))∥v-u∥𝒰∥𝒵∥𝒲(0,T)≤c1∥v-u∥𝒰,
where 𝒵=z-zxx and c0, c1 are constants which does not depend on g. And also for the solution g of (96), we can show that
(99)∥g∥𝒮(0,T)≤c2∥-4zx𝒵-2z𝒵x∥L2(0,T;L2(Ω))∥g∥𝒮(0,T)≤c3(∥zx∥L∞(Q)+∥z∥L∞(Q))∥𝒵∥L2(0,T;H01(Ω))∥g∥𝒮(0,T)≤c4∥z∥𝒮(0,T)∥𝒵∥𝒲(0,T)∥g∥𝒮(0,T)≤c5∥𝒵∥𝒲(0,T)2,
where c2,…,c5 are constants. Combining (98) with (99), we have (97).We prove the local uniqueness of the optimal control.Theorem 16.
WhenT is small enough, then there is a unique optimal control for the problem (29) for observations (72) and (73).Proof.
We prove the case (73). Then the same result will be followed for the case (72).We show the local uniqueness by proving the strict convexity of the mapv∈𝒰ad→J(v). Therefore as in [18], we need to show for all u,v∈𝒰ad(u≠v)(100)D2J(u+ξ(v-u))(v-u,v-u)>0(0<ξ<1).For simplicity, we denotey(u+ξ(v-u)), z(u+ξ(v-u)), and g(u+ξ(v-u)) by y(ξ), z(ξ), and g(ξ), respectively.We calculate(101)DJ(u+ξ(v-u))(v-u)=lim⁡l→0⁡J(u+(ξ+l)(v-u))-J(u+ξ(v-u))l=2∫0T‍(m(ξ)-Yd,𝒵(ξ))2ds+2(R(u+ξ(v-u)),v-u)𝒰,
where m(ξ)=y(ξ)-yxx(ξ) and 𝒵(ξ)=z(ξ)-zxx(ξ). From (101) we obtain the second order Gâteaux derivative of J as follows:
(102)D2J(u+ξ(v-u))(v-u,v-u)=lim⁡k→0⁡(k-1(DJ(u+(ξ+k)(v-u))(v-u)-DJ(u+ξ(v-u))(v-u))×k-1)=2∫0T‍(m(ξ)-Yd,𝒢(ξ))2ds+2∫0T‍|𝒵(ξ)|22ds+2(R(v-u),v-u)𝒰=2∫0T‍〈(I-∂x2)(m(ξ)-Yd),g(ξ)〉-2,2ds+2∫0T‍|𝒵(ξ)|22ds+2(R(v-u),v-u)𝒰,
where 𝒢(ξ)=g(ξ)-gxx(ξ).By Lemma15 and (102) we deduce that
(103)D2J(u+ξ(v-u))(v-u,v-u)≥-2∥g(ξ)∥L∞(0,T;H02(Ω))×∫0T‍∥(I-∂x2)(m(ξ)-Yd)∥H-2(Ω)ds+2∫0T‍|𝒵(ξ)|22ds+2d∥v-u∥𝒰2≥-2c0T∥g(ξ)∥𝒮(0,T)∥m(ξ)-Yd∥L2(0,T;L2(Ω))+2∫0T‍|𝒵(ξ)|22ds+2d∥v-u∥𝒰2≥2(d-c1T∥m(ξ)-Yd∥L2(0,T;L2(Ω)))∥v-u∥𝒰2+2∫0T‍|𝒵(ξ)|22ds,
where c0 and c1 are constants. Here we can take T>0 small enough so that the right hand side of (103) is strictly greater than 0. Therefore we obtain the strict convexity of the quadratic cost J(v), v∈𝒰ad, which prove this theorem.Remark 17.
If we assumed is large enough then we can obtain the strict convexity of the quadratic cost (29) in global sense. Therefore we can obtain the desired result of Theorem 16 in global sense for the cost (29).


## 4. Conclusions

In conclusion, in this paper we considered the optimal distributed control for the viscous Dullin-Gottwald-Holm equation due to Lions [10]. In order to apply the variational approach due to Lions [10] to our problem, we proposed the quadratic cost functional as studied in Lions [10] which is to be minimized within an admissible set of control variables. We showed the existence of optimal controls which minimizes the quadratic cost functional. Then, we established the necessary conditions of optimality of the optimal control for some physically meaningful observation cases employing the associate adjoint systems. For this we successfully proved the Gâteaux differentiability of the nonlinear solution mapping which is used to define the associate adjoint systems. Moreover, by proving strict convexity of the quadratic cost with respect to the control variable, we discussed the local uniqueness of optimal control.



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