Rev.Acad.Can ar .Cienc., III (Num. 1) , 87 - 102 (1991)
ALGEBRAIC DISCRETE SEMI - DYNAMICAL SYSTEMS
F. CASTILLO
University of MALAGA
2907 1 MALAGA. SPAIN
ABSTRACT
In the present paper we study the structure of discrete
semi-dynamical system on a set X without any kind of topological
structure.
In the fii-st section we do not impose any structure on the
set x. Some results then obta ined relative to the
c lassification of solutions and to the invariance ot subsets.
In the second section we assume X is a partially ordered
set. This allows us to introduce the weak notions which are
adequate for systems wit hout uniqueness. Ot particular interest in
the characterization of the weak positive invariance presented
here.
KEY WORDS: Dynamical systems , weak solution, weak invariance.
0. INTRODUCTION
In recent years multiple applications of the theory of
systems have been found, that motivate the wide expansion of the
abstract study of systems.
In 1970 , G. P. Szego and G. Treccani (9] have inj;roduced the
notion of discrete semi-dyna mical system without uniqueness in
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order to per- t arm an axiomatic study ot some algor-ithms o t
optimization. Later on new applications ot such a theory have been
found.
The semi.-dynamical systems used in such applications are
defined on the family ot the non-empty compact subsets of an
Euclidean space l~], or of a Hilbert space [4], where several
structures are then defined. Many ot the results that have been
found tor such semi-systems do not require so much structure. This
is the reason why in this paper we consider a discrete semidynami
ca l system on an arbitrary set X. We introduce the necessary
algebraic notjons and we obtain some results related to them. We
then impose on X a partial or-der and this allows us to introduce
the corresponding weak notions. Ot particular interest is the
study of the weak invariance and the characterization ot the
weakly positively invariant subsets, that play a so important role
in the applications.
1. OI SCRETE SEMI - DYNAMICAL SYSTEMS
1. 1 Definition and proJ?_grti_g2
Notation: X denotes an int inite set; l' denotes the set of
nonnegative integers, denotes the set of nonpositive integers;
n denotes a map tram the product set Xx1 · into X; the image O<.x, t>
of an element (X , t> in X>< l' will be written simply as xt.
The trip 1 et <X I 1 .. Ill ) is called a discrete semi-dynamical
. system on X, if the two following conditions hold:
i) xO=x, xEX.
ii) Cxt)s=x<t+s>, xEX and t,sEI•.
In the last definition, 1> can be replaced by
i') l mage n=X.
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Let be a map from X into X; we define the map ll, trom
X x I ' i n t o >: , w i t h n , (. ::< , t > = x t = i '· uo . Then the t r i p l et ( X , 1 ', l l ~· > i s
a discrete semi-dynami c al systE~ m on X, and we say that it is
induced bv t . Ree i. pr-oc a 11 y: J 1 ( x ' I I ' rl) is a discrete semi -
dynami cal system on X,
fn:X~X, with t 11 (x)=xl.
(X, I', fl ) ls the one induc ed by the map
In the fol lowllJ.g_ discussion we suppose that it is given a
discrete semi-dynamical system on X , which we denote by (X, r·. ll>.
Definition: 1 > ( X , l ·•· , ll ) , or ll , is said to have negative
existence, it for every xEX there is some yEX and tEI + , t >O , such
that yt=x.
2) ll is said to have [.!§_g_q_tive uni c ity_, if for any x -,,x.,,EX,
x, t = x -;,.t it and o nly it x -,=x::,,.
3) An element x t:X is said to be a st~oint, if x~yt tor
any yEX and tE I •· , t >O.
4) An element xC< is said to ~e a s ingular point, it there
exist x -,,x2 EX, x .,~x 2 , and tEI ~ with x 1 t=x2 t=x.
5> An element xEX is said to be a critical point, if xt=x
for every tE I•.
Notice that fl has: negative existence, if and only if for
every xEX there exists yt:X such that yl=x; negative existence, if
and only if it has no start points; negative unicity, if and only
if it has no singular points.
Consider Mc X, and AcI • . As usual, we represent:
ll <M,A >= {xt : xEM, t EA}' ll(x,A>={xt: tEA}.
Furthermore, the following relations hold:
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n(('.~Mi)' (~Aj)] = 1~~l(Mi,~j)
n [ (r,~ Mi ) , t] c 7n <M1 , t >
n [ <M-N) , t] '.) n <M, u -n ~N, t>.
1.2 Solutions and traiectories
Let xEX be arbitrary: The Q_Q_sitive traiecto_r_y___t_brough x_, is
the set xr -·, also denoted C(x> . The r i gh t ma x =i~m=a~l~~s~o~l =u~t~i~o~n~
throught x is the map ,Jl: i ·~x defined by the assignment ... n ( t ) =xt.
It is clear that the set of all right solutions determines I1,
Roughly spealdng, right maximal solutions are obtained by
fixing x in. xt. This also suggest fixing t and allowing x to vary.
This is how the translations, whose definitions follows, are
obtained.
For every ter · .. , we define the map nt. :x~x with nt,<x>=xt, and
call it a translation of n.
It is clear that the set of all translations of n determines ll,
Definition: Given xEX and s,tEI", with s <t,
The set T(x)={yEX: xE(; (y)} is called the !l§'!gatJve
traiectory funnel trough x.
2) The set Tst <x>={.y eX: xey[s,t]} is called a section of the
neg ative trajectory funnel through x.
t
3) The set Tt (x)={yeX: x=yt} is called a cross-sect ion of
the neg ative traiectory funnel through x.
4) The set T~<x)=T<x>vC(x) is called the com~lete trajectory
funn e l through x.
Finally, for any subset McX we define
Crn>=v{C<x>: xEM}, Tc<M)=u{T,~(x): xEM}.
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It1eorE;fil. Let. xC< and yEC(x>, then Ctx)=>i.., <y>, Ttx)cT ( y) and
T.= tx) cT.., <y).
The proof is quite inmediate.
Theorem. I t xEX, there exists an additive subgroup ot the
integers, G, and /...El"', such that xt,=xt<' • t,>t2 , it and only if
t,-t 2 EG, t 2 •A. . Furthermore, G and /... are determined uniquely by x .
Proof : Let us see that tor each t El "', the set A., .. with
At={sEJ ·•· : xt=x<t+s)}
is the trace on J• of a subgroup Gt ot the integers.
We only need to show that if s,,s ~ EAt with s 1 ~s 2 , then
and we have xt=xtt+s 1 >=x<t+s,+s-.?-s2 >=fxtt+s2 >] <s 1 - s ::.>=
<xt)(s 1 -s2 >=x(t+s,-s2 ).
Let us now show that if tEl'·, rEl'· with t ~ r, then A.tcA, .. .
If xt=xtt+s> and hence xr=xtt+r-t>=<xt> <r-t>=
=[x<t+s>] <r-t>=x(r+s>. Then sEA,. and A 1,cA,..
Let us see, finally, that if t ': r, and A,;t;{U ~ . then A 1.,=A, . .
We on 1 y need to prove that A 1.=>A,.. . Let aE A,.. . By assumption,
there is s >O, and hence tor some n with nE I•, we have
r<t+ns. If sEA .•. 1 also nsEAt, and xt=x<t+ns). Furthermore, it
r<t+ns then aEArcAt+ns• and x <t+s)= x<t+ns+cr>. So we have proved
that if At;t;{O} , and rEI+, A.t=Ar.
The result just established shows that there are two cases:
Either for every t is G1.={0 J,
fulfilled with G={O} and /...=O.
and then the theorem is
Or there exists a /...EI ... , and a nontrivial subgroup G of the
integers, such that
{o}, if O~t<A.
Gt {
G I if /...~ t
The above theorem leads to the following classification of
the positive trajectories:
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Defi.nition. Let xEX, and G and >-. be the same as in the last
theorem, if
1 ) >-.=O, and G={O}, C( x ) is called a non-sel f- intersecting
trajectory.
2) >--=O, and G a proper nontrivial subgroup of the integers,
CCx> is called a periodic o r- c i cl~si tive trajectory (with
primitive period the least positive element of G>.
3> >-=O , and G the group of the integers, C<x> is ca lled a
crit i cal positive trajectory, and we have C<x>=x.
4 ) >-.>O , hence G=F{ Ol, Ctx> i s said .!&_Jead to a cicl e , if G
is a p roper subgroup of the integers; and to lead to a critical
!2QJJl:L it G is the group of the integers.
Theorem. Let xEX, if fl has negative uni city
a> Only the cases 1 ) , 2> , 3> may occur.
b> If C<x) is periodi c , with primitive period T u , and
yE C( x ), there is a P v EI'·, P v<. -c ,., , such that the solutions of the
equation y=xt in I ·• are n-c .,,+pv wi th
c) If Ctx> is periodic, then
d ) lf x is a critical point,
The proof is quite simple.
Theorem. Let xEX,
n any element of I .. .
C <x> =T," ( X ).
then {x }=C <x>=T." ( x>.
a ) x is a cr itical poin t , if and only if x=xl, i.e. if x is
invariant under the map n,.
b) C(x) is periodic, if and only if xl#x, and there is tEI+
such that x=xt,
under n, .
i. e . if x is invariant under some nt, but not
Sometimes, it is said that a critical positive trajectory is
periodic with primitive period 1. This would produce a change of
notation.
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We have detined the rignt maximal so lu tion throuRh x as the
map ~ n, we shall give a more gen eral con cept.
An o bj ec t a is a so lJ:Lt..l_QJJ__QJ _ _D .•
conditi ons hold:
if the t wo following
i ) a:l~X, with I an interval 0 1 1 v!' .
ii) a<s+t>=(a<t>]s, if t ~ t+s, t,t+sEI.
Lemma. a) If a is a solution defined on (t 1 ,t :~ ] and k is an
integer, a -"· defined by the assignment a"~t)=a < t+k>, is a solution
defined on (t 1 -k, t ;,-k].
b > If {a1 } is a monotonous sequence of solutions, I ;. being
the domain ot a; ( i . e. for any i > j ,
every t in I .i> , then ua .. is a solution with domain I=vl 1 •
Both assertions are easy to verify.
Definition. Let xEX; a solution a of n is c alled:
1 ) A left-solution, it ldomaina]nr ··={O }. A left-solution
through x, if (domain a]nr ~ =~O} and a<O )=x.
2) A right - solution, if [domain a}nr -- ={O }. A righ t-solution
through x, if (domain a]nr - ={O} and aW>=x.
3) A left maximal solution, if it is a right-solution, and
is maximal with respect to the property ot being a left-solution.
4 ) A right maximal solution, if it is a right-solution, and
is maxima l with respect to the property ot being a right-solution.
5) A maximal solution, if it is a solution, and its
restriction to r · is a left maximal solution; and its restriction
to r· is a right maximal solution.
Remarks: A solution, maximal relative to the property of
being a solution, is called a maximal solution as above defined,
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only if its domain contains r-•. Let x be a start point of X; the
map a: [5,+ro)-+X with cr <.t)=x<.t-5) is a solution, maximal relative
to the property of being a solution, but a i s not a ma x i ma l
solution according to 5> because it s restriction to I' is not a
right maxima l solution.
A right maxima l solution through x is precisely "the" right
maximal'solution through x, xn , defined by vn<t>=xt.
For any x there exists some maximal solution.
A subset of X is called a negative trajectory, if it is the
range o f a left maximal solution; and a negative trajectory
through x if it is the range of a left maximal solution through x.
A left s~lution with domain r- is necessarily a left maximal
solution. Such left solutions, and the corresponding trajectories,
will be called ~rinci~al.
Notice that if N is a negat ive trajectory through x, then
NcT(x), and that if n has negative unicity N=T <x>, but if x is a
singular point, then N;tT(x).
Theorem. Let a be a left maxima l solution, and N=range a the
corresponding negative trajectory, then one and on 1 y one ot the
following alternatives holds:
a) a and N are principal.
b) Domaina = [o:,O], with -ro<o:~O. Then a(cx) is a start
point. We say that a and N lead from the start point a (cx).
The proof is quite simple .
Notice that if n has negative existence, the only
alternative is a).
It should be observed that different left maximal solutions
can define the same negative trajectory.
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If N is a negative trajectory through x, the set NuC ( x) is
called a complete Nr.C<x>~{x} is not
excluded. If N is principal, the complete trajectory NuC<x> is
also called principal.
Let xEX, and a be a let t maximal solution through x, we
write:
O:x - sup { t E I +· : x € n ( x' t)
o:.,,. inf {domain a}
and call o:x the negative escape time of x, and ex<,. the negative
escape time of a. It is clear that -oo ~ a ,,., ~ cx .... ~O. It may well happen
that o:x<o:.,,. for all left maximal solutions a through x.
1.3 Invariance
Let McX: Mis called positively invaria·nt, if C<M)cM; Mis called
negatively invariant, if T<M>cM; M is called invariant if T,=(M)c M,
i. e. if M is both, negatively and positively invariant.
Lemma. The following assertions are equivalent:
a) M is positively invariant.
b) xtEM, for any xEM, tEI'-.
c)D<M,t>cM.
d) M=C <M>.
e) xlEM, for any xEM.
The proof is quite simple.
Theorem. McX is positively invariant, if and only if X-M is
negatively invariant.
Proof: Let M be positively invariant. If xE.X-M, then we must
show that T(x)cX-M. Suppose not. Then there is yET(x), ~ith y~X-M,
but then there is tEI~ with yt=x, and by positive invariance of M,
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xe:M. This contradiction shows that X-M is negatively invariant.
The proof of the converse is entirely similar.
Theorem. A set McX is negatively invariant, if and only if
xlEM implies xe:M.
The proof is quite sJmple.
1~eorem. aJ X and 0 are both invariant.
b) A subset consisting of one element is positively
invariant, it and only if it is a critical pojnt.
c) If n has negative unicity, a subset consisting of one
element is negatively invariant,
critical or a start point.
if and only if it is either
d) If M;c:X are all positively invariant (or all negatively
invariant, invariant), then so are nM;_, and vM,.
e> The complement of a positively invariant
invariant, invariant) is negatively invariant
invariant, invariant>.
<negatively
<positively
f) The least positively invariant subset containing a given
McX is C (M>.
g> The least negatively invariant subset containing a given
McX is T<M>={ye:X: C<.y>nM:;t0"f.
h) A set McX is negatively invariant, if and only if for any
xe:M, each negative trajectory N through x verifies NcM.
The negative trajectories allow to define yet another kind
of invariance, which is called quasi-invariance .
Let McX; M is called: Negatively quasi-invariant, if for
every xe:M, there exists some negative trajectory through x which
is contained in M; and Quasi-invariant, if it is both positively
invariant and negatively quasi-invariant.
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Lemll@.. It M is negatively jnvariant, then M is negatively
quasi -i nvariant . It each M1 is negatively quasi-invariant, then so
is •JM,. If M, is negatively invariant, and M.2 is negatively quasiinvariant,
then M,nM .. is negatively quasi-invariant. A set McX is
negatively quasi-invariant,
there is y£M with yl=x.
it t or every non - start point xeM,
2. DISCRETE SEMI-DYNAMICAL SYSTEMS ON A PARTIALLY ORDERED SET
2. 1 Definition and properties
In this section we suppose that X is an infinite partia~
ordered set, i. e. X is an infinite set, and there is a relation R
defined in X such that:
i) xRx for every xeX.
ii) xRy, yRz implies :<il.'Z, to r every x,y,zeX
iii) xRy and ylt'x implies >:=y.
Let m be an element of X, we say that m is a minimal element
of X if meX and yRm implies m=y.
Let M be a subset of X, we define:
M,,={yeX: there exists xt::M, with · yRx}
M,..={yeX: y is a minimal element of X, yeMF}.
The triplet (X, I',0) is called a discrete semi-dinamical
system on the partially ordered set X, if the following conditions
hold:
i) xO=x, xeX.
ii) (xt>s=x<t+s>, xeX and t,sEI'.
iii) xRy implies xtRyt, te I .. and x, yeX.
Notice that a discrete semi-dynamical system on a partially
ordered set X is a discrete semi-dynamical system on X.
In the rest of this section we suppose that a discrete semi-
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dynamical system on the partially ordered set X,
given.
2.2 Weak concepts
(X,I + ,fl), is
An object 2:: is called a weak solution of n, if the two
following conditions hold:
i) 2::: I-+X, I being an interval of 1·· ur • .
ii) l::<s+t)fil:(t)s, if Ut+s; t, t+sEl.
Notice that a solution is a weak solution.
Lemma. If 2:: is a weak solution of II defined on [t , , t 2 ], k is
an integer and l::"'· (t>=l::<t+k), then 2::-*' is a weak solution of fl,
defined on [t .1-k, t 2 -k].
If {l::j_ } is a monotonous sequence of weak so 1 ut ions ot 11 with
(domain E 1 ]=I-11 then v2:; 1 is a weak solution of n defined in vI ; .
Definition. Let xe:X. A weak solution E of n is said to be:
1) A weak left solution if (domain l::]nI'={O}. A weak left
solution through x, if [domain E]nl'={O} and l::<O>=x.
2) A weak right solution, if [domain E]nl ={O }. A weak__r:_J_gll_i
solutiong through x, if [domain L:]nI ={O} and 2:<0 ) =x.
3) A weak left maximal solution, if it is a weak left
solution and is maximal with respect to the property of being a
weak left solution .
4) A weak right maximal solution, if it is a weak right
solution and is maximal with respect to the property of being a
weak right solution.
5) A weak maximal solution, if it is a weak solution, its
restriction to 1- is a weak left maximal solution, and the one to
r• is a weak right maximal solution .
Notice that not every weak solution, maximal with respect to
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the property ot being a weak solution, is a weak maximal so1uti.on.
Notice that tor every xt:X there is some weak ma,.:imal solution
through x.
Definition. A subset of X is said to be:
1) A weak negative trajectory through x, if it is the range
of a weak left maximal solution through x. A weak left solution,
defined on r - , is necessarily a weak left maximal solution. Such
solutions and the corresponding trajectories are calle principal.
2) A weak positive tra1ectory throu~, if it is the range
of a weak right maximal solution through x.
3) A weak complete trajectory, if it is the range of a weak
maximal solution.
Definition. Let McX; Mis called:
1) weal~ly negatively invariant_, it tor every xEM there is
some weak negative trajectory through x, contained in M.
2) weakly positively invariant_, if for every xeM there is
some weak positive trajectory through x contained in M.
3) weakly invariant, if for every xeM there exists some weak
complete trajectory through x contained in M.
Theorem. Let McX. M is a weakly positively invariant set, if
and only if for every xeM, there is some yeM with yRxl.
Proof: If M is a weakly positively invariant set, then the
condition holds. Since if xeM there is a E, weak right maximal
solution through x, with E<l+)cM, and therefore, if we take
y=E<l), it is clear that yRxl.
Conversely, assume the condition and take · x 0 EM. Now, using
the hypothesis, define x, EM with x 1 Rx0 1; and by recurrence, for
ner•, n>l define x,..eM with x,..Rx,.._ 1 1 . Now, consider the map 1:: r · .. ~x.
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defined by the assignment :i<n>=xn tor every n in l'. Let t,sEl'.
By definition we have xt.,_ .1 R"..<+. l 1 and therefore, I:<t+l> RL:<t>l. Now
we have L:<t+l>l R [I:<t>J 1, and therefore L:<t+l>l R L:<t>2. Now,
since X -t.+ 2 .Rx 1 , .... . ,1, we have that 2:(t+2) Rl:(t+l)l, and by transitive
property of R it holds that I:Ct+2J RI<t>2.
Now we have I:<t+2)1 RI:<t)3, and since L:<t+3) R2..:<t+2Jl, we
have L:<t+3) RL:<t>3.
Proceeding in this way, we have l.:(t+s> R I:<t>s, and
therefore, 2: is a weak right ma~drnal solution through x 0 , with
L:<I'JcM. This shows that M is a weakly positively invariant set.
Theorem. The union of weakly positively invariant sets is
also a weakly positively invariant set.
Theorem. Let McX be a positively invariant set, then MF is
also a positively invariant set.
Proof: Let xEMF, then there is some yEMF with xRy, and so
xlft'yl. Now, since M is a positively invariant set, ylEM, and
therefore xlEMF, and finally MF is a positively invariant set.
Definition. Let xEX; if xf..y holds only it y=x and t=O, x is
called a weak start point.
Notice that if x is a wealt start point, then x is a start
point, but x can be a start point without being a weak start
point.
Theorem" Let McX. If M has no weak start points, then it is
weakly negatively invariant, if and only if for every xEM there
exists yEM, such that xRyl.
Corollary. Let McX. M has no weak start points and is weakly
negatively invariant, if and only if for every xEM, there exists
yE M, such that xRy 1.
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Th e orem . L e t x,y E/, , and tEI' wi t h t .> u . lf yHxt hold s , then
t her e ex i s t s some I , weak maximal s olution thr o u g h x, with 2: Ct J =y.
Proo f : Let E~ be the ma p trom [o , t ] into X detined by
2:*< s >=xs it sE I' , with S \. t, and .l:*· ~s> =y it s = t. lt is c lear that
.l:" i s a weak right solution through x, and the theorem is proved,
if we take a weak maximal solution I with I * c2:.
Let xEX; x is called a weal{ critical point, if xRxt for
every tE J-•·,
Not.ice that if x is a critical point, then x is a weak critical
point, but x can be a weak critical point without being a critical
point.
Theorem. Let x EX. The following assertions are equivalent:
a > x i s a weak criti c al point.
b ) xRxl.
c ) The map 2:: I +--)x with l:<t>=x tor every t in r ~ is a weak
right maximal solution through x.
A weak solution of n is said to be of minimal type, if all
its elements are minimal elements of the partially ordered set X.
Theorem. Let X be a partially ordered set, such that if xEX,
there is some yEX with y a minimal element of X, and yRx. Let M be
a subset of X positively invariant, then MP is a weakly positively
invariant se t , and, evidently, all weak trajectories contained in
MP correspond with weak solutions of minimal type.
Proof: Le t yEMP , then there exists some xEM with yRx, and
ylRxl. Let z be a minimal element of X, with z .Ryl, then zRxl, and
therefore zEMP . Since zRyl, MP is a weak positively invariant set.
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REFERENCES
[1] BHATIA, N. P.; HAJEK, 0.: "local semi-dynamical systems".
Lecture Notes in Mathematics, Vol. 90, Springer, 1969.
[2] BHATIA, N.P.; SZEGO, G.P. "Stabi 1 i ty Theory or Dynamical
systems". Grund. Math. Wiss., Vol. 161, Springer, 1970.
[3] CASTILLO, F.: "Semi-sistemas dinami cos discretos". Tesis
Doc tor a 1 , Un i v er s i dad de Mad r i d ; A b r i 1 , 1 9 7 2 .
[ 4] CASTILLO, F. "Semi - Dynamical systems and optimization".
Towards Global Optimization, North Holland, Amsterdam, 1975.
[5] CASTILLO, F. LLORENS, E. "Semisis temas dinamicos
debiles en espacios de Hilbert". XIl Reuni6n Matematicos
Espaftoles, Malaga, 1976 <Actas 1983).
[6] HAJEK, 0.: "Dynamical Systems in the Plane". Academic
Press, 1968.
[7] ROXIN, E.: "Stability in general control systems". J.
Di ff. Eqs., pp. 115-150, 1965.
[8] SZEGO, G. P.; TRECCANI, G.: "Semigruppi di trasrormazioni
multivoche", Lecture Notes in Mathematics, Vol , 101, Springer 1969,
[9] SZEGO, G. P.; TRECCANI, G · "An abstract formulation or
minimization algorithms". In Differential Games and Related
Topics. Kuhn-Szego eds., North Holland Publishing Comp., 1971.
Recibido: 28 de Abri l de 1991
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