DIFFERENTIAL INCLUSIONS,
AND SENSITIVITY IN OPTIMIZATION
FRANKLEMPIO
Lehrstuhl fur Angewandte Mathematik an der Universitat Bayreuth
D-95440 Bayreuth
Federal Republic of Germany
Abstract.
Set-valued interpolation and integration methods are intro- duced with special emphasis on error representations and error estimates with respect to Hausdor distance. The connection between order of con- vergence results and sensitivity properties of nite-dimensional convex op- timization problems is discussed. The results are applied to the numerical approximation of reachable sets of linear control problems by quadrature formulae and interpolation techniques for set-valued mappings.(AMS) Subject Classication
: 34A60, 49M25, 65D05, 65D30, 65L05, 90C31, 93B03Keywords
: dierential inclusions, dierence methods, set-valued interpol- ation, set-valued integration, Aumanns's integral, sensitivity in optimiza- tion, attainable sets1. Introduction
Numerical methods for the solution of dierential inclusions follow three directions.
(i) Compute special solution trajectories with additional qualitative or quantitative properties:
Such trajectories have to be computed by dierence methods with addi- tional selection procedures choosing points from the set-valued right-hand side in an appropriate way. Common strategies result, e.g., in the discrete
analogue of heavy solutions, or slow solutions, or selections with a cer- tain kind of discrete minimal variation. These selection procedures require the solution of nite-dimensional optimization problems at every gridpoint.
Qualitative and quantitative sensitivity properties of this family of optim- ization problems determine qualitative and quantitative properties of the approximated solution, cp. in this connection [23] and the survey [19].
(ii) Compute all solution trajectories, or at least all belonging to a given class of functions:
This is theoretically and computationally an extremely dicult task.
Applying the abstract framework of general discretization theory requires correct notions of stability and consistency. Conditions assuring order of convergence higher than 2 are not available until now. In principle, a proper calculus of higher order derivatives is required for set-valued mappings, guaranteeing Taylor expansions with valid error estimates with respect to Hausdor distance. Some results concerning Euler's method resp. Euler- Cauchy method and order of convergence equal to 1 resp. equal to 2 are available, cp. [33], [34]. Every solution belonging to an appropriate Sobolev space can be approximated in a theoretical sense by a higher order linear multistep method, where the relevant notion of consistency is related to stability properties of a family of perturbed optimization problems, cp.
Denition 3.2 in [23].
(iii) Compute the reachable set of all solution trajectories at a prescribed point in time:
The techniques mentioned in (ii) like Euler's method resp. Euler-Cauchy method yield, as a by-product, rst resp. second order discrete approxim- ations of reachable sets of special classes of dierential inclusions. In [16]
even higher order of convergence is proven for a method exploiting fully the structure of special linear dierential inclusions with polyhedral control re- gion. In the sequel of papers [7], [6], [4], and in the thesis [5], the discrete approximation of reachable sets of linear dierential inclusions is totally re- duced to the numerical integration of set-valued mappings. The basis of this approach consists in adaptations of quadrature formulae and extrapolation methods to the calculation of Aumann's integral for set-valued mappings.
In principle, classical quadrature methods are applied to the support func- tional of the set-valued integrand. For every point in the integration interval and every unit vector in state space, the value of the support functional is determined by a convex optimization problem. Smoothness properties of this support functional as a function on the integration interval uni- formly with respect to the unit ball in state space, thus strong stability and sensitivity properties of an innite family of convex optimization prob- lems, determine the order of the integration method and, consequently, the
order of suitably dened discrete approximations of reachable sets. In this framework, higher order discrete approximations to reachable sets can be dened at least for special classes of linear dierential inclusions. Origin- ally, only the use of quadrature formulae with nonnegative weights seemed to be reasonable, like some open or closed Newton-Cotes formulae, Gauss quadrature, or Romberg's extrapolation method with Romberg's stepsize sequence. But exploiting some ideas in [8], compare also [9], depending on the geometry of the set-valued integrand, even quadrature fomulae with negative weights could be applied, thus opening the way to all kinds of extrapolation methods, error estimates by inclusion, and stepsize control for set-valued integration.
As outlined above, there exists an intrinsic relationship between nu- merical methods for dierential inclusions and questions of sensitivity and stability analysis of nite dimensional optimization problems. The main ob- jective of this paper is to describe this relationship. Hoping, that a numer- ical treatment of linear dierential inclusions in the very spirit of set-valued numerical analysis will also be of value for a more satisfactory numerical treatment of nonlinear dierential inclusions, we will concentrate on aspect (iii). Contrary to the thesis [5], where set-valued integration is the exclusive mathematical tool, we try to broaden the mathematical background to set- valued interpolation. The reader will easily recognize, that the techniques apply to set-valued mappings of several variables as well, thus opening the access to nite element methods for the discrete approximation of nonlinear dierential inclusions in the, hopefully, near future.
2. Set-Valued Interpolation
In the following, we introduce set-valued interpolation as a mathematical tool to approximate set-valued mappings by simpler set-valued mappings.
Deliberately, we avoid the technique of embedding spaces of convex sets into normed linear spaces, cp. the papers [28], [21], [30], [10], and [18]. This technique leaves the question unanswered how to interpret the results in the original spaces. Instead, we stay completely in the framework of set- valued mappings. Naturally, the problem arises how to dene dierences of sets in an appropriate way. This is done by a method already used in [8] for the proof of error estimates for set-valued quadrature formulae with negative weights, and in [5] for the derivation of inclusions of set-valued integrals by extrapolation methods. Only for simplicity we restrict ourselves to interpolation by set-valued polynomials, extensions to other function classes and even to interpolation of set-valued mappings of several variables by set-valued nite elements being rather obvious.
2.1. Interpolation Problem.
LetI
= [a;b
] witha < b
andF
:I
=)Rnbe a set-valued mapping with non-empty, convex and compact values.
Choose
N
2N and a grida
t
0< t
1< ::: < t
Nb ;
and compute for everyl
2Rn the polynomialp
N(l;
) of degreeN
withp
N(l;t
j) =?(l;F
(t
j)) (j
= 0;:::;N
):
Here, we denote by ?(l;A
) = supz2A
l
?z
(l
2Rn)the so-called support functional of the set
A
Rn. It is well-known, that ?(;A
) is a real-valued, positively homogenous continuous and convex func- tional on the whole of Rn for every non-empty convex and compact setA
. Moreover, the polynomialp
N(l;
) exists and is uniquely determined for everyl
2Rn. Naturally, except constant or linear interpolation or use of in- terpolation techniques with non-negative basis functions, cp. e.g. [24], [25], [35], the polynomialp
N(l;t
) is not for allt
2I
the support functional of a convex set. This can easily be seen by inspection of Lagrange's interpolation formula,p
N(l;t
) =XNj=0
?(l;F
(t
j)) YN=0
6=j
(
t t
)(
t
jt
);
(2.1) which, for xedt
2I
, is a linear combination of support functionals with, unfortunately, some negative weights in general. Hence,p
N(;t
) is real- valued, positively homogeneous and continuous for everyt
2I
, but in general not convex.A way out of this diculty consists in the replacement of
p
N(;t
) by its convexication resp. double conjugatep
??N(;t
) (t
2I
);
which can be computed as follows.By denition, cp. e.g. [29], we have
p
?N(z;t
) = supl2Rn[
z
?l p
N(l;t
)]=
8
<
:
0
;
ifz
?l
p
N(l;t
) for alll
2Rn;
1
;
ifz
?l > p
N(l;t
) for at least onel
2Rn:
Hence,p
?N(z;t
) is the indicator function of the setP
N(t
) =fz
2Rn :z
?l
p
N(l;t
) for alll
2Rng;
(2.2) and thereforep
??N(;t
) =?(;P
N(t
)) is the support functional ofP
N(t
) for everyt
2I
.2.2. Lemma.
On the standard assumptions of Interpolation Problem 2.1, the setP
N(t
) is closed, convex and bounded for everyt
2I
.Proof.
According to (2.2) the setP
N(t
) is the intersection of closed half spaces inRn, thereforeP
N(t
) is convex and closed. Moreover, (2.1) shows thatp
N(l;t
) is bounded uniformly for alll
2Rn withkl
k2 = 1,p
N(l;t
)c
(t
) (kl
k2 = 1);
(2.3) this implies forz
2P
N(t
)k
z
k22p
N(z;t
) and hence, forkz
k2 6= 0,k
z
k2 1k
z
k2p
N(z;t
)=
p
N
z
k
z
k2;t
c
(t
) (t
2I
):
Sincep
??N(l;t
j) =???(l;F
(t
j)) =?(l;F
(t
j)) andF
(t
j) is closed and convex,P
N(t
j) =F
(t
j) (j
= 0;:::;N
):
Therefore, in a very natural way, we can dene the set-valued interpolation
\polynomial" which solves Interpolation Problem 2.1.
2.3. Denition.
For everyl
2Rn letp
N(l;
)be the interpolation polynomial which solves Interpolation Problem 2.1.
Then the set-valued mapping
P
N :I
=)Rn;
dened byP
N(t
) =fz
2Rn :z
?l
p
N(l;t
) for alll
2Rng (t
2I
);
is called the set-valued solution of Interpolation Problem 2.1.At this point, we should add a warning: Neither is
P
N(t
) in general polynomial with respect tot
, nor isP
N(t
) necessarily non-empty for allt
2I
. Hence, it is crucial to give conditions which guaranteeP
N(t
) 6= ; for allt
2I
. In addition, these conditions should allow the proof of error estimates with respect to Hausdor distance betweenF
(t
) andP
N(t
) which are analogous to error estimates between the scalar functions?(l;F
(t
)) andp
N(l;t
). For this purpose, we use the following result which was already exploited in [8] for the proof of error estimates for set-valued quadrature formulae with negative weights.2.4. Lemma.
Consider a xedt
2I
wherep
N(;t
) is not itself a support functional. Assume moreover, that there exists a ballB
(m
(t
);r
(t
)) =fz
2Rn :kz m
(t
)k2r
(t
)gwith center
m
(t
) 2Rn and radiusr
(t
)>
0, which is contained entirely inP
N(t
),B
(m
(t
);r
(t
))P
N(t
):
Dene, as in (2.3),c
(t
) = supklk2=1
p
N(l;t
):
Then the following error estimate holdshaus(
F
(t
);P
N(t
)) 2c
(t
)r
(t
) supklk2=1j?(l;F
(t
))p
N(l;t
)j:
Here, haus(
;
) denotes Hausdor distance with respect to Euclidean norm kk2. The proof is contained in [8] and [5]. More convenient in ap- plications is the following condition onF
(t
) itself.2.5. Corollary.
Consider again a xedt
2I
wherep
N(;t
) is not itself a support functional. Assume moreover that the ballB
(m
(t
);r
(t
)) with centerm
(t
)2Rn and radiusr
(t
)>
0 is contained entirely inF
(t
).Then for every
(t
) = supklk2=1j
?(l;F
(t
))p
N(l;t
)j with0<
(t
)< r
(t
) the following error estimate holdshaus (
F
(t
);P
N(t
)) 2c
(t
)r
(t
) (t
)(t
): Proof.
SinceB
(m
(t
);r
(t
))F
(t
), it follows ?(l;B
(m
(t
);r
(t
))=
l
?m
(t
) +r
(t
)kl
k2?(
l;F
(t
));
hencel
?m
(t
) +r
(t
)kl
k2
p
N(l;t
) +(t
)kl
k2;
whence it followsl
?m
(t
) + (r
(t
) (t
))kl
k2p
N(l;t
) (l
2Rn):
This means that the ball
B
(m
(t
);r
(t
) (t
)) is contained inP
N(t
), and the estimate follows from Lemma 2.4.If for a xed
t
2I
the interpolating function is itself a support func- tional, which is clear for all grid points, and for linear interpolation or other interpolation techniques with non-negative basis functions, then the error estimate does not depend any longer on the geometry of the set-valued mappingF
(). Then the following estimate, cp. [11], [21], holds.2.6. Lemma.
Consider a xedt
2I
wherep
N(;t
) is itself a support func- tional of a non-empty convex and compact setP
N(t
). Thenhaus (
F
(t
);P
N(t
)) = supklk2=1j
?(l;F
(t
))p
N(l;t
)j:
The last representation of Hausdor distance is extremely useful for the direct proof of error estimates for set-valued quadrature formulae with non- negative weights without recourse to set-valued interpolation, cp. [7], [6], [4], [5], and Section 5.
By Lemma 2.4, Corollary 2.5 and Lemma 2.6, the error between
F
(t
) andP
N(t
) with respect to Hausdor distance is totally reduced to the classical error between?(l;F
(t
)) andp
N(l;t
) and, eventually, some upper bounds forc
(t
) and positive lower bounds forr
(t
) which depend on the geometry ofP
N(t
) resp.F
(t
). As we will see in Section 3, continuity and dierentiability properties of ?(l;F
(t
)) with respect tot
2I
uniformly for alll
2 Rn with kl
k2 = 1 play a crucial role for the classical error.But, we want to stress that such regularity properties of
?(l;F
()) can only be expected to hold for special classes of set-valued mappingsF
(), cp. Section 4. In any case, all subsequent error representations and error estimates have to be done very cautiously to exploit at least some absolute continuity properties for reasonably large classes of problems.3. Representation of the Interpolation Error
There are several methods, to prove estimates for the interpolation error
R
N(l;t
) =?(l;F
(t
))p
N(l;t
):
One could follow classical lines, cp. e.g. [32], which usually requires a little bit too strong smoothness assumptions on
?(l;F
()). One could also follow an approach exploiting systematically moduli of smoothness of?(l;F
()), cp. [31]. This approach yields the weakest estimates for problems in one variable. Instead, we present an approach which leads to weak error estim- ates for an especially important class of problems, and which can easily be extended to interpolation problems in several variables, cp. [15] and [14].For simplicity, in this section we use the abbreviation
f
(t
) =?(l;F
(t
))and suppress the explicit indication of
l
whenever possible.Hence,
f
(t
) satisesf
(t
) =p
N(t
) +R
N(t
) (t
2I
):
(3.1) We follow the idea in [15], cp. also [14], pp. 127{130. Taylor's theorem in [17] yields, forN
2,f
(t
j) =f
(t
) +f
0(t
)(t
jt
) +:::
+ 1(
N
2)!f
(N 2)(t
)(t
jt
)N 2 +Z 10
(1
)N 2(
N
2)!f
(N 1)(t
+(t
jt
))(t
jt
)N 1d
for
j
= 0;:::;N
and allt
2I
as long as at leastf
N () is continuous.If in addition
f
(N 1)() is absolutely continuous, thenf
(N)() exists almost everywhere and is integrable onI
, partial integration is justied and givesf
(t
j) =f
(t
) +f
0(t
)(t
jt
) +:::
+ 1(
N
2)!f
(N 2)(t
)(t
jt
)N 2 +"
(1
)N 1(
N
2)!(N
1)f
(N 1)(t
+(t
jt
))(t
jt
)N 1#=1 =0
Z
1
0
(1
)N 1(
N
1)!f
(N)(t
+(t
jt
))(t
jt
)Nd
=
f
(t
) +f
0(t
)(t
jt
) +:::
+ 1(
N
1)!f
(N 1)(t
)(t
jt
)N 1 +Z 10
(1
)N 1(
N
1)!f
(N)(t
+(t
jt
))(t
jt
)Nd :
Hence, for almost all
t
2I
, we have the representation, which holds forN
= 1 as well,f
(t
j) =f
(t
) +f
0(t
)(t
jt
) +:::
+ 1N
!f
(N)(t
)(t
jt
)N +Z 10
(1
)N 1 (N
1)!h
f
(N)(t
+(t
jt
))f
(N)(t
)i(t
jt
)Nd :
Considert
as a xed parameter, then the polynomial of degree at most equal toN
p
(z
) =f
(t
) +f
0(t
)(z t
) +:::
+ 1N
!f
(N)(t
)(z t
)Nsatises
d
dz
p
(z
)jz=t=f
()(t
) ( = 0;:::;N
) (3.2) and, forj
= 0;:::;N
,p
(t
j) =f
(t
j) Z 10
(1
)N 1 (N
1)!h
f
(N)(t
+(t
jt
))f
N(t
)i(t
jt
)Nd :
Therefore, it coincides with the Lagrange interpolation polynomial of degree at most equal toN
which attains the same values at the nodest
j,p
(z
) XNj=0
"
f
(t
j) Z 10
(1
)N 1 (N
1)!h
f
(N)(t
+(t
jt
))f
(N)(t
)i(t
jt
)Nd
#
N
Y
=0
6=j
z t
t
jt
:
Remembering that
p
N(z
)XNj=0
f
(t
j) YN=0
6=j
z t
t
jt
;
we get from (3.2) the following3.1. Error Representation.
Letf
(N 1)() be absolutely continuous onI
. Then for = 0;:::;N
and almost allt
2I
the following representation holdsf
()(t
) =p
(N)(t
)N
X
j=0
"
Z
1
0
(1
)N 1 (N
1)!h
f
(N)(t
+(t
jt
))f
(N)(t
)i(t
jt
)Nd
#
0
B
B
@
d
dz
N
Y
=0
6=j
z t
t
jt
1
C
C
A
z=t
:
This error representation clearly shows that the variation of
f
(N)() onI
plays a crucial r^ole, where this variation has to be dened in an appro- priate way, sincef
(N)() is only integrable. Fortunately,f
(N)() appears only in integrated form. Hence, the following denition is sucient for our purposes, cp. [5], p. 15,VarI
f
(N)() = inf varIg
() :g
() :I
!Rn isintegrableandg
(t
) =f
(N)(t
) foralmostallt
2I
;
(3.3) where var() denotes the usual variation of a vector valued function with respect to Euclidean norm. In the rest of this paper, the variation of integ- rable functions is to be understood in the sense of (3.3).
Assuming
f
() to be absolutely continuous andN
= 1;
= 0, we getf
(t
) =t t
1t
0t
1f
(t
0) +t t
0t
1t
0f
(t
1)Z
1
0
f
0(t
+(t
0t
))f
0(t
)(t
0t
)d
t t
1t
0t
1 (3.4)Z
1
0
f
0(t
+(t
1t
))f
0(t
)(t
1t
)d
t t
0t
1t
0for almost all
t
2I
as a very special case. This example is special in another sense as well. Since the basis functionst t
1t
0t
1; t t t
1t
00 are nonnegative onI
= [t
0;t
1],p
1(l;t
) =t t
1t
0t
1?(l;F
(t
0)) +t t
0t
1t
0?(l;F
(t
1))is itself a support functional of the non-empty convex and compact set
P
1(t
) =t t
1t
0t
1F
(t
0) +t t
0t
1t
0F
(t
1);
hence Lemma 2.6 applies, and (3.4) results directly in the following error estimate for the Hausdor distance between
F
(t
) andP
1(t
).3.2. Linear Interpolation.
Let ?(l;F
()) be absolutely continuous, and letd
dt
?(l;F
()) be of bounded variation inI
uniformly for alll
2Rn withk
l
k2 = 1.Then, for linear set-valued interpolation, the following error estimate holds
haus(
F
(t
);P
1(t
))= sup
klk2=1j
?(l;F
(t
))p
1(l;t
)jsup
klk2=1VarI
d
dt
?(l;F
())(t t
0)(t
1t
)t
1t
0:
Naturally, such error representations suggest the use of piecewise poly- nomial interpolation of set-valued mappings to get error estimates in terms of stepsize. Piecewise linear interpolation leads to corresponding error es- timates for the composite trapezoidal rule for set-valued mappings which is the basis for extrapolation methods for set-valued integration, cp. Section 5 and [7], [4], [5].
For later use, we add another special case, set-valued interpolation by polynomials of second degree. Assuming now
d
dtf
() to be absolutely con- tinuous andN
= 2;
= 0, we getf
(t
) = (t t
1)(t t
2)(
t
0t
1)(t
0t
2)f
(t
0)+ (t t
0)(t t
2)(
t
1t
0)(t
1t
2)f
(t
1)+ (t t
0)(t t
1) (t
2t
0)(t
2t
1)f
(t
2) (t t
1)(t t
2)(
t
0t
1)(t
0t
2)Z
1
0
(1
)hf
(2)(t
+(t
0t
))f
(2)(t
)i(t
0t
)2d
(t t
0)(t t
2)(
t
1t
0)(t
1t
2)Z
1
0
(1
)hf
(2)(t
+(t
1t
))f
(2)(t
)i(t
1t
)2d
(t t
0)(t t
1)(
t
2t
0)(t
2t
1)Z
1
0
(1
)hf
(2)(t
+(t
2t
))f
(2)(t
)i(t
2t
)2d :
Now, clearly, for xedt
2I
p
2(l;t
) =X2j=0
?(l;F
(t
j)) Y2=0
6=j
(
t t
) (t
jt
)is positively homogeneous with respect to
l
, but not any longer necessarily convex, since the Lagrangean elementary polynomials generally have dif- ferent signs. Therefore, applying Corollary 2.5, we get the following error representation for set-valued quadratic interpolation.3.3. Quadratic Interpolation.
Letd
dt
?(l;F
()) be absolutely continuous andd
2dt
2?(l;F
()) of bounded variation inI
uniformly for alll
2Rn withk
l
k2 = 1. Assume moreover that fort
2I
the ballB
(m
(t
);r
(t
)) with centerm
(t
)2Rn and radiusr
(t
)>
0 is contained inF
(t
), and that (t
) = supklk2=1j
?(l;F
(t
))p
2(l;t
)j is small enough, i.e. 0(t
)< r
(t
). Letc
(t
) = supklk2=1
p
2(l;t
) . Then the following error estimate holdshaus(
F
(t
);P
2(t
))
2
c
(t
)r
(t
) (t
) supklk2=1VarId
2dt
2?(l;F
())!
12
"
j
(
t t
0)2(t t
1)(t t
2)(
t
0t
1)(t
0t
2) j+j (t t
0)(t t
1)2(t t
2) (t
1t
0)(t
1t
2) j +j (t t
0)(t t
1)(t t
2)2(
t
2t
0)(t
2t
1) j#
:
Again, in concrete applications, one should use piecewise quadratic set- valued interpolation to get reasonable error estimates in terms of stepsize, compare in this connection Section 6.
All these representations of the interpolation error clearly show that in the case of interpolation by polynomials of degree at most equal to
N
the absolute continuity ofd
N 1dt
N 1?(l;F
()) and the variation ofd
Ndt
N?(l;F
())on
I
are essential for the error. For the special caseN
= 1, we only need that?(l;F
()) itself is absolutely continuous andd
dt
?(l;F
()) of bounded variation. Surprisingly enough, this property is satised automatically for set-valued mappings dened by a broad class of linear dierential inclusions, cp. [16] and Section 4.4. The R^ole of Sensitivity
As outlined in Section 2, cp. especially Lemma 2.4, Corollary 2.5, and Lemma 2.6, and exploited in Section 3, cp. Error Representation 3.1 and the special cases 3.2 and 3.3, the error between
F
(t
) andP
N(t
) with respect to Hausdor distance is reduced to the classical error between ?(l;F
(t
)) andp
N(l;t
) uniformly with respect to alll
2Rn withkl
k2 = 1.This is the point where sensitivity enters the scene, since the classical error
?(l;F
(t
))p
N(l;t
)is determined, for every xed
l
2Rn, by regularity properties of the value function?(l;F
()) of the following family of convex optimization problems.4.1. Perturbed Optimization Problems.
For every xedt
2I
, max- imizel
?z
subject toz
2F
(t
):
Here, the perturbation parameter is
t
2I
, the vectorl
2 Rn is con- sidered to be xed, and continuity and dierentiability properties of the corresponding value function ?(l;F
(t
)) with respect tot
2I
uniformly for alll
2Rn withkl
k2 = 1 play a crucial r^ole. Naturally, such additionalregularity properties of
?(l;F
()) can only be expected to hold for special classes of set-valued mappingsF
() or for some concrete problems. A rel- atively complete overview of such classes is contained in [5], pp. 81{106, which is based on the results of [26], [27], [1], [2], [16]. In the following, we cite only the most important cases.4.2. Theorem.
(a) LetU
Rm be compact and non-empty, and let the single-valued mappingf
:I
U
!Rn be a parametrization ofF
,F
(t
) =f
(t;U
) (t
2I
);
with compact values.Let
f
(t;
) be upper semicontinuous onU
for allt
2I
, and let there exist a Lipschitz constantL
withk
f
(t
1;u
)f
(t
2;u
)k2L
jt
1t
2j (t
1;t
2 2I;u
2U
):
Then, for everyu
2U; f
(;u
) is absolutely continuous, and the family@f
@t
(;u
)u2U
is integrable. Assume moreover, that this family is jointly of bounded vari- ation in the following sense:
There exist integrable functions
g
(;u
) :I
!Rn (u
2U
)with
g
(t;u
) =@f
@t
(t;u
) for almost allt
2I
, such that all the numbersmX1 i=0
k
g
(t
i+1;u
i)g
(t
i;u
i)k2 are bounded uniformly for all subdivisionsa
=t
0< t
1< ::: < t
m 1< t
m =b ;
allu
i 2U
, and allm
2N.Then,
?(l;F
()) is Lipschitz continuous, andd
dt
?(l;F
()) of bounded variation uniformly for alll
2Rn with kl
k2 = 1.(b) Let
U
Rm be compact and non-empty, and let then
m
-matrix functionA
() describe the following parametrization ofF
,F
(t
) =A
(t
)U
(t
2I
):
LetA
() be absolutely continuous, andd
dtA
() of bounded variation.Then,
?(l;F
()) is Lipschitz continuous, andd
dt
?(l;F
()) of bounded variation uniformly for alll
2Rn with kl
k2 = 1.Sometimes, it is possible to compute the support functional exactly for all
t
2I
, and to examine its regularity properties directly.4.3. Examples.
(i) LetF
(t
) =fz
2Rn :kz m
(t
)kpr
(t
)gbe a varying ball inRn with center
m
(t
) 2Rn and radiusr
(t
)0, where 1p
1. Then ?(l;F
(t
)) =l
?m
(t
) +r
(t
)kl
kq (t
2I
);
where 1p+ 1q = 1.(ii) Let
a
i(t
)b
i(t
) (i
= 1;:::;n
)and
F
(t
) =Yni=1[
a
i(t
);b
i(t
)]:
Then ?(l;F
(t
)) =Xni=1
l
i1 sign(
l
i)2
a
i(t
) + 1 + sign(l
i)2
b
i(t
) (t
2I
) for alll
= (l
1;:::;l
n)? 2Rn.(iii) Let
F
(t
) = cofp
1(t
);:::;p
r(t
)g (t
2I
)be a convex polyhedron with corners
p
1(t
);:::;p
r(t
)2Rn. Then ?(l;F
(t
)) = maxj=1;:::;r
l
?p
j(t
) (t
2I
):
If, for every
l
2Rn, there exists a cornerp
jl(t
)2fp
1(t
);:::;p
r(t
)g withl
?p
jl(t
) = maxj=1;:::;r
l
?p
j(t
) (t
2I
);
then dierentiability properties of
p
jl() onI
are inherited by ?(l;F
()).(iv) Consider a real function
:I
!R and a nonempty subsetU
Rn. Then ?(l;
(t
)U
) =8
<
:
(t
)?(l;U
) ((t
)0) (t
)?(l; U
) ((t
)<
0):
Hence, as long as
() does not change sign, dierentiability properties of () are inherited by ?(l;
()U
).If, moreover,
U
=U ;
then
?(l;
(t
)U
) =j(t
)j?(l;U
) (t
2I
):
Hence, dierentiability properties ofj
()j, especially those at zeros of(), determine the dierentiability properties of ?(l;
()U
).(v) Let () be an
n
m
-matrix function andB
(m
(t
);r
(t
)) =fz
2Rm :kz m
(t
)k2r
(t
)g (t
2I
) a varying ball inRm with centerm
(t
)2Rm and radiusr
(t
)0. DeneF
(t
) = (t
)B
(m
(t
);r
(t
)) (t
2I
):
Then
?(l;F
(t
)) =l
?(t
)m
(t
) +r
(t
)k?(t
)l
k2 (t
2I
):
Hence, as long as ?(
t
)l
6= 0Rm (which is, e.g., the case for alll
2 Rn with kl
k2 = 1 if the rows of (t
) are linearly independent for allt
2I
), dierentiability properties ofm
();r
(), and() are inherited by?(l;F
()).The situation is much worse with a varying ball in Rm with respect to innity norm,
B
1(m
(t
);r
(t
)) =fz
2Rm :kz m
(t
)k1r
(t
)g (t
2I
):
Again, let () be an
n
m
-matrix function onI
andF
(t
) = (t
)B
1(m
(t
);r
(t
)) (t
2I
):
Now, according to Example 4.3 (i), we have ?(l;F
(t
)) =l
?(t
)m
(t
) +r
(t
)k?(t
)l
k1 (t
2I
);
and only under rather special circumstances dierentiability properties of
m
();r
(), and () lead to the corresponding dierentiability properties of ?(l;F
()), cp. [6], Example 2.Summarizing, we want to stress that Theorem 4.2 just suces to justify the error estimate for (piecewise) linear set-valued interpolation for broader classes of parametrized set-valued mappings, whereas Examples 4.3 justify even higher order set-valued interpolation by (piecewise) polynomials for more restricted classes of set-valued mappings. In the following sections, these results are used for the derivation of error estimates for set-valued integration and discrete approximations of attainable sets.
5. Set-Valued Integration
Set-valued integration can be introduced in dierent ways either follow- ing [10] and [18] exploiting abstract embedding theorems for spaces of con- vex sets [28], [21], or in a direct way for quadrature formulae with non- negative weights, cp. [7], [4], [6], [5], resp. for quadrature formulae with negative weights, cp. [8] and [9]. First we give a motivation by a third approach following the classical introduction of interpolatory quadrature formulae: Interpolate the set-valued integrand by a set-valued mapping in the sense of Denition 2.3 and integrate this set-valued mapping in the sense of Aumann [3].
5.1. Denition.
LetI
= [a;b
] witha < b
andF
:I
=)Rn be a set-valued mapping. ThenZ
I
F
()d
=fz
2Rn : thereexistsanintegrableselectionf
() ofF
() onI
withz
=ZIf
()d
g is called Aumann's integral ofF
() overI
.The following theorem is fundamental, for proofs cp. e.g. [2] and [22].
5.2. Theorem.
LetF
:I
=)Rn be a measurable set-valued mapping with non-empty and closed images. ThenZ
I
F
()d
is convex.If, moreover,
F
() is integrably bounded, i.e., if there exists a functionk
()2L
1(I
) withf(tsup)2F(t)k
f
(t
)k2k
(t
) (5.1) for almost allt
2I
, thenZ
I
F
()d
=ZIco(F
())d
(5.2) is non-empty, compact, and convex.Here, co() denotes convex hull operation. The representation (5.2) sug- gests the interpolation of the set-valued mapping co(
F
()) in the sense of Interpolation Problem 2.1 and Denition 2.3. Just for simplicity, we assume in the following thatF
() itself is a measurable, integrably bounded set- valued mapping with non-empty compact convex values. Then we need not distinguish betweenF
();
co(F
());
co(F
());
where co(F
()) denotes the closed convex hull ofF
().According to Interpolation Problem 2.1, having set-valued interpolatory quadrature formulae in mind, choose
N
2N and a grida
t
0< t
1< ::: < t
Nb ;
and compute for everyl
2Rn the polynomialp
N(l;
) of degreeN
withp
N(l;t
j) =?(l;F
(t
j)) (j
= 0;:::;N
);
respectively the set-valued mappingP
N() =fz
2Rn :z
?l
p
N(l;
) foralll
2Rng with support functionalp
??N(l;t
)for every
t
2I
.According to the proof of Lemma 2.2,
P
N(t
) is closed, convex, and even uniformly bounded for allt
2I
withP
N(t
) 6=;. Moreover, the represent- ation (2.1) shows thatp
N(l;t
) is continuous with respect tot
2I
. Hence, following [2], ifP
N(t
)6=; for allt
2I
, then Aumann's integralZ
I
P
N()d
exists, is non-empty, convex, and compact, and satises
?(;
ZIP
N()d
) = ZI?(;P
N())d
= ZI
p
??N(;
)d :
Remembering the classical interpolatory quadrature formula dened by
p
N(l;
), i.e.Z
I
p
N(l;
)d
=XNj=0
?(l;F
(t
j))ZI YN=0
6=j
(
t
) (t
jt
)d ;
this suggests the use of this quadrature formula after convexication with respect to
l
2Rn :2
6
6
4
N
X
j=0
?(;F
(t
j))ZI YN=0
6=j
(
t
) (t
jt
)d
3
7
7
5
??
:
(5.3)For quadrature formulae with nonnegative weights, e.g. for closed New- ton-Cotes formulae with
N
nodes (N
= 2;:::;
8;
10), this representation simplyes, since a linear combination of support functionals with nonneg- ative coecients is again a support functional. Hence, if all the weightsc
j =ZI YN=0
6=j
(
t
)(
t
jt
)d
(j
= 0;:::;N
) are nonnegative, then (5.3) simplies to the support functionalN
X
j=0
