In this article, we present the optimal control for the modified Swift-Hohenberg equation, under certain boundary conditions, and show the existence of an optimal solution

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ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

OPTIMAL CONTROL OF A MODIFIED SWIFT-HOHENBERG EQUATION

NING DUAN, WENJIE GAO

Abstract. In this article, we present the optimal control for the modified Swift-Hohenberg equation, under certain boundary conditions, and show the existence of an optimal solution.

1. Introduction

This article concerns the 1-D modified Swift-Hohenberg equation that was pro- posed by Doelman et al [2]:

ut+kuxxxx+ 2uxx+au+b|ux|²+u³= 0, x∈Ω, t∈(0, T). (1.1) On the basis of physical considerations, (1.1) is supplemented with the boundary value condition

u(x, t) =uxx(x, t) = 0 forx∈∂Ω, (1.2) and the initial condition

u(x,0) =u₀(x), x∈Ω, (1.3)

where Ω is an open connected bounded domain in R, k, a and b are arbitrary constants. u₀(x) is a given function from a suitable phase space.

The Swift-Hohenberg equation is one of the universal equations used in the description of pattern formation in spatially extended dissipative systems, (see [15]), which arise in the study of convective hydrodynamics [16], plasma confinement in toroidal devices [5], viscous film flow and bifurcating solutions of the Navier-Stokes [12]. Note that, the usual Swift-Hohenberg equation [16] is recovered forb= 0. The additional termb|∇u|², reminiscent of the Kuramoto-Sivashinsky equation, which arises in the study of various pattern formation phenomena involving some kind of phase turbulence or phase transition, (see [4, 9, 14]), breaks the symmetryu→ −u.

During the past years, many authors have paid much attention to the Swift- Hohenberg equation (see, e.g. [6, 8, 16]). However, only a few people dovoted to the modified Swift-Hohenberg equation. It were A. Doelman et al.[2] who first studied the modified Swift-Hohenberg equation for a pattern formation system with two unbounded spatial directions that is near the onset to instability. M. Polat[9]

2000Mathematics Subject Classification. 35K55, 49A22.

Key words and phrases. Optimal control; modified Swift-Hohenberg equation;

optimal solution.

c

2012 Texas State University - San Marcos.

Submitted April 2, 2012. Published September 7, 2012.

Supported by grant 11271154 from NSFC..

1

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also considered the modified Swift-Hohenberg equation. In his paper, the existence of a global attractor is proved for the modified Swift-Hohenberg equation as (1.1)- (1.3). Recently, L. Song et al.[15] studied the long time behavior for modified Swift-Hohenberg equation in H^k (k ≥0) space. By using an iteration procedure, regularity estimates for the linear semigroups and a classical existence theorem of global attractor, they proved that problem (1.1)-(1.3) possesses a global attractor in Sobolev spaceH^k for allk≥0, which attracts any bounded subset ofH^k(Ω) in theH^k-norm.

The optimal control plays an important role in modern control theories, and has a wider application in modern engineering. Two methods are used for studying control problems in PDE: one is using a low model method, and then changing to an ODE model [3]; the other is using a quasi-optimal control method [1]. No matter which one is chosen, it is necessary to prove the existence of optimal solution and establish the optimality system. Many papers have already been published to study the control problems of nonlinear parabolic equations. For example, Yong and Zheng[19], Tian et al.[17, 18], Ryu and Yagi [10, 11], Zhao and Liu[20] and so on.

This article concerns the distributed optimal control problem minimize J(u, w) = 1

2kCu−z_dk²_S+δ

2kwk²_L2(Q0), (1.4) subject to

∂u

∂t +kuxxxx+ 2uxx+au+b|ux|²+u³=Bw, (x, t)∈Ω×(0, T), u(x, t) =uxx(x, t) = 0, x∈∂Ω,

u(x,0) =u₀(x), x∈Ω.

(1.5)

The control target is to match the given desired statez_din theL²-sense by adjusting the body forcewin a control volumeQ₀⊆Q= (0,1)×(0, T) in theL²-sense.

Assume that V = {u ∈ H²(0,1)

u(0, t) = u(1, t) = 0}, U = H₀¹(0,1) and H =L²(0,1). Assume further thatV⁰, U⁰ andH⁰ are dual spaces ofV,U andH. Then, we obtain

V ,→U ,→H =H⁰ ,→U⁰ ,→V⁰.

Each embedding being dense. The extension operatorB∈ L(L²(Q0), L²(0, T;H)) which is called the controller is introduced as

Bw=

(w, q∈Q0, 0, w∈Q\Q₀.

We supply H with the inner product (·,·) and the norm k · k, and define a space W(0, T;V) as

W(0, T;V) ={y:y∈L²(0, T;V), yt∈L²(0, T;V⁰)}, which is a Hilbert space endowed with common inner product.

This paper is organized as follows. In the next section, we prove the existence and uniqueness of weak solution to the equation in a special space. We also discuss the relation among the norms of weak solution, initial value and control item;

In section 3, we consider the optimal control problem and prove the existence of optimal solution; Finally in Section 4, conclusions are obtained.

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2. Existence and uniqueness of weak solutions

In this section, we prove the existence and uniqueness of weak solution for problem (1.5), wherex∈(0,1),t∈[0, T],Bw∈L²(0, T;H) and a controlw∈L²(Q₀).

Now, we give the definition of the weak solution in the spaceW(0, T;V).

Definition 2.1. For all η ∈ V, a function u(x, t) ∈ W(0, T;V) is called a weak solution to problem (1.5), if

(∂u

∂t, η) +k(uxx, ηxx)−2(ux, ηx) +a(u, η) +b(|ux|², η) + (u³, η) = (Bw, η). (2.1) We shall give a theorem on the existence and uniqueness of weak solution to problem (1.5).

Theorem 2.2. Suppose that k is sufficiently large, u0 ∈ V, Bw ∈ L²(0, T;H), then (1.5)admits a unique weak solutionu(x, t)∈W(0, T;V).

Proof. Galerkin’s method is applied for this proof. DenoteA= (−∂²_x)² as a differential operator, let{ψi}^∞_i=1 denote the eigenfunctions of the operatorA= (−∂_x²)². Forn∈N, define the discrete ansatz space by

Vn= span{ψ1, ψ2, . . . , ψn} ⊂V.

Letun =Pn

i=1uⁿ_i(t)ψi(x) requireun(0,·)→u0 inH to hold true.

By analyzing the limiting behavior of sequences of smooth function{un}, we can prove the existence of a weak solution to the modified Swift-Hohenberg equation.

Performing the Galerkin metod for (1.5), we obtain (x∂un

∂t , η) +k(u_n,xx, η_xx)−2(u_n,x, η_x) +a(u_n, η) +b(|u_n,x|², η) + (u³_n, η)

= (Bw, η), ∀η∈V, (x, t)∈Q,

(un(x,0), η) = (u0(x), η), ∀η∈V, x∈(0,1)

(2.2)

Then the equation of problem (2.2) is an ordinary differential equation and according to ODE theory, there exists a unique solution in the interval [0, tn).

what we should do is to show that the solution is uniformly bounded whentn→T. We need also to show that the timestn there are not decaying to 0 as n→ ∞.

Then, we shall prove the existence of solution in the following steps.

Step 1, multiplying the equation in (2.2) byu_n, integrating with respect toxon (0,1), we deduce that

1 2

d

dtkunk²+kkun,xxk²+kunk⁴₄

≤ |a|ku_nk²+ 2ku_nxk²+|b|(|u_nx|², u_n) + (Bw, u_n).

(2.3) By Nirenberg’s inequality,

kunxk8/3≤c₀kunxxk^1/2kunk^1/2₄ . Then

|b|(|unx|², un)≤ |b|kunxk²_8/3kuk4≤c²₀|b|kunxxkkunk²₄≤ kunk⁴₄+c⁴₀b²

4 kunxxk². On the other hand, we have

2ku_nxk²=−2(u_n, u_nxx)≤ ku_nk²+ku_nxxk²,

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(Bw, u_n)≤ kBwkkunk ≤ 1

2kBwk²+1 2kunk². Summing up, we have

d

dtkunk²+ (2k−c⁴₀b²

2 −2)kunxxk²≤(2|a|+ 3)kunk²+kBwk²,

whereksatisfies 2k−^c⁴⁰₂^b²−2>0. SinceBw∈L²(0, T;H) is the control item, we can assumekBwk ≤M, where M is a positive constant. Then

d

dtkunk²+ (2k−c⁴₀b²

2 −2)kunxxk²≤(2|a|+ 3)kunk²+M². (2.4) Using Gronwall’s inequality, we obtain

kunk²≤e^(2|a|+3)tkun,0k²+ M² 2|a|+ 3

≤e^(2|a|+3)Tku_n,0k²+ M²

2|a|+ 3 =c²₁, t∈[0, T].

(2.5)

Integrating (2.4) with on [0, T], Z T

0

kun,xxk²dt

≤ 2

4k−c⁴₀b²−4

(2|a|+ 3) Z T

0

kunk²dt+M²T+kun,0k²

≤ 2

4k−c⁴₀b²−4 (2|a|+ 3)c²₁T+M²T+kun,0k²

=c²₂.

(2.6)

Multiplying the equation in (2.2) byunxx, integrating with respect to xon (0,1), we deduce that

1 2

d

dtku_n,xk²+kku_n,xxxk²

= 2kunxxk²−akunxk²+ ((u_n)³, u_nxx) +b(|unx|², u_nxx)−(Bw, u_n,xx).

(2.7) Noticing that

2kunxxk²=−2(unx, unxxx)≤ k

12kunxxxk²+12

kkunxk², and

−(Bw, unxx)≤ kBwkkunxxk ≤ M² 2 +1

2kunxxk²

≤ M² 2 +1

2(k

6kunxxxk²+ 3

2kkunxk²).

By Nirenberg’s inequality,

kunk6≤c0kunxxxk^1/9kunk^8/9, kunxk4≤c0kunxxxk^5/12kunk^7/12. Hence

((un)³, unxx)≤2kunxxk²+1 8kunk⁶₆

≤ k

12kunxxxk²+12

k kunxk²+ k

12kunxxxk²+c(c₁)

=12

kku_nxk²+k

6ku_nxxxk²+c(c₁),

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and

|b|((unx)², unxx) =|b|

Z 1

0

(unx)²unxxdx≤|b|²

8 kunxk⁴₄+ 2kunxxk²

≤ k

12kunxxxk²+c(c1) + k

12kunxxxk²+12 kkunxk²

=k

6ku_nxxxk²+c(c₁) +12 kku_nxk². Summing up,

d

dtku_nxk²+kku_nxxxk²≤(72

k + 2|a|+ 3

2k)ku_nxk²+ 2c(c₁) +M². Using Gronwall’s inequality, we deduce that

kun,xk²≤e⁽⁷²^k^+2|a|+^2k³^)tkun,x(0)k²+2k(2c(c1) +M²) 144 + 4k|a|+ 3

≤e⁽⁷²^k^+2|a|+^2k³^)Tku_n,x(0)k²+2k(2c(c1) +M²)

144 + 4k|a|+ 3 =c²₃, t∈[0, T].

(2.8)

Then, by (2.5), (2.6) and (2.8), we obtain Z T

0

kun(x, t)k²_H2dt≤c.

Using Sobolev’s embedding theorem, we also have

kunk∞≤c₄. (2.9)

Step 2, we prove a uniformL²(0, T;V⁰) bound on a sequence{u_n,t}. In order to obtain the result, we first establish theH²-norm estimate for problem (2.2).

Multiplying the equation in (2.2) by u_nxxxx, integrating with respect to x on (0,1), we deduce that

1 2

d

dtkun,xxk²+kkun,xxxxk²

= 2kunxxxk²−akunxxk²−((un)³, unxxxx)−b(|unx|², unxxxx) + (Bw, unxxxx).

(2.10) By Nierenberg’s inequality,

ku_nxk₄≤c₀ku_nxxxxk^1/12ku_nxk^11/12. Therefore,

b((unx)², unxxxx)≤ k

10kunxxxxk²+5|b|²

2k kunxk⁴₄≤k

5kunxxxxk²+c(c2).

On the other hand, we have ((un)³, unxxxx)≤ sup

x∈[0,1]

|un|³· kunxxxxk1≤ k

10kunxxxxk²+c(c4), 2kunxxxk²=−2(unxx, unxxxx)≤ k

10kunxxxxk²+10

kkunxxk², (Bw, u_nxxxx)≤ kBwkku_nxxxxk ≤ k

10ku_nxxxxk²+5M² 2k .

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Summing up, d

dtkunxxk²+kkunxxxxk²≤(20

k + 2|a|)kunxxk²+5M²

k + 2c(c2) + 2c(c4).

Using Gronwall’s inequality, we derive that

kunxxk²≤e⁽²⁰^k^+2|a|)tkunxx(0)k²+5M²+ 2k(c(c2) +c(c4)) 20 + 2k|a|

≤e⁽²⁰^k^+2|a|)Tkunxx(0)k²+5M²+ 2k(c(c₂) +c(c₄)) 20 + 2k|a|

=c²₅, ∀t∈[0, T].

(2.11)

It then follows from (2.5), (2.6) and (2.11) that

kunxk∞≤c6. (2.12)

Notice that

(u_nxxxx, η) = (u_nxx, η_xx)≤ ku_nxxkkη_xxk ≤ ku_nxxkkηk_V, (|u_nx|², η)≤ sup

x∈[0,1]

|u_nx| ·(u_nx, η)≤c₆ku_nxkkηk ≤c₆ku_nxkkηk_V, ((u_n)³, η)≤ sup

x∈[0,1]

|un|²·(u_n, η)≤c²₄kunkkηk ≤c²₄kunkkηkV,

(unxx, η) = (un, ηxx)≤ kunkkηxxk ≤ kunkkηkV, (un, η)≤ kunkkηk ≤ kunkkηkV, Therefore,

kuntkV⁰ ≤kkunxxk+ 2kunk+|a|kunk+|b|c6kunxk+c²₄kunk+kBwk

≤(kc5+ 2c1+|a|c1+|b|c6c3+c²₄c1+M).

Hence,

ku_n,tk_L2(0,T;V)≤(kc₅+ 2c₁+|a|c₁+|b|c₆c₃+c²₄c₁+M)T =c₇. (2.13) Collecting the previous results, we obtain:

(1) For everyt∈[0, T], the sequence{un}n∈N is bounded inL²(0, T;V), which is independent of the dimension of ansatz spacen.

(2) For everyt∈[0, T], the sequence{u_n,t}_n∈N is bounded inL²(0, T;V⁰), which is independent of the dimension of ansatz spacen.

By the above discussion, we obtain u(x, t) ∈ W(0, T;V). It is easy to check thatW(0, T;V) is compactly embedded intoC(0, T;H) which denote the space of continuous functions. We concludes convergence of a subsequences, again denoted by{un}weak intoW(0, T;V), weak-star inL^∞(0, T;H) and strong inL²(0, T;H) to functionsu(x, t)∈W(0, T;V).

Since the proof of uniqueness is easy, we omit it. Then, Theorem 2.2 is proved.

Now, we shall discuss the relation among the norm of the weak solution, the initial value, and the control item.

Theorem 2.3. Suppose that k is sufficiently large, u0 ∈ V, Bw ∈ L²(0, T;H), then there exists positive constantsC₁ andC₂ such that

kuk²_W_(0,T;V₎≤C1(ku0k²_V +kwk²_L2(Q0)) +C2, (2.14)

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Proof. Clearly, (2.14) implies

kuk²_L2(0,T;V)+ku_tk²_L2(0,T;V⁰)≤C₁(ku₀k²_V +kBwk²_L2(H)) +C₂. (2.15) Passing to the limit in (2.3), we obtain

1 2

d

dtkuk²+kkuxxk²+kuk⁴₄≤ |a|kuk²+ 2kuxk²+|b|(|ux|², u) + (Bw, u). (2.16) Using the same method as in the proof of the above theorem, we derive that

d

dtkuk²+ (2k−c⁴₀b²

2 −2)kuxxk²≤(2|a|+ 3)kuk²+kBwk². (2.17) Then, by Gronwall’s inequality,

kuk²≤e^(2|a|+3)tku0k²+ 1

2|a|+ 3kBwk²

≤c8ku0k²+c9kBwk², ∀t∈[0, T].

(2.18)

Therefore,

kuk²_L2(0,T;H)≤c8Tku0k²+c9kBwk²_L2(0,T;H). (2.19) Integrating (2.17) with respect tot on [0, T], we obtain

ku(T)k²− ku0k²+ (2k−c⁴₀b²

2 −2)kuxxk²_L2(H)

≤ kBwk²_L2(H)+ (2|a|+ 3)kuk²_L2(H). By (2.19) and the above inequality,

kuxxk²_L2(H)

≤ 2

4k−c⁴₀b²−4

kBwk²_L2(H)+ (2|a|+ 3)(c₈Tku0k²+c₉kBwk²_L2(H)) +ku0k²

≤c10kBwk²_L2(H)+c11ku0k².

(2.20) Passing to the limit in (2.7), we obtain

1 2

d

dtku_xk²+kku_xxxk²

= 2kuxxk²−akuxk²+ ((u)³, u_xx) +b(|ux|², u_xx)−(Bw, u_xx).

Using the same method as in the proof of the above theorem, we derive that d

dtku_xk²+kku_xxxk²≤2c(c₁) +kBwk²+ (72

k + 2|a|+ 3

2k)ku_xk². By Gronwall’s inequality,

kuxk²≤e⁽⁷²^k^+2|a|+^2k³^)tkux0k²+ 4kc(c1)

144 + 4k|a|+ 3+ 2k

144 + 4k|a|+ 3kBwk²

≤c12kux0k²+c13kBwk²+c14.

(2.21)

Therefore,

kuk∞≤c, kuxk²_L2(H)≤c₁₂Tkux0k²+c₁₃kBwk²_L2(H)+c₁₄T. (2.22) Adding (2.19), (2.20) and (2.22) gives

kuk²_L2(0,T;V)≤c15(kBwk²_L2(0,T;H)+ku0k²_U) +c16. (2.23)

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On the other hand, passing to the limit in (2.10), a simple calculation shows that d

dtkuxxk²+kkuxxxxk²≤(20

k + 2|a|)kuxxk²+5

kkBwk²+ 2c(c2) + 2c(c4).

Using Gronwall’s inequality,

kuxxk²≤e⁽²⁰^k^+2|a|)tkuxx0k²+ 5kBwk²

20 + 2k|a| +kc(c2) +kc(c4) 10 +|k|a|

≤c17(kBwk²+kuxx0k²) +c18.

(2.24)

It then follows from (2.18), (2.21) and (2.24) that kux(x, t)k ≤c.

On the other hand, we have

(uxxxx, η) = (uxx, ηxx)≤ kuxxkkηxxk ≤ kuxxkkηkV, (|ux|², η)≤ sup

x∈[0,1]

|ux| ·(ux, η)≤ckuxkkηk ≤ckuxkkηkV, ((u)³, η)≤ sup

x∈[0,1]

|u|²·(u, η)≤c²kukkηk ≤c²kukkηk_V.

(uxx, η) = (u, ηxx)≤ kukkηxxk ≤ kukkηkV, (u, η)≤ kukkηk ≤ kukkηkV, Therefore,

kutkV⁰

≤kkuxxk+ 2kuk+|a|kuk+|b|ckuxk+c²kuk+kBwk

≤k(c17(kBwk²+kuxx0k²) +c18)^1/2+ (2 +|a|+c²)(c8ku0k²+c9kBwk²)^1/2 +|b|c(c₁₂ku_x0k²+c₁₃kBwk²+c₁₄)^1/2+kBwk

Hence,

kun,tk²_L2(0,T;V)≤c19(ku0k²_V +kBwk²) +c20. (2.25) By (2.23), (2.25) and the definition of extension operator B, we obtain (2.15).

Then, Theorem 2.3 is proved.

3. Optimal control problem

In this section, we consider the optimal control problem associated with the fourth-order parabolic equation and prove the existence of optimal solution basing on J. L. Lions’ theory (see [7]).

In the following, we supposeL²(Q0) is a Hilbert space of control variables, we also supposeB ∈ L(L²(Q0), L²(0, T;H)) is the controller and a controlw∈L²(Q0), consider the following control system

∂u

∂t +ku_xxxx+ 2u_xx+au+b|u_x|²+u³=Bw, (x, t)∈(0,1)×(0, T), u(x, t) =uxx(x, t) = 0, x= 0,1,

u(x,0) =u0(x), x∈(0,1).

(3.1)

Here, it is assumed thatu0∈V. By Theorem 2.2, we can define the solution map w → u(w) of L²(Q0) into W(0, T;V). The solution u is called the state of the control system (3.1). The observation of the state is assumed to be given by Cu.

HereC ∈ L(W(0, T;V), S) is an operator, which is called the observer,S is a real

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Hilbert space of observations. The cost function associated with the control system (3.1) is given by

J(u, w) = 1

2kCu−zdk²_S+δ

2kwk²_L2(Q₀), (3.2) where zd ∈ S is a desired state and δ > 0 is fixed. An optimal control problem about problem (3.1) is

minimize J(u, w). (3.3)

LetX=W(0, T;V)×L²(Q0) andY =L²(0, T;V)×H. We define an operator e=e(e1, e2) :X →Y, where

e1=G= (∆²)⁻¹(∂u

∂t +kuxxxx+ 2uxx+au+b|ux|²+u³−Bw), e₂=u(x,0)−u₀.

Here ∆²is an operator from V toV⁰. Then, we write (3.3) in the form minimize J(u, w) subject toe(u, w) = 0.

Theorem 3.1. Suppose that k is sufficiently large, u0 ∈ V, Bw ∈ L²(0, T;H), then there exists an optimal control solution (u^∗, w^∗) to problem (3.1).

Proof. Suppose (u, w) satisfye(u, w) = 0. In view of (3.2), we deduce that J(u, w)≥δ

2kwk²_L2(Q0).

By Theorem 2.3, we obtainkukW(0,T;V)→ ∞yieldskwkL²(Q₀)→ ∞. Therefore, J(u, w)→ ∞, when k(u, w)k_X→ ∞. (3.4) As the norm is weakly lower semi-continuous, we achieve that J is weakly lower semi-continuous. Since, for all (u, w)∈X, J(u, w)≥0, there existsλ≥0 defined by

λ= inf{J(u, w)|(u, w)∈X, e(u, w) = 0},

which imlies the existence of a minimizing sequence{(uⁿ, wⁿ)}_n∈N inX such that λ= lim

n→∞J(uⁿ, wⁿ) and e(uⁿ, wⁿ) = 0, ∀n∈N. From (3.4), there exists an element (u^∗, w^∗)∈X such that whenn→ ∞,

uⁿ→u^∗, weakly, u∈W(0, T;V), (3.5) wⁿ→w^∗, weakly, w∈L²(Q₀). (3.6) Using (3.5), we obtain

n→∞lim Z T

0

(uⁿ_t(x, t)−u^∗_t, ψ(t))V⁰,Vdt= 0, ∀ψ∈L²(0, T;V),

n→∞lim Z T

0

(uⁿ(x, t)−u^∗, ψ(t))_V⁰_,Vdt= 0, ∀ψ∈L²(0, T;V),

n→∞lim Z T

0

(uⁿ_xx(x, t)−u^∗_xx, ψ(t))V⁰,Vdt= 0, ∀ψ∈L²(0, T;V),

Since W(0, T;V) is compactly embedded intoL²(0, T;L^∞), we have uⁿ → u^∗ strongly in L²(0, T;L^∞). On the other hand, we know that un ∈ L^∞(0, T;V) and u_n,t ∈ L²(0, T;V^∗). Hence by [13, Lemma 4] we have uⁿ → u^∗ strongly in C(0, T;L^∞),uⁿ_x →u^∗_x strongly inC(0, T;H), asn→ ∞.

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As the sequence{uⁿ}n∈Nconverges weakly, thenkuⁿkW(0,T;V)is bounded. And kuⁿkL²(0,T;L^∞)is also bounded based on the embedding theorem.

Because uⁿ_x → u^∗_x in L²(0, T;L^∞) as n → ∞, we know that ku^∗_xkL²(0,T;L^∞) is bounded too.

By (3.5), we deduce that

Z T

0

Z 1

0

(uⁿ_x)²−(u^∗_x)² η dx dt

=

Z T

0

Z 1

0

(uⁿ_x+u^∗_x)(uⁿ_x−u^∗_x)η dx dt

≤

Z T

0

kuⁿ_x+u^∗_xk_L^∞kuⁿ_x−u^∗_xk_Hkηk_Hdt

≤ kuⁿ_x+u^∗_xk_L2(L^∞)kuⁿ_x−u^∗_xk_C(H)kηk_L2(H)

→0, n→ ∞, ∀η∈L²(0, T;H).

and

Z T

0

Z 1

0

(uⁿ)³−(u^∗)³ η dx dt

≤

Z T

0

Z 1

0

((uⁿ)²+u_nu^∗+ (u^∗)²)(uⁿ−u^∗)η dx dt

≤

Z T

0

k(uⁿ)²+unu^∗+ (u^∗)²kL^∞kuⁿ−u^∗kHkηkHdt

≤ k(uⁿ)²+u_nu^∗+ (u^∗)²k_L2(L^∞)kuⁿ−u^∗k_C(H)kηk_L2(H)

→0, n→ ∞, ∀η∈L²(0, T;H).

Using (3.6) again,

Z T

0

Z 1

0

(Bw−Bw^∗)η dx dt

→0, asn→ ∞, ∀η∈L²(0, T;H).

In view of the above discussions,

e1(u^∗, w^∗) = 0, ∀n∈N.

Noticing that u^∗ ∈W(0, T;V), we derive that u^∗(0)∈H. Since uⁿ →u^∗ weakly inW(0, T;V), we can infer thatuⁿ(0)→u^∗(0) weakly asn→ ∞. Thus,

(uⁿ(0)−u^∗(0), η)→0, asn→ ∞, ∀η∈H, which meanse₂(u^∗, w^∗) = 0. Therefore, we obtain

e(u^∗, w^∗) = 0, in Y.

So, there exists an optimal solution (u^∗, w^∗) to problem (3.1). Then, Theorem 3.1

is proved.

4. Conclusions

The modified Swift-Hohenberg equation is an important mathematical physical model. Because of the complexity of nonlinear parts of the equation, there has been no research on the optimal control and boundary control of this equation. In this paper, we study the distributed optimal control problem for problem (1.1)- (1.3) using a series of mathematical estimates. Our research is motivated by the study of the optimal control problem for the viscous Degasperis-Procesi equation,

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viscous Camassa-Holm equation [17, 18], and the existence theory of optimal control of distributed parameter systems. We also prove the existence of an optimal solution to problem (1.1)-(1.3). In order to realize optimal solutions of optimal control problems in practice one must be able to recompute the optimal solutions in the presence of disturbances in real time unless one gives up optimality. We will use mathematical theory and related numerical methods to solve that problem numerically, which is our intention in the future.

Acknowledgements. The author would like to thank the anonymous referee for the valuable comments and suggestions on the original manuscript.

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Ning Duan

College of Mathematics, Jilin University, Changchun 130012, China E-mail address:[email protected]

Wenjie Gao

College of Mathematics, Jilin University, Changchun 130012, China E-mail address:[email protected]