Internat. J. Math. & Math. Scl.
Vol. 9 No. 3 (1986) 459-469
459
EXISTENCE THEOREMS FOR DIFFERENTIAL INCLUSIONS
WITH NONCONVEX RIGHT HAND SIDE
NIKOLAOS S. PAPAGEORGIOU
University of Illinoi:
Department of Mathematics
1409 West Green Street
Urbana, Illinois 61801
(Received November 21, 1985 and in revised form February 26, 1986)
proe
ABSTRACT.
In this paper we
some new existence theorems for differential
inclusions with a nonconvex right -hand side, which is lower semicontinuous or
continuous in the state variable, measurable in the time variable and takes volucs in
a finite or infinite dimensional separable Pmch space.
1990 Mathemntics Subject Classifcation:
KEY WORDS AND PI{RASFS:
34GOZ
Orientor field, multfunct.on, measure of noncompactncss,
measurable n,ultifunction, lower scmicontnuous multiunction, Hausdorff metric, Kamke
funct ion.
I.
INTRODUCTION.
In the recent years there }as been an increase in interest in the investigation
of systems described by differential inclusions. In ay ordinary differential
equation the tangent at each point is prescribed by
single valued function. In a
differential inclusion the tangent is prescribed by a mu]tfunction
function)
which is usually called a: orientor field.
(set valued
.’Lany problems of appl
mathentics lead us to the study of d3q_nmical systems having velocities not miquc]y
determined by the state of the system, but depending only loosely upon it. In these
cases the classical equation
f(t.x(t)) describing the dy,xmics of the system
’(t)
is replaced by a relation of the form
multifunction
(the
orientor
field).
called "differential inclusion".
(t)
F(t,x(t)) where F(.,.)
is a
Such a "set valued differential equation" is
The initial impetus to study diffe.rential inclusions
came from control theory. Then the subject found additional impo,’tmt applications in
ma{hc,natical economics [I], nonsmooth dynamics [2], optimization [3], differential
equations with a discontinuous forcing term
[4]
etc.
N. S. PAPAGEORGIOU
460
The purpose of this paper is to prove existence theorems for differential
inclusions governed by nonconvex valued, lower semicontinuous orientor fields which
Until now, most of the existence theory for
take values in a separable Banach space.
differential inclusions was developed for upper semicontinuous, convex valued orientor
n.
fields with values in
However lower
semicontJnuous, nonconvex valued orienror
So
fields appear often in control theory in connection with the bang-bang principle.
it is important to have existence theorems for differential inclusions governed by
such orientor fields.
2.PRELIMINARIF.
Let
X
(,2) be
X be a separable Banach space, with
We will use the following notation.
a measurable space and let
being its topological dual.
{A _C X"
Pf(X)
A E
For
2X\c).
[A[
we set
closed}.
nonempty,
dA(.
and by
sup [[x[[
xEA
function from
A
i.e. for all
A multifunction F
x
e
X.dA(X
inf
aA
is said to be measurable if it satisfies any of
Pf(X)
d
we denote the distance
the following equivalent conditions.
is measurable for all
d (x)
(i)
X
x
FC)
(ii)
cl{fn()}n
F()
(iii)
(Castaing’s representation)
for all
X open
U
for all
of measurable functions s.t.
{fn(.)}n
there exists a sequence
F-(U)
inverse image of
U under
0 U
F()
E O
F (U)
language of measurable multifunction
} e 2 (in the
is called the
F(’)).
treatment of measurable multifuntions can be found
A detailed
[5]
Castaing-Valadier
We denote by
LebesNe-Bochner
S
n
ad Himmelberg
F(’)
the set of all selectors of
F
(n)
sNce
i.e. S
{f(.) e
F
that belong to the
f() e F()-a.e.}.
(n)
It
is
easy to see that this set is closed and it is nonempty if and oniy if
inf
Iixll
xE()
L+().
Assume that
Y,Z
are topological spaces and
is lower semicontinuous
V g }
(1.s.c.) if and only
F
Y
d
V
if for all
21a,{}.
We say that
Z open, {y E Y F(y)
is open too.
-{A-}n
Finally if
s-l__im
n-)co
By W(’)
are nonempty subsets of
An
X
(x
x
s-lira
X,
Xn,Xn
we define
6
A
n.n _>
I}.
B _C X
B can be covered by finitc.ly m.nny balls of
we will denote the Hausdorff measure of noncompactness i.e. if
is botmded, then
(B)
inf{r
>
0
EXISTENCE THEOREMS FOR DIFFERENTIAL INCLUSIONS
r}.
radius
+
Recall that by a Farc function we mean a function
[8]).
w
satisfying the Caratheodory conditions (i.e. it is measurable in
x),w(t,O)
and continuous in
u(t)
of the problem
3.
[7] (see
This is equivalent to the Kuratowski measure of nonconq>actness
also Banas-(;oebel
[O,TJ’xtR+
461
0
u(t)
a.e. and such that
z 0
t
is the only solution
O.
(s,u(s))ds,u(O)
F_XISTENCETHEOREIS.
e
The setting is the following.
we consider the Lebesgue measure
By X
space.
we ill denote
T
are given a fnite intervaI
dr.
X be
lso let
[O,b].
T
On
a separable reflexive Banach
e
X ith the weak topology.
Cauchy problem under
consideration is the following:
FCt,Ct))}
e
Co)
x
xCt)
By
o
) we understand
() for almost all t T.
a solution of
satisfying
an absolutely continuous function
X
T
x
Our first existence result is the following"
THEOII 3.1. I_f_f F
TxX Pf(X) is a multifunction .t.
1)
2)
for all
x
X, F(’,x)
is measurable
for all
t
T,F(t,-)
is 1.s.c. from
3)
for all
x
X,
4)
for all
B
X nonempty and bounded
[F(t,x)!
g,(t)
X
into
we have
CFCt,B)) _(wCt,c(B))
(.)
where
o
is the Hausdorff measure
El(T)
C’}
a.e. with
X
a.e.
nonco,npactness and
w(’,.)
is a
Kamke function.
()
then
Let
PROOF:
admits a solution.
1111.
r
mnd consider
X,Br(XO)
reflexivity of
Br(XO)
[10]
x)
Br..iXo
Our claim is tha
n-
SF(
x )"
’n
x
s t
n
d (fCt))
F(t,Xn)
lit(t)
fn(t)ll
Pf(I2.(T))_X
L(.)
o
the
weM: [opology (see
be the mtl] ti function
is 1.s.c.
tot any x
n
For that purpose let
A straightforward application of
a.e.
S(
C s-lm
Because
r}.
IlX-Xoll
In the sequel we will al,:ays consider
we know that it suffices to sl,ow t.hat
1
Sc
S(.,x).
L(x)
X
is w-compact md metrizable for the
anford-Schwartz [9], theorem 3, p. 3).
Br__fXo) with the were topology’. Let L
defined by
{x
t’(’)
x
SF(
in
x)"
From Delahaye-Deel
B
r
(Xo)
Then
we have
fCt)
Au,nknann’s selection theorem can give us
for all
t C
T.
Since
F(t .)
is
F(t,x)
f
s c
.)
E
462
N. S. PAPAGEORGIOU
Xw
from
O,
X,
into
-
F(t.x) C_ s-lim___
n-
F(t,Xn)
and so
lim d
n-
F(t,Xn)
which by the dominated convergence theorem implies that
f(.) E
s-lim
SF(" Xn ).
So we lmve. shown that
SF(,
f
C s-lim
x)
n-o
f(.) =>
n (.)
1
SF( ,Xn)
which as we
already said, implies the lower semicontinuity of L(-). Hence we can now apply
theorem 3.1 of Fryszkowski [11] and deduce that there exists $
Br(XO)
8C x)
continuous s.t.
6
LCx
x e
for
BrCXo).
set
f(t.x)
$Cx)C t)
[,(T)
and consider
the following single valued Cauchy problem
f(t,x(t) }
x(t)
(o)
Let
W
{xC- e
W
x
o
xCt e BrCxo)
CxCT
for al
T} and
t
consider the nap
W defined by
(x)Ct)
For
t,t’ E T,t _< t’
x +
0
fCs,xCs))ds.
we have tbt
t’
,,x +
0
IIO(x)(t’) -b(x)(t),,
0
t
f(s.x(s))ds
x
0
of(S,X(s))dsII
IItf(s,x(s))dsll <. ftll(s,x(s))llds < tq(s)ds
:> [[Cx)(t’) -Cx)Ct)l[ <
when
t’
t[
( 5,
x(’)
for a11
equicontinuous subset of
Cx(T
W.
Thus
v,e
a
deduce that
(W)
is an
and in fact it is uniformly equicontinuous since
T
is a compact interval.
Also we claim that
W.
(-)
is continuous.
For tha.t purpose let
n (’)
x
Then we have"
II(Xn.)(t
(x)(t)ll =lix0 +
_<
f(S,Xn(S))ds
O][fCS,XnCS))
x
0
fCs,xCs))[]ds.
Applying the dominated convergence theorem we get th-t
f(s.x(s))dsll
x(’)
in
EXISTENCE THEOREMS FOR DIFFERENTIAL INCLUSIONS
as
.
n
,Cxn)
o
Now consider the classical Caratheodory approximations
x (t)
n
+
0
Note that for all
n
0
>
llxnCt -OCx)Ct)ll
nf(s.Xn s))ds
x (-) E W
n
iiXn Ct) CXn)Ct)’l
while
+C)I
463
b
for
and
llCXn)Ct4)
(Xn)Ct)l
for
/,1/n
/,1/n
Jot)fCS.XnCS}},ds Jo, CS)ds
_<
for
l_n <0
<
_<
t
_<
t
b
1/n.
Thus we have that
ilx
as
R
n
.
Let
R
(XnC’)}n>_l.
is uniformly equicontinuous.
CRCt))
Note that given
a
0
R _c (I
Then since
Set
-* 0
0) (R) + (R)
{Xn(t)}n_>1
R(t)
for
t E
T.
we deduce that
Then we have
__1fcs’Rcs))ds
fCs,RCs))ds +
^
>
CXn) IIo
n
n
n()
we can find
(s)ds <
s.t.
a/2
for
t E
T,n _>
n
1.
Hence we have that
y
t_l.n CS,XnCS))ds
n )_
nCa)
(_ 2 sup
n>n()
Using this estinmte and the propoerties of
[R(t)] _<
Since for all
Since
R(O)
s
T,R(s)
(’)
t
l&CS)ds
y[f(s,R(s))]ds.
is bounded, using hypothesis
wCs.CRCs )))a.e.
=> ,[R(t)]
w(s,(R(s)))ds.
0
and
.
we get that
CfCs,RCs))) _<
Xo,(R(O))
(
_(
w(-,’)
4}
we have that"
is a Knmke function we must have that
N.S. PAPAGEORGIOU
464
for all
But recall (see [8]) that
T.
t
TC R)
So
O.
(R)
R
which means that
x(’)
IIxk
(Xk)ll0o
W.
6
IIx
IIxk
So
is a relatively compact subset of
IIx
b(Xk)llo
Xk(.)}k_l
Cx(T ).
{Xn(.)}n>_lS.t.
of
But we have already seen that
b(x)llm.
Thus finally we have that
-* O.
(x)ll
0
=> x(t}
Since the vector field of
solves
TCRCt)).
{%(-)
Therefore we can find a subsequence
Xk("
sup
{}
x(*)
f(s.x(s))ds
x +
0
F{’,’},
selector of
is a
solves
we conclude that
x(’)
{).
Q.E.D.
RENARK.
When
X
1)
I__f F
a(-),b(,)
for all
then
()
TxX
Pf(X}
x 6 X, F(.,x)
for all
a.e. with
2)
F(t..}
Xw
is 1.s.c. from
is a separable dual space ’ith
into
X.
is finite dimensional we can have a more general boundedncss hypothesis.
THEORF 3.2.
PROOF"
X
The theorem remains true if we essume that
a separable predual and
6
IF(t,:}]
is measurable and
a(t)llxll + b(t)
EI(T
T,F(t,.)
t 6
is a multifunction s.t.
is 1.s.c.
admits a solution.
[(llxoI12 + 1)e2[ilalll" + Ilblll)
Let N
1]
1/2
and define the following new
orientor field.
[F(t,x)
Mx
(t II-D
IF
F(t.x)
Then for every
t
T,F(t,’)
and of the M-radial retraction map
r(x)
It
is well known that
measurable in
t
r(,)
and for all
-
for
Ilxll
for
llxll
N
>
N
is the composition of the multiftmction
r
X
j
[]ll-Mx
if
f
BN{O}
Ilzll (_ N}
F(t.’)
defined by
Ilxll _< 14
Ilxll
is Lipschitz.
x 6 X,
{z
X
[F(t,x)[
>
M
So
F(t,.) is 1.s.c. Clearly
a(t)M + b(t)a.e. So F(.,-)
it is
theorem 3.1 (recall that X is finite dimensional) and so
(,x(t))
T X absolutely continuous s.t. (t)
by that theorem there exists x
Our goal is to show that for all t e T, IIx(t)ll _< M. We proceed by
a.e., x(O)
x
O.
satisfies the hypotheses of
contradiction.
Suppose that there
exist
tl,t 2
T
s.t. for
t 6
(tl,t2)llx(t)ll >
hi.
EXISTENCE THEOREMS FOR DIFFERENTIAL INCLUSIONS
’(t)
Then we bve thaz
Set
d--qtlx
E
2
z(t)
Ilx(t)ll + 1.
(t)ll _<
l(t)ll
-
FCt.[[xCt)i
Then
we get that
d
a.e. on
z(t)
(tl.t2) ----> [[(t)]
t IIx(t)ll
_< 211xCt)ll[aCt)ll
-zCt)
2
+
llx(t)ll
>
I
1
d
{tl,t2)
on
t
t
(tl,t2)
and get that for all
_
t E
on
So for all
we already have that
x(.)
{tl,t2)-
IIx(t)ll (_ I,
we can now integrate
T
z(t)
z(O)
/
2
[(llxol12 + l)ex’p
T, llx(t)[[ _( 1.
t
Since
(tl,t2).
on
(a(s) + b(s))z(s)ds.
Using Gronwall’s inequality we get that for all
llxCt)ll
Ix(t)la.e.
2
1) + bCt)CllxCt)ll + 1)]
2[a{t) + b(t)]z(t)a.e,
Since for
211x(t)ll
a.e.
we have that
2[aCt)Cllx(t)ll2 +
"zCt
+ 1]
aCt)hi + bCt
bCt)]
211xCt)llEaCt)llx(t)ll + bCt)]a.e,
Because
465
Then
C2JoCaCs
F(t,x{t))
t E
T
1]
+ bCs))ds)
F(t,x(t))
I/2
which implies that
solves the original Cauchy problem
Q.E.D.
Another existence result in this direction is the following.
X is finite dimensional.
THEOREN 3.3. ]__f F
TxX
1)
for all
with
x 6
d
Pf(X)
X,F(’,x)
L/ and (,)
a(’)
Again assume that
is a multifunction s.t.
is measurable and
IF(t,x)
a(t) @([Ixll)a.e.
a positive continuous function s t
+
ds
t
2)
for all
then
PROOF"
can find
()
Because
bI
t
T,F(t,,)
is l.s.c.
admits a solutiou.
a(,) E L+ and because of our integrability hypothesis on &(’)
> ]lXol,
s.t.
Oa(S)ds <
’
llx
011
(s)
ds.
we
N. S. PAPAGEORGIOU
466
Aain
introduce a new orientor field
F{t,x)
F(.,.) defined
IF(t.)
[(t
if
Ilxll
i
Ilxll
as before by
M
>
M
We have already seen in the proof o theorem 3.2 that F(.,-) satisfies all
hypotheses of theorem 3.1. So we can find x
T X absolutely continuous s.t.
e F(t,x(t)) a.e. x(O) x
Oar claim is that llx(t)ll
M for all t e T. Suppose
O.
t
Then we can find
not.
llxoll <
M.
[lX(to)ll >
T s.t.
o
t* e T
Hence we can find
s.t.
M.
On the other txnd we have
M for all
llx(t)ll
llxoll
llx(O)ll
[O,t’]. So
t
we can write
(t}
F(t,x(t)}a.e.
on
=> [[(t}l[ < a(t}#([[x(t}[I}a.e,
=>
-Itlx(t)ll
on
_< a(t)#(llx(t)ll)a.e,
=>
dlx(t )[ _(
2
[O,t’]
E0. t’]
on
(s)ds
(s)ds
J,x0,(s
IIx(t)ll g M for all
which implies tlt x(’) solves ().
a contraciction to our choice of
F(t,x(t))
F(t,x(t)),
M.
Thus
t 6
T and so
Q.E.D.
REMARKS"
T
1)
[0,1]
[1,2]
If
R+
T
then we divide it into subintervuls
we consider the Cauchy problem
we consider gain
() but
() and
find a solution
with initial condition
this way we obtain a sequence of partial solutions
x(1)
{Xn(’)}n
which vhen pieced together give us the global solution on
2)
I the domain
o F(’,’)
is
TXBr(XO),
T
[n-l,n].
Xl(- ).
Xl(1 ).
(h,
Then on
T2
Continuing
defined on
Tn n )_
+.
then local versions
o
those results
are valid.
3) The above theorems
as well as the one tl’t follows extend significantly
earlier results obtained by Bressan
e
[12], Kacz3mski-Olech [13] and Lojascwicz [14].
will conclude this work with an existence result concerning continuous
orientor-fields on
separable Ba_ch space
concept of a semi-inner product.
X.
1,
But first we need to introduce the
EXISTENCE THEOREMS FOR DIFFERENTIAL INCLUSIONS
467
x
X be a Banch spce and X its dual. Consider the map J X 2
defincd by J(x)
Thanks to the Hahn-ch
{ X (x.x) ,x, 2
X. Using J(-} we can define the semi-lnner product
theorem J(x)
for all x
by
(...)_ XxXLet
,xa,2}.
Inf{CY.X} y
Cx.Y}_
J(y)}.
For more details about semi-inner products the reader can consult Dimling [15]
(p. 33).
Now let T
[O.b] and let X be any separable Banach space. By h(’..) we
Pf(X).
will denote the Hausdorff metric on
I_[ F
THEOREM 3.4.
TxX
Pf(X)
is a multifunction s.t.
1) for all x X, F(..x) is measurable
2) for all x,y E X, h(F(t,x),F(t,y}) < k(t)llx- yl[
3)
SF(..Xo
4)
for all
then
()
y e F(t,x)
(y,x)_
<
cCt)EIIxll2+l]
M2
Let
F(’,’)
defined by
[.[]Xo]12+l]e
1
e
LI+cT)-
Again we introduce the new orientor field
1.
f
J](t’x)
F(t.x)
Ct.,-)
Clearly
FCt.x
FC’,x) is measurable. Also
FCt.r(x)). So for any x,y
_
h((t,x).(t,y))
Furthermore note that
S
g
llXol
if
Ilxll
if
Ilxll
r(’)
if
>
M
is the M-radial retraction map,
X we have"
6
r(y)ll
<
M and so
2k(t)lix-y[[.
F(t,Xo)
which is equivalent to saying that
F(-.Xo)
F(t,Xo}.
tnfllyll
ze(t,Xo
So we can apply theorem
LI(T)
"t"
(t.x(t))
x(O}
Let
ltence by hypothesis
of Muhsinov [16] and get that the Cauchy problem
(t)
admits a solution
M
h(F(t.r(x}),F(t,r(y)))
k(t) lir(x)
3)
coo
L+*(T)
admits a solution.
PROOF"
then
with
k(-) e
with
x(*) be such
the original Cauchy problem.
x
o
a solution.
I"o show that let
u(t}
We will show that
2
[Ix(t)ll + 1].
x(-)
solves
Then we have"
N. S. PAPAGEORGIOU
468
(().x())_
<_ c()[x()
=> u(t) _< u(O)
c()u()
z]
+
c(s}u(s)ds.
+
Applying Gronwa11’s inequality we get that
Ilcll
u(t) <_
e
0112
[llx
-->
lu(o
Ilcll
+
1]e
x()2 < 2
=> x(t) _<
But then from the definition of
FCt,x(t))
F (t0xCt))
=> x(’)
tET
we bve that for all
F(.,’)
solves
().
Q.E.D.
In this paper
we extended the works
o
Kaczyski-Olech
In particular theorems 3.1
Lojasicwicz [14].
[13], Brcssan [12]
and
and 3.4 provided infinite dimensional
versions of those results,which are important in studying distributed parameter
On the other
control systems, characteristic of mechanics and mathematical physics.
hand in theorems 3.2 -und 3.3 which are finite dimensional, we have less restrictive
hypotheses than [12], [13] and [14]. Specifically our orientor field
satisfies Caratheodory type conditions while in [12] and [14] F(*,*)
lower-semicontinuous and in
[13]
Furthermore our boundedness hypothesis is more general than thie of
o
radius
H
>
O,
[12]
mid
x.
[14]
[13]
while in
is integrably bounded.
ACKNOWLFADCHENT:
suggestions.
I would like to thank the referee fcr his constructive
o the University of"Pavia-Italy.
and N.S.F. Grnt D.H.S. 8q03135.
J.P.
criticism and
This work was done while the author was visiting the Hathematics
Department
1.
2.
is jointly
iz is Hausdorff continuous in zhe state variable
where the orientor field stays within a fixed ball
F(*,*)
F(’,,)
Financial support was provided by C.N.R.
Aubin-A. Cellina: "Differential Inclusions" Springer, Berlin (19S.i).
"Optimization and Hoasmoorh Anal;dts" Wiley, New York (19S3).
F.H. Clarke:
EXISTENCE THEOREMS FOR DIFFERENTIAL INCLUSIONS
3.
L. Cesari"
"Optimization-Theory and Applications"
469
Spriner, New York
(19S3).
4.
A. Filippov" "Differential equations with discontinuous right hand side"
Translations of the A.M.S. 42(1964) pp. 41-46.
5.
C. Castaing-M. Valadier" "Convex Analysis and Measurable Itifunctions"
Lecture Notes in Math, Vol. BSO, Springer, Berlin (1977).
Fund. Math $7(1975) pp. 52-71.
6.
C. Himmelberg"
"Measurable relations"
7.
K. Kuratowski"
Sur les espaces comp]etes"
8.
9.
Fund. lath
15(1930) pp. 301-309.
J. Banas-K. Goebe" "Measures of noncompactness in Banach spaces"
Dekker Inc., New York (1980).
N. Dunford-J. Schwartz"
"Linear Operators I"
Wiley, New York
Marcel
(1958).
10.
J. P. Delabaye-J. Denel" "The continuities of the point to set maps,
definitions and equivalences" th. Progr. Study 10(1979) pp. 8-12.
11.
A. Fryszkowski" "Continuous selections for a class of nonconvex multivalued
maps" Studia Fth IO(XVI(19B3) pp. 163-74.
A. Bressan: "On differential relations with lower continuous right hand
An existence theorem" J. Diff. Eq. _7(1980) pp. 89-97.
13. H. tczynski-C. Olech" "Existence of solutions of orientor fields with
12.
nonconvex right hand side"
14.
Annales Pol. th.
29__(1974)
side.
pp. 61-66.
S. Loasiewicz" "The existence of solutions for lower semicontinuous orientor
Bul. Acad. Pol. Sc. 2__.8(1980) pp. B3-87.
ields"
I5.
K. Deimling"
"rdinary Differential Equations in tkmach Spaces"
Springer Berlin’(19?).’
Lecture Notes
in Math, Vol. 596,
16.
A. Muhsinov"
15,_(1974)
"On differential inclusions in Banach spaces"
pp. 1122-25.
Soy. Math. Dokl.