# Fréchet Differentiability for a Damped Kirchhoff-Type Equation and Its Application to Bilinear Minimax Optimal Control Problems

Authors: Jin-soo Hwang
Journal: International Journal of Differential Equations
 (2019)

---

## Abstract

We consider a damped Kirchhoff-type equation with Dirichlet boundary conditions. The objective is to show the Fréchet differentiability of a nonlinear solution map from a bilinear control input to the solution of a Kirchhoff-type equation. We use this result to formulate the minimax optimal control problem. We show the existence of optimal pairs and find their necessary optimality conditions.


---

## Body

## 1. Introduction

LetΩ be an open bounded set of Rn(n≤3) with a smooth boundary Γ. We set Q=(0,T)×Ω, Σ=(0,T)×Γ for T>0. We consider a strongly damped Kirchhoff-type equation described by the following Dirichlet boundary value problem:(1)y′′-1+∫Ω∇y2dxΔy-μΔy′=Uy+finQ,y=0onΣ,y0,x=y0x,y′0,x=y1xinΩ,where ′=∂/∂t, y is the displacement of a string (or membrane), μ>0, f is a forcing function, and U is a bilinear forcing term, which is usually a bilinear control variable that acts as a multiplier of the displacement term. |·| denotes the Euclidean norm on Rn. As is well known by Kirchhoff [1], the nonlinear part of (1) represents an extension effect of a vibrating string (or membrane). Many kinds of Kirchhoff-type equations have been research subject of many researchers (see Arosio [2], Spagnolo [3], Pohozaev [4], Lions [5], Nishihara and Yamada [6], and references therein).From a physical perspective, the damping of (1) represents an internal friction in an elastic string (or membrane) that makes the vibration smooth. Therefore, we can obtain the well-posedness in the Hadamard sense under sufficiently smooth initial conditions (see [7]). Based on this result, Hwang and Nakagiri [8] set up optimal control problems developed by Lions [9] with (1) using distributed forcing controls. They proved the Gâteaux differentiability of the quasilinear solution map from the control variable to the solution and applied the result to derive the necessary optimality conditions for optimal control in some observation cases.It is important and challenging to extend the optimal control theory to practical nonlinear partial differential equations. There are several studies on semilinear partial differential equations (see [10]). Indeed, the extension of the theory to quasilinear equations is much more restrictive because the differentiability of a solution map is quite dependent on the model due to the strong nonlinearity. Only a few studies have investigated this topic (see [8, 11, 12]). Thus, the differentiability of a solution map in any sense is important to study optimal control or identification problems. In most cases, Gâteaux differentiability may be enough to solve a quadratic cost optimal control problem as in [8]. However, to study the problem in more general cost function like nonquadratic or nonconvex functions, the Fréchet differentiability of a solution map is more desirable.In this paper, we show the Fréchet differentiability of the solution map of (1): U→y from the bilinear control input variables to the solutions of (1). In the author’s knowledge, the Fréchet differentiability of a quasilinear solution map is not studied yet. Based on the result, we construct and solve a bilinear minimax optimal control problem on (1). For the study, we refer to the linear results from Belmiloudi [13], in which the author considered some linear parabolic partial differential equations as the state equations for the problem. Minimax control framework has been used by many researchers for various control problems. There are many literatures related to the minimax control problems. We can refer to just a few: Arada and Raymond [14], Lasiecka and Triggiani [15], and Li and Yong [16].In this paper, the minimax control framework was employed to take into account the undesirable effects of system disturbance (or noise) in control inputs such that a cost function achieves its minimum even when the worst disturbances of the system occur. For this purpose, we replace the bilinear multiplierU in (1) by u+v, where u is a control variable that belongs to the admissible control set Uad and v is a disturbance (or noise) that belongs to the admissible disturbance set Vad. We introduce the following cost function to be minimized within Uad and maximized within Vad:(2)Ju,v=12Cy-YdM2+α2uL2Q2-β2vL2Q2,where y is a solution of (1), M is a Hilbert space of observation variables, C is an operator from the solution space of (1) to M, Yd∈M is a desired value, and the positive constants α and β are the relative weights of the second and third terms on the RHS of (2).As mentioned, another goal of this paper is to find and characterize the optimal controls of the cost function (2) for the worst disturbances through control input in (1). This leads to the problem of finding the saddle points of the cost function (2). First, we prove the existence of an admissible control u∗∈Uad and disturbance (or noise) v∗∈Vad such that (u∗,v∗) is a saddle point of the functional J(u,v) of (2). That is,(3)Ju∗,v≤Ju∗,v∗≤Ju,v∗,∀u,v∈Uad×Vad.Secondly, we derive an optimality condition for (u∗,v∗) in (3). In this paper, we use the terminologyoptimal pair to represent such a saddle point (u∗,v∗) in (3). To prove the existence of an optimal pair (u∗,v∗) satisfying (3), we follow the arguments given by Belmiloudi [13], in which the author employed the minimax theorem in infinite dimensions given by Barbu and Precupanu [17]. Next, we derive the necessary optimal conditions for some observation cases that should be satisfied by the optimal pairs in these observation cases. To derive these conditions, we refer to the studies about bilinear optimal control problems where the state equation is linear partial differential equations such as the reaction diffusion equation or Kirchhoff plate equation (see [13, 18–20] and references therein).We now explain the content of this paper. In Section2, we prove the well-posedness of (1) in the Hadamard sense under sufficiently smooth initial conditions, including a stability estimate from the data space to the solution space. In Section 3, we shall show that the solution map of (1): U→y is Fréchet differentiable. In Section 4, we shall study the minimax optimal control problems: By using the Fréchet differentiability of the solution maps u→y and v→y, we prove that the maps u→J and v→J are convex and concave, respectively, under the assumptions that α,β are sufficiently large. And with an assumption on the operator C in (2), we prove the maps u→J and v→J are lower and upper semicontinuous, respectively. As a result, we can prove the existence of an optimal pair. Next, we derive the necessary optimal conditions for some practical observation cases by employing associate adjoint systems. Especially, we use a first-order Volterra integrodifferential equation as a proper adjoint equation in the velocity’s observation case, which is another novelty of this paper.


## 2. Preliminaries

Throughout this paper, we useC as a generic constant. Let X be a Banach space. We denote its topological dual as X′ and the duality pairing between X′ and X by 〈·,·〉X′,X. We also introduce the following abbreviations:(4)Lp=LpΩ,Hk=HkΩ,·p=·Lp,where p≥1. H0k is the completions of C0∞(Ω) in Hk for k≥1. Let the scalar product on L2 be ·,·2. From Poincare’s inequality and the regularity theory for elliptic boundary value problems (cf. Temam [21, p. 150]), the scalar products on H01 and D(Δ)=H2∩H01 can be endowed as follows:(5)ψ,ϕH01=∇ψ,∇ϕ2,∀ψ,ϕ∈H01;(6)ψ,ϕDΔ=Δψ,Δϕ2,∀ψ,ϕ∈DΔ.Then we know that(7)ψH01=∇ψ2,∀ψ∈H01,ψDΔ=Δψ2,∀ψ∈DΔ.The duality pairing between H01 and H-1 is denoted by 〈ϕ,ψ〉1,-1. It is clear that(8)DΔ↪H01↪L2↪H-1.Each space is dense in the following one, and the injections are continuous and compact. According to Adams [22], we know that the embeddings(9)H01↪Lp,i.e.,ψp≤C∇ψ2,∀ψ∈H01,1≤p<6,(10)DΔ↪C0Ω¯,i.e.,ϕC0Ω¯≤CΔϕ2,∀ϕ∈DΔare compact when n≤3.The solution spaceS(0,T) of (1) of strong solutions is defined by(11)S0,T=g∣g∈L20,T;DΔ,g′∈L20,T;DΔ,g′′∈L2Qwhich is endowed with the norm(12)gS0,T=gL20,T;DΔ2+g′L20,T;DΔ2+g′′L2Q21/2,where g′ and g′′ denote the first and second order distributional derivatives of g.Definition 1.
A functiony is said to be a strong solution of (1) if y∈S(0,T) and y satisfies(13)y′′t-1+∇yt22Δyt-μΔy′t=Utyt+ft,a.e.t∈0,T,y0=y0,y′0=y1.From Dautray and Lions [23, p.480] and Lions and Magnes [24], we remark that(14)S0,T↪C0,T;DΔ∩C10,T;H01.The following variational formulation is used to define theweak solution of (1).Definition 2.
A functiony is said to be a weak solution of (1) if y∈W(0,T)≡{g∣g∈L2(0,T;H01),g′∈L2(0,T;H01),g′′∈L2(0,T;H-1)} and y satisfies(15)y′′·,ϕ-1,1+1+∇y·22∇y·,∇ϕ2+μ∇y′·,∇ϕ2=U·y·+f·,ϕ-1,1∀ϕ∈H01inthesenseofD′0,T,y0=y0,y′0=y1.The following is the well-known Gronwall inequality.Lemma 3.
Letη(·) be a nonnegative, absolutely continuous function on [0,T], which satisfies the following differentiable inequality for a.e. t∈[0,T]:(16)η′t≤ϕtηt+ψt,where ϕ and ψ are nonnegative, summable functions on [0,T]. Then(17)ηt≤e∫0tϕsdsη0+∫0tψsds.Proof.
See Evans [25, p.624].Throughout this paper, we will omit writing the integral variables in the definite integral without any confusion. Referring to [7] and the previous result of [8], we can obtain the following theorem on existence, uniqueness, and regularity of a solution of (1).Theorem 4.
Assume that(y0,y1,f)∈D(Δ)×H01×L2(Q), and U∈L∞(Q). Then (1) has a unique strong solution y∈S(0,T). Moreover, the solution mapping p=(y0,y1,f,U)→y(p) of P≡D(Δ)×H01×L2(Q)×L∞(Q) into S(0,T) is locally Lipschitz continuous. Let p1=(y01,y11,f1,U1)∈P and p2=(y02,y12,f2,U2)∈P. The following is satisfied:(18)yp1-yp2S0,T≤CΔy01-y0222+∇y11-y1222+f1-f2L2Q2+U1-U2L∞Q21/2≡Cp1-p2P,where C>0 is a constant depending on the data.Proof.
From [7], for each fixed U∈L∞(Q) in (1), we can infer that (1) admits a unique strong solution y∈S(0,T) under the data condition (y0,y1,f)∈D(Δ)×H01×L2(Q).
Based on this result, for eachp1=(y01,y11,f1,U1)∈P and p2=(y02,y12,f2,U2)∈P, we prove the inequality (18). For that purpose, we denote y1-y2≡y(p1)-y(p2) by ψ. Then, from (1), we can know that ψ satisfies the following:(19)ψ′′-1+∇y122Δψ-μΔψ′=ϵψ+U1ψ+U1-U2y2+f1-f2inQ,ψ=0onΣ,ψ0=y01-y02,ψ′0=y11-y12inΩ,where(20)ϵψ=∇y122-∇y222Δy2=∇ψ,∇y1+∇y22Δy2.In estimating ψ in (19), we can refer to the previous results [8, Theorem 2.1] to obtain the following inequality:(21)∇ψ′t22+Δψt22+∫0tΔψ′22ds≤CΔy01-y0222+∇y11-y1222+U1-U2y2+f1-f2L2Q2.Since y2S(0,T)≤C(y02,y12,f2) and S(0,T)↪L2(Q), we have(22)U1-U2y2L2Q≤U1-U2L∞Qy2L2Q≤Cy2S0,TU1-U2L∞Q≤CU1-U2L∞Q.Together with (21) and (22), we can deduce the following:(23)∇ψ′t22+Δψt22+∫0tΔψ′22ds≤CΔy01-y0222+∇y11-y1222+f1-f2L2Q2+U1-U2L∞Q2≡Cp1-p2P2.Applying (23) to (19), we have(24)ψ′′L2Q≤Cp1-p2P.From (23) and (24), we can obtain(25)ψS0,T≤Cp1-p2P.
This completes the proof.Corollary 5.
Forp1=(y0,y1,f,U1),p2=(y0,y1,f,U2)∈P, the following inequality is satisfied:(26)yp1-yp2S0,T≤CU1-U2L2Q,where C>0 is a constant depending on the data and y(p1) and y(p2) are the solutions of (1) corresponding to p1 and p2, respectively.Proof.
We denotey(p1)-y(p2) by ψ. Then, as in the proof of Theorem 4, we can know that ψ satisfies the following:(27)ψ′′-1+∇y122Δψ-μΔψ′=ϵψ+U1ψ+U1-U2y2inQ,ψ=0onΣ,ψ0=0,ψ′0=0inΩ,where ϵ(ψ) is given in (20). Estimating ψ in (27) as in the proof of Theorem 4, we can arrive at(28)ψS0,T≤CU1-U2yp2L2Q.Thanks to the fact that y(p2)∈S(0,T)↪C([0,T];D(Δ)) and (10), we can know that S(0,T)↪C0(Q¯). Thus we have(29)RHSof28≤Cyp2C0Q¯U1-U2L2Q≤Cyp2S0,TU1-U2L2Q≤Cp2PU1-U2L2Q.Consequently, from (28) and (29), we have (26).
This completes the proof.


## 3. Fréchet Differentiability of the Nonlinear Solution Map

In this section, we study the Fréchet differentiability of the nonlinear solution map. The Fréchet differentiability of the solution map plays an important role in many applications. LetF=L∞(Q). We consider the nonlinear solution map from u∈F to y(u)∈S(0,T), where y(u) is the solution of(30)y′′u-1+∇yu22Δyu-μΔy′u=uyu+finQ,yu=0onΣ,yu;0,x=y0x,y′u;0,x=y1xinΩ.Based on Theorem 4, for fixed (y0,y1,f)∈D(Δ)×H01×L2(Q), we know that the solution map F→S(0,T), which maps from the term u∈F of (30) to y(u)∈S(0,T), is well defined and continuous. We define the Fréchet differentiability of the nonlinear solution map as follows.Definition 6.
The solution mapu→y(u) of F into S(0,T) is said to be Fréchet differentiable on F if for any u∈F there exists a T(u)∈L(F,S(0,T)) such that, for any w∈F,(31)yu+w-yu-TuwS0,TwF→0aswF→0.The operatorT(u) is called the Fréchet derivative of y at u, which we denote by Dy(u), and T(u)w=Dy(u)w∈S(0,T) is called the Fréchet derivative of y at u in the direction of w∈F.Theorem 7.
The solution mapu→y(u) of F to S(0,T) is Fréchet differentiable on F and the Fréchet derivative of y(u) at u in the direction w∈F, that is to say z=Dy(u)w, is the solution of(32)z′′-1+∇yu22Δz-2∇yu,∇z2Δyu-μΔz′=uz+wyuinQ,z=0onΣ,z0,x=0,z′0,x=0inΩ.We prove this theorem by two steps:(i)
For anyw∈F, (32) admits a unique solution z∈S(0,T). That is, there exists an operator T∈L(F,S(0,T)) satisfying Tw=z(=z(w)).(ii)
We show thaty(u+w)-y(u)-zS(0,T)=o(wF) as wF→0.Proof.
(i) Let(33)Gyu,z≔1+∇yu22Δz+2∇yu,∇z2Δyu.Then from Theorem 4 and (14), we can estimate the above as follows:(34)Gyu,z2≤1+yuC0,T;H012Δz2+2yuC0,T;H01∇z2yuC0,T;DΔ≤with14and8≤1+yuS0,T2Δz2+CyuS0,T2Δz2≤C1+yuS0,T2Δz2≤C1+y0,y1,f,uP2Δz2.Hence, by (34) we know that(35)Gyu,·∈LDΔ,L2.To estimate the solution z of (32), we take the scalar product of (32) with -Δz′-Δz in L2:(36)12ddt∇z′22+μ2ddtΔz22+μΔz′22=z′′,Δz2-Gyu,z,Δz′+Δz2-uz+wyu,Δz′+Δz2.Integrating (36) over [0,t], we obtain(37)12∇z′t22+μ2Δzt22+μ∫0tΔz′22ds=-∇z′t,∇zt2+∫0t∇z′22ds-∫0tGyu,z,Δz′+Δz2ds-∫0tuz+wyu,Δz′+Δz2ds.The right hand side of (37) can be estimated as follows:(38)∇z′t,∇zt2=∇z′t,∫0t∇z′ds2≤∇z′t2∫0t∇z′ds2≤T∇z′t2∇z′L20,t;L2≤withtheYounginequality≤ϵ∇z′t22+Tϵ∫0t∇z′22ds;(39)∫0tGyu,z,Δz′+Δz2ds≤∫0tGyu,z2Δz′2+Δz2ds≤with35≤C∫0tΔz2Δz′2+Δz22ds≤withtheYounginequality≤ϵ∫0tΔz′t22ds+C∫0tΔz22ds;(40)∫0tuz,Δz′+Δz2ds≤∫0tuz2Δz′2+Δz2ds≤uF∫0tz2Δz′2+Δz2ds≤with8≤C∫0tΔz2Δz′2+Δz22ds≤withtheYounginequality≤ϵ∫0tΔz′t22ds+C∫0tΔz22ds;(41)∫0twyu,Δz′+Δz2ds≤∫0twyu2Δz′2+Δz2ds≤withtheYounginequality≤ϵ∫0tΔz′t22ds+C∫0tΔz22ds+C∫0twyu22ds.Considering (38)-(41) and taking ϵ=1/6min⁡{1/2,μ/2}, we can obtain the following from (37):(42)∇z′t22+Δzt22+∫0tΔz′22ds≤C∫0t∇z′22+Δz22ds+CwyuL2Q2.Applying Lemma 3 to (42), we obtain(43)∇z′t22+Δzt22+∫0tΔz′22ds≤CwyuL2Q2.In view of (32), (43) implies that(44)z′′L2Q≤CwyuL2Q.Therefore, from (43) and (44), we can know that z∈S(0,T), and the solution z(=z(w)) of (32) satisfies(45)zwS0,T≤CwyuL2Q≤CwFyuL2Q≤CyuS0,TwF≤Cy0,y1,f,uPwF.Hence, from (45), the mapping w∈F↦z(w)∈S(0,T) is linear and bounded. From this, we can infer that there exists T∈L(F,S(0,T)) such that Tw=z(w) for each w∈F.
(ii) We set the differenceδ=y(u+w)-y(u)-z. Then, from (30) and (32), we can have the following:(46)
δ
′
′
-
μ
Δ
δ
′
=
u
+
w
y
u
+
w
-
u
y
u
-
u
z
-
w
y
u
+
1
+
∇
y
u
+
w
2
2
Δ
y
u
+
w
-
1
+
∇
y
u
2
2
Δ
y
u
-
1
+
∇
y
u
2
2
Δ
z
-
2
∇
y
u
,
∇
z
2
Δ
y
u
=
u
δ
+
w
y
u
+
w
-
y
u
+
1
+
∇
y
u
2
2
Δ
δ
+
∇
y
u
+
w
2
2
-
∇
y
u
2
2
Δ
y
u
+
w
-
2
∇
y
u
,
∇
z
2
Δ
y
u
=
u
+
w
δ
+
w
z
+
1
+
∇
y
u
2
2
Δ
δ
+
∇
y
u
+
w
-
∇
y
u
,
∇
y
u
+
w
+
∇
y
u
2
Δ
y
u
+
w
-
2
∇
y
u
,
∇
z
2
Δ
y
u
=
u
+
w
δ
+
w
z
+
1
+
∇
y
u
2
2
Δ
δ
+
∇
δ
,
∇
y
u
+
w
+
∇
y
u
2
Δ
y
u
+
w
+
∇
z
,
∇
y
u
+
w
+
∇
y
u
2
Δ
y
u
+
w
-
2
∇
y
u
,
∇
z
2
Δ
y
u
=
u
+
w
δ
+
w
z
+
1
+
∇
y
u
2
2
Δ
δ
+
∇
δ
,
∇
y
u
+
w
+
∇
y
u
2
Δ
y
u
+
w
+
∇
z
,
∇
y
u
+
w
-
∇
y
u
2
Δ
y
u
+
w
+
2
∇
z
,
∇
y
u
2
Δ
y
u
+
w
-
Δ
y
u
i
n
Q
.Thus, we know from (46) that δ satisfies(47)δ′′-1+∇yu22Δδ-∇δ,∇yu+w+∇yu2Δyu+w-μΔδ′=u+wδ+wz+I1+I2inQ,δ=0onΣ,δ0,x=0,δ′0,x=0inΩ,where (48)I1=∇z,∇yu+w-∇yu2Δyu+w,I2=2∇z,∇yu2Δyu+w-Δyu.If we let (49)Hyu+w,yu,z≔1+∇yu22Δδ+∇δ,∇yu+w+∇yu2Δyu+w,then by similar arguments used for (34), we have(50)Hyu+w,yu,·∈LDΔ,L2.Thanks to (50), if we follow similar arguments as in (i), then we can arrive at(51)δS0,T≤Cwz+I1+I2L2Q.From (14), Theorem 4, and (45), we can deduce the following:(52)wzL2Q≤wFzL2Q≤CwFzS0,T≤CwF2;(53)I1L2Q≤zC0,T;H01yu+w-yuC0,T;H01×Δyu+wL2Q≤CzS0,Tyu+w-yuS0,Tyu+wS0,T≤CwFu+w-uFy0,y1,f,u+wP≤CwF2;(54)I2L2Q≤2zC0,T;H01yuC0,T;H01×Δyu+w-ΔyuL2Q≤CzS0,TyuS0,Tyu+w-yuS0,T≤CwFy0,y1,f,uPu+w-uF≤CwF2.Hence, from (51) to (54), we can obtain(55)δS0,T≤Cwz+I1+I2L2Q≤CwzL2Q+I1L2Q+I2L2Q≤CwF2,which immediately implies that δS(0,T)=o(wF) as wF→0.
This completes the proof.The following result plays an important role in proving the existence of optimal controls in the next section.Proposition 8.
Givenw∈F, the Fréchet derivative Dy(u)w is locally Lipschitz continuous on F with L2(Q) topology. Indeed, it is satisfied that(56)Dyu1w-Dyu2wS0,T≤Cu1-u2L2QwL2Q,where C>0 is a constant depending on the data.Proof.
Letzi=Dy(ui)w, (i=1,2) be the solutions of (32) corresponding to ui,(i=1,2), and we set ϕ=z1-z2. Then, by similar calculations as in (46), we can deduce that ϕ satisfies(57)ϕ′′-1+∇yu122Δϕ-2∇ϕ,∇yu12Δyu1-μΔϕ′=u1ϕ+∑i=14IiinQ,ϕ=0onΣ,ϕ0,x=0,ϕ′0,x=0inΩ,where (58)I1=2∇z2,∇yu1-∇yu22Δyu1,I2=2∇z2,∇yu22Δyu1-Δyu2,I3=∇yu1-∇yu2,∇yu1+∇yu22Δz2,I4=u1-u2z2+wyu1-yu2.By similar arguments as in the proof of (i) of Theorem 7, ϕ in (57) can be estimated as follows:(59)ϕS0,T≤C∑i=14IiL2Q.From Theorem 4, the embedding S(0,T)↪C0(Q¯), and the first inequality of (45), we can deduce(60)z2S0,T≤Cwyu2L2Q≤Cyu2C0Q¯wL2Q≤with10and14≤Cyu2S0,TwL2Q≤Cy0,y1,f,u2PwL2Q≤CwL2Q.We can estimate Ii(i=1,…,4) of (57) as follows:(61)I1L2Q≤2z2C0,T;H01yu1-yu2C0,T;H01Δyu1L2Q≤with14≤Cz2S0,Tyu1-yu2S0,Tyu1S0,T≤withCorollary5,Theorem4and60≤CwL2Qu1-u2L2Qy0,y1,f,u1P≤Cu1-u2L2QwL2Q;(62)I2L2Q≤2z2C0,T;H01yu2C0,T;H01Δyu1-Δyu2L2Q≤withanargumentssimilarto61≤Cz2S0,Tyu2S0,Tyu1-yu2S0,T≤CwL2Qy0,y1,f,u2Pu1-u2L2Q≤Cu1-u2L2QwL2Q;(63)I3L2Q≤yu1-yu2C0,T;H01yu1+yu2C0,T;H01×Δz2L2Q≤withanargumentssimilarto61≤Cyu1-yu2S0,Tyu1+yu2S0,Tz2S0,T≤Cu1-u2L2Qy0,y1,f,u1P+y0,y1,f,u2PwL2Q≤Cu1-u2L2QwL2Q;(64)I4L2Q≤u1-u2z2L2Q+wyu1-yu2L2Q≤z2C0Q¯u1-u2L2Q+wL2Qyu1-yu2C0Q¯≤with10andS0,T↪C0Q¯≤Cz2S0,Tu1-u2L2Q+wL2Qyu1-yu2S0,T≤withCorollary5and60≤Cu1-u2L2QwL2Q.From (61) to (64), we can obtain the following from (59):(65)ϕS0,T≤Cu1-u2L2QwL2Q.
This completes the proof.


## 4. Quadratic Cost Minimax Control Problems

In this section, we study the quadratic cost minimax optimal control problems for a damped Kirchhoff-type equation. Let the following be the set of the admissible controls:(66)Uad=u∈F∣a≤u≤ba.e.inQ.Let the following be the set of the admissible disturbance or noises:(67)Vad=v∈F∣c≤v≤da.e.inQ.To perform our variational analysis, L2(Q) norms of Uad and Vad are preferable, even though Uad and Vad are subsets of F. For simplicity, let Fad be a product space defined by Fad=Uad×Vad.Using Theorem4, we can uniquely define the solution mapping Fad→S(0,T), which maps the term q=(u,v)∈Fad to the solution y(q)∈S(0,T), which satisfies the following equation:(68)y′′q-1+∇yq22Δyq-μΔy′q=u+vyq+finQ,yq=0onΣ,yq;0,x=y0x,y′q;0,x=y1xinΩ,The solution y(q) of (68) is the state of the control system (68). From Theorem 7, we can deduce that the map q=(u,v)→y(q) of Fad to S(0,T) is Fréchet differentiable at q=q∗=(u∗,v∗), and the Fréchet derivative of y(q) at q=q∗ in the direction w=(h,l)∈F2, say z=Dy(q∗)w is a unique solution of the following problem:(69)z′′-1+∇yq∗22Δz-2∇yq∗,∇z2Δyq∗-μΔz′=u∗+v∗z+h+lyq∗inQ,z=0onΣ,z0,x=0,z′0,x=0inΩ.The quadratic cost function associated with the control system (68) is(70)Ju,v=12Cyq-YdM2+α2uL2Q2-β2vL2Q2,where M is a Hilbert space of observation variables, the operator C∈L(S(0,T),M) is an observer, Yd∈M is a desired value, and the positive constants α and β are the relative weights of the second and the third terms on the RHS of (70).To pursue our objective, we assume that the observerC(∈L(S(0,T),M)) in (70) is a compact operator. As mentioned in the introduction, the minimax optimal control problem can be summarized as follows:(i)
Find an admissible controlu∗∈Uad and a noise (or disturbance) v∗∈Vad such that (u∗,v∗) is a saddle point of the functional J(u,v) of (70). That is,(71)Ju∗,v≤Ju∗,v∗≤Ju,v∗,∀u,v∈Fad.(ii)
Characterize(u∗,v∗) (optimality condition).Such a pair(u∗,v∗) in (71) is called an optimal pair (or an optimal strategy pair) for the problem (70).


### 4.1. Existence of Optimal Pairs

To study the existence of optimal pairs, we present the following results.Proposition 9.
The solution mapping fromFad to S(0,T) is continuous from the weakly-star topology of Fad to the weak topology of S(0,T).In proving the Proposition9, we need the following compactness lemma.Lemma 10.
LetX,Y and Z be Banach spaces such that the embeddings X↪Y↪Z are continuous and the imbedding X↪Y is compact. Then a bounded set of W1,∞(0,T;X,Z)={g∣g∈L∞(0,T;X),g′∈L∞(0,T;Z)} is relatively compact in C([0,T];Y).Proof.
See Simon [26].Proof of Proposition9.
Letq=(u,v)∈Fad and let qn=(un,vn)∈Fad be a sequence such that(72)qn⇀qweakly-starinFadasn→∞.For simplicity, we let each state yn=y(qn) be a solution of(73)yn′′-1+∇yn22Δyn-μΔyn′=un+vnyn+finQ,yn=0onΣ,yn0,x=y0x,yn′0,x=y1xinΩ.We conduct the scalar product of (73) with -Δyn′-Δyn in L2:(74)12ddt∇yn′22+μ+12ddtΔyn22+μΔyn′22+1+∇yn22Δyn22+∇yn2212ddtΔyn22=yn′′,Δyn2-un+vnyn+f,Δyn′+Δyn2,which immediately implies(75)12ddt∇yn′22+μ+12ddtΔyn22+μΔyn′22+∇yn2212ddtΔyn22≤yn′′,Δyn2-un+vnyn+f,Δyn′+Δyn2.The integration of (75) over [0,t] implies(76)12∇yn′t22+μ+12Δynt22+μ∫0tΔyn′22ds+12∇ynt22Δynt22≤Iy0,y1-∇yn′t,∇ynt2+∫0t∇yn′22ds+∫0t∇yn,∇yn′2Δyn22ds-∫0tun+vnyn+f,Δyn′+Δyn2ds,where(77)Iy0,y1=12∇y122+μ+12Δy022+12∇y022Δy022+∇y1,∇y02.By conducting similar calculations to the proof of (i) of Theorem 7, we can obtain the following from (76):(78)∇yn′t22+Δynt22+∫0tΔyn′22ds≤CIy0,y1+fL2Q2+∫0t∇yn′22+Δyn22ds+∫0t∇yn,∇yn′2Δyn22ds.Since we know from Theorem 4 that yn∈S(0,T), we can note that(79)∇yn·,∇yn′·2≤ynC0,T;H01yn′C0,T;H01≤with14≤CynS0,T2≤Cy0,y1,f,un+vnP2.From (78) and (79), we can infer(80)∇yn′t22+Δynt22+∫0tΔyn′22ds≤C1+∫0t∇yn′22+Δyn22ds.Applying Lemma 3 to (80), we have(81)∇yn′t22+Δynt22+∫0tΔyn′22ds≤C.Theorem 4 and (81) imply that yn remains in a bounded set of S(0,T)∩W1,∞(0,T;D(Δ),H01). Therefore, by using Rellich’s extraction theorem, we can find a subsequence of {yn} also called {yn}, and find y∈S(0,T)∩W1,∞(0,T;D(Δ),H01) such that(82)yn⇀yweaklyinS0,Tasn→∞,(83)yn⇀yweakly-starinL∞0,T;DΔasn→∞,(84)yn′⇀y′weakly-starinL∞0,T;H01asn→∞.Since the embedding D(Δ)↪H01 is compact, we can apply Lemma 10 to (83) and (84) with X=D(Δ) and Y=Z=H01 in Lemma 10 to verify that(85)ynispre-compactinC0,T;H01.Hence, we can find a subsequence {ynk}⊂{yn} if necessary such that(86)ynkt→ytinH01for∀t∈0,Task→∞.Therefore, (82) and (86) imply(87)∇ynk22Δynk⇀∇y22ΔyweaklyinL2Qask→∞.From (72) and (85), we can also extract a subsequence, if necessary, denoted again by qn≡(un,vn) such that(88)un+vnyn⇀u+vyweaklyinL2Q.We replace yn by ynk, if necessary, and take k→∞ in (73). Then, by the standard argument in Dautray and Lions [23, pp.561-565], we conclude that the limit y is a solution of(89)y′′-1+∇y22Δy-μΔy′=u+vy+finQ,y=0onΣ,y0,x=y0x,y′0,x=y1xinΩ.Moreover, from the uniqueness of solutions of (89), we conclude that y=y(q) in S(0,T), which implies that y(qn)⇀y(q) weakly in S(0,T).
This completes the proof.We now study the existence of optimal pairs.Theorem 11.
Let the observerC in (70) be a compact operator. Then, for sufficiently large α and β in (70), there exists (u∗,v∗)∈Fad such that (u∗,v∗) satisfies (71).Proof.
LetPv be the map u→J(u,v) and let Qu be the map v→J(u,v). To obtain the existence of optimal pairs in the minimax control problem, we follow the steps given by [13]: We prove that Pv is convex and lower semicontinuous for all v∈Vad and that Qu is concave and upper semicontinuous for all u∈Uad. Then, we employ the minimax theorem in infinite dimensions (see Barbu and Precupanu [17]).
For sufficiently largeα and β in (70), we first prove the convexity of Pv and the concavity of Qu. To prove the convexity of Pv, which is a differentiable map, it is sufficient to show that(90)DPvu1-DPvu2u1-u2≥0,∀u1,u2∈Uad.From Fréchet differentiability of the solution map u→y(u,v), where v is fixed, (90) can be rewritten as(91)Cyu1,v-Yd,CDuyu1,vu1-u2M+α∫0Tu1,u1-u22dt-Cyu2,v-Yd,CDuyu2,vu1-u2M-α∫0Tu2,u1-u22dt≥0,∀u1,u2∈Uad,where Duy(ui,v)(u1-u2),(i=1,2) are solutions of (69), in which (u∗+v∗)z+(h+l)y(p) is replaced by (ui+v)z+(u1-u2)y(ui,v),(i=1,2), respectively. We can easily deduce that (91) is equivalent again to(92)Cyu1,v-yu2,v,CDuyu1,vu1-u2M+Cyu2,v-Yd,CDuyu1,vu1-u2-Duyu2,vu1-u2M+αu1-u2L2Q2≥0,∀u1,u2∈Uad.From Corollary 5, Proposition 8, and (60), we can estimate the left hand side of (92) as follows:(93)Cyu1,v-yu2,v,CDuyu1,vu1-u2M≤Cyu1,v-yu2,vMCDuyu1,vu1-u2M≤CLS0,T,M2yu1,v-yu2,vS0,TDuyu1,vu1-u2S0,T≤withCorollary5and60≤Cu1-u2L2Q2;(94)Cyu2,v-Yd,CDuyu1,vu1-u2-Duyu2,vu1-u2M≤Cyu2,v-YdMCDuyu1,vu1-u2-Duyu2,vu1-u2M≤CLS0,T,MCLS0,T,Myu2,vS0,T+YdM×Duyu1,vu1-u2-Duyu2,vu1-u2S0,T≤withProposition8≤Cyu2,vS0,T+YdMu1-u2L2Q2≤Cy0,y1,f,u2+vP+YdMu1-u2L2Q2.Considering from (92) to (94), we can deduce that there exists a sufficiently large αl(P,Fad,Yd,C) such that, for any α>αl(P,Fad,Yd,C), (92) holds true. Therefore, the map Pv is convex.
Similarly, we can also show that there exist a sufficiently largeβl(P,Fad,Yd,C) such that the following inequality is satisfied for any β>βl(P,Fad,Yd,C):(95)DQuv1-DQuv2v1-v2≤0,∀v1,v2∈Vad.This also indicates the concavity of Qu.
Next, we prove the existence of an optimal pair(u∗,v∗)∈Fad by verifying that Pv is lower semicontinuous for all v∈Vad and Qu is upper semicontinuous for all u∈Uad. Let {un}⊂Uad be a minimizing sequence of J. Thus(96)liminfn→∞Jun,v=minu∈UadJu,v.Since Uad defined by (66) is a closed, bounded, and convex in F, we can extract a subsequence {unk}⊂{un} such that(97)unk⇀u∗weakly-starinUadask→∞.Then, by Proposition 9, we have ∀v∈Vad,(98)yunk,v⇀yu∗,vweaklyinS0,Task→∞.Thus, by the assumption that C∈L(S(0,T),M) is a compact operator, we can extract a subsequence of {unk}, if necessary, denoted again by {unk}, such that(99)Cyunk,v→Cyu∗,vstronglyinMask→∞,∀v∈Vad. From (97), it can be easily verified for the same subsequence {unk} in (97) that(100)unk⇀u∗weaklyinL2Qask→∞.Due to the weakly lower semicontinuity in the L2(Q) norm topology, we can determine from (99) and (100) that the map Pv:u→J(u,v) is lower semicontinuous for all v∈Vad. By similar arguments, we can prove that Qu is upper semicontinuous for all u∈Uad.
Hence, we know that(101)J0v=liminfn→∞Jun,v≥Ju∗,v,∀v∈Vad.But since J0(v)≤J(u∗,v), we have(102)J0v=Ju∗,v=minu∈UadJu,v,∀v∈Vad.Similarly, we also know that there exists v∗∈Vad such that(103)J0v∗=maxv∈VadJ0v.From (102) and (103), we can conclude that (u∗,v∗)∈Fad is an optimal pair for the cost (70).
This completes the proof.


### 4.2. Necessary Conditions of Optimal Pairs

We now turn to the necessary optimality conditions that have to be satisfied by optimal pairs with the cost (70). For this purpose, we consider the following two types of observations Ci, (i=1,2) of distributive and terminal values:(1)
we takeM1=L2Q×L2 and C1∈LS0,T,M1 and observe C1yq=yq;·,yq;T∈L2Q×L2;(2)
we takeM2=L2(Q) and C2∈L(S(0,T),M2) and observe C2y(q)=y′(q;·)∈L2(Q).Remark 12.
Clearly, the embeddingS(0,T)↪L2(Q) is compact. From the embedding (14) we can utilize Lemma 10 in which X=D(Δ) and Y=Z=L2 to obtain the embedding S(0,T)↪C([0,T];L2) is also compact. Consequently, the observer C1 is a compact operator. Thus, C1 satisfies the requirement for the existence of optimal pairs given in Theorem 11.Remark 13.
Sincey′(q)∈H1(0,T;D(Δ),L2)≡{g|g∈L2(0,T;D(Δ)),g′∈L2(Q)}, and the embedding D(Δ)↪L2 is compact, we can employ the Aubin-Lions-Temam’s compact embedding theorem (cf. Temam [27, p. 274]) to determine that the embedding H1(0,T;D(Δ),L2)↪L2(Q) is compact. Consequently, the observer C2 is a compact operator. Therefore, C2 satisfies the requirement for the existence of optimal pairs given in Theorem 11.


#### 4.2.1. Case of Distributive and Terminal Values ObservationsC1

In this observation case, we consider the cost function associated with the control system (68):(104)Ju,v=12yq-YdL2Q2+12yq;T-YdT22+α2uL2Q2-β2vL2Q2,where Yd∈L2(Q) and YdT∈L2 are desired values, and the positive constants α and β are the relative weight of the second and the third terms on the RHS of (104).Now we formulate the following adjoint equation to describe the necessary optimality conditions for this observation:(105)p′′-Gyq∗,p+μΔp′=u∗+v∗p+yq∗-YdinQ,p=0onΣ,pT,x=0,p′T,x=-yq∗;T,x+YdTxinΩ,where G(·,·) is defined in (33). Using a similar estimation to (34), we can have(106)Gyq∗,·∈LH01,H-1.Remark 14.
By considering the observation conditionsy(q∗)-Yd∈L2(Q)⊂L2(0,T;H-1) and y(q∗;T)-YdT∈L2 and (106), we can refer to the well-posedness result of Dautray and Lions [23, pp.558-570] to verify that (105), reversing the direction of time t→T-t, admits a unique weak solution p∈W(0,T), which is given in Definition 2.We now discuss the first-order optimality conditions for the minimax optimal control problem (71) for the quadratic cost function (104).Theorem 15.
Ifα and β in the cost (104) are large enough, then an optimal control u∗∈Uad and a disturbance v∗∈Vad, namely, an optimal pair q∗=(u∗,v∗)∈Fad satisfying (71), can be given by(107)u∗=max⁡a,min⁡-yq∗pα,b,v∗=max⁡c,min⁡yq∗pβ,d,where p is the weak solution of (105).Proof.
Letq∗=(u∗,v∗)∈Fad be an optimal pair in (71) with the cost (104) and let y(q∗) be the corresponding weak solution of (68).
From Theorem7, we know that the map q=(u,v)→y(q) is Fréchet differentiable at q=q∗=(u∗,v∗) in the direction w=(h,l)∈F2, which satisfies q∗+ϵw∈Fad for sufficiently small ϵ>0. Thus, the map q=(u,v)→y(q) is also (strongly) Gâteaux differentiable at q=q∗ in the direction w=(h,l)∈F2. Thus, we have(108)yq∗+ϵw-yq∗ϵ→z=zwstronglyinS0,Tasϵ→0+,where z=Dy(q∗)w is a unique solution of (69). Therefore we can obtain the Gâteaux derivative of the cost (104) at q=q∗ in the direction w=(h,l) as follows:(109)DJu∗,v∗h,l=limϵ→0+⁡Ju∗+ϵh,v∗+ϵl-Ju∗,v∗ϵ=limϵ→0+⁡12∫0Tyq∗+ϵw+yq∗-2Yd,yq∗+ϵw-yq∗ϵ2dt+limϵ→0+⁡12yq∗+ϵw;T+yq∗;T-2YdT,yq∗+ϵw;T-yq∗;Tϵ2+limϵ→0+α2∫0T2u∗,h2+ϵh22dt-β2∫0T2v∗,l2+ϵl22dt=∫0Tyq∗-Yd,z2dt+yq∗;T-YdT,zT2+α∫0Tu∗,h2dt-β∫0Tv∗,l2dt,where z=Dy(q∗)w is a solution of (69).
Before we proceed to the calculations, we note that(110)Gyq∗,φ,ϕ-1,1=-1+∇yq∗22∇φ,∇ϕ2-2∇yq∗,∇φ2∇yq∗,∇ϕ2=1+∇yq∗22Δϕ,φ-1,1+2∇yq∗,∇ϕ2Δyq∗,φ2=φ,Gyq∗,ϕ1,-1,∀φ,ϕ∈H01.We multiply both sides of the weak form of (105) by z, which is a solution of (69), and integrate it over [0,T]. Then, we have(111)∫0Tp′′,z-1,1dt-∫0T(Gyq∗,p+u∗+v∗p,z-1,1dt+μ∫0TΔp′,z-1,1dt=∫0Tyq∗-Yd,z2dt.By integration by parts and the terminal value of the weak solution p of (105), (111) can be rewritten as(112)∫0Tp,z′′2dt+p′T,zT2-∫0TGyq∗,p,z-1,1dt-μ∫0Tp,Δz′2dt-∫0Tp,u∗+v∗z2dt=by110andp′T=-yq∗;T+YdT=∫0Tp,z′′2dt-yq∗;T-YdT,zT2-∫0Tp,Gyq∗,z2dt-μ∫0Tp,Δz′2dt-∫0Tp,u∗+v∗z2dt=∫0Tyq∗-Yd,z2dt.Since z is the solution of (69), we can obtain the following from (112):(113)∫0Tyq∗-Yd,z2dt+yq∗;T-YdT,zT2=∫0Th+lyq∗,p2dt.Therefore, we can deduce that (109) and (113) imply(114)DJu∗,v∗h,l=∫0Tαu∗+yq∗p,h2dt+∫0T-βv∗+yq∗p,l2dt.Since q∗=(u∗,v∗)∈Fad is an optimal pair in (71), we know that(115)DuJu∗,v∗h≥0,DvJu∗,v∗l≤0,h,l∈F2.Therefore, we can obtain the following from (114) and (115):(116)∫0Tαu∗+yq∗p,h2dt≥0,∫0T-βv∗+yq∗p,l2dt≤0,where (h,l)∈F2. By considering the signs of the variations h and l in (116), which depend on u∗ and v∗, respectively, we can deduce the following from (116) (possibly not unique):(117)u∗=max⁡a,min⁡-yq∗pα,b,v∗=max⁡c,min⁡yq∗pβ,d.
This completes the proof.


#### 4.2.2. Case of Velocity ObservationC2

In this observation case, we consider the cost function associated with the control system (68):(118)Ju,v=12y′q-YdL2Q2+α2uL2Q2-β2vL2Q2,where Yd∈L2(Q) is a desired value and the positive constants α and β are the relative weight of the second and the third terms on the RHS of (118). Now we turn to the necessary optimality conditions that have to be satisfied by each solution of the minimax optimal control problem with the cost (118). For this purpose, as proposed in a previous study [8], we introduce the following adjoint equation corresponding to (68), in which q=(u,v) is replaced by q∗=(u∗,v∗):(119)p′+∫tTGyq∗,p+u∗+v∗pds+μΔp=y′q∗-YdinQ,p=0onΣ,pT,x=0inΩ,where G(·,·) is defined in (33).Remark 16.
Usually, adjoint systems of second order problems are also second order (cf. Lions [9]) as long as they are meaningful. However, we have a barrier in this quasilinear (68). If we derive a formal second order adjoint system related to the velocity observation with the cost (118), then it is hard to explain the well-posedness. To overcome this difficulty, we follow the idea given in [8, 11], in which it is adopted that the first-order integrodifferential system as an appropriate adjoint system instead of the formal second order adjoint system.Proposition 17.
Equation (119) admits a unique weak solution p satisfying(120)p∈H10,T;H01,L2∩C0,T;H01,where H1(0,T;H01,L2) is the solution space of (119) given by(121)H10,T;H01,L2=ϕ∣ϕ∈L20,T;H01,ϕ′∈L2Q.Proof.
Since(122)∫T-tTGyq∗,p+u∗+v∗psds=∫0tGyq∗,p+u∗+v∗pT-σdσ,the time reversed equation of (119) (t→T-t in (119)) is given by(123)-ψ′+∫0tGyq∗,ψ+u∗+v∗ψdσ+μΔψ=-y′q∗-YdinQ,ψ=0onΣ,ψ0,x=0inΩ,where ψ(·)=p(T-·). From (106) and -y′(q∗)-Yd∈L2(Q), it is verified that all requirements of Dautray and Lions [23, pp.656-661] are satisfied with (123). Therefore, it readily follows that there exists a unique weak solution ψ∈H1(0,T;H01,L2)∩C([0,T];H01) of (123).
This completes the proof.We now discuss the first-order optimality conditions for the minimax optimal control problem (71).Theorem 18.
Ifα and β in the cost (118) are large enough, then an optimal control u∗∈Uad and a disturbance v∗∈Vad, namely, an optimal pair q∗=(u∗,v∗)∈Fad satisfying (71), can be given by:(124)u∗=max⁡a,min⁡yq∗pα,b,v∗=max⁡c,min⁡-yq∗pβ,d, where p is the weak solution of (119).Proof.
Letq∗=(u∗,v∗)∈Fad be an optimal pair in (71) with the cost (118) and y(q∗) be the corresponding weak solution of (68).
By analogy with the proof of Theorem15, the Gâteaux derivative of the cost (118) at q∗=(u∗,v∗) in the direction w=(h,l)∈F2 that satisfies q∗+ϵw∈Fad for sufficiently small ϵ>0 is given by(125)DJu∗,v∗h,l=limϵ→0+⁡Ju∗+ϵh,v∗+ϵl-Ju∗,v∗ϵ=∫0Ty′q∗-Yd,z′2dt+α∫0Tu∗,h2dt-β∫0Tv∗,l2dt,where z=Dy(q∗)w is a solution of (69). We multiply both sides of the weak form of (119) by z′ and integrate it over [0,T]. Then, we have (126)∫0Tp′,z′2dt+∫0T∫tTGyq∗,p+u∗+v∗pds,z′-1,1dt-μ∫0T∇p,∇z′2dt=∫0Ty′q∗-Yd,z′2dt.By integration by parts and the terminal value of the weak solution p of (119), (126) can be rewritten as(127)-∫0Tp,z′′2dt+∫0TGyq∗,p+u∗+v∗p,z-1,1dt+μ∫0Tp,Δz′2dt=By110=-∫0Tp,z′′2dt+∫0Tp,Gyq∗,z+u∗+v∗z2dt+μ∫0Tp,Δz′2dt=∫0Ty′q∗-Yd,z′2dt.Since z is the solution of (69), we can obtain the following from (127):(128)∫0Ty′q∗-Yd,z′2dt=-∫0Th+lyq∗,p2dt.Therefore, we can deduce that (125) and (128) imply(129)DJu∗,v∗h,l=∫0Tαu∗-yq∗p,h2dt+∫0T-βv∗-yq∗p,l2dt.Since q∗=(u∗,v∗)∈Fad is an optimal pair in (71), we know that(130)DuJu∗,v∗h≥0,DvJu∗,v∗l≤0,h,l∈F2.Therefore, we can obtain the following from (129) and (130):(131)∫0Tαu∗-yq∗p,h2dt≥0,∫0T-βv∗-yq∗p,l2dt≤0,where (h,l)∈F2. By considering the signs of the variations h and l in (131), which depend on u∗ and v∗, respectively, we can deduce from (131) that (possibly not unique)(132)u∗=max⁡a,min⁡yq∗pα,b,v∗=max⁡c,min⁡-yq∗pβ,d.
This completes the proof.


## 4.1. Existence of Optimal Pairs

To study the existence of optimal pairs, we present the following results.Proposition 9.
The solution mapping fromFad to S(0,T) is continuous from the weakly-star topology of Fad to the weak topology of S(0,T).In proving the Proposition9, we need the following compactness lemma.Lemma 10.
LetX,Y and Z be Banach spaces such that the embeddings X↪Y↪Z are continuous and the imbedding X↪Y is compact. Then a bounded set of W1,∞(0,T;X,Z)={g∣g∈L∞(0,T;X),g′∈L∞(0,T;Z)} is relatively compact in C([0,T];Y).Proof.
See Simon [26].Proof of Proposition9.
Letq=(u,v)∈Fad and let qn=(un,vn)∈Fad be a sequence such that(72)qn⇀qweakly-starinFadasn→∞.For simplicity, we let each state yn=y(qn) be a solution of(73)yn′′-1+∇yn22Δyn-μΔyn′=un+vnyn+finQ,yn=0onΣ,yn0,x=y0x,yn′0,x=y1xinΩ.We conduct the scalar product of (73) with -Δyn′-Δyn in L2:(74)12ddt∇yn′22+μ+12ddtΔyn22+μΔyn′22+1+∇yn22Δyn22+∇yn2212ddtΔyn22=yn′′,Δyn2-un+vnyn+f,Δyn′+Δyn2,which immediately implies(75)12ddt∇yn′22+μ+12ddtΔyn22+μΔyn′22+∇yn2212ddtΔyn22≤yn′′,Δyn2-un+vnyn+f,Δyn′+Δyn2.The integration of (75) over [0,t] implies(76)12∇yn′t22+μ+12Δynt22+μ∫0tΔyn′22ds+12∇ynt22Δynt22≤Iy0,y1-∇yn′t,∇ynt2+∫0t∇yn′22ds+∫0t∇yn,∇yn′2Δyn22ds-∫0tun+vnyn+f,Δyn′+Δyn2ds,where(77)Iy0,y1=12∇y122+μ+12Δy022+12∇y022Δy022+∇y1,∇y02.By conducting similar calculations to the proof of (i) of Theorem 7, we can obtain the following from (76):(78)∇yn′t22+Δynt22+∫0tΔyn′22ds≤CIy0,y1+fL2Q2+∫0t∇yn′22+Δyn22ds+∫0t∇yn,∇yn′2Δyn22ds.Since we know from Theorem 4 that yn∈S(0,T), we can note that(79)∇yn·,∇yn′·2≤ynC0,T;H01yn′C0,T;H01≤with14≤CynS0,T2≤Cy0,y1,f,un+vnP2.From (78) and (79), we can infer(80)∇yn′t22+Δynt22+∫0tΔyn′22ds≤C1+∫0t∇yn′22+Δyn22ds.Applying Lemma 3 to (80), we have(81)∇yn′t22+Δynt22+∫0tΔyn′22ds≤C.Theorem 4 and (81) imply that yn remains in a bounded set of S(0,T)∩W1,∞(0,T;D(Δ),H01). Therefore, by using Rellich’s extraction theorem, we can find a subsequence of {yn} also called {yn}, and find y∈S(0,T)∩W1,∞(0,T;D(Δ),H01) such that(82)yn⇀yweaklyinS0,Tasn→∞,(83)yn⇀yweakly-starinL∞0,T;DΔasn→∞,(84)yn′⇀y′weakly-starinL∞0,T;H01asn→∞.Since the embedding D(Δ)↪H01 is compact, we can apply Lemma 10 to (83) and (84) with X=D(Δ) and Y=Z=H01 in Lemma 10 to verify that(85)ynispre-compactinC0,T;H01.Hence, we can find a subsequence {ynk}⊂{yn} if necessary such that(86)ynkt→ytinH01for∀t∈0,Task→∞.Therefore, (82) and (86) imply(87)∇ynk22Δynk⇀∇y22ΔyweaklyinL2Qask→∞.From (72) and (85), we can also extract a subsequence, if necessary, denoted again by qn≡(un,vn) such that(88)un+vnyn⇀u+vyweaklyinL2Q.We replace yn by ynk, if necessary, and take k→∞ in (73). Then, by the standard argument in Dautray and Lions [23, pp.561-565], we conclude that the limit y is a solution of(89)y′′-1+∇y22Δy-μΔy′=u+vy+finQ,y=0onΣ,y0,x=y0x,y′0,x=y1xinΩ.Moreover, from the uniqueness of solutions of (89), we conclude that y=y(q) in S(0,T), which implies that y(qn)⇀y(q) weakly in S(0,T).
This completes the proof.We now study the existence of optimal pairs.Theorem 11.
Let the observerC in (70) be a compact operator. Then, for sufficiently large α and β in (70), there exists (u∗,v∗)∈Fad such that (u∗,v∗) satisfies (71).Proof.
LetPv be the map u→J(u,v) and let Qu be the map v→J(u,v). To obtain the existence of optimal pairs in the minimax control problem, we follow the steps given by [13]: We prove that Pv is convex and lower semicontinuous for all v∈Vad and that Qu is concave and upper semicontinuous for all u∈Uad. Then, we employ the minimax theorem in infinite dimensions (see Barbu and Precupanu [17]).
For sufficiently largeα and β in (70), we first prove the convexity of Pv and the concavity of Qu. To prove the convexity of Pv, which is a differentiable map, it is sufficient to show that(90)DPvu1-DPvu2u1-u2≥0,∀u1,u2∈Uad.From Fréchet differentiability of the solution map u→y(u,v), where v is fixed, (90) can be rewritten as(91)Cyu1,v-Yd,CDuyu1,vu1-u2M+α∫0Tu1,u1-u22dt-Cyu2,v-Yd,CDuyu2,vu1-u2M-α∫0Tu2,u1-u22dt≥0,∀u1,u2∈Uad,where Duy(ui,v)(u1-u2),(i=1,2) are solutions of (69), in which (u∗+v∗)z+(h+l)y(p) is replaced by (ui+v)z+(u1-u2)y(ui,v),(i=1,2), respectively. We can easily deduce that (91) is equivalent again to(92)Cyu1,v-yu2,v,CDuyu1,vu1-u2M+Cyu2,v-Yd,CDuyu1,vu1-u2-Duyu2,vu1-u2M+αu1-u2L2Q2≥0,∀u1,u2∈Uad.From Corollary 5, Proposition 8, and (60), we can estimate the left hand side of (92) as follows:(93)Cyu1,v-yu2,v,CDuyu1,vu1-u2M≤Cyu1,v-yu2,vMCDuyu1,vu1-u2M≤CLS0,T,M2yu1,v-yu2,vS0,TDuyu1,vu1-u2S0,T≤withCorollary5and60≤Cu1-u2L2Q2;(94)Cyu2,v-Yd,CDuyu1,vu1-u2-Duyu2,vu1-u2M≤Cyu2,v-YdMCDuyu1,vu1-u2-Duyu2,vu1-u2M≤CLS0,T,MCLS0,T,Myu2,vS0,T+YdM×Duyu1,vu1-u2-Duyu2,vu1-u2S0,T≤withProposition8≤Cyu2,vS0,T+YdMu1-u2L2Q2≤Cy0,y1,f,u2+vP+YdMu1-u2L2Q2.Considering from (92) to (94), we can deduce that there exists a sufficiently large αl(P,Fad,Yd,C) such that, for any α>αl(P,Fad,Yd,C), (92) holds true. Therefore, the map Pv is convex.
Similarly, we can also show that there exist a sufficiently largeβl(P,Fad,Yd,C) such that the following inequality is satisfied for any β>βl(P,Fad,Yd,C):(95)DQuv1-DQuv2v1-v2≤0,∀v1,v2∈Vad.This also indicates the concavity of Qu.
Next, we prove the existence of an optimal pair(u∗,v∗)∈Fad by verifying that Pv is lower semicontinuous for all v∈Vad and Qu is upper semicontinuous for all u∈Uad. Let {un}⊂Uad be a minimizing sequence of J. Thus(96)liminfn→∞Jun,v=minu∈UadJu,v.Since Uad defined by (66) is a closed, bounded, and convex in F, we can extract a subsequence {unk}⊂{un} such that(97)unk⇀u∗weakly-starinUadask→∞.Then, by Proposition 9, we have ∀v∈Vad,(98)yunk,v⇀yu∗,vweaklyinS0,Task→∞.Thus, by the assumption that C∈L(S(0,T),M) is a compact operator, we can extract a subsequence of {unk}, if necessary, denoted again by {unk}, such that(99)Cyunk,v→Cyu∗,vstronglyinMask→∞,∀v∈Vad. From (97), it can be easily verified for the same subsequence {unk} in (97) that(100)unk⇀u∗weaklyinL2Qask→∞.Due to the weakly lower semicontinuity in the L2(Q) norm topology, we can determine from (99) and (100) that the map Pv:u→J(u,v) is lower semicontinuous for all v∈Vad. By similar arguments, we can prove that Qu is upper semicontinuous for all u∈Uad.
Hence, we know that(101)J0v=liminfn→∞Jun,v≥Ju∗,v,∀v∈Vad.But since J0(v)≤J(u∗,v), we have(102)J0v=Ju∗,v=minu∈UadJu,v,∀v∈Vad.Similarly, we also know that there exists v∗∈Vad such that(103)J0v∗=maxv∈VadJ0v.From (102) and (103), we can conclude that (u∗,v∗)∈Fad is an optimal pair for the cost (70).
This completes the proof.


## 4.2. Necessary Conditions of Optimal Pairs

We now turn to the necessary optimality conditions that have to be satisfied by optimal pairs with the cost (70). For this purpose, we consider the following two types of observations Ci, (i=1,2) of distributive and terminal values:(1)
we takeM1=L2Q×L2 and C1∈LS0,T,M1 and observe C1yq=yq;·,yq;T∈L2Q×L2;(2)
we takeM2=L2(Q) and C2∈L(S(0,T),M2) and observe C2y(q)=y′(q;·)∈L2(Q).Remark 12.
Clearly, the embeddingS(0,T)↪L2(Q) is compact. From the embedding (14) we can utilize Lemma 10 in which X=D(Δ) and Y=Z=L2 to obtain the embedding S(0,T)↪C([0,T];L2) is also compact. Consequently, the observer C1 is a compact operator. Thus, C1 satisfies the requirement for the existence of optimal pairs given in Theorem 11.Remark 13.
Sincey′(q)∈H1(0,T;D(Δ),L2)≡{g|g∈L2(0,T;D(Δ)),g′∈L2(Q)}, and the embedding D(Δ)↪L2 is compact, we can employ the Aubin-Lions-Temam’s compact embedding theorem (cf. Temam [27, p. 274]) to determine that the embedding H1(0,T;D(Δ),L2)↪L2(Q) is compact. Consequently, the observer C2 is a compact operator. Therefore, C2 satisfies the requirement for the existence of optimal pairs given in Theorem 11.


### 4.2.1. Case of Distributive and Terminal Values ObservationsC1

In this observation case, we consider the cost function associated with the control system (68):(104)Ju,v=12yq-YdL2Q2+12yq;T-YdT22+α2uL2Q2-β2vL2Q2,where Yd∈L2(Q) and YdT∈L2 are desired values, and the positive constants α and β are the relative weight of the second and the third terms on the RHS of (104).Now we formulate the following adjoint equation to describe the necessary optimality conditions for this observation:(105)p′′-Gyq∗,p+μΔp′=u∗+v∗p+yq∗-YdinQ,p=0onΣ,pT,x=0,p′T,x=-yq∗;T,x+YdTxinΩ,where G(·,·) is defined in (33). Using a similar estimation to (34), we can have(106)Gyq∗,·∈LH01,H-1.Remark 14.
By considering the observation conditionsy(q∗)-Yd∈L2(Q)⊂L2(0,T;H-1) and y(q∗;T)-YdT∈L2 and (106), we can refer to the well-posedness result of Dautray and Lions [23, pp.558-570] to verify that (105), reversing the direction of time t→T-t, admits a unique weak solution p∈W(0,T), which is given in Definition 2.We now discuss the first-order optimality conditions for the minimax optimal control problem (71) for the quadratic cost function (104).Theorem 15.
Ifα and β in the cost (104) are large enough, then an optimal control u∗∈Uad and a disturbance v∗∈Vad, namely, an optimal pair q∗=(u∗,v∗)∈Fad satisfying (71), can be given by(107)u∗=max⁡a,min⁡-yq∗pα,b,v∗=max⁡c,min⁡yq∗pβ,d,where p is the weak solution of (105).Proof.
Letq∗=(u∗,v∗)∈Fad be an optimal pair in (71) with the cost (104) and let y(q∗) be the corresponding weak solution of (68).
From Theorem7, we know that the map q=(u,v)→y(q) is Fréchet differentiable at q=q∗=(u∗,v∗) in the direction w=(h,l)∈F2, which satisfies q∗+ϵw∈Fad for sufficiently small ϵ>0. Thus, the map q=(u,v)→y(q) is also (strongly) Gâteaux differentiable at q=q∗ in the direction w=(h,l)∈F2. Thus, we have(108)yq∗+ϵw-yq∗ϵ→z=zwstronglyinS0,Tasϵ→0+,where z=Dy(q∗)w is a unique solution of (69). Therefore we can obtain the Gâteaux derivative of the cost (104) at q=q∗ in the direction w=(h,l) as follows:(109)DJu∗,v∗h,l=limϵ→0+⁡Ju∗+ϵh,v∗+ϵl-Ju∗,v∗ϵ=limϵ→0+⁡12∫0Tyq∗+ϵw+yq∗-2Yd,yq∗+ϵw-yq∗ϵ2dt+limϵ→0+⁡12yq∗+ϵw;T+yq∗;T-2YdT,yq∗+ϵw;T-yq∗;Tϵ2+limϵ→0+α2∫0T2u∗,h2+ϵh22dt-β2∫0T2v∗,l2+ϵl22dt=∫0Tyq∗-Yd,z2dt+yq∗;T-YdT,zT2+α∫0Tu∗,h2dt-β∫0Tv∗,l2dt,where z=Dy(q∗)w is a solution of (69).
Before we proceed to the calculations, we note that(110)Gyq∗,φ,ϕ-1,1=-1+∇yq∗22∇φ,∇ϕ2-2∇yq∗,∇φ2∇yq∗,∇ϕ2=1+∇yq∗22Δϕ,φ-1,1+2∇yq∗,∇ϕ2Δyq∗,φ2=φ,Gyq∗,ϕ1,-1,∀φ,ϕ∈H01.We multiply both sides of the weak form of (105) by z, which is a solution of (69), and integrate it over [0,T]. Then, we have(111)∫0Tp′′,z-1,1dt-∫0T(Gyq∗,p+u∗+v∗p,z-1,1dt+μ∫0TΔp′,z-1,1dt=∫0Tyq∗-Yd,z2dt.By integration by parts and the terminal value of the weak solution p of (105), (111) can be rewritten as(112)∫0Tp,z′′2dt+p′T,zT2-∫0TGyq∗,p,z-1,1dt-μ∫0Tp,Δz′2dt-∫0Tp,u∗+v∗z2dt=by110andp′T=-yq∗;T+YdT=∫0Tp,z′′2dt-yq∗;T-YdT,zT2-∫0Tp,Gyq∗,z2dt-μ∫0Tp,Δz′2dt-∫0Tp,u∗+v∗z2dt=∫0Tyq∗-Yd,z2dt.Since z is the solution of (69), we can obtain the following from (112):(113)∫0Tyq∗-Yd,z2dt+yq∗;T-YdT,zT2=∫0Th+lyq∗,p2dt.Therefore, we can deduce that (109) and (113) imply(114)DJu∗,v∗h,l=∫0Tαu∗+yq∗p,h2dt+∫0T-βv∗+yq∗p,l2dt.Since q∗=(u∗,v∗)∈Fad is an optimal pair in (71), we know that(115)DuJu∗,v∗h≥0,DvJu∗,v∗l≤0,h,l∈F2.Therefore, we can obtain the following from (114) and (115):(116)∫0Tαu∗+yq∗p,h2dt≥0,∫0T-βv∗+yq∗p,l2dt≤0,where (h,l)∈F2. By considering the signs of the variations h and l in (116), which depend on u∗ and v∗, respectively, we can deduce the following from (116) (possibly not unique):(117)u∗=max⁡a,min⁡-yq∗pα,b,v∗=max⁡c,min⁡yq∗pβ,d.
This completes the proof.


### 4.2.2. Case of Velocity ObservationC2

In this observation case, we consider the cost function associated with the control system (68):(118)Ju,v=12y′q-YdL2Q2+α2uL2Q2-β2vL2Q2,where Yd∈L2(Q) is a desired value and the positive constants α and β are the relative weight of the second and the third terms on the RHS of (118). Now we turn to the necessary optimality conditions that have to be satisfied by each solution of the minimax optimal control problem with the cost (118). For this purpose, as proposed in a previous study [8], we introduce the following adjoint equation corresponding to (68), in which q=(u,v) is replaced by q∗=(u∗,v∗):(119)p′+∫tTGyq∗,p+u∗+v∗pds+μΔp=y′q∗-YdinQ,p=0onΣ,pT,x=0inΩ,where G(·,·) is defined in (33).Remark 16.
Usually, adjoint systems of second order problems are also second order (cf. Lions [9]) as long as they are meaningful. However, we have a barrier in this quasilinear (68). If we derive a formal second order adjoint system related to the velocity observation with the cost (118), then it is hard to explain the well-posedness. To overcome this difficulty, we follow the idea given in [8, 11], in which it is adopted that the first-order integrodifferential system as an appropriate adjoint system instead of the formal second order adjoint system.Proposition 17.
Equation (119) admits a unique weak solution p satisfying(120)p∈H10,T;H01,L2∩C0,T;H01,where H1(0,T;H01,L2) is the solution space of (119) given by(121)H10,T;H01,L2=ϕ∣ϕ∈L20,T;H01,ϕ′∈L2Q.Proof.
Since(122)∫T-tTGyq∗,p+u∗+v∗psds=∫0tGyq∗,p+u∗+v∗pT-σdσ,the time reversed equation of (119) (t→T-t in (119)) is given by(123)-ψ′+∫0tGyq∗,ψ+u∗+v∗ψdσ+μΔψ=-y′q∗-YdinQ,ψ=0onΣ,ψ0,x=0inΩ,where ψ(·)=p(T-·). From (106) and -y′(q∗)-Yd∈L2(Q), it is verified that all requirements of Dautray and Lions [23, pp.656-661] are satisfied with (123). Therefore, it readily follows that there exists a unique weak solution ψ∈H1(0,T;H01,L2)∩C([0,T];H01) of (123).
This completes the proof.We now discuss the first-order optimality conditions for the minimax optimal control problem (71).Theorem 18.
Ifα and β in the cost (118) are large enough, then an optimal control u∗∈Uad and a disturbance v∗∈Vad, namely, an optimal pair q∗=(u∗,v∗)∈Fad satisfying (71), can be given by:(124)u∗=max⁡a,min⁡yq∗pα,b,v∗=max⁡c,min⁡-yq∗pβ,d, where p is the weak solution of (119).Proof.
Letq∗=(u∗,v∗)∈Fad be an optimal pair in (71) with the cost (118) and y(q∗) be the corresponding weak solution of (68).
By analogy with the proof of Theorem15, the Gâteaux derivative of the cost (118) at q∗=(u∗,v∗) in the direction w=(h,l)∈F2 that satisfies q∗+ϵw∈Fad for sufficiently small ϵ>0 is given by(125)DJu∗,v∗h,l=limϵ→0+⁡Ju∗+ϵh,v∗+ϵl-Ju∗,v∗ϵ=∫0Ty′q∗-Yd,z′2dt+α∫0Tu∗,h2dt-β∫0Tv∗,l2dt,where z=Dy(q∗)w is a solution of (69). We multiply both sides of the weak form of (119) by z′ and integrate it over [0,T]. Then, we have (126)∫0Tp′,z′2dt+∫0T∫tTGyq∗,p+u∗+v∗pds,z′-1,1dt-μ∫0T∇p,∇z′2dt=∫0Ty′q∗-Yd,z′2dt.By integration by parts and the terminal value of the weak solution p of (119), (126) can be rewritten as(127)-∫0Tp,z′′2dt+∫0TGyq∗,p+u∗+v∗p,z-1,1dt+μ∫0Tp,Δz′2dt=By110=-∫0Tp,z′′2dt+∫0Tp,Gyq∗,z+u∗+v∗z2dt+μ∫0Tp,Δz′2dt=∫0Ty′q∗-Yd,z′2dt.Since z is the solution of (69), we can obtain the following from (127):(128)∫0Ty′q∗-Yd,z′2dt=-∫0Th+lyq∗,p2dt.Therefore, we can deduce that (125) and (128) imply(129)DJu∗,v∗h,l=∫0Tαu∗-yq∗p,h2dt+∫0T-βv∗-yq∗p,l2dt.Since q∗=(u∗,v∗)∈Fad is an optimal pair in (71), we know that(130)DuJu∗,v∗h≥0,DvJu∗,v∗l≤0,h,l∈F2.Therefore, we can obtain the following from (129) and (130):(131)∫0Tαu∗-yq∗p,h2dt≥0,∫0T-βv∗-yq∗p,l2dt≤0,where (h,l)∈F2. By considering the signs of the variations h and l in (131), which depend on u∗ and v∗, respectively, we can deduce from (131) that (possibly not unique)(132)u∗=max⁡a,min⁡yq∗pα,b,v∗=max⁡c,min⁡-yq∗pβ,d.
This completes the proof.


## 4.2.1. Case of Distributive and Terminal Values ObservationsC1

In this observation case, we consider the cost function associated with the control system (68):(104)Ju,v=12yq-YdL2Q2+12yq;T-YdT22+α2uL2Q2-β2vL2Q2,where Yd∈L2(Q) and YdT∈L2 are desired values, and the positive constants α and β are the relative weight of the second and the third terms on the RHS of (104).Now we formulate the following adjoint equation to describe the necessary optimality conditions for this observation:(105)p′′-Gyq∗,p+μΔp′=u∗+v∗p+yq∗-YdinQ,p=0onΣ,pT,x=0,p′T,x=-yq∗;T,x+YdTxinΩ,where G(·,·) is defined in (33). Using a similar estimation to (34), we can have(106)Gyq∗,·∈LH01,H-1.Remark 14.
By considering the observation conditionsy(q∗)-Yd∈L2(Q)⊂L2(0,T;H-1) and y(q∗;T)-YdT∈L2 and (106), we can refer to the well-posedness result of Dautray and Lions [23, pp.558-570] to verify that (105), reversing the direction of time t→T-t, admits a unique weak solution p∈W(0,T), which is given in Definition 2.We now discuss the first-order optimality conditions for the minimax optimal control problem (71) for the quadratic cost function (104).Theorem 15.
Ifα and β in the cost (104) are large enough, then an optimal control u∗∈Uad and a disturbance v∗∈Vad, namely, an optimal pair q∗=(u∗,v∗)∈Fad satisfying (71), can be given by(107)u∗=max⁡a,min⁡-yq∗pα,b,v∗=max⁡c,min⁡yq∗pβ,d,where p is the weak solution of (105).Proof.
Letq∗=(u∗,v∗)∈Fad be an optimal pair in (71) with the cost (104) and let y(q∗) be the corresponding weak solution of (68).
From Theorem7, we know that the map q=(u,v)→y(q) is Fréchet differentiable at q=q∗=(u∗,v∗) in the direction w=(h,l)∈F2, which satisfies q∗+ϵw∈Fad for sufficiently small ϵ>0. Thus, the map q=(u,v)→y(q) is also (strongly) Gâteaux differentiable at q=q∗ in the direction w=(h,l)∈F2. Thus, we have(108)yq∗+ϵw-yq∗ϵ→z=zwstronglyinS0,Tasϵ→0+,where z=Dy(q∗)w is a unique solution of (69). Therefore we can obtain the Gâteaux derivative of the cost (104) at q=q∗ in the direction w=(h,l) as follows:(109)DJu∗,v∗h,l=limϵ→0+⁡Ju∗+ϵh,v∗+ϵl-Ju∗,v∗ϵ=limϵ→0+⁡12∫0Tyq∗+ϵw+yq∗-2Yd,yq∗+ϵw-yq∗ϵ2dt+limϵ→0+⁡12yq∗+ϵw;T+yq∗;T-2YdT,yq∗+ϵw;T-yq∗;Tϵ2+limϵ→0+α2∫0T2u∗,h2+ϵh22dt-β2∫0T2v∗,l2+ϵl22dt=∫0Tyq∗-Yd,z2dt+yq∗;T-YdT,zT2+α∫0Tu∗,h2dt-β∫0Tv∗,l2dt,where z=Dy(q∗)w is a solution of (69).
Before we proceed to the calculations, we note that(110)Gyq∗,φ,ϕ-1,1=-1+∇yq∗22∇φ,∇ϕ2-2∇yq∗,∇φ2∇yq∗,∇ϕ2=1+∇yq∗22Δϕ,φ-1,1+2∇yq∗,∇ϕ2Δyq∗,φ2=φ,Gyq∗,ϕ1,-1,∀φ,ϕ∈H01.We multiply both sides of the weak form of (105) by z, which is a solution of (69), and integrate it over [0,T]. Then, we have(111)∫0Tp′′,z-1,1dt-∫0T(Gyq∗,p+u∗+v∗p,z-1,1dt+μ∫0TΔp′,z-1,1dt=∫0Tyq∗-Yd,z2dt.By integration by parts and the terminal value of the weak solution p of (105), (111) can be rewritten as(112)∫0Tp,z′′2dt+p′T,zT2-∫0TGyq∗,p,z-1,1dt-μ∫0Tp,Δz′2dt-∫0Tp,u∗+v∗z2dt=by110andp′T=-yq∗;T+YdT=∫0Tp,z′′2dt-yq∗;T-YdT,zT2-∫0Tp,Gyq∗,z2dt-μ∫0Tp,Δz′2dt-∫0Tp,u∗+v∗z2dt=∫0Tyq∗-Yd,z2dt.Since z is the solution of (69), we can obtain the following from (112):(113)∫0Tyq∗-Yd,z2dt+yq∗;T-YdT,zT2=∫0Th+lyq∗,p2dt.Therefore, we can deduce that (109) and (113) imply(114)DJu∗,v∗h,l=∫0Tαu∗+yq∗p,h2dt+∫0T-βv∗+yq∗p,l2dt.Since q∗=(u∗,v∗)∈Fad is an optimal pair in (71), we know that(115)DuJu∗,v∗h≥0,DvJu∗,v∗l≤0,h,l∈F2.Therefore, we can obtain the following from (114) and (115):(116)∫0Tαu∗+yq∗p,h2dt≥0,∫0T-βv∗+yq∗p,l2dt≤0,where (h,l)∈F2. By considering the signs of the variations h and l in (116), which depend on u∗ and v∗, respectively, we can deduce the following from (116) (possibly not unique):(117)u∗=max⁡a,min⁡-yq∗pα,b,v∗=max⁡c,min⁡yq∗pβ,d.
This completes the proof.


## 4.2.2. Case of Velocity ObservationC2

In this observation case, we consider the cost function associated with the control system (68):(118)Ju,v=12y′q-YdL2Q2+α2uL2Q2-β2vL2Q2,where Yd∈L2(Q) is a desired value and the positive constants α and β are the relative weight of the second and the third terms on the RHS of (118). Now we turn to the necessary optimality conditions that have to be satisfied by each solution of the minimax optimal control problem with the cost (118). For this purpose, as proposed in a previous study [8], we introduce the following adjoint equation corresponding to (68), in which q=(u,v) is replaced by q∗=(u∗,v∗):(119)p′+∫tTGyq∗,p+u∗+v∗pds+μΔp=y′q∗-YdinQ,p=0onΣ,pT,x=0inΩ,where G(·,·) is defined in (33).Remark 16.
Usually, adjoint systems of second order problems are also second order (cf. Lions [9]) as long as they are meaningful. However, we have a barrier in this quasilinear (68). If we derive a formal second order adjoint system related to the velocity observation with the cost (118), then it is hard to explain the well-posedness. To overcome this difficulty, we follow the idea given in [8, 11], in which it is adopted that the first-order integrodifferential system as an appropriate adjoint system instead of the formal second order adjoint system.Proposition 17.
Equation (119) admits a unique weak solution p satisfying(120)p∈H10,T;H01,L2∩C0,T;H01,where H1(0,T;H01,L2) is the solution space of (119) given by(121)H10,T;H01,L2=ϕ∣ϕ∈L20,T;H01,ϕ′∈L2Q.Proof.
Since(122)∫T-tTGyq∗,p+u∗+v∗psds=∫0tGyq∗,p+u∗+v∗pT-σdσ,the time reversed equation of (119) (t→T-t in (119)) is given by(123)-ψ′+∫0tGyq∗,ψ+u∗+v∗ψdσ+μΔψ=-y′q∗-YdinQ,ψ=0onΣ,ψ0,x=0inΩ,where ψ(·)=p(T-·). From (106) and -y′(q∗)-Yd∈L2(Q), it is verified that all requirements of Dautray and Lions [23, pp.656-661] are satisfied with (123). Therefore, it readily follows that there exists a unique weak solution ψ∈H1(0,T;H01,L2)∩C([0,T];H01) of (123).
This completes the proof.We now discuss the first-order optimality conditions for the minimax optimal control problem (71).Theorem 18.
Ifα and β in the cost (118) are large enough, then an optimal control u∗∈Uad and a disturbance v∗∈Vad, namely, an optimal pair q∗=(u∗,v∗)∈Fad satisfying (71), can be given by:(124)u∗=max⁡a,min⁡yq∗pα,b,v∗=max⁡c,min⁡-yq∗pβ,d, where p is the weak solution of (119).Proof.
Letq∗=(u∗,v∗)∈Fad be an optimal pair in (71) with the cost (118) and y(q∗) be the corresponding weak solution of (68).
By analogy with the proof of Theorem15, the Gâteaux derivative of the cost (118) at q∗=(u∗,v∗) in the direction w=(h,l)∈F2 that satisfies q∗+ϵw∈Fad for sufficiently small ϵ>0 is given by(125)DJu∗,v∗h,l=limϵ→0+⁡Ju∗+ϵh,v∗+ϵl-Ju∗,v∗ϵ=∫0Ty′q∗-Yd,z′2dt+α∫0Tu∗,h2dt-β∫0Tv∗,l2dt,where z=Dy(q∗)w is a solution of (69). We multiply both sides of the weak form of (119) by z′ and integrate it over [0,T]. Then, we have (126)∫0Tp′,z′2dt+∫0T∫tTGyq∗,p+u∗+v∗pds,z′-1,1dt-μ∫0T∇p,∇z′2dt=∫0Ty′q∗-Yd,z′2dt.By integration by parts and the terminal value of the weak solution p of (119), (126) can be rewritten as(127)-∫0Tp,z′′2dt+∫0TGyq∗,p+u∗+v∗p,z-1,1dt+μ∫0Tp,Δz′2dt=By110=-∫0Tp,z′′2dt+∫0Tp,Gyq∗,z+u∗+v∗z2dt+μ∫0Tp,Δz′2dt=∫0Ty′q∗-Yd,z′2dt.Since z is the solution of (69), we can obtain the following from (127):(128)∫0Ty′q∗-Yd,z′2dt=-∫0Th+lyq∗,p2dt.Therefore, we can deduce that (125) and (128) imply(129)DJu∗,v∗h,l=∫0Tαu∗-yq∗p,h2dt+∫0T-βv∗-yq∗p,l2dt.Since q∗=(u∗,v∗)∈Fad is an optimal pair in (71), we know that(130)DuJu∗,v∗h≥0,DvJu∗,v∗l≤0,h,l∈F2.Therefore, we can obtain the following from (129) and (130):(131)∫0Tαu∗-yq∗p,h2dt≥0,∫0T-βv∗-yq∗p,l2dt≤0,where (h,l)∈F2. By considering the signs of the variations h and l in (131), which depend on u∗ and v∗, respectively, we can deduce from (131) that (possibly not unique)(132)u∗=max⁡a,min⁡yq∗pα,b,v∗=max⁡c,min⁡-yq∗pβ,d.
This completes the proof.


## 5. Conclusion

The Fréchet differentiability from a bilinear control input into the solution space of a damped Kirchhoff-type equation is verified. As an application of this result, we proposed a minimax optimal control problem for the above state equation by using quadratic cost functions that depend on control and disturbance (or noise) variables. By utilizing the Fréchet differentiability of the solution map and the continuity of the solution map in a weak topology, we have proven existence of the optimal control of the worst disturbance, called the optimal pair under some hypothesis. And we derived necessary optimality conditions that any optimal pairs must satisfy in some observation cases.



---

*Source: 3238462-2019-02-03.xml*