Gliklikh Y.E. Global and Stochastic Analysis with Applications to Mathematical Physics

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96 3 Ordinary Diﬀerential Equations

so the integral curve of X

with coordinates (q

(t),...,q

(t),p

(t),...,p

(t))

satisﬁes the system

∂H

∂p

= −

∂H

∂q

. The latter system is called the

Hamiltonian system with Hamiltonian H.

Hamiltonian systems describe a broad class of processes in physics and

mechanics. For conservative mechanical systems (see Section 11.3 below) the

Hamiltonian H is the total energy expressed in terms of momenta.

The Hamiltonian H is constant along the integral curves of X

. Indeed,

H =dH(X

)=Ω(X

)=0sinceΩ is skew symmetric. Taking into

account that the usual interpretation of the Hamiltonian in mechanics is the

total energy, this is a version of the conservation of energy law.

Consider two smooth functions f and g as Hamiltonians and (as usual) de-

note by X

and X

, respectively, their Hamiltonian vector ﬁelds. The function

{f,g} = Ω(X

) is called the Poisson bracket of f and g. Its coordinate

representation is easily derived in the form {f,g} =

∂f

∂p

∂g

∂q

−

∂f

∂q

∂g

∂p

The Poisson bracket is evidently skew-symmetric. It is easily shown that it

satisﬁes the Jacobi identity (1.8). So, the Poisson bracket deﬁnes the structure

of a Lie algebra on the space of smooth functions on T

∗

The Poisson bracket is useful for applications. In physical applications it

is more important to ﬁnd the values of a function along a solution than the

solution itself (for example, the solution can describe a process in a chemical

reactor and the function of the solution, the temperature of the wall of the

reactor). Let γ(t) be a solution of a Hamiltonian system with Hamiltonian

H and f : T

∗

M → R be a smooth function. Then one can easily show that

the real-valued function of the real variable f(γ(t)) satisﬁes the equation

f = {H, f}.

The notion of a Hamiltonian system can be generalized to a class of man-

ifolds much broader than cotangent bundles. A manifold M,onwhicha

smooth closed non-degenerate 2-form Ω is given, is called a symplectic man-

ifold and in this case Ω is called the symplectic form.

Every 2-form on an odd-dimensional manifold is degenerate. Hence all

symplectic manifolds are even-dimensional. The construction and results on

Hamiltonian systems can be translated to symplectic manifolds. Locally (i.e.,

in the charts) every symplectic manifold is organized as the cotangent bundle

of a certain manifold. This follows from a famous theorem of Darboux.

There is also a broad generalization of Hamiltonian theory to so-called

Poisson manifolds, i.e., manifolds in the space of smooth functions on which

a bracket operation {·, ·} satisfying the Jacobi identity is given ({·, ·} is the

Poisson bracket). In this case equations of the type

f = {H, f} are consid-

ered instead of ordinary Hamiltonian systems.

Wereferthereader,say,to[212] where a detailed geometrical description

of Hamiltonian theory is given.

Chapter 4

Elements of the Theory of Set-Valued

Mappings

In this chapter we survey some notions in the theory of set-valued mappings

which will be used below for the description of complicated mechanical sys-

tems such as systems with discontinuous forces, with control, etc.

More details can be found, for example, in [31, 155], where in particular

the proofs of many of the results presented here are given.

4.1 Set-Valued Mappings and Diﬀerential Inclusions

A set-valued mapping F from a set X intoasetY is a correspondence that

assigns a non-empty subset F (x) ⊂ Y to each point x ∈ X. F(x) is called

the value of x.

In order to distinguish set-valued mappings from single-valued mappings

we shall denote a set-valued mapping F sending X to Y by the symbol

F : X  Y while for a single-valued mapping we shall retain the notation

f : X → Y .

If X and Y are metric spaces, for set-valued mappings there are several

diﬀerent analogs of continuity that in the case of single-valued mappings

coincide with the usual deﬁnition of continuity (here we do not deal with the

description of such a notion for set-valued mappings of topological spaces,

see, e.g., [31]).

Deﬁnition 4.1. A set-valued mapping F is upper semicontinuous at the

point x ∈ X if for each ε>0 there exists a neighborhood U(x)ofx such

that from x



∈ U (x) it follows that F(x



) belongs to the ε-neighborhood of

the set F (x). F is upper semicontinuous on X if it is upper semicontinuous

at every point of X.

Deﬁnition 4.2. A set-valued mapping F is lower semicontinuous at the

point x ∈ X if for each ε>0 there exists a neighborhood U(x)ofx such that

from x



∈ U(x) it follows that F (x) belongs to the ε-neighborhood of F (x



Y.E. Gliklikh, Global and Stochastic Analysis with Applications

to Mathematical Physics, Theoretical and Mathematical Physics,

DOI 10.1007/978-0-85729-163-9

4, © Springer-Verlag London Limited 2011

98 4 Elements of the Theory of Set-Valued Mappings

F is lower semicontinuous on X if it is lower semicontinuous at every point

of X.

Deﬁnition 4.3. If F is both upper and lower semicontinuous, it is said to

be continuous (sometimes it is also called Hausdorﬀ continuous).

A continuous set-valued mapping F with the property that, for each x,its

value F (x) is a closed bounded set, is continuous with respect to the so-called

Hausdorﬀ metric on the space of all non-empty closed bounded subsets in

Y . In order to describe the Hausdorﬀ metric we ﬁrst introduce the submetric

H(A, B)=sup

a∈A

ρ(a, B)whereρ isthemetricinY . Then the Hausdorﬀ metric

is deﬁned by the formula

H(A, B)=max(

H(A, B),

H(B, A)). (4.1)

A set-valued mapping is said to be closed if its graph is a closed subset in

X ×Y .IfF is closed and for each point x ∈ X there exists a neighborhood

U(x) such that F (U(x)) is relatively compact, F is upper semicontinuous.

Deﬁnition 4.4. We say that F (t, x) satisﬁes the upper Carath´eodory condi-

tion if:

1) for every x ∈ X the map F (·,x):I  Y is measurable;

2) for almost all t ∈ I the map F(t, ·):X  Y is upper semicontinuous.

Deﬁnition 4.5. Let I =[0,l] ⊂ R. The set-valued mapping F : I ×X  Y

said to be almost lower semicontinuous if there exists a countable sequence

of disjoint compact sets {I

}, I

⊂ I such that:

(i) the measure of I\∪

is equal to zero;

(ii) the restriction of F on each I

× X is lower semicontinuous.

An important technical role in the investigation of set-valued mappings is

played by single-valued mappings that approximate the set-valued mappings

in some sense. We describe two kinds of such single-valued mappings: selectors

and ε-approximations.

Deﬁnition 4.6. Let F : X  Y be a set-valued mapping. A single-valued

mapping f : X → Y such that for each x ∈ X the inclusion f(x) ∈ F (x)

holds is called a selector of F .

Not every set-valued mapping has a continuous selector. However, for lower

semicontinuous set-valued mappings with convex closed values, their exis-

tence is proved in the following classical Theorem.

Theorem 4.7 (Michael’s Theorem) If X is an arbitrary metric space and Y

is a Banach space, then a lower semicontinuous mapping such that the value

of every point of X is a convex closed set has a continuous selector.

4.1 Set-Valued Mappings and Diﬀerential Inclusions 99

If the values of a lower semicontinuous set-valued mapping are not, in gen-

eral, convex, it may not have continuous selectors. In this case the following

construction is often very useful.

Deﬁnition 4.8. Let E be a separable Banach space. A non-empty set M⊂

([0,l]; E) is called decomposable if f ·χ

+g ·χ

[0,l]\M

∈Mfor all f,g ∈M

and for every measurable subset M in [0,l]whereχ is the characteristic

function of the corresponding set.

The reader can ﬁnd more details about decomposable sets in [48]and[155].

Theorem 4.9 (Bressan-Colombo Theorem) Let (Ω,d) be a separable metric

space, X be a Banach space and (J, A,μ) be a measurable space with a σ-

algebra A and a non-atomic measure μ such that μ(J)=1. Consider the

space Y = L

(J, A,μ) of integrable mappings from (J, A,μ) into X.Ifa

set-valued mapping F : Ω  Y is lower semicontinuous and has closed

decomposable values, F has a continuous selector.

The assertion of Theorem 4.9 is proved, for example, as Lemma 9.2 in [48].

Upper semicontinuous mappings arise in applications more often than

lower semicontinuous mappings. Generally speaking, they do not have con-

tinuous selectors (but they do have measurable selectors). The so-called ε-

approximations are very useful for investigating upper semicontinuous map-

pings.

Deﬁnition 4.10. For a given ε>0 a continuous single-valued mapping f

X → Y is called an ε-approximation of a set-valued mapping F : X  Y if

the graph of f,asasetinX ×Y , belongs to the ε-neighborhood of the graph

of F .

We mention the following classes of upper semicontinuous set-valued map-

pings of ﬁnite-dimensional spaces, for which the existence of ε-approximations

is proved for each ε>0:

(i) the mappings with convex closed values;

(ii) the mappings with values that are aspheric in all dimensions from 1

to n − 1 and weakly aspheric in the dimension n (see [32]). This class

of set-valued mappings was ﬁrst considered by A.D. Myshkis in 1954

[184]. In [32]and[87] topological characteristics of topological index and

Lefschetz number types were constructed for such mappings. Later (in

the 1980s) this class was rediscovered and described as “the mappings

whose values at every point have the so-called uv

-property for k =

1,...,n” (for the exact deﬁnition see, e.g., [166]).

Let X be a Banach space and F : X  X be an upper semicontinuous set-

valued mapping with convex closed values. For each bounded subset Ω ⊂ X,

let the image F (Ω) be relatively compact. Then if F sends a ball B of X

100 4 Elements of the Theory of Set-Valued Mappings

into itself, in B there exists a ﬁxed point x ∈ F (x)ofF (this is an analog of

Schauder’s principle known as the Glicksberg-Ky Fan Theorem ).

Let F : R ×R

 R

be a set-valued mapping. A diﬀerential inclusion

˙x ∈ F (t, x) (4.2)

is an analog of a diﬀerential equation and transforms into the latter if F is

single-valued.

A solution of (4.2) is an absolutely continuous curve x(t) such that (4.2)

is satisﬁed for x(t) almost everywhere.

If F is upper semicontinuous and has convex closed bounded values, for

each pair x

∈ R

, t

∈ R there exists a local in time solution of (4.2) with the

initial condition x(t

)=x

. It is also known that for an upper semicontinuous

F with closed bounded (not necessarily convex) values there exists a solution

of the Cauchy problem for the diﬀerential inclusion

˙x ∈

coF (t, x),

where

coF (t, x) is the convex closure of F (t, x).

The existence of solutions of (4.2) for lower semicontinuous F is possible

also for non-convex values. Often such existence can be proved by applying

the Bressan-Colombo Theorem (Theorem 4.9).

4.2 Special Approximations

Here, following [10, 11], we prove the existence of special approximations

for upper semicontinuous mappings in ﬁnite-dimensional spaces with convex

closed values which point-wise converge to a Borel measurable selector of the

set-valued mapping as ε → 0. These results will often be used below.

Theorem 4.11 Let Φ : R

 R

be an upper semi-continuous set-valued

mapping with convex closed bounded values. For a sequence ε

→ 0 there exists

a sequence of continuous ε

-approximations for Φ that point-wise converges

to a Borel measurable selector of Φ.IfΦ takes values in a convex set Ξ in

, these ε-approximations also take values in Ξ.

Proof. It is shown in [79, Theorem 2] that, in the case under consideration,

for any ε

there exists a lower semi-continuous set-valued map Ψ

: R

 R

with closed convex bounded values such that: (i) for any x ∈ R

the inclusion

Φ(x) ⊂ Ψ

(x) holds and (ii) the graph of Ψ

belongs to the ε

-neighborhood

of the graph of Φ. From the construction it follows that if Φ takes values

in a convex set Ξ in R

, then the values of all Ψ

(x) belong to Ξ. Notice

that for an upper semi-continuous mapping with compact values the sum of

such mappings and the products with a continuous function are upper semi-

continuous. Hence, from the proof of Theorem 2 of [79] it follows that, in

4.2 Special Approximations 101

the case under consideration, all Ψ

are continuous set-valued mapping and,

in particular in our case, they are continuous with respect to the Hausdorﬀ

metric.

Consider the minimal selector ψ

(·)ofΨ

(·), i.e., ψ

(x) is the closest point

to the origin in Ψ

(x), x ∈ R

. We refer the reader to [5] for a complete

description of minimal selectors. In particular, it is shown there that minimal

selectors are continuous. Thus, ψ

is an ε

-approximation of Φ.

Let x ∈ R

.SinceΦ(x) ⊂ Ψ

(x)foreachi, for the Hausdorﬀ submetric

H we have

H(Φ(x),Ψ

(x)) = 0. Hence for the Hausdorﬀ metric H we obtain

that H(Ψ

(x),Φ(x)) =

H(Ψ

(x),Φ(x)) for each i.

Now specify ε

. By the deﬁnition of upper semi-continuity, for any x ∈ R

there exists a δ

> 0 such that for any x



in the δ

-neighborhood of x the

value Φ(x



) belongs to the ε

-neighborhood of Φ(x). Since ε

→ 0, ε

k+l

<δ

for some l = l(k, x) and without loss of generality we may take l(k, x) ≥ 0.

Thus

H(Φ(x



),Φ(x)) <ε

for each x



in the ε

k+l

-neighborhood of x.

Since the graph of Ψ

k+l

belongs to the ε

k+l

-neighborhood of the graph of

Φ, there exists a point x



in the ε

k+l

-neighborhood of x such that Ψ

k+l

(x)

belongs to the ε

k+l

-neighborhood of Φ(x



), i.e.,

H(Ψ

k+l

(x),Φ(x



)) <ε

k+l

Thus

H(Ψ

k+l

(x),Φ(x)) =

H(Ψ

k+l

(x),Φ(x))

≤

H(Ψ

k+l

(x),Φ(x



)) +

H(Φ(x



),Φ(x))

<ε

k+l

+ ε

< 2ε

Hence at each x the convex set Ψ

(x) tends to the convex set Φ(x) with respect

to the Hausdorﬀ metric as i →∞. It follows that ψ

(x) tends to the point

ϕ(x) ∈ Φ(x) that is the closest to the origin. The well-known fact that the

point-wise limit ϕ(·) of the sequence of continuous mappings ψ

(·)isaBorel

measurable mapping completes the proof. 

We introduce

Ω = C

([0,T], R

) – the Banach space of continuous curves

in R

given on [0,T], with the usual uniform norm – and the σ-algebra

Ω generated by cylinder sets. By P

we denote the σ-subalgebra of F

generated by cylinder sets with bases over [0,t] ⊂ [0,T]. Recall that

F is the

Borel σ-algebra on

Ω (see [208]).

Let B :[0,T] ×

Ω → Z be a mapping to some metric space Z.Below

we shall often suppose that such mappings, for various spaces Z, satisfy the

following condition:

Condition 4.12 For each t ∈ [0,T], from the fact that the curves x

(·) and

(·) ∈

Ω coincide for 0 ≤ s ≤ t, it follows that B(t, x

(·)) = B(t, x

(·)).

Remark 4.13. The fact that a mapping B satisﬁes Condition 4.12 is equiv-

alent to the fact that B is measurable at each t with respect to a Borel

σ-algebra in Z and P

Ω (see [83], cf. Condition 6.19(ii) below).

102 4 Elements of the Theory of Set-Valued Mappings

Theorem 4.14 Let (ε

) be a sequence of positive numbers such that ε

→ 0

as k →∞.LetB be an upper semi-continuous set-valued mapping with com-

pact convex values sending [0,T] ×

Ω to a ﬁnite-dimensional Euclidean space

Y and satisfying Condition 4.12. Then there exists a sequence of continuous

single-valued mappings B

:[0,T] ×

Ω → Y with the following properties:

(i) each B

satisﬁes Condition 4.12;

(ii) the sequence B

point-wise converges to a selector of B that is measur-

able with respect to the Borel σ-algebra in Y and the product σ-algebra

of the Borel σ-algebra on [0,T] and

F on

Ω;

(iii) at each (t, x(·)) ∈ [0,T] ×

Ω the inequality B

(t, x(·))≤B(t, x(·))

holds for all k;

(iv) if B takes values in a closed convex set Ξ ⊂ Y , the values of all B

belong to Ξ.

Proof. In this proof we combine and modify the ideas used in the proofs of

[79, Theorem 2] by Gel’man and Theorem 4.11 above.

For t ∈ [0,T] deﬁne the mapping f

Ω →

Ω by the formula

x(·)=



x(s)if0≤ s ≤ t

x(t)ift ≤ s ≤ T.

(4.3)

Clearly f

x(·) is jointly continuous in t ∈ [0,T]andx(·) ∈

Ω.SinceB satisﬁes

Condition 4.12, B(t, x(·)) = B(t, f

x(·)) for each x(·) ∈

Ω and t ∈ [0,T].

Choose an element ε

from the sequence. Since B is upper semi-continuous,

for every (t, x(·)) ∈ [0,T] ×

Ω there exists a δ

(t, x) > 0 such that for ev-

ery (t

∗

(·)) from the δ

(t, x)-neighborhood of (t, x(·)) the set B(t

∗

(·))

is contained in the

-neighborhood of the set B(t, x(·)). Without loss of

generality we can suppose 0 <δ

(t, x) <ε

for every (t, x(·)). Consider the

(t,x)

-neighborhood of (t, x(·)) in [0,T] ×

Ω and construct the open covering

of [0,T]×

Ω by such neighborhoods for all (t, x(·)). Since [0,T]×

Ω is paracom-

pact, there exists a locally ﬁnite reﬁnement {V

} of this covering. Without

loss of generality we can consider each V

as an η

)-neighborhood of

some (t

(·)) where by construction the radius η

) ≤

)

Consider a continuous partition of unity {ϕ

} adapted to {V

} and in-

troduce the set-valued mapping Φ

(t, x(·)) =



(t, x(·))coB(V

)where

co denotes the convex closure. Since B(t, x(·)) is upper semi-continuous and

has compact values, without loss of generality we can suppose δ

(t, x)tobe

such that the images B(V

) are bounded in Y and so the sets coB(V

)are

compact. Denote by

(t, x(·)) the closure of Φ

(t, x(·)). Then one can easily

see that

:[0,T] ×

Ω → Y is a Hausdorﬀ continuous set-valued mapping

with compact convex values.

Deﬁne Ψ

:[0,T] ×

Ω → Y by the formula Ψ

(t, x(·)) = Φ

(t, f

x(·)) and

consider the set-valued mapping

(t, x(·)). Since f

is continuous, every Ψ

4.2 Special Approximations 103

is a Hausdorﬀ continuous set-valued mapping with compact convex values

and by construction it satisﬁes Condition 4.12.

The pair (t, f

x(·)) belongs to a ﬁnite collection of neighborhoods V

with centers at (t

(·)), i =1,...,n, and so by construction B(t, x(·)) =

B(t, f

x(·)) ⊂ B(V

)foreachi. Hence B(t, x(·)) = B(t, f

x(·)) ⊂ Ψ

(t, x(·))

for every pair (t, x(·)).

Let l be the index from the collection of indices j

above such that η

)

takes the greatest value among η

). Then all (t

(·)) are con-

tained in the 2η

)-neighborhood of (t

(·)) and so every V

is con-

tained in the 3η

)-neighborhood of (t

(·)) that is contained in

the δ

(·))-neighborhood of (t

) by construction. Then, also by con-

struction, Ψ

(t, x(·)) belongs to the

-neighborhood of B(t

(·)). Since

both Ψ

(t, x(·)) and B(t

(·)) are convex, this means that Ψ

(t, x(·)) also

belongs to the

-neighborhood of B(t

(·)). Notice that this is true for

each k.

Since B(t, x(·)) ⊂ Ψ

(t, x(·)) ⊂ Ψ

(t, x(·)), for the Hausdorﬀ submetric

we have

H (B(t, x(·)) ,

(t, x(·))) = 0.

Hence for the Hausdorﬀ metric H we obtain that



(t, x(·)), B(t, x(·))





(t, x(·)), B(t, x(·))



Since ε

→ 0, for (t, x(·)) there exists an integer θ = θ(t, x(·)) > 0such

that ε

k+θ

<δ

(t, x(·)). Without loss of generality we can suppose that θ ≥ 1.

Thus B(t

k+θ

(·)) belongs to the

-neighborhood of B(t, x(·)) and



k+θ

(·)



, B(t, x(·))



Since

k+θ

(t, x(·)) belongs to the

k+θ

-neighborhood of B(t

k+θ

(·))

(see above), we obtain that



k+θ

(t, x(·)), B



k+θ

(·)



k+θ

Thus



k+θ

(t, x(·)), B(t, x(·))





k+θ

(t, x(·)), B(t, x(·))



≤



k+θ

(t, x(·)), B



k+θ

(·)





k+θ

(·)



, B(t, x(·))



k+θ

<ε

So, at each (t, x(·)) we have that H(

(t, x(·)), B(t, x(·))) → 0ask →∞

and B(t, x(·)) ⊂ Ψ

(t, x(·)) for all k.

104 4 Elements of the Theory of Set-Valued Mappings

Consider the minimal selector B

(t, x(·)) of Ψ

(t, x(·)), i.e., B

(t, x(·)) is

the closest point to the origin in

(t, x(·)). We again refer the reader to [5]

for a complete description of minimal selectors and, in particular, recall that

the minimal selectors we shall be considering are all continuous. One can

easily see that all B

satisfy Condition 4.12.

By construction the minimal selectors B

(t, x(·)) of Ψ

(t, x(·)) point-wise

converge to the minimal selector B(t, x(·)) of B(t, x(·)) as k →∞since

at any (t, x(·)) we have that H(

(t, x(·)), B(t, x(·))) → 0ask →∞and

B(t, x(·)) ⊂ Ψ

(t, x(·)) for all k (see above). It is a well-known fact that the

point-wise limit B of the sequence of continuous mappings B

is measurable

with respect to the Borel σ-algebras in Y and in [0,T] ×

Ω (see [194]). The

latter coincides with the product σ-algebra of the Borel σ-algebra on [0,T]

and

F on

Ω (see [208]). Properties (iii) and (iv) immediately follow from the

construction. 

Remark 4.15. Unlike

(t, x(·)), the set-valued mapping

(t, x(·)) may not

satisfy Condition 4.12 since two diﬀerent curves x

(·)andx

(·) coinciding on

[0,t] may have diﬀerent neighborhoods V

to which they belong, and so the

values

(t, x

(·)) and

(t, x

(·)) may be diﬀerent. On the other hand, it

follows from [79]that

is an ε

-approximation of B while the same is not

true for

Chapter 5

Analysis on Groups of

Diﬀeomorphisms

5.1 General Concepts

By H

we denote the Sobolev space of functions such that the functions and

their generalized derivatives up to order s belong to the functional space

. A detailed description of Sobolev spaces can be found, e.g., in [62]. An

introduction to the manifold structure in functional sets can be found in [64].

The reader may wish to consult [61] for details on the remaining material of

this section.

The case of a compact manifold without boundary

Let M,dimM = n, be a compact oriented manifold without boundary

equipped with a Riemannian metric ·, · and its Levi-Civit´a connection H.

Denote by H

= H

(M,M) the set of Sobolev H

-mappings from M to

M with s>

n + 1. Recall that for s>

n + k the maps from H

are

-smooth. Sometimes we shall also deal with sets H

(M,N)whereN is

a manifold with the same dimension as M. There is an inﬁnite-dimensional

manifold structure on H

(M,M)andH

(M,N)(see[61]).

Consider the open neighborhood D

(M) of the identical mapping in

(M,M) that consists of all H

-diﬀeomorphisms. Consider also its subset

(M) comprising the H

-diﬀeomorphisms which preserve the Riemannian

volume form.

(M)andD

(M) have the structures of smooth (and separable) Hilbert

manifolds as well as the natural multiplicative group structures (with respect

to composition). A detailed description of these structures and their inter-

connections can be found in [61]. The tangent space T

(M) at the unit

e = id is the space of all vector ﬁelds on M belonging to H

,andT

(M)

is the space of all divergent-free vector ﬁelds on M belonging to H

.The

tangent space T

(M), g ∈D

(M), consists of the compositions of the

ﬁelds from T

(M) with g, i.e., T

(M)={X ◦ g|X ∈ T

(M)}.This

Y.E. Gliklikh, Global and Stochastic Analysis with Applications

to Mathematical Physics, Theoretical and Mathematical Physics,

DOI 10.1007/978-0-85729-163-9

105