# On the Control of Coefficient Function in a Hyperbolic Problem with Dirichlet Conditions

Authors: Seda İğret Araz; Murat Subaşi
Journal: International Journal of Differential Equations
 (2018)

---

## Abstract

This paper presents theoretical results about control of the coefficient function in a hyperbolic problem with Dirichlet conditions. The existence and uniqueness of the optimal solution for optimal control problem are proved and adjoint problem is used to obtain gradient of the functional. However, a second adjoint problem is given to calculate the gradient on the spaceW210,l. After calculating gradient of the cost functional and proving the Lipschitz continuity of the gradient, necessary condition for optimal solution is constructed.


---

## Body

## 1. Introduction

Hyperbolic boundary value problems have appeared as mathematical modelling of physical phenomena like small vibration of a string, in the fields of science and engineering. There has been much attention to studies related to optimal control problems involving hyperbolic problems [1]. There have been many studies about optimal control for hyperbolic systems which are considered [2–4].Some of these important studies can be summarized as follows.Hasanov [5] has considered problem of controlling the function w≔Fx,t;ft for the following problem:(1)utt=kxuxx+Fx,t,x,t∈0,l×0,Tux,0=u0x,utx,0=u1x,x∈0,l-k0ux0,t=ft,kluxl,t=0t∈0,Twith the conditions(2)ux,T=μx,utx,T=vx,using the functionals(3)J1w=∫0lux,T;w-μx2dx,J2w=∫0lutx,T;w-vx2dx,J3w=J1w+J2w.Majewski [6] has controlled the function ux,y∈L2P,RM for hyperbolic equation:(4)∂2z∂x∂yx,y=f~x,y,zx,y,∂z∂xx,y,∂z∂yx,y,ux,yzx,0=z0,y=0,∀x,y∈0,1using the functional (5)Jkz·,u·=∫PFkx,y,zx,y,ux,ydxdy.k=0,1,2,…Yeloğlu and Subaşı [7] have dealt with determination pair w≔fx,t,hx in the following problem:(6)pxutt=kxuxx+fx,t,x,t∈ΩTux,0=gx,utx,0=hx,x∈0,lu0,t=0,ul,t=0,t∈0,Tfor the functional(7)Jαw=∫0lux,T;w-yx2dx+αwW2.Kröner [8] has specified the function ut∈L20,T for nonlinear hyperbolic equation:(8)ytt-Au,y=fy0=y0u,yt0=y1uusing the functional(9)Ju,y1=∫0TJ1y1tdt+J2y1T+α2uU2.Tagiyev [9] has studied the problem of controlling the coefficients v=kx,qx,t∈L∞Ω×L∞QT for linear hyperbolic equation:(10)∂2u∂t2-∑i=1n∂∂xikix∂u∂xi+qx,tu=fx,t,x,t∈QTut=0=φ0x,∂u∂tt=0=φ1x,uST=0using the functional(11)Jv=α0∫QTux,t,v-z0x,t2dxdt+α1∫Ωux,T,v-z1x2dx.


## 2. Statement of the Problem

In this study, we deal with the process of vibration in finite homogeneous string, occupying the interval0,l. As the control function, we take the transverse elastic force which is in the coefficient of the vibration problem. Also, we propose the usage of a more regular space than the space of square integrable functions in the cost functional. In general, this process exposes some difficulties in the stage of acquiring the gradient. This study offers a second adjoint problem to overcome this case.In the domainΩ≔x,t∈0,l×0,T, we consider the functional(12)Jαq=∫0lux,T;q-yx2dx+αq-rW210,l2on the set(13)Q=qx:qx∈W210,l,0<q1≤qx≤q2,qxW210,l≤q3subject to the hyperbolic problem(14)utt-uxx+qxu=0,x,t∈Ω(15)ux,0=φ1x,utx,0=φ2x,x∈0,l(16)u0,t=0,ul,t=0,t∈0,T.Here yx∈L20,l is the desired target function to which ux,T must be close enough. The function rx∈W210,l is an initial guess for optimal solution. α>0 is regularization parameter. q1,q2,q3>0 are given positive numbers.The initial status functions are in the following spaces:(17)φ1x∈W210,l,φ2x∈L20,l.The aim of this study is to deal with the problem of(18)Jα∗=infq∈Q⁡Jαq=Jαq∗under conditions (12)-(17).Namely, we want to control the transverse elastic force on the spaceW210,l and the solution ux,T corresponding to this control function must be close enough to yx in L20,l. In order to get a stable solution, we choose the space W210,l which is more regular than L20,l.The inner product and norm inW210,l are defined, respectively, as(19)f,gW210,l=∫0lfx.gx+dfxdx.dgxdxdx,fW210,l2=∫0lfx2+dfxdx2dx.The paper is organized as follows: in Section 3, we obtain the generalized solution for hyperbolic problem. In Section 4, we prove the existence and uniqueness of the optimal solution. In Section 5, we obtain the adjoint problem for the optimal control problem and find the gradient of the functional. The main contribution of this paper is executed in this section. Because the controls are chosen in the space W210,l, getting the gradient of the functional necessitates finding a second adjoint problem. In the last section, we demonstrate the Lipschitz continuity of the gradient and state the necessary condition for optimal solution.


## 3. Solvability of the Problem

In this section, we first give the definition of the generalized solution for hyperbolic problem.The generalized solution of problem (14)-(15) is the function u∈W21,1oΩ satisfying the following integral equality:(20)∫Ω-utηt+uxηx+qxuηdxdt=∫0lφ2xηx,0dxfor ∀η∈W21,1oΩ, ηx,T=0.It can be seen in [10] that solution in the sense of (20) exists, is unique, and satisfies the following inequality:(21)max0≤t≤T⁡u·,tL20,l2+ut·,tL20,l2+ux·,tL20,l2≤c1φ1W210,l2+φ2L20,l2where c1=max⁡3c0,3c0/q1 and c0=max⁡1,q2 or(22)uW21,1Ω2≤c2φ1W210,l2+φ2L20,l2.c2=c1TSince φ1 and φ2 are given functions, it can be written as follows:(23)uW21,1Ω2≤c3.Now, we give an increment δqx∈W210,l to the control function qx such as q+δq∈Q. Then the difference function δu=δux,t=ux,t;q+δq-ux,t;q is the solution of the following difference initial-boundary problem:(24)δutt-δuxx+qx+δqxδu+δqxu=0(25)δux,0=0,δutx,0=0(26)δu0,t=0,δul,t=0.By considering (23), we obtain that the solution of above difference initial-boundary problem holds the following inequality:(27)max0≤t≤T⁡δu.,tL20,l2≤c4δqW210,l2.Here c4=t32l/3c3 is independent from δq.


## 4. Existence and Uniqueness of the Optimal Solution

To demonstrate the existence and the uniqueness of optimal solution for problem (12)-(17), it is enough to show that conditions of the following theorem given by Goebel [11] hold.Theorem 1.
LetH be a uniformly convex Banach space and the set Q be a closed, bounded, and convex subset of H. If α>0 and β≥1 are given numbers and the functional Jq is lower semicontinuous and bounded from below on the set Q, then there is a dense set G of H that the functional(28)Jαq=Jq+αq-rHβtakes its minimum on the set Q for ∀r∈G. If β>1 then minimum is unique.Before showing that these conditions have been satisfied, we prove that the functional(29)Jq=∫0lux,T;q-yx2dxis continuous. For this, we write the following increment of the functional:(30)δJq=Jq+δq-Jq=∫0l2ux,T-yxδux,Tdx+∫0lδux,T2dx.Since yx∈L20,l, if we consider inequalities (22) and (27), we conclude that this increment satisfies the following continuity inequality on the set Q:(31)δJq≤c5δqW210,l+δqW210,l2.Here c5 is independent of δq.Thanks to this inequality, we can say that this functional is also lower semicontinuous and bounded from below on the setQ.On the other hand, the setW210,l is a uniformly convex Banach space [12], the set Q is a closed, bounded, and convex subset of W210,l, and β=2.Therefore the conditions of above theorem hold and optimal solution to the problem (18) is unique.


## 5. Adjoint Problem and Gradient of the Functional

In this section, we write the Lagrange functional used for finding adjoint problem, before we show the Frechet differentiability of the functionalJαq on the set Q. Lagrange functional to the problem is(32)Lu,q,η=∫0lux,T;q-yx2dx+αq-rW210,l2+∫0T∫0lutt-uxx+qxuηdxdt.The first variation of this functional according to the function u is obtained such as(33)δL=∫0l2ux,T-yx-ηtx,Tδux,Tdx+∫0l∫0Tηtt-ηxx+qxηδudtdx=0.By means of stationary condition δL=0, the following adjoint boundary value problem is found:(34)ηtt-ηxx+qxη=0(35)ηx,T=0,ηtx,T=2ux,T-yx(36)η0,t=0,ηl,t=0.For ∀γ∈W21,1o(Ω), the function η∈C10,T,L20,l∩C00,T,W210,l which satisfies the following equality(37)∫0T∫0l-ηtγt+ηxγx+qxηγdxdt=∫0lηtx,0γx,0dx-∫0l2ux,T-yxγx,Tdxis the solution of adjoint boundary value problem (34)-(36).This solution satisfies the following inequality:(38)ηL20,l≤c6ux,T-yxL20,l,∀t∈0,T.Now, we can pass the calculation of the gradient. In order to do this, we must evaluate the increment of the functional Jαq. The increment can be written such as(39)δJαq=Jαq+δq-Jαq=∫0l2ux,T-yxδudx+∫0lδu2dx+2αq-r,δqW210,l.The difference problem (24)-(26) and the adjoint problem (34)-(36) give together the equality of(40)2∫0lux,T-yxδudx=∫0T∫0lδqδuη+δquηdxdt.Inserting (40) in (39), we have(41)δJαq=∫0T∫0luηδqdxdt+∫0T∫0lηδuδqdxdt+∫0lδux,T2dx+2αq-r,δqW210,l.By (27) and (38), the second and third integrals of the above equality give the following inequality: (42)∫0T∫0lηδuδqdxdt+∫0lδux,T2dx≤c7δqW210,l2.The statement (41) can be rewritten as(43)δJαq=uη,δqL2Ω+2αq-r,δqW210,l+oδqW210,l2or(44)δJαq=∫0Tuηdt,δqL20,l+2αq-r,δqW210,l+oδqW210,l2.In order to pass the inner product in W210,l, we rearrange (44) such as(45)δJαq=ξ+2αq-r,δqW210,l+oδqW210,l2.Here function ξx is the solution of the second adjoint problem:(46)-ξ′′+ξ=∫0Tuηdtξ′0=0,ξ′l=0.Therefore, we have the following gradient:(47)Jα′q=ξ+2αq-r.


## 6. Lipschitz Continuity of the Gradient

In this section, we introduce a theorem about Lipschitz continuity of the gradient. By this means, we can express the necessary condition for optimal solution.Theorem 2.
GradientJα′q satisfies the following Lipschitz inequality:(48)Jα′q+δq-Jα′qW210,l2≤c8δqW210,l2.Here c8 is independent from δq.Hence, it has been proven that the gradientJα′q is continuous on the set Q and it can be seen that it holds the Lipschitz condition with constant c8>0.Proof.
Increment of the functionalJα′q by giving the increment of δq to the control q∈Q is obtained:(49)Jα′q+δq-Jα′q=ξδ+2αq+δq-r-ξ+2αq-r=δξ+2αδqwhere the function δξx is the solution of the increment problem:(50)δξ′′x-δξx=∫0Tuδδη+δuηdt.Taking the norm of (49) in the space W210,l, we acquire the following inequality belonging to the functional δJα′q:(51)δJα′qW210,l2≤2δξW210,l2+8α2δqW210,l2.There is a solution of problem (50) in W210,l and this solution satisfies the following inequality:(52)δξxW210,l≤∫0Tuδδη+δuηdtL20,l.The function (53)δηx,t=ηδx,t-ηx,t=ηx,t;q+δq-ηx,t;qin the right hand side of inequality (52) is the solution of the following problem:(54)∂2δη∂t2-∂2δη∂x2+qx+δqxδη+δqxη=0,x,t∈Ωδηx,T=0,δηtx,T=2δux,T,δη0,t=δηl,t=0and this function holds the following inequality:(55)max0≤t≤T⁡δη.,tL20,l2≤c9δqW210,l2.Here c9 is independent of δq.
So, the functionuδ that takes place in the right hand side of (52) holds the same inequality given as follows:(56)uδL2Ω2≤c3.Hence inequality (52) has the following property:(57)δξxW210,l2≤2uδL2Ω2max0≤t≤T⁡δη.,tL20,l2+2ηL2Ω2max0≤t≤T⁡δu.,tL20,l2.If inequalities (27), (38), (55), and (56) about functions uδ, δη.,t, η, and δu.,t are written in (57), then the following assessment is obtained:(58)δξxW210,l2≤c10δqW210,l2.Here c10 is independent of δq.
Considering inequality (58), the following is written: (59)δJα′qW210,l2≤2δξxW210,l2+8α2δqxW210,l2≤2c10δqW210,l2+8α2δqW210,l2≤c11δqW210,l2.So the following inequality for the gradient Jα′q is obtained:(60)Jα′q+δq-Jα′qW210,l2≤c11δqW210,l2.Once we take as c8=c11, then the proof is obtained.


## 7. The Necessary Condition for Optimal Solution

After showing Lipschitz continuity of the gradient, it can be said that the gradientJα′q is continuous on the set Q and it holds the Lipschitz constant c8>0. The fact that the functional Jαq is continuously differentiable on the set Q and the set Q is convex, in that case the following inequality is valid according to theorem in [13]:(61)Jα′q∗,q-q∗W210,l≥0,∀q∈Q.Therefore, the following inequality is written for optimal control problem:(62)ξ+2αq∗-r,q-q∗W210,l≥0,∀q∈Q.



---

*Source: 7417590-2018-09-23.xml*