# Bilinear Minimax Optimal Control Problems of the Velocity Term in an Extensible Beam Equation with Rotational Inertia

Authors: Jinsoo Hwang
Journal: Journal of Mathematics
 (2023)

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## Abstract

The author studies the minimax optimal control problem for an extensible beam equation that takes into account rotational inertia effects. A velocity term multiplied by the bilinear control and disturbance functions to construct a minimax optimal control strategy is added. The existence of an optimal pair, meaning optimal control and perturbation, is proved by assuming some conditions on the considered quadratic cost function. The optimal conditions for optimal pairs are provided by the adjoint systems that correspond to some physically meaningful observation cases.


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## Body

## 1. Introduction

This study is devoted to the study of the minimax optimal control problem for the following model of an extensible beam with rotational force:(1)1−γΔy″+Δ2y−1+∫Ω∇y2dxΔy+gy=Uy′+finQ≔Ω×0,T,where ′=∂/∂t,Ω is a bounded domain in RN with a sufficiently smooth boundary ∂Ω, where N∈1,2,3 (mainly N=3), the positive constant γ represents the effect of the rotational inertia of the beam, gy is a nonlinear term that is explicitly explained later, U is a control input function, and f is a forcing term. We consider either hinged boundary condition(2)y=Δy=0onΣ≔∂Ω×0,T,or clamped boundary condition(3)y=∂y∂ν=0onΣ,where ν is the outward unit normal vector tailing on ∂Ω. And we consider the initial condition(4)yx,0=y0x,y′x,0=y1xinΩ.Equation (1) with γ=0 has been extensively studied in many articles. Initially, Woinowsky [1] proposed a one-dimensional version with the goal of describing the transverse deflection of an extensible beam. From a more physical point of view, including mathematical analysis of the extensible beam model, we can refer to Ball [2], Dickey [3], and Eisley [4].We can also find many pioneering studies on the same equation, such as the well posedness of the equation with an additional damping term, or the stability of a solution to the equation (see [5–9]).By the way, the case whereγ>0 in Equation (1) is called a Rayleigh beam that is introduced to consider the effect of rotational inertia on a large deflection beam model. For the introduction and studies on Rayleigh beam models, we can refer to Chueshov and Lasiecka [10, 11].As a contribution to control theory to nonlinear beam equations without rotational inertiaγ=0, we studied in [12] the optimal control problem by the framework of Lions [13] (cf. [14], [15]) using distributed forced control variables. In [12], the optimal control problems were studied for Equation (1) in which γ=0 and without gy and U under the framework of Lions [13]. And quite recently, we studied in [16] that the nonlinear solution map of equation (1) with γ=0 is Fréchet differentiable and applied our results to a bilinear robust control problem in which the control variable is taken as a multiplier of the displacement term rather than the velocity term. That is to say, Uy′ is replaced by Uy in Equation (1). By the minimax optimal control strategy (cf. [17]), the existence of an optimal pair was shown and the necessary optimality condition of the optimal pair was studied, which satisfies(5)Jc¯,d≤Jc¯,d¯≤Jc,d¯∀c,d∈Uad×Vad,where J is a quadratic cost, c and d are distributed control and disturbance, respectively, and Uad and Vad are admissible sets.As is noted in [18], [19], in a hyperbolic control system, a control strategy with a velocity term rather than a displacement term seems preferable. The author in this study has already pointed this out and tried to apply it to Equation (1) with γ=0. However, we found that this requires more regularity of the control variable or solution, which is not favorable in control theory. With this in mind, in this study, the author is motivated to study the minimax optimal control problems for the bilinear control input of the velocity term of Equation (1) with γ>0.A description of the minimax optimal control strategy is given in [16]. To set up the control problem, the multiplier function U in Equation (1) is replaced with c+d. Here, c and d are control and disturbance variables, respectively, and c and d belong to the set of admissible control and disturbance sets Uad and Vad, respectively. The following quadratic cost function is considered:(6)Jc,d=12Cy−YdM2+α2cU2−β2dV2,where y is a solution satisfying Equation (1), C is an observation operator, M,U,V are the Hilbert spaces of observation, control, and disturbance (or noise) variables, respectively, Yd∈M is the aiming value, and α and β are the positive constants related to the weights of the second and third terms of (6).In this study, the author tries to find and characterize control and noise variables to minimize and maximize the quadratic cost (6) within Uad and Vad, respectively. In other words, it is a problem of finding and characterizing a saddle point that satisfies (5). As stated in [16], without any confusion, the term optimal pair is still used to denote these saddle points c¯,d¯ in (5). For the study of the existence of an optimal pair c¯,d¯ satisfying (5), the minimax theorem in infinite dimensions proposed by Barbu and Precupanu [20] (cf. [17]) is used. To do this, we need to ensure that the solution map is differentiable and that its derivative is continuous in the Hilbert norm topology. Therefore, the Fréchet differentiability of the solution map was verified from U in Equation (1) to the solution of Equation (1), and the local Lipschitz continuity of the Fréchet derivative was proved by imposing some conditions on the nonlinear function g in Equation (1).Next, the necessary optimality conditions for the optimal pairsc¯,d¯ are derived, corresponding to physically meaningful observation cases. In this paper, the author mainly considers two observation cases: The first observation case is the distribution of velocities, and the other is the distributive and terminal value observation case. In order to derive the optimal conditions for the optimal pair, we need to find and use the relevant adjoint equation corresponding to the observed case. The optimal pair can then be given explicitly through the adjoint system.The novelty of this paper is summarized as follows: In the case of velocity distribution observation, the author is faced with the difficulties of regularity in the process of deriving the optimal condition for the optimal pair through the adjoint equation, but this problem is overcome by the double regularization method. For the distributive and terminal value observation case, due to the lack of regularity of the control and noise variables, it is not possible to explicitly construct the related adjoint system. In general, control can be irregular; instead of assuming regularity in admissible control and noise sets, the transposition method is used to derive the necessary optimality condition for the optimal pair.


## 2. Notations and Preliminaries

Given a Banach spaceX, its topological dual is denoted by X′, and the duality pairing between X′ and X by ⋅,⋅X′,X. For simplicity, the following abbreviations are used:(7)⋅p=⋅LpΩ,H=L2Ω,where p≥1.⋅,⋅2 and ⋅2 represent the scalar product and norm on H (or HNN≤3). As is known, it is denoted by HkΩ that the Sobolev space of order k≥1 on Ω with the L2-based Sobolev norm. H0kΩ is the completion of C0∞Ω for the HkΩ norm.It is denoted as(8)V2=H2Ω∩H01Ω,for condition,H02Ω,for condition,and the operator A:V2⟶V2′ is defined by(9)Aϕ,ψV2′,V2=Δϕ,Δψ2,∀ϕ,ψ∈V2,(10)DA=ϕ∈V2|Aϕ∈H.SinceA is self-adjoint from V2 into V2′, it is strictly positive on V2 due to (9), and the injection of V2 in H is compact. Thus, from the spectral theory of self-adjoint compact operators in the Hilbert space as given in ([27], Theorem 7.7) (cf. [21]), a complete orthonormal basis of H, wkk=1∞, can be found,which consists of eigenvectors of A, such that(11)Awk=λkwk∀k,0<λ1≤λ2≤⋯,λk⟶∞ask⟶∞.Thus, the powersAs of A can be defined for any s∈R, such that for s≥0,(12)DAs=ϕ∈H|∑k=1∞λk2sϕ,wk22<∞.Fors<0, DAs is the completion of H for the following norm:(13)∑k=1∞λk2sϕ,wk221/2.Fors∈R, the scalar product and norm of DAs can be written alternatively as(14)ϕ,ψDAs=Asϕ,Asψ2=∑k=1∞λk2sϕ,wk2ψ,wk2,ϕDAs=ϕ,ϕDAs1/2=∑k=1∞λk2sϕ,wk221/2.Embedding is as follows:(15)DAs1↪DAs2is continuous ifs1≥s2and compact ifs1>s2.Furthermore, fors1≥s2,(16)ϕDAs2≤λ1s2−s1ϕDAs1∀ϕ∈DAs1.For simplicity, throughout the paper, the author denotesVs=DAs/4. Then, for s∈R, Vs are Hilbert spaces with the following scalar products and norms:(17)ϕ,ψs≔ϕ,ψVs=As/4ϕ,A4/sψ2,ϕVs=As/4ϕ2.For allϕ,ψ∈Vs. Thus,(18)ϕV2=A1/2ϕ2=Δϕ2,ψV1=A1/4ψ2=∇ψ2.For allϕ,ψ∈V2×V1.From the well-known embedding theorem, as given by Adams [22], the following embedding(19)V2↪C0Ω,is compact when N≤3.The following assumptions forgy in Equation (1) are given as follows. g⋅ in Equation (1) is a C1 function with g0=0:(A1) LetG be the function given by Gs=∫0sgrdr, and we make the following assumption:(20)liminfs⟶∞Gss2≥0.(A2) There exists a constantc1>0 such that(21)g′s≤c11+sr0≤r<∞.It is deduced from (20) that, for every η>0, there exists a constant Cη>0 such that(22)Gs+ηs2>−Cη∀s∈R.We can convert Equation (1) with the boundary condition (2) or (3) and the initial condition (4) to an absolute evolution equation, which is given by(23)Mγy″+Ay+1+yV12A1/2y+gy=Uy′+fin0,T,y0=y0,y′0=y1,where Mγ=I+γA1/2.Remark 1.
From (A1), (A2), and (19), the following conditions are deduced which are suitable for the nonlinear operator g in Equation (23):(i)
It follows from (21) and (19) that the nonlinear operator g in Equation (23) is a C1 bounded operator from V2 into H, a Fréchet differentiable with the differential g′, and Lipschitzian from the bounded sets of V2 into H. Indeed, for every R>0, there exists cR such that(24)gϕ1−gϕ22≤cRϕ1−ϕ22∀ϕi∈V2i=1,2,whereϕiV2≤Ri=1,2.(ii)
As a consequence of (A1) and (22), it is deduced that there exists G∈C1V2,R,G0=0, such that gϕ=G′ϕ,∀ϕ∈V2 and that for every η>0, there exists a constant Cη>0 such that(25)Gϕ+ηϕ22>−Cη∀ϕ∈V2.
The Hilbert spaceW0,T is defined by(26)W0,T=ϕ|ϕ∈L20,T;V2,ϕ′∈L20,T;V1,ϕ″∈L20,T;H,equipped with the norm(27)ϕW0,T=ϕL20,T;V22+ϕ′L20,T;V12+ϕ″L20,T;H21/2,where ϕ′ and ϕ″ represent the distributional derivatives of ϕ with respect to time variables.
In the following, for simplicity,C is frequently used to denote the generic constant and the integral variable is omitted from all definite integrals.Definition 1.
The author callsy a weak solution of Equation (23) if y∈W0,T, and the following holds(28)Mγy″⋅,ϕV2′,V2+y⋅,ϕ2+1+y⋅V12y⋅,ϕ1+gy⋅,ϕ2=U⋅y′⋅+f⋅,ϕ2∀ϕ∈V2in the sense ofD′0,T,y0=y0,y′0=y1.
The following existence theorem can be given by referring to Chueshov and Lasiecka [10, 11].Theorem 1.
Let (A1) and (A2) be fulfilled and y0,y1,f∈V2×V1×L20,T;V1′, and U∈L20,T;H. Then, there exists a weak solution y of Equation (23), satisfying(29)y∈W0,T∩L∞0,T;V2∩W1,∞0,T;V1.To prove the regularity and uniqueness of a weak solution of Equation (23), the steps mentioned by Lions and Magenes are followed ([23], pp. 275–278). First, the following lemma provided by Lions and Magenes is exploited [23].Lemma 1.
LetX,Y be two Banach spaces, X⊂Y with dense and X being reflexive.(30)Cw0,T;Y=f∈L∞0,T;Y|f⋅,φY,Y′∈C0,T,∀φ∈Y′,and then,(31)L∞0,T;X∩Cw0,T;Y=Cw0,T;X.Then, the following improved regularity for the weak solutiony of Equation (23) can be proved.Corollary 1.
Lety be a weak solution of Equation (23). Then, it can be seen that(32)y∈Cw0,T;V2,y′∈Cw0,T;V1.Proof.
From Dautray and Lions ([26], p. 480), it is clear that W0,T ↪ C0,T;V1∩C10,T;H. Therefore, since y∈W0,T∩L∞0,T;V2∩W1,∞0,T;V1, the proof is the immediate consequence of Lemma 1 obtained by setting X=V2,Y=V1 to have y∈Cw0,T;V2 and by setting X=V1,Y=H to have y′∈Cw0,T;V1.
Hence, the proof is completed.
The following lemma is frequently used to obtain many estimates throughout the study.Lemma 2.
Ify is the weak solution of Equation (23), then the equation is obtained as follows:(33)y′t22+γy′tV12+ytV22+121+ytV122+2Gyt=2∫0tUy′,y′2ds+2∫0tf,y′V1′,V1ds+y122+γy1V12+y0V22+121+y0V122+2Gy0.Proof.
Based on (32), we follow the proof given in Lions and Magenes ([23], pp. 276–279). By regarding f in ([23], pp. 276–279) as 1+yV12A1/2y−gy+Uy′+f, the following equation is obtained through the double regularization, as given in ([23], pp. 276–279):(34)y′t22+γy′tV12+ytV22=y122+γy1V12+y0V22+2∫0tUy′,y′2ds+2∫0tf,y′V1′,V1ds−2∫0tgy,y′2ds−2∫0t1+yV12A1/2y,y′2ds.
Since(35)12ddt1+ytV122=1+ytV12ddtytV12=21+ytV12A1/2yt,y′t2,ddtGyt=gyt,y′t2,(33) can be obtained by combining (34) with (35).
Hence, the proof is completed.
Now, it paves the way to state the following theorem.Theorem 2.
Lety be the weak solution of Equation (23). Then, y∈C0,T;V2∩C10,T;V1. Moreover, the mapping p=y0,y1,f,U⟶yp of P:=V2×V1×L20,T;V1′×L20,T;H into S0,T=W0,T∩C0,T;V2∩C10,T;V1 is locally Lipschitz continuous. Let pi=y0i,y1i,fi,Ui∈Pi=1,2. Then, the equation is obtained as follows:(36)yp1−yp2S0,T≤Cy01−y02V22+y11−y12V12+f1−f2L20,T;V1′2+U1−U2L20,T;H21/2≡Cp1−p2P.Proof.
Using the energy equality method by Dautray and Lions ([26], pp. 578–581) combined with (32) and (34); then, the equation is obtained as follows:(37)y∈C0,T;V2∩C10,T;V1.
Therefore, the author focuses on showing (36). For each pi=y0i,y1i,fi,Ui∈Pi=1,2, the author denotes y1−y2≡yp1−yp2 by y. Then, it is deduced from Equation (23) that y satisfies(38)Mγy″+Ay+1+y1V12A1/2y+gy1−gy2=ηy+U1y′+U1−U2y2′+f1−f2in0,T,y0=y01−y02,y′0=y11−y12,in the weak sense where(39)ηy=−y1V12−y2V12A1/2y2=−y,y1+y21A1/2y2.
By analogy with (34), the energy equality for the weak solution y of Equation (38) can be deduced as follows:(40)y′t22+γy′tV12+ytV22=y′022+γy′0V12+y′0V22+2∫0tηy+U1y′,y′2ds+2∫0tU1−U2y2′,y′2ds+2∫0tf1−f2,y′V1′,V1ds−2∫0tgy1−gy2,y′2ds−2∫0t1+y1V12A1/2y,y′2ds.
From (24) and the fact that S0,T ↪ CQ¯,(41)gy1−gy222≤Cy22,where C depends only on pii=1,2. Thanks to (41),(42)2∫0tgy1−gy2,y′2ds≤2∫0tgy1−gy22y′2ds≤C∫0ty2y′2ds≤C∫0tyV2y′2ds≤C∫0tyV22+y′22ds.
SinceV1 ↪ L4Ω when N≤3,(43)2∫0tU1−U2y2′,y′2ds≤2∫0tU1−U22y2′4y′4ds≤C∫0tU1−U22y2′V1y′V1ds≤C∫0tU1−U22y′V1ds≤C∫0tU1−U222+y′V12ds,where C depends only on p2. Then, for the other terms to the right of (40), the author refers to [16] and uses Gronwall’s lemma to obtain(44)yC0,T;V2∩C10,T;V1≤Cp1−p2P.
SinceA∈LV2,V2′ is isomorphism, it is inferred from Equations (38) and (44) that(45)Mγy″L20,T;V2′≤Cp1−p2P,where Mγ∈LV2,H∩LH,V2′ is also isomorphism. Thus, finally from (44) and (45), the following is obtained:(46)yS0,T≤Cp1−p2P.
Hence, the proof is completed.


## 3. Statement of the Main Results

The goal of this section is to describe the minimax optimal control problem for the proposed equation and to state the main results. As stated before, the control problem is set up as follows:U in equation (23) is replaced by c+d, where c and d belong to the admissible control set Uad and the admissible disturbance (or noise) set Vad, respectively.The assumptions forUad and Vad are given as follows:(A3) The control set ofu is given by(47)Uad=c∈L∞Q|c1≤c≤c2a.e.inQ.Here,ci∈R,i=1,2.(A4) The set ofd (noises) is given by(48)Vad=d∈L∞Q|d1≤d≤d2a.e.inQ.Here,di∈R,i=1,2.For simplicity, the author denotesWad=Uad×Vad. Thanks to Theorem 1, for fixed y0,y1,f∈V2×V1×L20,T;V1′, there exists a unique weak solution yq∈S0,T satisfying(49)Mγy″q+Ayq+1+yqV12A1/2yq+gyq=c+dy′q+fin0,T,y0=y0,y′0=y1,where q=c,d∈Wad. Thus, it is deduced that the solution map Wad⟶S0,T is well defined. The quadratic cost function is considered as follows:(50)Jq=Jc,d=12Cyq−YdM2+α2cL20,T;H2−β2dL20,T;H2,where Yd∈M is the aiming state value and M and C∈LW0,T,M denote the Hilbert space of observation variables and the observation operator, respectively.As stated before, the control strategy is to find and characterize optimal controlc¯∈Uad even at the worst disturbance (or noise) d¯∈Vad, satisfying(51)Jc¯,d≤Jc¯,d¯≤Jc,d¯∀c,d∈Wad,where c¯,d¯ is called the saddle point of the function J⋅,⋅ in (50). Then, using the methods given in [16] (cf. [18]), the author characterizes the optimal pair c¯,d¯ by stating the necessary optimal conditions through the adjoint equations related to Equation (49) and the cost (50).The main results of this paper are as follows.


### 3.1. Case ofM=L20,T;H and Observe Cyq=y′q

If we takeM=L20,T;H and observe Cyq=y′q in (50), then the optimal pair q¯=c¯,d¯∈Wad can be characterized as follows.Theorem 3.
Let (A1)– (A5) be fulfilled. For sufficiently largeα and β in the cost (50), there exists an optimal pair q¯=c¯,d¯∈Wad satisfying (50) such that it can be given by(52)c¯=maxc1,miny′q¯p′α,c2,d¯=maxd1,min−y′q¯p′β,d2,where yq¯ is the weak solution of Equation (49), corresponding to q¯=c¯,d¯∈Wad, and p is the weak solution of(53)Mγp″+Ap−∫tTGyq¯,p′+g′0p′ds=−c¯+d¯p′+y′q¯−Ydin0,T,pT=0,p′T=0,where(54)Gyq¯,p′=1+yq¯V12A1/2p′+2yq¯,p′1A1/2yq¯.Yd∈M=L20,T;H is an aiming observation value.


### 3.2. Case ofM=L20,T;H×H and Observe Cyq=yq,yq;T

If we takeM=L20,T;H×H and observe Cyq=yq,yq;T in (50), then the optimal pair q¯=c¯,d¯∈Wad can be characterized as follows.Theorem 4.
Let (A1)–(A5) be fulfilled. For sufficiently largeα and β in the cost (50), there exists an optimal pair q¯=c¯,d¯∈Wad satisfying (50) such that it can be given by(55)c¯=maxc1,min−y′q¯pα,c2,d¯=maxd1,miny′q¯pβ,d2,where yq¯ is the weak solution of Equation (49), corresponding to q¯=c¯,d¯∈Wad, and p is the weak solution satisfying(56)∫0Tp,LϕV1,V1′dt=∫0Tyq¯−Yd,ϕ2dt+yq¯;T−YdT,ϕT2∀ϕsuch thatLϕ∈L20,T;V1′,ϕ0=ϕ′0=0,where(57)Lϕ=Mγϕ″+Aϕ+Gyq¯,ϕ+g′yq¯ϕ−c¯+d¯ϕ′.Gyq¯,⋅is defined in (54), and Yd,YdT∈M=L20,T;H×H is an aiming observation value.Remark 2.
(i) Assumption (A5) is specified in the next section. (ii) The weak solutionp satisfying (56) is usually called the transposition solution (cf. ([13], pp. 291–295)) and formally satisfies the following equation:(58)Mγp″+Ap+Gyq¯,p+g′yq¯p=−c¯+d¯p′+yq¯−Ydin0,T,pT=0,p′T=−Mγ−1yq¯;T−YdT.
The reason for using the transposition solutionp in the necessary optimality conditions is explained later.


## 3.1. Case ofM=L20,T;H and Observe Cyq=y′q

If we takeM=L20,T;H and observe Cyq=y′q in (50), then the optimal pair q¯=c¯,d¯∈Wad can be characterized as follows.Theorem 3.
Let (A1)– (A5) be fulfilled. For sufficiently largeα and β in the cost (50), there exists an optimal pair q¯=c¯,d¯∈Wad satisfying (50) such that it can be given by(52)c¯=maxc1,miny′q¯p′α,c2,d¯=maxd1,min−y′q¯p′β,d2,where yq¯ is the weak solution of Equation (49), corresponding to q¯=c¯,d¯∈Wad, and p is the weak solution of(53)Mγp″+Ap−∫tTGyq¯,p′+g′0p′ds=−c¯+d¯p′+y′q¯−Ydin0,T,pT=0,p′T=0,where(54)Gyq¯,p′=1+yq¯V12A1/2p′+2yq¯,p′1A1/2yq¯.Yd∈M=L20,T;H is an aiming observation value.


## 3.2. Case ofM=L20,T;H×H and Observe Cyq=yq,yq;T

If we takeM=L20,T;H×H and observe Cyq=yq,yq;T in (50), then the optimal pair q¯=c¯,d¯∈Wad can be characterized as follows.Theorem 4.
Let (A1)–(A5) be fulfilled. For sufficiently largeα and β in the cost (50), there exists an optimal pair q¯=c¯,d¯∈Wad satisfying (50) such that it can be given by(55)c¯=maxc1,min−y′q¯pα,c2,d¯=maxd1,miny′q¯pβ,d2,where yq¯ is the weak solution of Equation (49), corresponding to q¯=c¯,d¯∈Wad, and p is the weak solution satisfying(56)∫0Tp,LϕV1,V1′dt=∫0Tyq¯−Yd,ϕ2dt+yq¯;T−YdT,ϕT2∀ϕsuch thatLϕ∈L20,T;V1′,ϕ0=ϕ′0=0,where(57)Lϕ=Mγϕ″+Aϕ+Gyq¯,ϕ+g′yq¯ϕ−c¯+d¯ϕ′.Gyq¯,⋅is defined in (54), and Yd,YdT∈M=L20,T;H×H is an aiming observation value.Remark 2.
(i) Assumption (A5) is specified in the next section. (ii) The weak solutionp satisfying (56) is usually called the transposition solution (cf. ([13], pp. 291–295)) and formally satisfies the following equation:(58)Mγp″+Ap+Gyq¯,p+g′yq¯p=−c¯+d¯p′+yq¯−Ydin0,T,pT=0,p′T=−Mγ−1yq¯;T−YdT.
The reason for using the transposition solutionp in the necessary optimality conditions is explained later.


## 4. Differentiability of the Control-to-State Map

The goal of this section is to show the Fréchet differentiability of the nonlinear solution map fromv∈L20,T;H to yv∈S0,T, where yv satisfies(59)Mγy″v+Ayv+1+yvV12A1/2yv+gyv=vy′v+fin0,T,yv;0=y0,y′v;0=y1,where y0,y1,f∈V2×V1×L20,T;V1′ is fixed. Then, from Theorem 2, it is deduced that the control-to-state map L20,T;H⟶S0,T from the term v∈L20,T;H of Equation (59) to yv∈S0,T is uniquely defined and continuous.The following definitions of functional differentiation are presented.Definition 2.
The control-to-state mapv⟶yv of L20,T;H into S0,T is said to be Fréchet differentiable at v if there exists an operator Fv∈LL20,T;H,S0,T and a mapping rv,⋅:L20,T;H⟶S0,T with the following properties. For any h∈L20,T;H, there exists(60)yv+h=yv+Fvh+rv,h,rv,hS0,T∈ohL20,T;H,as hL20,T;H⟶0, where yv and yv+h are the weak solutions of Equation (59), corresponding to v and v+h, respectively.Definition 3.
Letyv be a weak solution of Equation (59), corresponding to v∈U⊂L20,T;H. Suppose that the first variation δyvh at v in the direction h∈L20,T;H exists, and there exists a continuous linear operator G:L20,T;H⟶S0,T such that(61)δyvh=Gh∀h∈L20,T;H.
Then, the control-to-state mapv⟶yv is said to be Gâteaux differentiable at v in the direction h, and we write G=Dyv.
As is well known, Fréchet differentiability also implies Gâteaux differentiability, and the Gâteaux derivative then coincides with the Fréchet derivative. Therefore, the author gives priority to the study on the Fréchet differentiability of the control-to-state map.Theorem 5.
Letyv be the weak solution of Equation (59). Then, the Fréchet derivative of yv at v in the direction of h∈L20,T;H, denoted by z=Dyvh, is given by the weak solution(62)Mγz″+Az+1+yvV12A1/2z+2yv,z1A1/2yv+g′yvz=vz′+hy′vin0,T,z0=0,z′0=0.Proof.
Let(63)Gyv,z≔1+yvV12A1/2z+2yv,z1A1/2yv.
Then, from Theorem2 and Theorem 3 in [16], the following equation is obtained:(64)Gyv,⋅∈LV2,H.
Due to (A2) withyv∈S0,T ↪ CQ¯ and (64), it can be seen from Theorem 2 that there exists a unique weak solution z∈S0,T to linearized system Equation (62) and satisfies(65)zS0,T≤ChL20,T;H.
Hence, from (65), the mapping h∈L20,T;H↦zh∈S0,T is linear and bounded. Therefore, it is verified that the existence of an operator F∈LL20,T;H,S0,T with Fh=zh for any h∈L20,T;H.
For convenience of calculation, the author denotesyv+h−yv−z by ϕ. Then, from Theorem 3 given in [16],(66)gyv+h−gyv−g′yvz=G1yv+h,yvyv+h−yv−g′yvz=G1yv+h,yvϕ+G1yv+h,yv−g′yvz,where(67)G1yv+h,yv=∫01g′θyv+h+1−θyvdθ.
It is known thatϕ satisfies(68)Mγϕ″+Aϕ+1+yvV12A1/2ϕ+ϕ,yv+h+yv1A1/2yv+h+G1yv+h,yvϕ=v+hϕ′+hz′+I1+I2+I3in0,T,ϕ0=0,ϕ′0=0.
In the weak sense,(69)I1=−2yv,z1A1/2yv+h−A1/2yv,I2=−z,yv+h−yv1A1/2yv+h,I3=−G1yv+h,yv−g′yvz.
Regardingf1−f2 in Equation (38) as hz′+I1+I2+I3 and estimating the weak solution ϕ of Equation (68) through the energy equation for ϕ, the following inequality is obtained:(70)ϕS0,T≤Chz′+I1+I2+I3L20,T;V1′.
First, sinceV1↪L4Ω when N≤3, the following inequality is obtained with (65) that(71)∫0Thz′,φ2dt≤∫0Th2z′4φ4dt≤C∫0Th2z′V1φV1dt≤CzS0,T∫0Th2φV1dt≤ChL20,T;H2φL20,T;V1.
For allφ∈L20,T;V1. Thus,(72)hz′L20,T;V1′≤ChL20,T;H2.
By similar arguments in [16] together with Theorem 2 and (65), it can be deduced that(73)IiL20,T;H≤ChL20,T;H2,where i=1,2. And also,(74)I3L20,T;H≤zC0,T;HG1yv+h,yv−g′yvL20,T;H≤CzS0,TG1yv+h,yv−g′yvL20,T;H≤CwL20,T;HG1yv+h,yv−g′yvL20,T;H.(75)Sinceyu+h⟶yvstrongly inS0,T↪CQ¯.
AshL20,T;H⟶0 and g is a C1- class nonlinear operator, the author infers with (A2) and the Lebesgue-dominated convergence theorem that(76)g∈C1S0,T,L20,T;H.
Therefore, the author deduces with (74), (75), and (76) that(77)I3L20,T;H∈ohL20,T;HashL20,T;H⟶0.
Hence, from (70)–(77), the following equation can be obtained:(78)ϕS0,T∈ohL20,T;HashL20,T;H⟶0.
Hence, the proof is concluded.
The following results are used to prove the existence of optimal pairs in the next section. Another assumption forg is needed to prove the following result. In addition to (A1) and (A2), we give another assumption for gy in equation (1) as follows:(A5)g⋅ in Equation (1) is a C2 function, and there exists a constant c2>0 such that(79)g″s≤c21+sp0≤p<∞.Proposition 1.
Let (A1), (A2), and (A5) hold. Givenh∈L20,T;H, the Fréchet derivative Dyvh is locally Lipschitz continuous on L20,T;H. Indeed, it is satisfied that(80)Dyv1h−Dyv2hS0,T≤Cv1−v2L20,T;HhL20,T;H.Proof.
Letzi=Dyvihi=1,2 be the weak solutions of Equation (62), corresponding to vii=1,2. The author sets z=z1−z2. Then, it is deduced that z satisfies(81)Mγz″+Az+1+yv1V12A1/2z+2yv1,z1A1/2yv1+g′yv1z=v1z′+∑i=15Iiin0,T,z0=0,z′0=0.
In the weak sense,(82)I1=−yv1−yv2,yv1+yv21A1/2z2,I2=−2yv1−yv2,z21A1/2yv1,I3=−2yv2,z21A1/2yv1−A1/2yv2,I4=−g′yv1−g′yv2z2,I5=v1−v2z2′+hy′v1−y′v2.
By analogy with (70), it is deduced that the weak solution z of Equation (81) can be estimated as(83)zS0,T≤C∑i=15IiL20,T;V1′.
From (65), the following equation can be obtained:(84)ziS0,T≤ChL20,T;Hi=1,2.
Hence, by Theorem2 and (84), I1 can be estimated as(85)I1L20,T;H≤Cv1−v2L20,T;HhL20,T;H.
Similarly,Iii=2,3 can be estimated as follows:(86)IiL20,T;H≤Cv1−v2L20,T;HhL20,T;Hi=2,3.
Thanks to (A5) and (84) with S0,T↪CQ¯, I4 can be estimated by(87)I4L20,T;H≤g′yv1−g′yv2L20,T;Hz2C0,T;H≤Cv1−v2L20,T;HhL20,T;H.
By analogy with (71) and (72), the following equation can be obtained:(88)I5L20,T;V1′≤v1−v2z2′L20,T;V1′+hy′v1−y′v2L20,T;V1′≤Cv1−v2L20,T;Hz2S0,T+Cyv1−yv2S0,ThL20,T;H≤Cv1−v2L20,T;HhL20,T;H.
Finally, by (83)–(88), the following equation can be obtained:(89)zS0,T≤Cv1−v2L20,T;HhL20,T;H.
Hence, the proof is concluded.


## 5. Proof of the Existence

The goal of this section is to study the existence of an optimal pairc∗,d∗ satisfying (5) under some reasonable conditions for the constants α and β in the quadratic cost. First, the following result is needed.Proposition 2.
The mappingc,d↦yc,d is sequentially continuous from Wad, endowed with weak-L20,T;H topology, to W0,T.Proof.
Letc¯,d¯=q¯∈Uad×Vad, and let cn,dn=qn∈Wad=Uad×Vad be a sequence such that(90)cn,dn⇀c¯,d¯weakly inL20,T;H2,respectively, as n⟶∞. From now on, ycn,dn is denoted by yn, which is the solution of Equation (49), in which c and d are replaced by cn and dn, respectively. From Theorem 2, the following equation can be obtained:(91)ynS0,T≤Cy0,y1,f,cn+dnP.
Hence, by Rellich’s extraction theorem, a subsequence can be extracted fromyn, say again yn, and y¯∈W0,T can be found such that(92)yn⇀y¯weaklyinW0,Tasn⟶∞,(93)yn⇀y¯weakly∗inL∞0,T;V2asn⟶∞,(94)yn′⇀y¯′weakly∗inL∞0,T;V1asn⟶∞.
From the fact that embeddingV2↪V1orCΩ¯ is compact, we can apply Simon’s compact embedding theorem [24] to (93) and (94) to verify that(95)ynis pre−compact inC0,T;V1∩CQ¯.
Hence, if necessary, a subsequenceynk⊂yn can be found such that(96)ynk⟶y¯strongly inC0,T;V1∩CQ¯.
Therefore, from (21), (92), and (96),(97)ynkV12Aynk⇀y¯V12Ay¯weakly inL20,T;Hask⟶∞,gynk⟶gy¯strongly inL20,T;Hask⟶∞.
Since embeddingV1↪H is compact, we can apply again Simon’s compact embedding theorem [24] to (92) and (94) to verify that(98)yn′is pre−compact inC0,T;H.
Hence, a subsequenceynk′⊂yn′ can also be found if necessary such that(99)ynk′⟶y¯′strongly inC0,T;Hask⟶∞.
For allφ∈L20,T;V1, the following equation can be obtained:(100)∫0Tcnk+dnkynk′−c¯+d¯y¯′,φ2dt≤∫0Tcnk+dnkynk′−y¯′,φ2dt+∫0Tcnk+dnk−c¯−d¯y¯′,φ2dt≤C∫0Tynk′−y¯′2φ2dt+∫0Tcnk+dnk−c¯−d¯y¯′,φ2dt.
By (99) and (90) with y¯′φ∈L20,T;H, it is evident that (100) converges to 0 as k⟶∞. Thus,(101)cnk+dnk=ynk′⟶c¯+d¯y¯′weakly inL20,T;V1′,ask⟶∞. Finally, y in (28) is replaced by ynk, U in (28) is replaced by cnk+dnk, and k⟶∞is taken. Then, the limit y¯ satisfies(102)Mγy¯″+Ay¯+1+y¯V12A1/2y¯+gy¯=c¯+d¯y¯+fin0,T,y¯0=y0,y¯′0=y1.
From Theorem2, it is finally deduced that the unique solution y¯ of Equation (102) equals to yq¯ in W0,T. Thus,(103)yqnk⟶yq¯weakly inW0,Task⟶∞.
Hence, the proof is concluded.Remark 3.
Since we are considering observation spacesMin which W0,T is compactly embedded, the author infers from (103) that(104)Cyqnk⟶Cyq¯strongly inMask⟶∞,where C∈LW0,T,M is the observation operator in (50).
From Theorem5, it is known that the Fréchet derivative of yq at q=q¯ in the direction w=l,h∈L20,T;H2, which is denoted by z=Dyq¯w, is given as the weak solution of(105)Mγz″+Az+1+yq¯V12A1/2z+2yq¯,z1A1/2yq¯+g′yq¯z=c¯+d¯z′+l+hy′q¯in0,T,z0=0,z′0=0.Proposition 3.
Let (A1)– (A5) be satisfied. For sufficiently largeα and β in (50),(106)DcJc1,d−DcJc2,dc1−c2≥0,∀c1,c2∈Uad,DdJc,d1−DdJc,d2d1−d2≤0,∀d1,d2∈Vad,where DcJci,dc1−c2,DdJc,dii=1,2 are the Gâteaux derivatives of J at ci,dii=1,2 in the directions of c1−c2 and d1−d2, respectively.Indeed, the mapsc⟶Jc,d and d⟶Jc,d are convex for all d∈Vad and concave for all c∈Uad, respectively.Proof.
From the Fréchet differentiability of the solution mapc⟶yc,d where d is fixed, (106) can be written again by(107)Cyc1,d−Yd,CDcyc1,dc1−c2M+α∫0Tc1,c1−c22dt−Cyc2,d−Yd,CDcyc2,dc1−c2M−α∫0Tc2,c1−c22dt≥0,∀c1,c2∈Uad,where Dcyci,dc1−c2i=1,2 is the weak solution of Equation (105), in which c¯+d¯z′+l+hy′q¯ is replaced by ci+dz′+c1−c2y′ci,di=1,2. It can easily be verified that (107) equals to(108)Cyc1,d−Cyc2,d,CDcyc1,dc1−c2M+Cyc2,d−Yd,CDcyc1,dc1−c2−CDcyc2,dc1−c2M+αc1−c2L20,T;H2≥0∀c1,c2∈Uad.
By Theorem2, (65), and Proposition 1, the left hand side of (108) can be estimated as follows:(109)Cyc1,d−Cyc2,d,CDcyc1,dc1−c2M≤Cyc1,d−Cyc2,dMCDcyc1,dc1,c2M≤CLW0,T,M2yc1,d−yc2,dW0,TDcyc1,dc1,c2W0,T≤Cyc1,d−yc2,dS0,TDcyc1,dc1−c2S0,T≤Cc1−c2L20,T;H2,(110)Cyc2,d−Yd,CDcyc1,dc1−c2−CDcyc2,dc1−c2M≤Cyc2,d−YdMCDcyc1,dc1−c2−CDcyc2,dc1−c2M≤CCLW0,T,MDcyc1,dc1−c2−Dcyc2,dc1−c2W0,T≤CDcyc1,dc1−c2−Dcyc2,dc1−c2S0,T≤Cc1+d−c2+dL20,T;Hc1−c2L20,T;H=Cc1−c2L20,T;H2.
From (108)–(110), sufficiently large αl can be found, depending on P,Wad, Yd, and C, so that (106) is satisfied for any α>αl. Thus, we know the map c⟶Jc,d, where d is fixed is convex for such α.
Similarly, sufficiently largeβl can also be found, depending on P,Wad, Yd, and C, so that (106) is satisfied for any β>βl. This implies the concavity of the map d⟶Jc,d, where c∈Uad is fixed.
Hence, the proof is concluded.
Finally, the following existence theorem of optimal pairs is presented (cf. [17] and [20]).Theorem 6.
Assume that (A1)–(A5) is fulfilled. For sufficiently largeα and β in (50), there exists c¯,d¯∈Wad such that c¯,d¯ satisfies (3.3).Proof.
By Proposition3, for sufficiently large α and β, we know that the maps c⟶Jc,d with fixed d and d⟶J⋅,d with fixed c are convex and concave, respectively. Together with (104), this theorem can be proved by referring to Theorem 5 of [16].
Hence, the proof is concluded.


## 6. Proofs of Necessary Optimality Conditions

This section is dedicated to deriving the optimal conditions required for each of the optimal pairs of minimax optimal control problems along with the costs from Theorem3 and Theorem 4.


### 6.1. Proof for the Case ofM=L20,T;H and Cyq=y′q

For this purpose, adjoint Equation (53)is introduced. The well posedness of Equation (53) is clarified by the following proposition.Proposition 4.
Equation (53) admits a unique weak solution p∈S0,T.Proof.
By reversing the direction of timet⟶T−t in Equation (53), it can be seen that Equation (53) is changed as follows:(111)Mγp″+Ap−∫0tGyq¯,p′+g′yq¯p′ds=c¯+d¯p′−y′q¯−Ydin0,T,p0=0,p′0=0,where Gyq¯,p′ is defined in (54) and Yd∈M=L20,T;H.
Estimating the weak solutionp of Equation (111) through energy equality, as given in the proof of Theorem 2, the following equation is obtained:(112)p′t22+p′tV12+ptV22≤C∫0t∫0sGyq¯;σ,p′σ+g′yq¯;σp′σ,p′sV1′,V1dσds+y′q¯+YdL20,T;H2.
We note(113)Gyq¯,⋅∈LV1,V1′.
By (21) with (19), it is deduced that(114)∫0t∫0sGyq¯;σ,p′σ+g′yq¯;σp′σ,p′sV1′,V1dσds≤C∫0tp′sV1∫0sp′σV1dσds+∫0tp′s2∫0sp′σ2dσds≤C∫0tp′sV1∫0sp′σV1dσds≤CTp′L20,t;V12.
Therefore, if we combine (112) and (114) and apply Gronwall’s lemma to the combined inequality, we obtain(115)pS0,T≤C.
Based on this, using Faedo–Galerkin’s approximation procedure, it can be verified that Equation (111) has a unique weak solution p∈S0,T.
Hence, the proof is concluded.
Before discussing the first-order optimality condition for the minimax optimal control problem (51) for the observation case of Theorem 3, the following remark is noted.Remark 4.
In deriving the optimality condition for the observation case of Theorem3, the weak form of Equation (53) needs to be multiplied by z′. However, as is often encountered in hyperbolic problems, it is the only formal procedure because z′∈C0,T;V1 and Mγp″,Ap∈L20,T;V2′. In [25], this difficulty can be eliminated by employing the regularization method proposed by Lions ([13], pp. 286–288).Proof of Theorem 1.
Thanks to the assumptions of Theorem3, it is evident from Theorem 6 that there exists an optimal pair in (51) for the observation case of Theorem 3.
Letq¯=c¯,d¯∈Wad be an optimal pair in (51) with the cost (50) through the observation case of Theorem 3 and yq¯ be the corresponding weak solution of Equation (49).
The Gâteaux derivative of the cost (50) through the observation case of Theorem 3 at q=q¯ in the direction w=l,h∈L20,T;H2, with q¯+εw∈Wad for sufficiently small ε>0, yields(116)DJc¯,d¯l,h=limε⟶0+Jc¯+εl,d¯+εh−Jc¯,d¯ε=y′q¯−Yd,z′L20,T;H+αc¯,lL20,T;H−βd¯,hL20,T;H,where z=Dyq¯w is a solution of Equation (105).
To eliminate the difficulty caused by the regularity of weak solutions, Lions’ regularization method is employed [13] (cf. [25]). For that, the author extends the time domains of Equation (105) and Equation (53) to Rt as follows:(117)Mγϕ″+Aϕ+Gyq¯,ϕ+g′yq¯ϕ−c¯+d¯ϕ′=l+hy′q¯,in0,T,0,otherwise,ϕ0=0,ϕ′0=0.
And(118)Mγξ″+Aξ−∫tTGyq¯,ξ′+g′yq¯ξ′ds+c¯+d¯ξ′=y′q¯−Yd,in0,T,0,otherwise,ξT=0,ξ′T=0.
In fact,(119)ψ=zin0,T,ξ=pin0,T,and(120)y′q¯−Yd,z′L20,T;H=∫RtMγξ″+Aξ−∫ttGyq¯,ξ′+g′yq¯ξ′dsc¯+d¯+ξ′,ϕ′2dt.
For convenience of representation, the scalar product onH or antiduality Vi,Vi′i=1,2 is denoted as ⋅,⋅. Let ρn be a mollifying sequence on Rt. Then, the right hand side of (120) equals(121)limn⟶∞∫RtMγξ″+Aξ−∫tTGyq¯,ξ′+g′yq¯ξ′ds+c¯+d¯ξ′∗ρn,ϕ′∗ρndt=limn⟶∞Xn.
Xn may be integrated by parts to obtain(122)Xn=∫Rt−ξ′∗ρn,Mγϕ″∗ρn+Aξ∗ρn,ϕ′∗ρn−Gyq¯,ξ′+g′yq¯ξ′∗ρn,ϕ∗ρn+c¯+d¯ξ′∗ρn,ϕ′∗ρndt.
Using(123)Mγϕ″=−Aϕ−Gyq¯,ϕ−g′yq¯ϕ+c¯+d¯ϕ′+l+hy′q¯.
(122) can be given again by(124)Xn=∫Rt−ξ′∗ρn,l+hy′q¯∗ρndt+∑i=14Yni,where(125)Yn1=∫Rtξ′∗ρn,Aϕ∗ρn+Aξ∗ρn,ϕ′∗ρndt,Yn2=∫Rtξ′∗ρn,Gyq¯,ϕ∗ρn−Gyq¯,ξ′∗ρn,ϕ∗ρndt,Yn3=∫Rtξ′∗ρn,g′yq¯ϕ∗ρn−g′yq¯ξ′∗ρn,ϕ∗ρndt,Yn4=∫Rtc¯+d¯ξ′∗ρn,ϕ′∗ρn−ξ′∗ρn,c¯+d¯ϕ′∗ρndt.
We immediately knowYn1=0 by integration by parts. Since(126)∫Rtξ′∗ρn,Gyq¯,ϕ∗ρndt⟶∫Rtξ′,G[yq¯,ϕ]dt=∫RtGyq¯,ξ′,ϕdt,as n⟶∞. It can be deduced that Yn2⟶0 as n⟶∞. In a similar manner, Yn3⟶0 and Yn4⟶0 as n⟶∞. This justifies(127)limn⟶∞Xn=∫Rt−ξ′,l+hy′q¯dt=−p′,l+hy′q¯L20,T;H.
Sinceq¯=c¯,d¯∈Wad is an optimal pair in (51), the following equation is obtained:(128)DcJc¯,d¯l≥0,DdJc¯,d¯h≤0,l,h∈L20,T;H2.
From (116), (120), and (127), (128) implies(129)αc¯−y′q¯p′,lL20,T;H≥0,−βd¯−y′q¯p′,hL20,T;H≤0,where l,h∈L20,T;H2. By referring to the proof of ([16], Theorem 4.2), it can be deduced from (129) that the optimal pair q¯=c¯,d¯ can be given by(130)c¯=maxc1,miny′q¯p′α,c2,d¯=maxd1,min−y′q¯p′β,d2.
Hence, the proof is concluded.


### 6.2. Proof for the Case ofM=L20,T;H×H and Cyq=yq,yq;T

Before proceeding the proof of Theorem4, the following remark is noted.Remark 5.
In the case of observation in Theorem4, adjoint equations cannot be constructed directly, because they must involve the term −c¯+d¯p′, as shown in (58), where p is an adjoint state. However, the regularity of the admissible set is insufficient to guarantee the existence of adjoint equations. This difficulty is overcome by employing the modified transposition method.
The transposition method is explained. For any givenyq¯=yc¯,d¯∈S0,T, the following equation is considered:(131)Mγϕ″+Aϕ+Gyq¯,ϕ+g′yq¯ϕ=c¯+d¯ϕ′+fin0,T,ϕ0=0,ϕ′0=0,where f∈L20,T;V1′, c¯,d¯∈L20,T;H2, and(132)Gyq¯,ϕ≔2yq¯,ϕ1A1/2yq¯+1+yq¯V12A1/2ϕ.
From Theorem2, it is evident that Equation (131) has a unique weak solution ϕ∈S0,T satisfying(133)ϕS0,T≤CfL20,T;V1′.
Thus, the spaceX=ϕ|ϕsatisfies Equation (131) as f ranges over L20,T;V1′.
Then,(134)X⊂S0,T.
Endowed with the normzX=fL20,T;V1′, X is the Hilbert space, and the map L defined by(135)Lϕ=Mγϕ″+Aϕ+Gyq¯,ϕ+g′yq¯ϕ−c¯+d¯ϕ′,is an isomorphism from X onto L20,T;V1′. Therefore, for any continuous linear functional l on X, there exists unique η∈L20,T;V1 such that(136)∫0Tη,LϕV1,V1′dt=lϕ∀,ϕ∈X.
We call such aη to be a solution by the transposition method associated with the bounded linear functional l on X for the transposed equation with respect to L.
To discuss the existence of a solution to (formal) adjoint system Equation (58), l in (136) is taken as follows:(137)lϕ=yq¯−Yd,ϕL20,T;H+yq¯;T−YdT,ϕT2∀ϕ∈X.
It is easily verified that(138)yq¯−Yd,ϕL20,T;H≤yq¯−YdL20,T;HϕL20,T;H≤Cyq¯−YdL20,T;HϕS0,T≤Cyq¯−YdL20,T;HLϕL20,T;V1′≡Cyq¯−YdL20,T;HϕX,yq¯;T−YdT,ϕT2≤yq¯;T−YdT2ϕT2≤yq¯;T−YdT2ϕC0,T;H≤Cyq¯;T−YdT2ϕS0,T≤Cyq¯;T−YdT2LϕL20,T;V1′≡Cyq¯;T−YdT2ϕX.
Hence, from (137)–(139), the following equation is obtained:(139)lϕ≤Cyq¯−YdL20,T;H+yq¯;T−YdT2ϕX∀z∈X.
As a consequence, the functionall is linear and bounded on X. Also, it is deduced that the map yq¯−Yd,yq∗;T−YdT⟶p is continuous from L20,T;H×H to L20,T;V1. Hence, the following proposition is obtained.Proposition 5.
For givenyq¯−Yd∈L20,T;H and yq¯;T−YdT∈H, there exists a unique solution p∈L20,T;V1 of Equation (58) such that(140)∫0Tp,LϕV1,V1′dt=lϕ∀ϕ∈X,where L is defined in (135) and lϕ is defined in (137). Moreover, the solution p of (141) satisfies(141)pL20,T;V1≤Cyq¯−YdL20,T;H+yq¯;T−YdT2.Proof of Theorem 2.
Letq¯=c¯,d¯∈Wad be an optimal pair in (51) with the cost (50) through the observation case of Theorem 4 and yq¯ be the corresponding weak solution of (49).
The Gâteaux derivative of the cost (50) through the observation case of Theorem 4 at q=q¯ in the direction w=l,h∈L20,T;H2, with q¯+εw∈Wad for sufficiently small ε>0, yields(142)DJc¯,d¯l,h=limε⟶0+Jc¯+εl,d¯+εh−Jc¯,d¯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∫0T2c¯,l2+εl22dt−β2∫0T2d¯,h2+εh22dt=yq¯−Yd,zL20,T;H+yq¯;T−YdT,zT2+αc¯,lL20,T;H−βd¯,hL20,T;H,where z=Dyq¯w is a weak solution of Equation (105). Hence, if the author takes ϕ=z in (141), then it is inferred from (142) that(143)DJc¯,d¯l,h=p,l+hy′q¯L20,T;H+αc¯,lL20,T;H−βd¯,hL20,T;H.
Sinceq¯=c¯,d¯∈Wad is an optimal pair in (50), the following equation is obtained:(144)DcJc¯,d¯l≥0,DdJc¯,d¯h≤0,l,h∈L20,T;H2.
Thus, from (144) and (145), the following equation is obtained:(145)αc¯+y′q¯p,lL20,T;H≥0,−βd¯+y′q¯p,hL20,T;H≤0,where l,h∈L20,T;H2. By analogy with the case of (130), it can be inferred that from (145), the optimal pair q¯=c¯,d¯ can be given as (possibly not unique)(146)c¯=maxc1,min−y′q¯pα,c2,d¯=maxd1,miny′q¯pβ,d2.
Hence, the proof is concluded.


## 6.1. Proof for the Case ofM=L20,T;H and Cyq=y′q

For this purpose, adjoint Equation (53)is introduced. The well posedness of Equation (53) is clarified by the following proposition.Proposition 4.
Equation (53) admits a unique weak solution p∈S0,T.Proof.
By reversing the direction of timet⟶T−t in Equation (53), it can be seen that Equation (53) is changed as follows:(111)Mγp″+Ap−∫0tGyq¯,p′+g′yq¯p′ds=c¯+d¯p′−y′q¯−Ydin0,T,p0=0,p′0=0,where Gyq¯,p′ is defined in (54) and Yd∈M=L20,T;H.
Estimating the weak solutionp of Equation (111) through energy equality, as given in the proof of Theorem 2, the following equation is obtained:(112)p′t22+p′tV12+ptV22≤C∫0t∫0sGyq¯;σ,p′σ+g′yq¯;σp′σ,p′sV1′,V1dσds+y′q¯+YdL20,T;H2.
We note(113)Gyq¯,⋅∈LV1,V1′.
By (21) with (19), it is deduced that(114)∫0t∫0sGyq¯;σ,p′σ+g′yq¯;σp′σ,p′sV1′,V1dσds≤C∫0tp′sV1∫0sp′σV1dσds+∫0tp′s2∫0sp′σ2dσds≤C∫0tp′sV1∫0sp′σV1dσds≤CTp′L20,t;V12.
Therefore, if we combine (112) and (114) and apply Gronwall’s lemma to the combined inequality, we obtain(115)pS0,T≤C.
Based on this, using Faedo–Galerkin’s approximation procedure, it can be verified that Equation (111) has a unique weak solution p∈S0,T.
Hence, the proof is concluded.
Before discussing the first-order optimality condition for the minimax optimal control problem (51) for the observation case of Theorem 3, the following remark is noted.Remark 4.
In deriving the optimality condition for the observation case of Theorem3, the weak form of Equation (53) needs to be multiplied by z′. However, as is often encountered in hyperbolic problems, it is the only formal procedure because z′∈C0,T;V1 and Mγp″,Ap∈L20,T;V2′. In [25], this difficulty can be eliminated by employing the regularization method proposed by Lions ([13], pp. 286–288).Proof of Theorem 1.
Thanks to the assumptions of Theorem3, it is evident from Theorem 6 that there exists an optimal pair in (51) for the observation case of Theorem 3.
Letq¯=c¯,d¯∈Wad be an optimal pair in (51) with the cost (50) through the observation case of Theorem 3 and yq¯ be the corresponding weak solution of Equation (49).
The Gâteaux derivative of the cost (50) through the observation case of Theorem 3 at q=q¯ in the direction w=l,h∈L20,T;H2, with q¯+εw∈Wad for sufficiently small ε>0, yields(116)DJc¯,d¯l,h=limε⟶0+Jc¯+εl,d¯+εh−Jc¯,d¯ε=y′q¯−Yd,z′L20,T;H+αc¯,lL20,T;H−βd¯,hL20,T;H,where z=Dyq¯w is a solution of Equation (105).
To eliminate the difficulty caused by the regularity of weak solutions, Lions’ regularization method is employed [13] (cf. [25]). For that, the author extends the time domains of Equation (105) and Equation (53) to Rt as follows:(117)Mγϕ″+Aϕ+Gyq¯,ϕ+g′yq¯ϕ−c¯+d¯ϕ′=l+hy′q¯,in0,T,0,otherwise,ϕ0=0,ϕ′0=0.
And(118)Mγξ″+Aξ−∫tTGyq¯,ξ′+g′yq¯ξ′ds+c¯+d¯ξ′=y′q¯−Yd,in0,T,0,otherwise,ξT=0,ξ′T=0.
In fact,(119)ψ=zin0,T,ξ=pin0,T,and(120)y′q¯−Yd,z′L20,T;H=∫RtMγξ″+Aξ−∫ttGyq¯,ξ′+g′yq¯ξ′dsc¯+d¯+ξ′,ϕ′2dt.
For convenience of representation, the scalar product onH or antiduality Vi,Vi′i=1,2 is denoted as ⋅,⋅. Let ρn be a mollifying sequence on Rt. Then, the right hand side of (120) equals(121)limn⟶∞∫RtMγξ″+Aξ−∫tTGyq¯,ξ′+g′yq¯ξ′ds+c¯+d¯ξ′∗ρn,ϕ′∗ρndt=limn⟶∞Xn.
Xn may be integrated by parts to obtain(122)Xn=∫Rt−ξ′∗ρn,Mγϕ″∗ρn+Aξ∗ρn,ϕ′∗ρn−Gyq¯,ξ′+g′yq¯ξ′∗ρn,ϕ∗ρn+c¯+d¯ξ′∗ρn,ϕ′∗ρndt.
Using(123)Mγϕ″=−Aϕ−Gyq¯,ϕ−g′yq¯ϕ+c¯+d¯ϕ′+l+hy′q¯.
(122) can be given again by(124)Xn=∫Rt−ξ′∗ρn,l+hy′q¯∗ρndt+∑i=14Yni,where(125)Yn1=∫Rtξ′∗ρn,Aϕ∗ρn+Aξ∗ρn,ϕ′∗ρndt,Yn2=∫Rtξ′∗ρn,Gyq¯,ϕ∗ρn−Gyq¯,ξ′∗ρn,ϕ∗ρndt,Yn3=∫Rtξ′∗ρn,g′yq¯ϕ∗ρn−g′yq¯ξ′∗ρn,ϕ∗ρndt,Yn4=∫Rtc¯+d¯ξ′∗ρn,ϕ′∗ρn−ξ′∗ρn,c¯+d¯ϕ′∗ρndt.
We immediately knowYn1=0 by integration by parts. Since(126)∫Rtξ′∗ρn,Gyq¯,ϕ∗ρndt⟶∫Rtξ′,G[yq¯,ϕ]dt=∫RtGyq¯,ξ′,ϕdt,as n⟶∞. It can be deduced that Yn2⟶0 as n⟶∞. In a similar manner, Yn3⟶0 and Yn4⟶0 as n⟶∞. This justifies(127)limn⟶∞Xn=∫Rt−ξ′,l+hy′q¯dt=−p′,l+hy′q¯L20,T;H.
Sinceq¯=c¯,d¯∈Wad is an optimal pair in (51), the following equation is obtained:(128)DcJc¯,d¯l≥0,DdJc¯,d¯h≤0,l,h∈L20,T;H2.
From (116), (120), and (127), (128) implies(129)αc¯−y′q¯p′,lL20,T;H≥0,−βd¯−y′q¯p′,hL20,T;H≤0,where l,h∈L20,T;H2. By referring to the proof of ([16], Theorem 4.2), it can be deduced from (129) that the optimal pair q¯=c¯,d¯ can be given by(130)c¯=maxc1,miny′q¯p′α,c2,d¯=maxd1,min−y′q¯p′β,d2.
Hence, the proof is concluded.


## 6.2. Proof for the Case ofM=L20,T;H×H and Cyq=yq,yq;T

Before proceeding the proof of Theorem4, the following remark is noted.Remark 5.
In the case of observation in Theorem4, adjoint equations cannot be constructed directly, because they must involve the term −c¯+d¯p′, as shown in (58), where p is an adjoint state. However, the regularity of the admissible set is insufficient to guarantee the existence of adjoint equations. This difficulty is overcome by employing the modified transposition method.
The transposition method is explained. For any givenyq¯=yc¯,d¯∈S0,T, the following equation is considered:(131)Mγϕ″+Aϕ+Gyq¯,ϕ+g′yq¯ϕ=c¯+d¯ϕ′+fin0,T,ϕ0=0,ϕ′0=0,where f∈L20,T;V1′, c¯,d¯∈L20,T;H2, and(132)Gyq¯,ϕ≔2yq¯,ϕ1A1/2yq¯+1+yq¯V12A1/2ϕ.
From Theorem2, it is evident that Equation (131) has a unique weak solution ϕ∈S0,T satisfying(133)ϕS0,T≤CfL20,T;V1′.
Thus, the spaceX=ϕ|ϕsatisfies Equation (131) as f ranges over L20,T;V1′.
Then,(134)X⊂S0,T.
Endowed with the normzX=fL20,T;V1′, X is the Hilbert space, and the map L defined by(135)Lϕ=Mγϕ″+Aϕ+Gyq¯,ϕ+g′yq¯ϕ−c¯+d¯ϕ′,is an isomorphism from X onto L20,T;V1′. Therefore, for any continuous linear functional l on X, there exists unique η∈L20,T;V1 such that(136)∫0Tη,LϕV1,V1′dt=lϕ∀,ϕ∈X.
We call such aη to be a solution by the transposition method associated with the bounded linear functional l on X for the transposed equation with respect to L.
To discuss the existence of a solution to (formal) adjoint system Equation (58), l in (136) is taken as follows:(137)lϕ=yq¯−Yd,ϕL20,T;H+yq¯;T−YdT,ϕT2∀ϕ∈X.
It is easily verified that(138)yq¯−Yd,ϕL20,T;H≤yq¯−YdL20,T;HϕL20,T;H≤Cyq¯−YdL20,T;HϕS0,T≤Cyq¯−YdL20,T;HLϕL20,T;V1′≡Cyq¯−YdL20,T;HϕX,yq¯;T−YdT,ϕT2≤yq¯;T−YdT2ϕT2≤yq¯;T−YdT2ϕC0,T;H≤Cyq¯;T−YdT2ϕS0,T≤Cyq¯;T−YdT2LϕL20,T;V1′≡Cyq¯;T−YdT2ϕX.
Hence, from (137)–(139), the following equation is obtained:(139)lϕ≤Cyq¯−YdL20,T;H+yq¯;T−YdT2ϕX∀z∈X.
As a consequence, the functionall is linear and bounded on X. Also, it is deduced that the map yq¯−Yd,yq∗;T−YdT⟶p is continuous from L20,T;H×H to L20,T;V1. Hence, the following proposition is obtained.Proposition 5.
For givenyq¯−Yd∈L20,T;H and yq¯;T−YdT∈H, there exists a unique solution p∈L20,T;V1 of Equation (58) such that(140)∫0Tp,LϕV1,V1′dt=lϕ∀ϕ∈X,where L is defined in (135) and lϕ is defined in (137). Moreover, the solution p of (141) satisfies(141)pL20,T;V1≤Cyq¯−YdL20,T;H+yq¯;T−YdT2.Proof of Theorem 2.
Letq¯=c¯,d¯∈Wad be an optimal pair in (51) with the cost (50) through the observation case of Theorem 4 and yq¯ be the corresponding weak solution of (49).
The Gâteaux derivative of the cost (50) through the observation case of Theorem 4 at q=q¯ in the direction w=l,h∈L20,T;H2, with q¯+εw∈Wad for sufficiently small ε>0, yields(142)DJc¯,d¯l,h=limε⟶0+Jc¯+εl,d¯+εh−Jc¯,d¯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∫0T2c¯,l2+εl22dt−β2∫0T2d¯,h2+εh22dt=yq¯−Yd,zL20,T;H+yq¯;T−YdT,zT2+αc¯,lL20,T;H−βd¯,hL20,T;H,where z=Dyq¯w is a weak solution of Equation (105). Hence, if the author takes ϕ=z in (141), then it is inferred from (142) that(143)DJc¯,d¯l,h=p,l+hy′q¯L20,T;H+αc¯,lL20,T;H−βd¯,hL20,T;H.
Sinceq¯=c¯,d¯∈Wad is an optimal pair in (50), the following equation is obtained:(144)DcJc¯,d¯l≥0,DdJc¯,d¯h≤0,l,h∈L20,T;H2.
Thus, from (144) and (145), the following equation is obtained:(145)αc¯+y′q¯p,lL20,T;H≥0,−βd¯+y′q¯p,hL20,T;H≤0,where l,h∈L20,T;H2. By analogy with the case of (130), it can be inferred that from (145), the optimal pair q¯=c¯,d¯ can be given as (possibly not unique)(146)c¯=maxc1,min−y′q¯pα,c2,d¯=maxd1,miny′q¯pβ,d2.
Hence, the proof is concluded.


## 7. Conclusion

In this study, the author investigated the bilinear minimax optimal control problems for an extensible beam equation with rotational inertia effects. The well posedness of the solution to the Cauchy problem and the improved regularity theorem were given for the underlying equation. The author formulated the bilinear minimax optimal control problem for the state equation by adding the bilinear control and noise inputs of the velocity term to the state equation. Using and analyzing the properties of the Fréchet derivative of the nonlinear solution map from the bilinear control and noise inputs to the solution of the equation, the author proved the existence of an optimal pair and found its necessary optimality conditions corresponding to the following physically meaningful observation cases. The main novelties of this study can be summarized as follows: In the case of velocity distribution observation, the double regularization method was used to derive the optimality condition for the optimal pair through the adjoint equation. For the distributive and terminal value observations, instead of assuming regularity for the data and control sets required to construct an explicit adjoint equation, the transposition method was used to derive the necessary optimality condition for the optimal pair.



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*Source: 9903316-2023-01-05.xml*