Zuicha DENGLiu YANG

1 Introduction

In this paper,we study an inverse problem ofidentifying the radiative coefficient in a degenerate parabolic equation from the final overspecified data.Problems of this type have important applications in several fields of applied science and engineering.The problem can be stated in the following form.

Problem 1.1Consider the following parabolic equation:

whereaandϕare two given smooth functions,which satisfy

and

respectively,andq(x)is an unknown coefficient in(1.1).In this paper,we always assume thata(x)is at leastC1continuous,i.e.,a(x)∈C1[0,l].Assume that an additional condition is given as follows:

wheregis a known function.We shall determine the functionsuandqsatisfying(1.1)and(1.4),respectively.

If the principle coefficienta(x)is required to be strictly positive,i.e.,

a(x)≥a0>0,x∈[0,l],

then the equation should be rewritten as an initial-boundary value problem,e.g.,the homogeneous Dirichlet boundary value problem as follows:

which is often referred as the classical parabolic equation.The mathematical model(1.5)arises in various physical and engineering settings.If(1.5)is used to describe the heat transfer system,the coefficientq(x)is called the radiative coefficient which is often dependent on the medium property.

Being different from the ordinary parabolic equation(1.5),(1.1)belongs to the second order differential equations with non-negative characteristic form.The main character of such kinds of equations is degeneracy.It can be easily seen that atx=0 andx=l,(1.1)degenerates into two hyperbolic equations

By the well-known Fichera’s theory(see[33])for degenerate parabolic equations,we know that whether or not boundary conditions should be given at the degenerate,boundaries are determined by the sign of the Fichera function.By simple calculations,one can easily check that boundary conditions for(1.1)on the lateral boundariesx=0,x=land the terminal boundaryt=Tshould not be given,while ont=0 they are indispensable.In other words,the parabolic problem(1.1)is well-defined.

In general,most physical and industrial phenomenons can be described by the classical parabolic model,such as(1.5).However,with the development of the modern financial mathematics,more and more degenerate elliptic or parabolic equations arising in derivatives pricing have to be taken into account.For example,the well-known Black-Scholes equation

is such the case,where the degenerate parabolic boundary isS=0.

For a given coefficientq(x),the degenerate parabolic equation(1.1)which is referred as a direct problem consists of the determination of the solution from the given initial condition.It is well-known that in all cases,the inverse problem is ill-posed or improperly posed in the sense of Hadamard,while the direct problem is well-posed(see[23,30,32]).The ill-posedness,particularly the numerical instability,is the main difficulty for Problem 1.1.Since data errors in the extra conditiong(x)are inevitable,arbitrarily small changes ing(x)may lead to arbitrarily large changes inq(x),which may make the obtained results meaningless(see,e.g.,[20,36]).

Inverse coefficient problems for parabolic equations are well studied in the literature.However,most of these inverse problems are governed by classical parabolic equations in which the principle coefficients are assumed to be strictly positive definite.The inverse problem ofidentifying the diffusion coefficienta(x)in the following parabolic equation:

from some additional conditions was investigated by several authors(see,e.g.,[14,21,24,29]).In[29,21],the output least-squares method with Tikhonov regularization is applied to the inverse problem and the numerical solution is obtained by the finite element method.The determination ofa(x)with two Neumann measured data

has been considered carefully in[14]by the semigroup approach.In[24],the inverse problem is reduced to a nonlinear equation and the uniqueness,as well as the conditional stability of the solution is proved.

The inverse problem ofidentification of the radiative coefficientq(x)in the following heat conduction equation:

from the final overdetermination datau(x,T)was considered by several authors(see,e.g.,in[8,10–11,34,39]).Moreover,treatments on the case of purely time dependentq=q(t)can be found in[6–7,12–13].For the general case in which the unknown coefficient(s)depend(s)on both spatial and temporal variables,we refer the readers to the references,e.g.,in[16,27–28,35].

Compared with classical parabolic equations,the main difficulty for degenerate equations lies in the degeneracy of the principle coefficients which may lead to the corresponding solution has no sufficient regularity,even if the initial value and the coefficients are sufficiently smooth functions.Many effective tools,e.g.,the Schauder’s type a priori estimate which was extensively applied in classical parabolic equations,are no longer applicable for the degenerate parabolic equations.The documents concerned with inverse degenerate problems are quite few in contrast with those dealt with non-degenerate problems.In[2],the authors investigated an inverse problem of determining the source termgin the following degenerate parabolic equation:

whereα∈[0,2).The uniqueness and Lipschitz stability of the solution are obtained by the global Carleman estimates,which was introduced in[22]in 1998.Recently,in[37],analogous methods were applied to a nonlinear inverse coefficient problem arising in the field of climate evolution,where the diffusion coefficient is assumed to vanish at both extremities of the domain.For other topics of degenerate parabolic equations,e.g.,the null controllability,we may refer the reader to[3–5]and the references therein.

The most important inverse problem in which the underlying model is degenerate may be the reconstruction of local volatility in the Black-Scholes equation(1.6).In[25–26],the inverse problem ofidentifying the implied volatilityσ=σ(S)from current market prices of options was considered carefully.Based on the optimal control framework,the existence,the uniqueness ofσ(S)and a well-posed algorithm are obtained.Similar results were derived in[15],where a new extra condition,i.e.,the average option premium,was assumed to be known.In[19],on the basis of the parameter-to-solution mapping,the stability and convergence of approximations forσ(S)are gained by Tikhonov regularization.

It should be mentioned that the degeneracy in the Black-Scholes equation can be removed by some change of variable(see[19]).However,the degeneracy in our problem can not be removed by any method,which is also the main difficulty in this paper.

To our knowledge,this paper is the first one concerning uniqueness,stability and convergence of optimal solution in inverse problem for degenerate parabolic equations such as(1.1).In this paper,we use an optimal control framework(see,e.g.,[16–17,25,39])to discuss Problem 1.1 mainly from the theoretical analysis angle.The outline of the manuscript is as follows:In Section 2,the inverse Problem 1.1 is transformed into an optimal control Problem 2.1 and the existence of minimizer of the cost functional is proved.The necessary condition of the minimizer is established in Section 3.By assuming thatTis relatively small,the local uniqueness and stability of the minimizer are shown in Section 4.The convergence of the minimizer with noisy input data is obtained in Section 5 by some a priori regularity conditions imposed on the forward operator.In Section 6,we complete this paper with concluding remarks.

2 Optimal Control Problem

In general,uniqueness is very important for the inverse problems.It illustrates if the extra condition is sufficient to identify the unknown information.There are many mathematical tools can be used to derive the uniqueness,such as maximum principle,energy estimate,unique continuation,integral equation,Carleman estimate,and so on.It should be mentioned that the Carleman estimate is an effective tool to derive uniqueness and conditional stability for inverse problems(see[22]).But unfortunately,it fails in treating the terminal control problems such as inverse Problem 1.1.We have obtained a uniqueness results of the inverse Problem 1.1 in a sense of partial order.It seems that the partial order imposed on the uniqueness is rather disgusting,but until now we do not know how to remove it due to the coefficient degeneration on the lateral boundaries.The details can be found in[18].

Since the original problem is ill-posed,we would like to discuss the regularization of Problem 1.1.Before this,let us to discuss the forward problem(2.1)and give some basic definitions,lemmas and estimations.We would like to consider the more general equation:

Definition 2.1Define B to be the closure ofunder the following norm:

Definition 2.2A function u(x,t)is called the weak solution to(2.1),if u∈C([0,T];L2(0,l))∩B,and for anythe followingintegration identity holds:

Remark 2.1Assumeu∈C([0,T];L2(0,l))∩BandThen(2.2)can be rewritten as

whereusatisfiesu|t=0=ϕ(x)in the sense of trace.

Theorem 2.1For any given f∈L∞(Q),ϕ∈L∞(0,l),there exists a unique weak solution to(2.1),which satisfies the following estimate:

Furthermore,if a|∇ϕ|2∈L1(0,l),thenand

ProofFirstly,we prove the existence.For any given 0<ε<1,we consider the following regularized problem:

where

From the well-known theory for parabolic equations(see[31]),there exists a unique weak solutionuε(x,t)to(2.3).

Then,we will give some a priori estimates foruε(x,t).Without loss of generality,we assume thatuε(x,t)is the classical solution to(2.3).Otherwise,one can smooth the coefficients of(2.3),and then consider the solution to the approximation problem.

Multiplying both sides of(2.3)byuεand integrating onQt=[0,l]×(0,t),we have

Integration by parts,we get

From(2.4)and the Gronwall’s inequality,we have

On the other hand,ifathen by multiplyingon both sides of(2.3)and integrating onQt,we obtain

Integrating by parts,we have

From(2.5),we get

From(2.6),we have

Moreover,it follows from the maximum principle that

From the estimations above,it can be derived that there exists a subsequence of{uε}(denoted by itself)and

such that

Lettingu=uεin(2.2),we have

Lettingε→0,one can immediately obtain

which implies the existence of weak solutions.

Next,we prove the uniqueness of weak solutions.Suppose thatu1,u2be two solutions to(2.1),and let

It can be easily seen thatU∈C([0,T];L2(0,l))∩B,and for anyψ∈L∞((0,T);L2(0,l))∩B,the following integration identity holds:

For any givenby the existence obtained above,we know that there exists a weak solutionv∈L∞((0,T);L2(0,l))∩Bandfor the following equation:

Lettingψ=vin(2.7),we obtain

Noting the arbitrariness ofg,we have

i.e.,

This completes the proof of Theorem 2.1.

Remark 2.2The weak solution defined above is on the whole domainQ.If we only consider the spatial case,we can modify the Definition 2.1 as follows.

Definition 2.1′Defineto be the closure ofunder the following norm:

For the case off≡0,the Definition2.2can also be rewritten as follows.

Definition 2.2′A function uis called the weaksolution to(2.1),if u satisfies

and the following integration identity

holds for a.e.t∈(0,T].Then,by analogously arguments,one can establish the existence,the uniqueness and the regularity for such kind of weak solution,which are similar to those of Theorem2.1.

Remark 2.3 .We recall that the principle coefficienta(x)∈C1[0,1].Due to the degeneracy atx=0 andx=l,fromone can only deriverather thanu∈H1(0,l),which is different from the case of non-degenerate.However,we may derive

In fact,if(2.10)is not true,i.e.,then we havewhereBδ(0)is a ball withδ-radius centered atx=0.Note that

Hence,

which is contradicts withBy analogous arguments,we have

It should be mentioned that these conclusions are no longer valid fora/∈C1[0,l].For example,let

It can be easily seen thatcan not guaranteeaux→0 asxtends to 0 orl.In some references(see,e.g.,[3–4]),the case(2.11)is called the weak degeneracy and the boundary conditions are indispensable for corresponding mathematical model,e.g.,we shall replace(1.1)by the following initial-boundary value problem:

Since the inverse Problem 1.1 is ill-posed,i.e.,its solution depends unstably on the data,we turn to consider the following optimal control Problem 2.1.

Problem 2.1Find(x)∈A,such that

where

u(x,t;q)is the solution to(1.1)for a given coefficientq(x)∈A,Nis the regularization parameter,andα,βare two given positive constants.

For the extra condition(1.4),we shall assume that

From(2.15)and Theorem 2.1,it can be easily seen that the control functional(2.13)is well-defined for anyq∈A.

We are now going to show the existence of minimizers to the problem(2.12).Firstly,we assert that the functionalJ(q)is of some continuous property inAin the following sense.

Lemma 2.1For any sequence{qn}in A which converges to some q∈A in L1(0,l)as n→∞,we have

Proof Step 1By takingq=qnand choosing the test function asu(qn)(·,t)in(2.9)and then integrating with respect tot,we derive that

for anyt∈(0,T].

From(2.17),we know that the sequence{u(qn)}is uniformly bounded in the spaceL2((0,T);So we may extract a subsequence,still denoted by{u(qn)},such that

Step 2Proveu∗(x,t)=u(q)(x,t).

By takingq=qnin(2.9)and multiplying both sides by a functionη(t)∈C1[0,T]withη(T)=0,we have

Then integrating with respect tot,we get

Lettingn→∞in(2.20)and using(2.18),we obtain

By noticing that(2.21)is valid for anyη(t)∈C1[0,T]withη(T)=0,we have

andu∗(x,0)=ϕ(x).

Therefore,u∗=u(q)by the de finition ofu(q).

Step 3Prove

We rewrite(2.9)forq=qnin the form

Takingψ=u(qn)−gin(2.23),we have

Similar relations hold foru(q),namely,

Subtracting(2.25)from(2.24),we obtain

Takingψ=u(qn)−u(q)in(2.9),we have

Similarly,for(u(qn)−u(q))t(u(q)−g),we have

Substituting(2.27)–(2.28)into(2.26),and after some manipulations,we derive

Then by rewriting the third term on the left-hand side of(2.29),we have

Integrating over the interval(0,t)for anyt≤T,we get

By the convergence of{qn}and the weak convergence of{u(qn)},one can easily get

Combining(2.31)and(2.32),we have

On the other hand,from the Hölder’s inequality,we have

From(2.15),(2.17)and(2.33)–(2.34),we obtain

This completes the proof of Lemma 2.1.

Theorem 2.2There exists a minimizer∈A of J(q),i.e.,

ProofIt is obvious thatJ(q)is non-negative,and thusJ(q)has the greatest lower boundLet{qn}be a minimizing sequence,i.e.,

By noticing thatJ(qn)≤C,we deduce

whereCis independent ofn.Noticing the boundedness of{qn}and(2.35),we also have

So we can extract a subsequence,still denoted by{qn},such that

By the Sobolev imbedding theorem(see[1]),we obtain

It can be easily seen that{qn(x)}∈A.So we get asn→∞that

inL1(0,l).

Moreover,from(2.37),we have

From Lemma 2.1 and the convergence of{qn},we know that there exists a subsequence of{qn},still denoted by{qn},such that

From(2.39)–(2.41),we get

Hence,

This completes the proof of Theorem 2.2.

3 Necessary Condition

Theorem 3.1Let q be the solution to the optimal control problem(2.12).Then there exists a triple offunctions(u,v;q)satisfying the following system:

and

for any h∈A.

ProofFor anyh∈A,0≤δ≤1,we have

Then

Letuδbe the solution to(1.1)with givenq=qδ.Sinceqis an optimal solution,we have

LetDirect calculations lead to the following equation:

LetThenξsatisfies

From(3.5),we have

LetLξ=ξt−(aξx)x+qξ,and suppose thatvis the solution to the following problem:

whereL∗is the adjoint operator of the operatorL.

By the well-known Green’s formula,we have

which implies

Combining(3.8)and(3.11),one can easily obtain that

This completes the proof of Theorem 3.1.

4 Uniqueness and Stability

The optimal control Problem 1.1 is non-convex.So,in general one may not expect a unique solution.In fact,it is well-known that the optimization technique is a classical tool to yield“general solution”for inverse problems without unique solution.However,we find that if the terminal timeTis relatively small,the minimizer of the cost functional can be proved to be local unique and stable.

Throughout this paper,if there is no specific illustration,Cwill be denoted the different constants.

Lemma 4.1Supposingwe have that for any k≥0,

(u−k)+=sup(u−k,0)

(u+k)−=sup(−(u+k),0)

Moreover,for a.e.x∈(0,l),we have

and

ProofForwe know

Notinga(x)>0,x∈(0,l),we have that for allδ>0,

By the definition of weak derivative(see[38]),it can be easily seen that

and for a.e.x∈(δ,l−δ),

Then we have

whereSince the quantityis bounded from the abovewhich does not depend onδ,by passing to the limit asδ→0,we get

Moreover,the following inequality

is obvious.Hence,Similar arguments can be used to treat the case of(u+k)−.

This completes the proof of Lemma 4.1.

Now,we can give a weak maximum principle for the weak solution to(1.1).

Lemma 4.2Supposingwe have for u the following estimate:

ProofLetMultiplying(1.1)by(u−k)+,we get from Lemma 5.1 that

DenotingE={x∈(0,l)|u(x)>k},one has

From(4.2)–(4.3),we have that for allt∈[0,T],

which implies thatis decreasing on[0,T].Since(ϕ−k)+≡0,we deduce that for allt∈[0,T]and for a.e.x∈(0,l),u(x,t)≤k.

By analogous arguments for(u+k)−,we can obtain that for allt∈[0,T]and for a.e.x∈(0,l),u(x,t)≥−k.

This completes the proof of Lemma 4.2.

Lemma 4.3For(3.2),we have the following estimate:

ProofLetτ=T−t.Then(3.2)is reduced to

The rest of the proofis similar to that of Lemma 4.2.

Suppose thatg1(x)andg2(x)are two given functions which satisfy the condition(2.15).Letq1(x)andq2(x)be the minimizers of Problem 2.1 corresponding tog=gi(i=1,2),respectively,and let{ui,vi}(i=1,2)be solutions to(3.1)–(3.2)in whichq=qi(i=1,2),respectively.

Set

ThenUandVsatisfy

Lemma 4.4For any bounded continuous function k(x)∈C(0,l),we have

where x0is a fixed point in(0,l).

ProofFor 0

This completes the proof of Lemma 4.4.

Lemma 4.5For(4.5),we have the following estimate:

where C is independent of T.

ProofFrom(4.5),we have that for 0

Integrating by parts,we obtain

which implies

From the Gronwall’s inequality and(4.10),we have

This completes the proof of Lemma 4.5.

Lemma 4.6For(4.6),we have the following estimate:

where C is independent of T.

ProofFrom(4.6),we have

Integrating by parts,we obtain

From Lemma 4.5 and(4.12),we have

From the Gronwall’s inequality,we have

This completes the proof of Lemma 4.6.

Theorem 4.1Let q1(x),q2(x)be the minimizers of the optimal control Problem2.1corresponding to g1(x),g2(x),respectively.If there exists a point x0∈(0,l),such that

q1(x0)=q2(x0),

then for relatively small T,we have

where the constant C is independent of T,l and N.

ProofBy takingh=q2whenq=q1,and takingh=q1whenq=q2in(3.3),we have

where{ui,vi}(i=1,2)are solutions to(3.1)–(3.2)withq=qi(i=1,2),respectively.

From(4.14)–(4.15),we have

From the assumption of Theorem 4.1,there exists a pointx0∈(0,l),such that

From Lemma 4.4 and(4.17),we have

From(4.16),(4.18)and the Young’s inequality,we obtain that

where we have used estimates(4.7)and(4.11).

From Lemmas 4.2–4.3,we have

From(4.19)–(4.20),we have

ChooseT≪1,such that

Combining(4.21)and(4.22),one can easily get

This completes the proof of Theorem 4.1.

Remark 4.1It should be mentioned that the regularization parameter plays a major role in the numerical simulation ofill-posed problems.From Theorem 4.1,we can obtain that if there exists a constantδ,such that

then the reconstructed optimal solution is unique and stable,which is consistent with the existed results(see,e.g.,[20]).Note that the estimate(4.23)is based on(4.22),from which we can seeSince the parameterNis often taken to be very small,particularly in numerical computations,Theorem 4.1 is indeed the local well-posedness of the optimal solution.For more detailed discussion on the regularization parameter,we refer the readers to the references(see,e.g.,[9,20]).

5 Convergence Analysis

In this section,we would like to discuss the convergence of the optimal solution.It has been shown in previous section that the optimal solution is stable and unique,which is very important in numerical process.However,the optimization problem is just a“modified problem”rather than the original one.Therefore,it is necessary to investigate what about the difference between the optimal solution to the optimization problem and the exact solution to the original problem.

We assume that the“real solution”g(x)is attainable,i.e.,there exists aq∗(x)∈H1(0,l),such that

and that an upper boundδfor the noisy level

of the observation is known a priori.

It should be mentioned that for terminal control problems,it is rather difficult to derive the convergence.To the authors’knowledge,there is no convergence result for the optimal control problem with the cost functional whose form is similar to(2.13).

In this paper,we introduce the following auxiliary control problems with observations averaged over the given terminal time interval[T−σ,T]:

Note that asσ→0+,

which impliesJσ(q)→J(q).Analogously,instead of(5.2),we assume that for the real solutionq∗(x),we have

Define the following forward operatoru(q):

whereu(x,t;q(x))is the solution to the variational problem(2.9)forq∈A.For anyq∈Aandp∈H1(0,l),one can easily deduce that the Gˆateaux directional differentialu′(q)psatisfies a homogeneous initial condition and solves

for anyφ∈L2(0,l)∩H1(0,l).For the remainder termR(q)=u(p+q)−u(q)−u′(q)p,we have the following variational characterization.

Lemma 5.1For any q∈A and p∈H1(0,l),such that p+q∈A,the remainder R(q)=u(p+q)−u(q)−u′(q)p solves

for any φ∈L2(0,l)∩H1(0,l).

ProofNote thatu(q+p)satisfies

Subtracting(5.7)from(2.9)and denotingW=u(q+p)−u(q),we obtain

Now(5.6)follows by subtracting(5.5)from(5.8).

This completes the proof of Lemma 5.1.

To obtain the convergence,we shall require some source conditions.We introduce the following linear operatorF(q):

whereu(q)is the solution to(2.9).Using(5.5),we immediately see that for anyp∈H1(0,l)and anythe following holds:

where⟨·,·⟩denote the scalar product inL2(0,l).Since∇is a linear operator,we can define its adjoint operator∇∗by

It can be easily seen that ifthen∇∗is equivalent to∇.In this paper,we will only need a weak form of∇∗∇.

Theorem 5.1Assume that there exists a function

such that the following source condition holds in the weak sense:

with F(q∗)defined by(5.9),i.e.,for any p∈H1(0,l),

Furthermore,assume that

andsatisfies

Then,with N∼δ,we have

and

whereis a minimizer of(5.3)with g replaced by gδ,is the solution to the variational problem(2.9)withand C is a positive constant independent of δ,N and T.

ProofNoting thatis a minimizer of(5.3),we have

which implies

From(5.18),we can derive

Using(5.10)and(5.13),we have for the last term in(5.19)that

Let

Using this notation,we obtain

Now,we need to estimate I1–I4.The main idea is to control I1–I4by the left-hand side item ofinequality(5.19).

For I1,we use(5.6)to get

From(5.23)and theinequality,we have

Using(2.14)and the Young’s inequality,we obtain

where we have used the assumption(5.4).

For I2,using integration by parts with respect totand noticingwe derive

For I3,using integration by parts with respect toxand noticinga(0)=a(l)=0,we obtain

The last term I4can be estimated similarly as follows by using the Young’s inequality:

where we have used the bound ofq∗.

Combining(5.19),(5.22)and(5.25)–(5.28),we obtain

From(5.29)and noticing the regularity ofφ,we have

By choosingN∼δ,one can easily get

The estimate(5.16)follows immediately from(5.31)and the Poincar´e’s inequality.

This completes the proof of Theorem 5.1.

Remark 5.1The motivation of replacing the cost functional(2.13)by(5.3)mainly lies in the difficulty in treating the second integration term in(5.22).In fact,if we choose the functional form(2.13),then we can deduce the second term in(5.22)(denoted by)to be

Since we have no information regarding to thet-derivative of the real and approximate solution,it is quite difficult,even impossible,to control the termby the left-hand side of(5.19),and thus we can not obtain any convergence.

6 Concluding Remarks

The inverse problem ofidentifying the coefficient in parabolic equations from some extra conditions is very important in some engineering texts and many industrial applications.Classical parabolic models are plentifully discussed and developed well,while documents dealt with degenerate parabolic models are quite few.

In this paper,we solve the inverse Problem 1.1 of recovering the radiative coefficientq(x)in the following degenerate parabolic equation:

ut−(aux)x+q(x)u=0

in an optimal control framework.Being different from other works(see,e.g.,[24,29]),which also treat with inverse radiative coefficient problems,the mathematical model discussed in this paper contains degeneracy on the lateral boundaries.Furthermore,unlike the well-known Black-Scholes equation whose degeneracy can be removed by some change of variable,the degeneracy in our problem can not be removed by any method.On the basis of the optimal control framework,the existence,the uniqueness,the stability and the convergence of the minimizer for the cost functional are established.

This paper focuses on the theoretical analysis of the 1-D inverse problem.For the multidimensional case,i.e.,the determination ofq(x)in the following equation:

ut−∇·(a(x)∇u)+q(x)u=0,(x,t)∈Q=Ω×(0,T],

where the principle coefficienta(x)satisfies

a(x)≥0,x∈

and Ω⊂Rm(m≥1)is a given bounded domain,the method proposed in this paper is also applicable.

AcknowledgementsThe authors would like to thank professor Jin Cheng of School of Mathematical Sciences at Fudan University for his valuable comments and helpful suggestions to improve the earlier version of the paper.

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