Electronic Journal of Differential Equations
Vol. 1994(1994), No. 08, pp. 1-9. Published November 29, 1994.
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 ,
u(T ) = f
0<t<T
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 quasiboundary-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
Let A be a self-adjoint operator on a Hilbert space H such that −A generates
a compact contraction semi-group on H. We consider the problem of finding a
u : [0, T ] −→ H such that
0
u (t) + Au(t) = 0 , 0 < t < T
(F V P )
u(T ) = f
for some prescribed final value f in H. 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.
c
1994
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
2
Quasireversibility Methods
EJDE–1994/08
final value f . 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
0
vα (t) + Avα (t) − αA2 vα (t) = 0 , 0 < t < T
vα (T ) = f ,
where the operator A is replaced by a perturbation, in this case by A − αA2 .
For each α > 0, they use the initial value u0 = va (0) in
0
uα (t) + Auα (t) = 0 , 0 < t < T
uα (0) = vα (0) .
Finally they show that the uα (T ) converge to f as α tends to zero. The method
does not consider u(t) for t < T and the operator carrying f into vα (0) has
c
large norm for small α (on the order of e α )[3].
In [6], Showalter approximates (FVP) with
0
vα (t) + αAvα0 (t) + Avα (t) = 0 , 0 < t < T
vα (T ) = f
and as above for each α > 0, uses the initial value u0 = vα (0) in
0
uα (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 solution u(t) of (FVP) if and only if such exists, but again the norm of the
function carrying f to vα (0) is quite large for small α.
Miller [3] addresses this problem of large norm by finding optimal perturbations of the operator A. He states that it should be possible to make the norm
on the order of αc rather than exp( αc ) and derives conditions on the perturbation
f (A) to achieve best possible results. As in the methods above he approximates
(FVP) with
0
v (t) + f (A)v(t) = 0 , 0 < t < T
v(T ) = f
and again solves the problem forward using v(0) as an initial condition. Miller
calls this stabilized quasi reversibility.
Finally Showalter [7] addresses a more general problem in a different way.
He approximates the problem
0
u (t) + Au(t) − Bu(t) = 0 , 0 < t < T
u(0) = f .
EJDE–1994/08
with
G.W. Clark & S. F. Oppenheimer
3
u0 (t) + Au(t) − Bu(t) = 0 , 0 < t < T
u(0) + αu(T ) = f .
He calls this the quasi-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 the quasi-boundary value problem
0
u (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 to solve 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 each α > 0, and that the approximations uα are stable. We show that
uα (T ) converges to f as α goes to zero and that the values uα (t) converge on
[0, T ] if and only if (FVP) has a solution.
In the following, assume that H is a separable Hilbert space and A is 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
eigenbasis φn for H and eigenvalues λ1n of A−1 such that A−1 φn = λ1n φn . Then
the eigenvalues of −A are −λn and those for S(t) are e−tλn (and possibly
zero)P[5]. In particular, for each
P∞positive α, αI + S(T ) is invertible. Also, if
∞
u = i=1 ai φi , then S(T )u = i=1 e−T λi ai φi and
(S(T )u, u) =
∞
X
e−T λi a2i ≥ 0 .
i=1
From this accretive type condition we obtain
k (αI + S(T ))−1 k ≤
1
.
α
It is useful to know exactly when (FVP) has a solution. The following lemma
answers this question.
P∞
Lemma
1 If f = i=1 bi φi , then (FVP) has a solution if and only if
P∞ 2 2T
λi
converges.
i=1 bi e
4
Quasireversibility Methods
EJDE–1994/08
P
P
2 2T λi
(T −t)λi
Proof. If ∞
converges, we merely define u(t) = ∞
bi φi .
i=1 bi e
i=1 e
Let
u
be
a
solution
to
(FVP).
Then
u(0)
has
an
eigenfunction
expansion
u=
P∞
a
φ
,
and
i
i
i=1
∞
∞
X
X
S(T )u =
e−T λi ai φi = f =
bi φi .
i=1
i=1
This implies that e−T λi ai = bi and thus ai = bi eT λi . Since u(0) is in H, we have
P∞
2
||u|| = i=1 a2i < ∞ 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 ))−1 f , for f in H, α > 0 and t in
[0, T ].
Theorem 1 The function uα (t) is the unique solution of (QBVP) and it depends continuously on f.
Proof. Since (αI + S(T ))−1 f 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 ))−1 f + S(T )(αI + S(T ))−1 f
= (αI + S(T ))(αI + S(T ))−1 f = f.
To see the continuous dependence of uα on f , compute
kS(t)(αI + S(T ))−1 f1 − S(t)(αI + S(T ))−1 f2 k
= kS(t)(αI + S(T ))−1 (f1 − f2 )k
1
≤
kf1 − f2 k .
α
Uniqueness follows from the fact that any solution v must satisfy v(0) = (αI +
S(T ))−1 f and the uniqueness of solutions to the forward problem. 2
We make two observations at this point which will be useful later.
P∞ First,
1
from the above it is clear
that
ku
(t)k
≤
kf
k.
Secondly,
if
u
=
α
i=1 ai φi ,
α
P
−T λi
then (αI + S(T ))u = ∞
)ai φi and
i=1 (α + e
(αI + S(T ))−1 u =
∞
X
i=1
ai
φi .
α + e−T λi
Theorem 2 For all f in H, α > 0, and t in [0, T ] we have that
kuα (t)k ≤ α
t−T
T
kf k .
EJDE–1994/08
Proof.
If f =
G.W. Clark & S. F. Oppenheimer
P∞
i=1 bi φi ,
kuα (t)k2
=
∞
X
we have
e−2tλi b2i α + e−T λi
i=1
≤
∞
X
i=1
≤
∞
X
5
e−2tλi b2i
h
−2
α + e−T λi
Tt
α + e−T λi
1− Tt i−2
−2
t
b2i α1− T
i=1
∞
t−T 2 X
=
α T
b2i
i=1
and we are done.
2
Theorem 3 For all f in H, ||uα (T ) − f || tends to zero as α tends to zero.
That is uα (T ) converges to f in H.
Proof.
If f =
P∞
i=1 bi φi ,
then
= kS(T )(αI + S(T ))−1 f − f k2
= α2 k(αI + S(T ))−1 f k2
kuα (T ) − f k2
=
∞
X
α2 b2i α + e−T λi
−2
i=1
Fix > 0. Choose N so that
kuα (T ) − f k2
.
P∞
<
b2i < /2. Thus
i=N
N
X
α2 b2i α + e−T λi
i=1
≤ α2
N
X
b2i e2λi T +
i=1
−2
+
2
.
2
P
−2
2 2λi T
b
e
and we are done.
Now let α be such that α2 < 2 N
i=1 i
2
Theorem 4 For all f in H, (FVP) has a solution u if and only if the sequence
uα (0) converges in H. Furthermore, we then have that uα (t) converges to u(t)
as α tends to zero uniformly in t.
Proof. Assume that limα↓0 uα (0) = u0 exists. Let u(t) = S(t)u0 . Since
limα↓0 uα (T ) = f ,
lim ku(t) − uα (t)k
α↓0
= kS(t)u0 − uα (t)k
6
Quasireversibility Methods
EJDE–1994/08
= lim kS(t) u0 − (αI + S(T ))−1 f k
α↓0
≤ lim ku0 − (αI + S(T ))−1 f k
α↓0
= lim ku0 − uα (0)k = 0 .
α↓0
Thus, u(T ) = f and u(t) = S(t)u0 solves (FVP). We also see that uα (t) converges to u(t) uniformly in t.
(FVP). Let > 0 and f =
P∞Now let us assume that u(t) is the solution2 to P
∞
bi φi . From Lemma 1 we have that ku(0)k = i=1 b2i e2T λi . Choose N so
i=1P
∞
that i=N b2i e2T λi < 2 . Let α, γ > 0. Then
kuα (0) − uγ (0)k2
= k(αI + S(T ))−1 f − (γI + S(T ))−1 f k
∞ X
1
1
= k
−
bi φi k
α + e−T λi
γ + e−T λi
i=1
=
∞
X
−2 2
(γ − α)2 αγ + (α + γ)e−T λi + e−2T λi
bi
i=1
N
X
−2 2
=
(γ − α)2 αγ + (α + γ)e−T λi + e−2T λi
bi
i=1
+
∞
X
(γ − α)2 αγ + (α + γ)e−T λi + e−2T λi
−2
b2i
i=N +1
2
N
∞
X
X
γ−α
2 4T λi 2
≤
(γ − α) e
bi +
b2i e2T λi
α
+
γ
i=1
i=N +1
N
X
≤
(γ − α)2 e4T λi b2i + .
2
i=1
Now if we choose δ > 0 so that δ 2 < and γ be less than δ, we have that
P
N
i=1
e4T λi b2i
−1
and require that α
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 that uα (t) converges to u(t) uniformly in t. 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.
P∞
Theorem
i=1 bi φi is in H and there exists an > 0 so that
P∞ 2 λi5T If f =
converges, then kuα (T ) − f k converges to zero with order α −2 .
i=1 bi e
EJDE–1994/08
G.W. Clark & S. F. Oppenheimer
P∞
2 λi T
i=1 bi e
Proof. Let be in (0, 2) such that
Fix a natural number n. Define
gn (α) =
7
is finite and let k be in (0, 2).
αk
.
(α + e−λn T )2
Differentiating with respect to α yields
gn0 (α) = αk−1
(k − 2)α + ke−T λn
.
(α + e−λn T )3
Thus gn0 (α) = 0 when either α = 0 or
α=
k −T λn
e
.
2−k
k
e−T λn
Since gn (α) > 0, gn (0) = 0, and limα→∞ gn (α) = 0 we have that α0 = 2−k
is the critical value at which gn achieves its maximum. Thus we have the
inequality
k
k
e−kT λn
2−k
gn (α) ≤
.
(α0 + e−λn T )2
We now calculate
∞
∞
X
X
2
2 2
−λn T −2
2−k
kuα (T ) − f k =
bn α (α + e
) =α
b2n gn (α)
n=1
≤ α2−k
≤ α2−k
≤ α
2−k
∞
X
n=1
∞
X
n=1
∞
X
b2n
b2n
b2n
n=1
2−k
= α
k
2−k
k
2−k
k
2−k
k
k
n=1
e−kT λn (α0 + e−λn T )−2
e(2−k)T λn (α20 + 2α0 eλn T + 1)−1
k
k
e(2−k)T λn
2−k
k X
∞
b2n e(2−k)T λn .
n=1
If we choose k = 2 − we arrive at
2 X
∞
2
kuα (T ) − f k2 ≤
α
b2n eT λn = cα −2 .
n=1
If we assume that
P∞
2
2 (2+)λi T
converges, working as above, we have that
i=1 bi e
kuα (0) − u(0)k2
= α2−k
∞
X
n=1
b2n gn (α)e2T λn
8
Quasireversibility Methods
≤ α
2−k
∞
X
n=1
b2n
k
2−k
EJDE–1994/08
k
e(4−k)T λn .
As above, letting k = 2 − , we arrive at the following.
P∞
Corollary 1 If f =
i=1 bi φi is in H and there exists an > 0 so that
P
∞
2 (2+)λi T
converges, then kuα (t) − u(t)k converges to zero with order
i=1 bi e
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 Applications”, 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, SpringerVerlag, 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, SpringerVerlag, Berlin, 1973, pp 1-30
[5] Pazy, A., “Semigroups of Linear Operators and Applications to Partial Differential 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: clark@math.msstate.edu Seth F. Oppenheimer
EJDE–1994/08
G.W. Clark & S. F. Oppenheimer
Department of Mathematics and Statistics
Mississippi State University
Drawer MA MSU, MS 39762
E-mail address: seth@math.msstate.edu
9