Quasireversibility Methods for Non-Well-Posed Problems ∗

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ISSN 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp (login: ftp) 147.26.103.110 or 129.120.3.113

Quasireversibility Methods for Non-Well-Posed Problems ^∗

Gordon W. Clark and

Seth F. Oppenheimer

Abstract The final value problem,

ut+Au= 0, 0< t < T u(T) =f

with positive self-adjoint unbounded A is known to be ill-posed. One approach to dealing with this has been the method of quasireversibility, where the operator is perturbed to obtain a well-posed problem which approximates the original problem. In this work, we will use a quasi- boundary-value method, where we perturb the final condition to form an approximate non-local problem depending on a small parameter α.

We show that the approximate problems are well posed and that their solutions uα converge on [0, T] if and only if the original problem has a classical solution. We obtain several other results, including some explicit convergence rates.

1 Introduction

LetAbe a self-adjoint operator on a Hilbert space H such that−A generates a compact contraction semi-group onH. We consider the problem of finding a u: [0, T]−→H such that

u⁰(t) +Au(t) = 0, 0< t < T

u(T) =f (F V P)

for some prescribed final valuef inH. Such problems are not well posed, that is, even if a unique solution exists on [0, T] it need not depend continuously on the

∗1991 Mathematics Subject Classifications: 35A35, 35R25.

Key words and phrases: Quasireversibility, Final Value Problems, Ill-Posed Problems.

c1994 Southwest Texas State University and University of North Texas.

Submitted: November 14, 1994.

Partially supported by Army contract DACA 39-94-K-0018 (S. F. O.)

1

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final valuef. One method for approaching such problems is quasi reversibility, introduced by Lattes and Lions in the 1960’s. The idea is to replace (FVP) with an approximate problem which is well posed, then use the solutions of this new problem to construct approximate solutions to (FVP). In the original method of quasi reversibility [2] Lattes and Lions approximate (FVP) with

vα⁰(t) +Avα(t)−αA²vα(t) = 0, 0< t < T vα(T) =f ,

where the operatorA is replaced by a perturbation, in this case by A−αA². For eachα >0, they use theinitial value u0=va(0) in

u⁰_α(t) +Auα(t) = 0, 0< t < T uα(0) =vα(0).

Finally they show that theuα(T) converge tof asαtends to zero. The method does not consider u(t) for t < T and the operator carrying f into vα(0) has large norm for smallα(on the order ofe^α^c)[3].

In [6], Showalter approximates (FVP) with

v⁰_α(t) +αAv_α⁰(t) +Avα(t) = 0, 0< t < T vα(T) =f

and as above for eachα >0, uses the initial valueu0=vα(0) in u⁰_α(t) +Auα(t) = 0, 0< t < T

ua(0) =vα(0).

The solutions ua are shown to approximate (FVP) in the sense that uα(T) converges to f as α tends to zero. Also the uα(t) are shown to converge to the solutionu(t) of (FVP) if and only if such exists, but again the norm of the function carryingf tovα(0) is quite large for smallα.

Miller [3] addresses this problem of large norm by finding optimal perturba- tions of the operator A. He states that it should be possible to make the norm on the order of_α^c rather thanexp(_α^c) and derives conditions on the perturbation f(A) to achievebest possible results. As in the methods above he approximates

(FVP) with

v⁰(t) +f(A)v(t) = 0, 0< t < T v(T) =f

and again solves the problem forward usingv(0) as an initial condition. Miller calls thisstabilized quasi reversibility.

Finally Showalter [7] addresses a more general problem in a different way.

He approximates the problem

u⁰(t) +Au(t)−Bu(t) = 0, 0< t < T u(0) =f .

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with

u⁰(t) +Au(t)−Bu(t) = 0, 0< t < T u(0) +αu(T) =f .

He calls this thequasi-boundary-value method, and he suggests that this method gives a better approximation than many other quasireversibility type methods.

In this work we study this method to approximate (FVP) and prove results analogous to the ones stated in [7]. We note that (FVP) is a special case of the problem studied in [7]. However, ore results are proved directly and this allows us to obtain explicit estimates for the convergence rate of the approximations.

2 Perturbing the final conditions

We approximate (FVP) with thequasi-boundary value problem u⁰(t) +Au(t) = 0, 0< t < T

αu(0) +u(T) =f . (QBV P)

One superficial advantage of this method is that there is no need tosolve forward here. More importantly, the error introduced by small changes in the final value f is not exponential, but of the order _α¹. We will show that this problem is well posed for eachα >0,and that the approximationsuαare stable. We show that uα(T) converges tof as αgoes to zero and that the valuesuα(t) converge on [0, T] if and only if (FVP) has a solution.

In the following, assume thatHis a separable Hilbert space andAis as above and that 0 is in the resolvent set of A. Let S(t) be the compact contraction semi-group generated by−A. Since A⁻¹ is compact, there is an orthonormal eigenbasisφnforH and eigenvalues _λ¹

n ofA⁻¹such thatA⁻¹φn =_λ¹

nφn.Then the eigenvalues of −A are −λn and those for S(t) are e⁻^tλⁿ (and possibly zero) [5]. In particular, for each positive α, αI +S(T) is invertible. Also, if u=P_∞

i=1aiφi,then S(T)u=P_∞

i=1e⁻^{T λ}ⁱaiφi and (S(T)u, u) =

X∞ i=1

e⁻^{T λi}a²_i ≥0. From this accretive type condition we obtain

k(αI+S(T))⁻¹k ≤ 1 α.

It is useful to know exactly when (FVP) has a solution. The following lemma answers this question.

Lemma 1 Iff =P_∞

i=1biφi, then (FVP) has a solution if and only if P_∞

i=1b²_ie^{2T λ}ⁱ converges.

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Proof. IfP_∞

i=1b²_ie^{2T λ}ⁱ converges, we merely defineu(t) =P_∞

i=1e^(T⁻^t)λⁱbiφi. Letube a solution to (FVP). Then u(0) has an eigenfunction expansion u= P_∞

i=1aiφi,and

S(T)u= X∞ i=1

e⁻^{T λ}ⁱaiφi=f = X∞

i=1

biφi.

This implies thate⁻^{T λ}ⁱai=biand thusai=bie^{T λ}ⁱ.Sinceu(0) is inH, we have

||u||²=P_∞

i=1a²_i <∞and we are done. ²

We wish to show that our approximate problem is well-posed and the following gives us what we need.

Definition. Define uα(t) =S(t)(αI+S(T))⁻¹f, forf in H, α >0 and t in [0, T].

Theorem 1 The function uα(t) is the unique solution of (QBVP) and it de- pends continuously on f.

Proof. Since (αI+S(T))⁻¹f is in the domain of A, it is clear that uα is a classical solution of the differential equation. Furthermore,

αuα(0) +uα(T) = α(αI+S(T))⁻¹f +S(T)(αI+S(T))⁻¹f

= (αI+S(T))(αI+S(T))⁻¹f =f.

To see the continuous dependence ofuαonf, compute

kS(t)(αI+S(T))⁻¹f1−S(t)(αI+S(T))⁻¹f2k

= kS(t)(αI+S(T))⁻¹(f1−f2)k

≤ 1

αkf1−f2k.

Uniqueness follows from the fact that any solutionv must satisfyv(0) = (αI+ S(T))⁻¹f and the uniqueness of solutions to the forward problem. ²

We make two observations at this point which will be useful later. First, from the above it is clear thatkuα(t)k ≤ α¹kfk. Secondly, if u = P_∞

i=1aiφi, then (αI+S(T))u=P_∞

i=1(α+e⁻^{T λ}ⁱ)aiφi and (αI+S(T))⁻¹u=

X∞ i=1

ai

α+e⁻^{T λ}ⁱφi. Theorem 2 For allf inH, α >0, andtin [0, T]we have that

kuα(t)k ≤α^t−T^T kfk.

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Proof. Iff =P_∞

i=1biφi,we have kuα(t)k² =

X∞ i=1

e⁻^2tλⁱb²_i α+e⁻^{T λ}ⁱ−2

≤ X^∞

i=1

e⁻^2tλⁱb²_i h

α+e⁻^{T λ}ⁱ_T^t

α+e⁻^{T λ}ⁱ¹−T^ti−2

≤ X∞ i=1

b²i

α¹⁻^T^t

₋2

=

α^t−T^T 2X∞

i=1

b²i

and we are done. ²

Theorem 3 For all f in H, ||uα(T)−f|| tends to zero as α tends to zero.

That isuα(T)converges to f in H.

Proof. Iff =P_∞

i=1biφi, then

kuα(T)−fk² = kS(T)(αI+S(T))⁻¹f−fk²

= α²k(αI+S(T))⁻¹fk²

= X∞ i=1

α²b²_i α+e⁻^{T λ}ⁱ−2

. Fix >0.ChooseN so thatP_∞

i=Nb²_i < /2.Thus kuα(T)−fk² <

XN

i=1

α²b²_i α+e⁻^{T λ}ⁱ−2

+ 2

≤ α² XN i=1

b²_ie^2λⁱ^T + 2. Now letαbe such thatα²<

2PN

i=1b²_ie^2λⁱ^T−2

and we are done. ²

Theorem 4 For allf inH,(FVP) has a solutionuif and only if the sequence uα(0)converges in H. Furthermore, we then have thatuα(t) converges tou(t) asαtends to zero uniformly int.

Proof. Assume that limα↓0uα(0) = u0 exists. Let u(t) = S(t)u0. Since limα↓0uα(T) =f,

limα↓0ku(t)−uα(t)k = kS(t)u0−uα(t)k

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= lim

α↓0kS(t) u0−(αI+S(T))⁻¹f k

≤ lim

α↓0ku0−(αI+S(T))⁻¹fk

= lim

α↓0ku0−uα(0)k= 0.

Thus, u(T) =f and u(t) = S(t)u0 solves (FVP). We also see that uα(t) converges tou(t) uniformly int.

Now let us assume thatu(t) is the solution to (FVP). Let > 0 andf = P_∞

i=1biφi. From Lemma 1 we have thatku(0)k²=P_∞

i=1b²_ie^{2T λ}ⁱ. ChooseN so thatP_∞

i=Nb²_ie^{2T λ}ⁱ< ₂. Letα, γ >0. Then

kuα(0)−uγ(0)k² = k(αI+S(T))⁻¹f−(γI+S(T))⁻¹fk

= kX^∞

i=1

1

α+e⁻^{T λ}ⁱ − 1 γ+e⁻^{T λ}ⁱ

biφik

= X∞ i=1

(γ−α)² αγ+ (α+γ)e⁻^{T λ}ⁱ+e⁻^{2T λ}ⁱ−2

b²i

= XN

i=1

(γ−α)² αγ+ (α+γ)e⁻^{T λ}ⁱ+e⁻^{2T λ}ⁱ−2

b²_i

+ X∞ i=N+1

(γ−α)² αγ+ (α+γ)e⁻^{T λ}ⁱ+e⁻^{2T λ}ⁱ−2

b²_i

≤ XN i=1

(γ−α)²e^{4T λ}ⁱb²_i + X∞ i=N+1

γ−α α+γ

2

b²_ie^{2T λ}ⁱ

≤ XN i=1

(γ−α)²e^{4T λ}ⁱb²i + 2. Now if we chooseδ >0 so thatδ²< PN

i=1e^{4T λ}ⁱb²_i ₋1

and require thatα andγbe less thanδ, we have that

kuα(0)−uγ(0)k²< .

We therefore have that{uα(0)} is Cauchy and thus converges. From the first part of the theorem, we have thatuα(t) converges tou(t) uniformly int. ²

We end this paper with a result that gives explicit convergence rates in the case that (FVP) is soluble for some positive final time.

Theorem 5 If f = P_∞

i=1biφi is in H and there exists an > 0 so that P_∞

i=1b²_ie^λⁱ^T converges, thenkuα(T)−fkconverges to zero with orderα⁻².

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Proof. Letbe in (0,2) such thatP_∞

i=1b²_ie^λⁱ^T is finite and letkbe in (0,2).

Fix a natural numbern. Define

gn(α) = α^k (α+e⁻^λⁿ^T)². Differentiating with respect toαyields

g⁰n(α) =α^k⁻¹(k−2)α+ke⁻^{T λ}ⁿ (α+e⁻^λⁿ^T)³ . Thusg⁰_n(α) = 0 when eitherα= 0 or

α= k

2−ke⁻^{T λ}ⁿ.

Sincegn(α)>0,gn(0) = 0, and limα→∞gn(α) = 0 we have thatα0= ₂₋^k_ke⁻^{T λ}ⁿ is the critical value at which gn achieves its maximum. Thus we have the inequality

gn(α)≤ k

2−k

k

e⁻^{kT λ}ⁿ (α0+e⁻^λⁿ^T)² . We now calculate

kuα(T)−fk² = X∞ n=1

b²_nα²(α+e⁻^λⁿ^T)⁻²=α²⁻^k X∞ n=1

b²_ngn(α)

≤ α²⁻^k X∞ n=1

b²_n k

2−k k

e⁻^{kT λ}ⁿ(α0+e⁻^λⁿ^T)⁻²

≤ α²⁻^k X∞ n=1

b²_n k

2−k k

e⁽²⁻^{k)T λ}ⁿ(α²₀+ 2α0e^λⁿ^T + 1)⁻¹

≤ α²⁻^k X∞ n=1

b²_n k

2−k k

e⁽²⁻^{k)T λ}ⁿ

= α²⁻^k k

2−k kX∞

n=1

b²_ne⁽²⁻^{k)T λ}ⁿ.

If we choosek= 2−we arrive at kuα(T)−fk²≤

2

α X∞ n=1

b²ne^{T λ}ⁿ=cα⁻².

2

If we assume thatP_∞

i=1b²_ie^(2+)λⁱ^T converges, working as above, we have that kuα(0)−u(0)k² = α²⁻^k

X∞ n=1

b²_ngn(α)e^{2T λ}ⁿ

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≤ α²⁻^k X∞ n=1

b²_n k

2−k k

e⁽⁴⁻^{k)T λ}ⁿ.

As above, lettingk= 2−, we arrive at the following.

Corollary 1 If f = P_∞

i=1biφi is in H and there exists an > 0 so that P_∞

i=1b²_ie^(2+)λⁱ^T converges, then kuα(t)−u(t)k converges to zero with order α⁻² uniformly in t.

References

[1] Conway, J.B., “A Course in Functional Analysis, Springer-Verlag, New York, 1990

[2] Lattes, R. and Lions, J.L., “Methode de Quasi-Reversibility et Applica- tions”, Dunod, Paris, 1967 (English translation R. Bellman, Elsevier, New York, 1969)

[3] Miller, K.,Stabilized quasireversibility and other nearly best possible methods for non-well-posed problems, “Symposium on Non-Well-Posed Problems and Logarithmic Convexity”, Lecture Notes in Mathematics, Vol. 316, Springer- Verlag, Berlin, 1973, pp 161-176

[4] Payne, L.E., Some general remarks on improperly posed problems for partial differential equations, “Symposium on Non-Well-Posed Problems and Logarithmic Convexity”, Lecture Notes in Mathematics, Vol. 316, Springer- Verlag, Berlin, 1973, pp 1-30

[5] Pazy, A., “Semigroups of Linear Operators and Applications to Partial Dif- ferential Equations”, Springer-Verlag, New York, 1983

[6] Showalter, R.E.,The Final Value Problem for Evolution Equations, J. Math.

Anal. Appl. 47, 1974, pp 563-572

[7] Showalter, R.E., Cauchy Problem for Hyper-Parabolic Partial Differential Equations, “Trends in the Theory and Practice of Non-Linear Analysis”, Elsevier, 1983

[8] Yosida, K.,“Functional Analysis”, Springer-Verlag, Berlin, 1980 Gordon W. Clark

Department of Mathematics Kennesaw State College P O Box 444

Marietta, GA 30061

E-mail address: [email protected] F. Oppenheimer

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Department of Mathematics and Statistics Mississippi State University

Drawer MA MSU, MS 39762 E-mail address: [email protected]