IJDE International Journal of Differential Equations 1687-9651 1687-9643 Hindawi 10.1155/2019/3238462 3238462 Research Article Fréchet Differentiability for a Damped Kirchhoff-Type Equation and Its Application to Bilinear Minimax Optimal Control Problems http://orcid.org/0000-0001-8746-8463 Hwang Jin-soo 1 Messaoudi Salim Department of Mathematics Education College of Education Daegu University Jillyang Gyeongsan Gyeongbuk Republic of Korea daegu.ac.kr 2019 322019 2019 22 10 2018 29 11 2018 322019 2019 Copyright © 2019 Jin-soo Hwang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

Daegu University Research 2015
1. Introduction

Let Ω be an open bounded set of Rn(n3) 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+Ωy2dxΔy-μΔy=Uy+finQ,y=0onΣ,y0,x=y0x,y0,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 , 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 , Spagnolo , Pohozaev , Lions , Nishihara and Yamada , 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 ). Based on this result, Hwang and Nakagiri  set up optimal control problems developed by Lions  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 ). 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 . 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): Uy 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 , 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 , Lasiecka and Triggiani , and Li and Yong .

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 multiplier U 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, YdM 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 uUad and disturbance (or noise) vVad such that (u,v) is a saddle point of the functional J(u,v) of (2). That is,(3)Ju,vJu,vJu,v,u,vUad×Vad.Secondly, we derive an optimality condition for (u,v) in (3). In this paper, we use the terminology optimal 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 , in which the author employed the minimax theorem in infinite dimensions given by Barbu and Precupanu . 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, 1820] and references therein).

We now explain the content of this paper. In Section 2, 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): Uy is Fréchet differentiable. In Section 4, we shall study the minimax optimal control problems: By using the Fréchet differentiability of the solution maps uy and vy, we prove that the maps uJ and vJ 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 uJ and vJ 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 use C 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 p1. H0k is the completions of C0(Ω) in Hk for k1. 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)=H2H01 can be endowed as follows:(5)ψ,ϕH01=ψ,ϕ2,ψ,ϕH01;(6)ψ,ϕDΔ=Δψ,Δϕ2,ψ,ϕDΔ.Then we know that(7)ψH01=ψ2,ψH01,ψDΔ=Δψ2,ψDΔ.The duality pairing between H01 and H-1 is denoted by ϕ,ψ1,-1. It is clear that(8)DΔH01L2H-1.Each space is dense in the following one, and the injections are continuous and compact. According to Adams , we know that the embeddings(9)H01Lp,i.e.,ψpCψ2,ψH01,1p<6,(10)DΔC0Ω¯,i.e.,ϕC0Ω¯CΔϕ2,ϕDΔare compact when n3.

The solution space S(0,T) of (1) of strong solutions is defined by(11)S0,T=ggL20,T;DΔ,gL20,T;DΔ,gL2Qwhich is endowed with the norm(12)gS0,T=gL20,T;DΔ2+gL20,T;DΔ2+gL2Q21/2,where g and g denote the first and second order distributional derivatives of g.

Definition 1.

A function y is said to be a strong solution of (1) if yS(0,T) and y satisfies(13)yt-1+yt22Δyt-μΔyt=Utyt+ft,a.e.t0,T,y0=y0,y0=y1.

From Dautray and Lions [23, p.480] and Lions and Magnes , we remark that(14)S0,TC0,T;DΔC10,T;H01.

The following variational formulation is used to define the weak solution of (1).

Definition 2.

A function y is said to be a weak solution of (1) if yW(0,T){ggL2(0,T;H01),  gL2(0,T;H01),  gL2(0,T;H-1)} and y satisfies(15)y·,ϕ-1,1+1+y·22y·,ϕ2+μy·,ϕ2=U·y·+f·,ϕ-1,1ϕH01inthesenseofD0,T,y0=y0,y0=y1.

The following is the well-known Gronwall inequality.

Lemma 3.

Let η(·) be a nonnegative, absolutely continuous function on [0,T], which satisfies the following differentiable inequality for a.e. t[0,T]:(16)ηtϕtηt+ψt,where ϕ and ψ are nonnegative, summable functions on [0,T]. Then(17)ηte0tϕsdsη0+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  and the previous result of , 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 UL(Q). Then (1) has a unique strong solution yS(0,T). Moreover, the solution mapping p=(y0,y1,f,U)y(p) of PD(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,TCΔy01-y0222+y11-y1222+f1-f2L2Q2+U1-U2LQ21/2Cp1-p2P,where C>0 is a constant depending on the data.

Proof.

From , for each fixed UL(Q) in (1), we can infer that (1) admits a unique strong solution yS(0,T) under the data condition (y0,y1,f)D(Δ)×H01×L2(Q).

Based on this result, for each p1=(y01,y11,f1,U1)P and p2=(y02,y12,f2,U2)P, we prove the inequality (18). For that purpose, we denote y1-y2y(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,ψ0=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)ψt22+Δψt22+0tΔψ22dsCΔy01-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-U2y2L2QU1-U2LQy2L2QCy2S0,TU1-U2LQCU1-U2LQ.Together with (21) and (22), we can deduce the following:(23)ψt22+Δψt22+0tΔψ22dsCΔy01-y0222+y11-y1222+f1-f2L2Q2+U1-U2LQ2Cp1-p2P2.Applying (23) to (19), we have(24)ψL2QCp1-p2P.From (23) and (24), we can obtain(25)ψS0,TCp1-p2P.

This completes the proof.

Corollary 5.

For p1=(y0,y1,f,U1),p2=(y0,y1,f,U2)P, the following inequality is satisfied:(26)yp1-yp2S0,TCU1-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 denote y(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,ψ0=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,TCU1-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)RHSof28Cyp2C0Q¯U1-U2L2QCyp2S0,TU1-U2L2QCp2PU1-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. Let F=L(Q). We consider the nonlinear solution map from uF to y(u)S(0,T), where y(u) is the solution of(30)yu-1+yu22Δyu-μΔyu=uyu+finQ,yu=0onΣ,yu;0,x=y0x,yu;0,x=y1xinΩ.Based on Theorem 4, for fixed (y0,y1,f)D(Δ)×H01×L2(Q), we know that the solution map FS(0,T), which maps from the term uF 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 map uy(u) of F into S(0,T) is said to be Fréchet differentiable on F if for any uF there exists a T(u)L(F,S(0,T)) such that, for any wF,(31)yu+w-yu-TuwS0,TwF0aswF0.

The operator T(u) is called the Fréchet derivative of y at u, which we denote by Dy(u), and T(u)w=Dy(u)wS(0,T) is called the Fréchet derivative of y at u in the direction of wF.

Theorem 7.

The solution map uy(u) of F to S(0,T) is Fréchet differentiable on F and the Fréchet derivative of y(u) at u in the direction wF, that is to say z=Dy(u)w, is the solution of(32)z-1+yu22Δz-2yu,z2Δyu-μΔz=uz+wyuinQ,z=0onΣ,z0,x=0,z0,x=0inΩ.

We prove this theorem by two steps:

For any wF, (32) admits a unique solution zS(0,T). That is, there exists an operator TL(F,S(0,T)) satisfying Tw=z(=z(w)).

We show that y(u+w)-y(u)-zS(0,T)=o(wF) as wF0.

Proof.

(i) Let(33)Gyu,z1+yu22Δz+2yu,z2Δyu.Then from Theorem 4 and (14), we can estimate the above as follows:(34)Gyu,z21+yuC0,T;H012Δz2+2yuC0,T;H01z2yuC0,T;DΔwith14and81+yuS0,T2Δz2+CyuS0,T2Δz2C1+yuS0,T2Δz2C1+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)12ddtz22+μ2ddtΔz22+μΔz22=z,Δz2-Gyu,z,Δz+Δz2-uz+wyu,Δz+Δz2.Integrating (36) over [0,t], we obtain(37)12zt22+μ2Δzt22+μ0tΔz22ds=-zt,zt2+0tz22ds-0tGyu,z,Δz+Δz2ds-0tuz+wyu,Δz+Δz2ds.The right hand side of (37) can be estimated as follows:(38)zt,zt2=zt,0tzds2zt20tzds2Tzt2zL20,t;L2withtheYounginequalityϵzt22+Tϵ0tz22ds;(39)0tGyu,z,Δz+Δz2ds0tGyu,z2Δz2+Δz2dswith35C0tΔz2Δz2+Δz22dswiththeYounginequalityϵ0tΔzt22ds+C0tΔz22ds;(40)0tuz,Δz+Δz2ds0tuz2Δz2+Δz2dsuF0tz2Δz2+Δz2dswith8C0tΔz2Δz2+Δz22dswiththeYounginequalityϵ0tΔzt22ds+C0tΔz22ds;(41)0twyu,Δz+Δz2ds0twyu2Δz2+Δz2dswiththeYounginequalityϵ0tΔzt22ds+C0tΔz22ds+C0twyu22ds.Considering (38)-(41) and taking ϵ=1/6min{1/2,μ/2}, we can obtain the following from (37):(42)zt22+Δzt22+0tΔz22dsC0tz22+Δz22ds+CwyuL2Q2.Applying Lemma 3 to (42), we obtain(43)zt22+Δzt22+0tΔz22dsCwyuL2Q2.In view of (32), (43) implies that(44)zL2QCwyuL2Q.Therefore, from (43) and (44), we can know that zS(0,T), and the solution z(=z(w)) of (32) satisfies(45)zwS0,TCwyuL2QCwFyuL2QCyuS0,TwFCy0,y1,f,uPwF.Hence, from (45), the mapping wFz(w)S(0,T) is linear and bounded. From this, we can infer that there exists TL(F,S(0,T)) such that Tw=z(w) for each wF.

(ii) We set the difference δ=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,δ0,x=0inΩ,where (48)I1=z,yu+w-yu2Δyu+w,I2=2z,yu2Δyu+w-Δyu.If we let (49)Hyu+w,yu,z1+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,TCwz+I1+I2L2Q.From (14), Theorem 4, and (45), we can deduce the following:(52)wzL2QwFzL2QCwFzS0,TCwF2;(53)I1L2QzC0,T;H01yu+w-yuC0,T;H01×Δyu+wL2QCzS0,Tyu+w-yuS0,Tyu+wS0,TCwFu+w-uFy0,y1,f,u+wPCwF2;(54)I2L2Q2zC0,T;H01yuC0,T;H01×Δyu+w-ΔyuL2QCzS0,TyuS0,Tyu+w-yuS0,TCwFy0,y1,f,uPu+w-uFCwF2.Hence, from (51) to (54), we can obtain(55)δS0,TCwz+I1+I2L2QCwzL2Q+I1L2Q+I2L2QCwF2,which immediately implies that δS(0,T)=o(wF) as wF0.

This completes the proof.

The following result plays an important role in proving the existence of optimal controls in the next section.

Proposition 8.

Given wF, the Fréchet derivative Dy(u)w is locally Lipschitz continuous on F with L2(Q) topology. Indeed, it is satisfied that(56)Dyu1w-Dyu2wS0,TCu1-u2L2QwL2Q,where C>0 is a constant depending on the data.

Proof.

Let zi=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-2ϕ,yu12Δyu1-μΔϕ=u1ϕ+i=14IiinQ,ϕ=0onΣ,ϕ0,x=0,ϕ0,x=0inΩ,where (58)I1=2z2,yu1-yu22Δyu1,I2=2z2,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,TCi=14IiL2Q.From Theorem 4, the embedding S(0,T)C0(Q¯), and the first inequality of (45), we can deduce(60)z2S0,TCwyu2L2QCyu2C0Q¯wL2Qwith10and14Cyu2S0,TwL2QCy0,y1,f,u2PwL2QCwL2Q.We can estimate Ii(i=1,,4) of (57) as follows:(61)I1L2Q2z2C0,T;H01yu1-yu2C0,T;H01Δyu1L2Qwith14Cz2S0,Tyu1-yu2S0,Tyu1S0,TwithCorollary5,Theorem4and60CwL2Qu1-u2L2Qy0,y1,f,u1PCu1-u2L2QwL2Q;(62)I2L2Q2z2C0,T;H01yu2C0,T;H01Δyu1-Δyu2L2Qwithanargumentssimilarto61Cz2S0,Tyu2S0,Tyu1-yu2S0,TCwL2Qy0,y1,f,u2Pu1-u2L2QCu1-u2L2QwL2Q;(63)I3L2Qyu1-yu2C0,T;H01yu1+yu2C0,T;H01×Δz2L2Qwithanargumentssimilarto61Cyu1-yu2S0,Tyu1+yu2S0,Tz2S0,TCu1-u2L2Qy0,y1,f,u1P+y0,y1,f,u2PwL2QCu1-u2L2QwL2Q;(64)I4L2Qu1-u2z2L2Q+wyu1-yu2L2Qz2C0Q¯u1-u2L2Q+wL2Qyu1-yu2C0Q¯with10andS0,TC0Q¯Cz2S0,Tu1-u2L2Q+wL2Qyu1-yu2S0,TwithCorollary5and60Cu1-u2L2QwL2Q.From (61) to (64), we can obtain the following from (59):(65)ϕS0,TCu1-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=uFauba.e.inQ.Let the following be the set of the admissible disturbance or noises:(67)Vad=vFcvda.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 Theorem 4, 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)yq-1+yq22Δyq-μΔyq=u+vyq+finQ,yq=0onΣ,yq;0,x=y0x,yq;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+yq22Δz-2yq,z2Δyq-μΔz=u+vz+h+lyqinQ,z=0onΣ,z0,x=0,z0,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 CL(S(0,T),M) is an observer, YdM 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 observer C(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:

Find an admissible control uUad and a noise (or disturbance) vVad such that (u,v) is a saddle point of the functional J(u,v) of (70). That is,(71)Ju,vJu,vJu,v,u,vFad.

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 from Fad to S(0,T) is continuous from the weakly-star topology of Fad to the weak topology of S(0,T).

In proving the Proposition 9, we need the following compactness lemma.

Lemma 10.

Let X,Y and Z be Banach spaces such that the embeddings XYZ are continuous and the imbedding XY is compact. Then a bounded set of W1,(0,T;X,Z)={ggL(0,T;X),gL(0,T;Z)} is relatively compact in C([0,T];Y).

Proof.

See Simon .

Proof of Proposition <xref ref-type="statement" rid="prop4.1">9</xref>.

Let q=(u,v)Fad and let qn=(un,vn)Fad be a sequence such that(72)qnqweakly-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,yn0,x=y1xinΩ.We conduct the scalar product of (73) with -Δyn-Δyn in L2:(74)12ddtyn22+μ+12ddtΔyn22+μΔyn22+1+yn22Δyn22+yn2212ddtΔyn22=yn,Δyn2-un+vnyn+f,Δyn+Δyn2,which immediately implies(75)12ddtyn22+μ+12ddtΔyn22+μΔyn22+yn2212ddtΔyn22yn,Δyn2-un+vnyn+f,Δyn+Δyn2.The integration of (75) over [0,t] implies(76)12ynt22+μ+12Δynt22+μ0tΔyn22ds+12ynt22Δynt22Iy0,y1-ynt,ynt2+0tyn22ds+0tyn,yn2Δyn22ds-0tun+vnyn+f,Δyn+Δyn2ds,where(77)Iy0,y1=12y122+μ+12Δy022+12y022Δy022+y1,y02.By conducting similar calculations to the proof of (i) of Theorem 7, we can obtain the following from (76):(78)ynt22+Δynt22+0tΔyn22dsCIy0,y1+fL2Q2+0tyn22+Δyn22ds+0tyn,yn2Δyn22ds.Since we know from Theorem 4 that ynS(0,T), we can note that(79)yn·,yn·2ynC0,T;H01ynC0,T;H01with14CynS0,T2Cy0,y1,f,un+vnP2.From (78) and (79), we can infer(80)ynt22+Δynt22+0tΔyn22dsC1+0tyn22+Δyn22ds.Applying Lemma 3 to (80), we have(81)ynt22+Δynt22+0tΔyn22dsC.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 yS(0,T)W1,(0,T;D(Δ),H01) such that(82)ynyweaklyinS0,Tasn,(83)ynyweakly-starinL0,T;DΔasn,(84)ynyweakly-starinL0,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)ynktytinH01 for t0,Task.Therefore, (82) and (86) imply(87)ynk22Δynky22ΔyweaklyinL2Qask.From (72) and (85), we can also extract a subsequence, if necessary, denoted again by qn(un,vn) such that(88)un+vnynu+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,y0,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 observer C 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.

Let Pv be the map uJ(u,v) and let Qu be the map vJ(u,v). To obtain the existence of optimal pairs in the minimax control problem, we follow the steps given by : We prove that Pv is convex and lower semicontinuous for all vVad and that Qu is concave and upper semicontinuous for all uUad. Then, we employ the minimax theorem in infinite dimensions (see Barbu and Precupanu ).

For sufficiently large α 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-u20,u1,u2Uad.From Fréchet differentiability of the solution map uy(u,v), where v is fixed, (90) can be rewritten as(91)Cyu1,v-Yd,CDuyu1,vu1-u2M+α0Tu1,u1-u22dt-Cyu2,v-Yd,CDuyu2,vu1-u2M-α0Tu2,u1-u22dt0,u1,u2Uad,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-u2L2Q20,u1,u2Uad.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-u2MCyu1,v-yu2,vMCDuyu1,vu1-u2MCLS0,T,M2yu1,v-yu2,vS0,TDuyu1,vu1-u2S0,TwithCorollary5and60Cu1-u2L2Q2;(94)Cyu2,v-Yd,CDuyu1,vu1-u2-Duyu2,vu1-u2MCyu2,v-YdMCDuyu1,vu1-u2-Duyu2,vu1-u2MCLS0,T,MCLS0,T,Myu2,vS0,T+YdM×Duyu1,vu1-u2-Duyu2,vu1-u2S0,TwithProposition8Cyu2,vS0,T+YdMu1-u2L2Q2Cy0,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 large βl(P,Fad,Yd,C) such that the following inequality is satisfied for any β>βl(P,Fad,Yd,C):(95)DQuv1-DQuv2v1-v20,v1,v2Vad.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 vVad and Qu is upper semicontinuous for all uUad. Let {un}Uad be a minimizing sequence of J. Thus(96)liminfnJun,v=minuUadJu,v.Since Uad defined by (66) is a closed, bounded, and convex in F, we can extract a subsequence {unk}{un} such that(97)unkuweakly-starinUadask.Then, by Proposition 9, we have vVad,(98)yunk,vyu,vweaklyinS0,Task.Thus, by the assumption that CL(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,vCyu,vstronglyinMask,vVad. From (97), it can be easily verified for the same subsequence {unk} in (97) that(100)unkuweaklyinL2Qask.Due to the weakly lower semicontinuity in the L2(Q) norm topology, we can determine from (99) and (100) that the map Pv:uJ(u,v) is lower semicontinuous for all vVad. By similar arguments, we can prove that Qu is upper semicontinuous for all uUad.

Hence, we know that(101)J0v=liminfnJun,vJu,v,vVad.But since J0(v)J(u,v), we have(102)J0v=Ju,v=minuUadJu,v,vVad.Similarly, we also know that there exists vVad such that(103)J0v=maxvVadJ0v.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:

we take M1=L2Q×L2 and C1LS0,T,M1 and observe C1yq=yq;·,yq;TL2Q×L2;

we take M2=L2(Q) and C2L(S(0,T),M2) and observe C2y(q)=y(q;·)L2(Q).

Remark 12.

Clearly, the embedding S(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.

Since y(q)H1(0,T;D(Δ),L2){g    |    gL2(0,T;D(Δ)),  gL2(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 Observations <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M408"><mml:mrow><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>

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 YdL2(Q) and YdTL2 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+vp+yq-YdinQ,p=0onΣ,pT,x=0,pT,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 conditions y(q)-YdL2(Q)L2(0,T;H-1) and y(q;T)-YdTL2 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 tT-t, admits a unique weak solution pW(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.

If α and β in the cost (104) are large enough, then an optimal control uUad and a disturbance vVad, namely, an optimal pair q=(u,v)Fad satisfying (71), can be given by(107)u=maxa,min-yqpα,b,v=maxc,minyqpβ,d,where p is the weak solution of (105).

Proof.

Let q=(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 Theorem 7, 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+ϵwFad 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-yqϵz=zwstronglyinS0,Tasϵ0+,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,vh,l=limϵ0+Ju+ϵh,v+ϵl-Ju,vϵ=limϵ0+120Tyq+ϵw+yq-2Yd,yq+ϵw-yqϵ2dt+limϵ0+12yq+ϵw;T+yq;T-2YdT,yq+ϵw;T-yq;Tϵ2+limϵ0+α20T2u,h2+ϵh22dt-β20T2v,l2+ϵl22dt=0Tyq-Yd,z2dt+yq;T-YdT,zT2+α0Tu,h2dt-β0Tv,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+yq22φ,ϕ2-2yq,φ2yq,ϕ2=1+yq22Δϕ,φ-1,1+2yq,ϕ2Δyq,φ2=φ,Gyq,ϕ1,-1,φ,ϕH01.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+vp,z-1,1dt+μ0TΔp,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,z2dt+pT,zT2-0TGyq,p,z-1,1dt-μ0Tp,Δz2dt-0Tp,u+vz2dt=by110andpT=-yq;T+YdT=0Tp,z2dt-yq;T-YdT,zT2-0Tp,Gyq,z2dt-μ0Tp,Δz2dt-0Tp,u+vz2dt=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,vh,l=0Tαu+yqp,h2dt+0T-βv+yqp,l2dt.Since q=(u,v)Fad is an optimal pair in (71), we know that(115)DuJu,vh0,DvJu,vl0,h,lF2.Therefore, we can obtain the following from (114) and (115):(116)0Tαu+yqp,h2dt0,0T-βv+yqp,l2dt0,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=maxa,min-yqpα,b,v=maxc,minyqpβ,d.

This completes the proof.

4.2.2. Case of Velocity Observation <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M462"><mml:mrow><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>

In this observation case, we consider the cost function associated with the control system (68):(118)Ju,v=12yq-YdL2Q2+α2uL2Q2-β2vL2Q2,where YdL2(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 , 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+vpds+μΔp=yq-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 ) 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)pH10,T;H01,L2C0,T;H01,where H1(0,T;H01,L2) is the solution space of (119) given by(121)H10,T;H01,L2=ϕϕL20,T;H01,ϕL2Q.

Proof.

Since (122)T-tTGyq,p+u+vpsds=0tGyq,p+u+vpT-σdσ,the time reversed equation of (119) (t        T-t in (119)) is given by(123)-ψ+0tGyq,ψ+u+vψdσ+μΔψ=-yq-YdinQ,ψ=0onΣ,ψ0,x=0inΩ,where ψ(·)=p(T-·). From (106) and -y(q)-YdL2(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 ψH1(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.

If α and β in the cost (118) are large enough, then an optimal control uUad and a disturbance vVad, namely, an optimal pair q=(u,v)Fad satisfying (71), can be given by:(124)u=maxa,minyqpα,b,v=maxc,min-yqpβ,d, where p is the weak solution of (119).

Proof.

Let q=(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 Theorem 15, the Gâteaux derivative of the cost (118) at q=(u,v) in the direction w=(h,l)F2 that satisfies q+ϵwFad for sufficiently small ϵ>0 is given by(125)DJu,vh,l=limϵ0+Ju+ϵh,v+ϵl-Ju,vϵ=0Tyq-Yd,z2dt+α0Tu,h2dt-β0Tv,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,z2dt+0TtTGyq,p+u+vpds,z-1,1dt-μ0Tp,z2dt=0Tyq-Yd,z2dt.By integration by parts and the terminal value of the weak solution p of (119), (126) can be rewritten as(127)-0Tp,z2dt+0TGyq,p+u+vp,z-1,1dt+μ0Tp,Δz2dt=By110=-0Tp,z2dt+0Tp,Gyq,z+u+vz2dt+μ0Tp,Δz2dt=0Tyq-Yd,z2dt.Since z is the solution of (69), we can obtain the following from (127):(128)0Tyq-Yd,z2dt=-0Th+lyq,p2dt.Therefore, we can deduce that (125) and (128) imply(129)DJu,vh,l=0Tαu-yqp,h2dt+0T-βv-yqp,l2dt.Since q=(u,v)Fad is an optimal pair in (71), we know that(130)DuJu,vh0,DvJu,vl0,h,lF2.Therefore, we can obtain the following from (129) and (130):(131)0Tαu-yqp,h2dt0,0T-βv-yqp,l2dt0,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=maxa,minyqpα,b,v=maxc,min-yqpβ,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.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares no conflicts of interest.

Authors’ Contributions

The author read and approved the final manuscript.

Acknowledgments

This research was supported by the Daegu University Research Grant 2015.

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