HAL Id: inria-00001089
https://hal.inria.fr/inria-00001089
Submitted on 3 Feb 2006
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Weighted Petri nets and polynomial dynamical systems
Mikhail Foursov, Christiane Hespel
To cite this version:
Mikhail Foursov, Christiane Hespel. Weighted Petri nets and polynomial dynamical systems. [Re- search Report] PI 1781, 2006, pp.17. �inria-00001089�
I R I
S A
IN STITUT D
E R
ECHERCHEE N IN
FORMATIQU
EET SYSTÈ MES
ALÉ
ATOIRES
P U B L I C A T I O N I N T E R N E
N
o1781
WEIGHTED PETRI NETS AND POLYNOMIAL DYNAMICAL SYSTEMS
MIKHAIL V. FOURSOV AND CHRISTIANE HESPEL
INSTITUT DERECHERCHE EN INFORMATIQUE ETSYSTÈMESALÉATOIRES
Campus de Beaulieu – 35042 Rennes Cedex – France Tél. : (33) 02 99 84 71 00 – Fax : (33) 02 99 84 71 71 http://www.irisa.fr
! ! "
# ! " !
!! $ %
&! "" " $
"'& ""'"$ %(
" )
" " !
""" $
* ! " ! "
$
+ , - --" ".
, # # / , - --"0 1
"-#""-- ",,"0-
" " 0 - 2$ 3 -(
-""-""2",$ +"#
", #"'-"2" ","'-$
3 (, " " -#" "
#- " )" "0- " ", "
, " " - " ", "#
"-""$
- ", - -- -#"
"--$
5 ! ! "" 6$$
78 9:; " $ 6
!< #
""#1 $
5! <
" "$
5! ! "" < "
" 8" $ 5!
""6$$ 78 !!"
" 8$ ="
,! 8"!$
> ! "$
6$*9?; """ " "
@$ A!
"" 8"! "($
!"" B" !"1" ".
!""""9B 4;$ 5
""!" "
" ! ""!
$ 6 "1
"C! &" " !&$
9; " ! ""$
!! "! "1 "
"! "1$ 59;"!
""!! "
$
! " " " $
4 !"! "" $ : !
"< !! ""
" $ ? " ! " ! "
" & $ *
! ! !
" "$
⎧⎪
⎪⎨
⎪⎪
⎩
˙
x(t) =v0(x) + n j=1
vj(x)uj(t), s(t) =h(x(t)),
D$E
u(t) = (u1(t), . . . , un(t))∈Rn
x(t)∈ M M Rm
! {v0,v1, . . . ,vn} " M
# h:M →R
$ s(t)∈R
% ! !& ! n
D""Eu(t)" y(t)$ 5.
" u1(t), u2(t), . . . , un(t) ! % Z ={z0, z1, . . . , zn} (n+ 1) . z0 " " D! ! ! ""
u0(t)≡1E$ 51"# I = (i1, i2, . . . , ik)! !"
w=zI =zi1zi2· · ·zik$ 5!"Z∗ "Z$ D5!"
""λ$E
5,""!!
< "F$
5 G=
w∈Z∗G|zIzI 9?;
"""$ GG|zI"
) "! (" vi "
#x0<
G|zI=G|zi1zi2· · ·zik=vik◦ · · · ◦vi2◦viq◦h
x0.
D5 ) " f(x1, . . . , xn) ! (" v = (v1, . . . , vn) "(" v(f) =
ivi∂x∂f
i$E 5 "
$ 6 !!"(
",9 ?;$
5& Cu(t) =
w∈Z∗Cu(t)|zIzI 94 :; "
""$ 5GF "
! !<
• Cu(t)|ε= 1
• Cu(t)|w= t
0 Cu(τ)|vuj(τ)dτ !"w=zjv$
5y(t)""
"F9;<
y(t) =G||Cu(t)=
w∈Z∗
G|wCu(t)|w D$E
5 &! '() ! *
$
+ W P N = (P, T, K, F, W, M0)
• P ={p1, . . . , pm}"
• T ={t1, . . . , tn} "
• K:T →R
• F ⊆(P×T)∪(T ×P)
• W :F → {1,2,3, . . .}
• M0:P → {0,1,2, . . .} )
• P∩T =∅ P∪T =∅
5 ( ! "!& ! H"I 1! "
# ! $ J !
$ + ( t !
" ! " p t Cnk
! n & p" k ! p
("t$
> " " ! "
! $
!! !! #HI "(
! "' $$ !" HI "
H1I "( T $ >! &
, " ( ,$
! ,W P N % - G
" .
• T = {t1, . . . , tn} %
{z1, . . . , zn}/ %tj
zj0
• ) pi /0 xi / %
%0
• " " ti1, . . . , tik ) M(p) = (k1, . . . , km) m
i=1xkii k
j=1K(tij)
zi1· · ·zik
• G
%
#
! G " & $
8 !& "'!
!"$
" W P N S = (P, T, K, F, W)
%+ ) "
! '
#$ ! -
+
- 1
• pi xi
• ti ui(t) /
0
• ) (k1, . . . , km)
xj(t)kj
/ % %
0
• Auk(t)
xj(t)nj ' xl
% tk .
A pl pj j
nj= 0 pj + nj
• ) xi 2
xi(0) =xi
5"$ %HI !
"'$ 5
"" $
5 G !" λ $ 3! & ( , ti1, . . . , tik k " & M = (l1, . . . , lm)$
C m
i=1xlii
zi1· · ·zik C$ J
" " "!"
(vtk◦ · · · ◦vt1◦s(t)
t=0)zi1· · ·zik$
3! ! ! G !" zi1· · ·zikzik+1$ 5 !
$ tik+1 ( G8$ 5) "
w(t) =Cm
i=1xi(t)li !vik+1 $ tik+1 ("
& M = (p1, . . . , pm) GC·K(tii+1)m
i=1xpii$ > vik+1
w(t) !C·K(tii+1)m
i=1xi(t)pi " $
!
•
K
L
•
M
t1 t2
t3 t4
⇐⇒
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
x =u1xy+u2z2 y =u3x2 z =u4yz
s=xz
!! N & "
" " ! !
""! ("$
!
"' ! " ! "
! " # $
" " " & $ 5 !
" & $
%$"
$ "& &
$ !kziD!k! ("
E ""&
4$ " "D!(
""E !sn "Sn
:$ " Sn=
Xjnj "sn(t) = xj(t)nj
?$ sn(t) "#
$ !
! """',"
$ "' "D xj(t)≡0E
$ "" "' , !
" " !
"#$
" 5" "
(" & " !
!& $ "
$
" ! ? # ! .
! " " $ 5
! ,$
" F"! <
•
L t1
L
L
2
L
3
L
z1 2z1 3z1 4 4z1
5" ("
(y(t)n) =nu1(t)y(t)n+1, n≥1
! " ,y(t) =u1(t)y(t)2! "
s=y(t)$ & & "',
"" $
"! F"! <
••
L t1
4
L
2
L
3
L
4
L
z1 3z1 6z1 5 10z1
5" ("
(y(t)n)=n(n−1)u1(t)y(t)n+1/2, n≥2
! " {y(t) = 0, u1(t)y(t) = 0}$ 5 ".
" $ 5 & $
"" D ) %
) E3! "! <
K L
M
%
t1 t2
t3 t4
5 & Y Zn
LM
n
KM
n
%M
n−1 M
n
u1 u2
nu3 u4
"
⎧⎪
⎪⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎪⎪
⎩
x(t) =u2(t)y(t)
z(t)y(t) =u1(t)x(t)z(t) +nu3(t)w(t) z(t) = 0
w(t) =u4(t)z(t)
! $ =! z(t)
& , "$ =!
! & "
", !
$ "" & Y2Zn !<
L
2
M
n
KLM
n
K
2
M
n
L%M
n−1
K%M
n−1
%
2
M
n−2
LM
n
KM
n
%M
n−1 M
n
u1 u2 2u1
u2
u1 2u2
u1 u2
nu3 u4
2nu3
nu3
u4
u4 (n−1)u3
2u4
5 ,u3(t) = 0$ 5 ,
& !|X|+|Y|>1$ 5 ",
"$
"& D E 3! " !
<
K
L M
t1
t2 t3
t4 t5
*& X
K
L M
u2
u4 u3
u5
"
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
x(t) =u2(t)y(t) +u3(t)z(t) y(t) =u4(t)x(t)
z(t) =u5(t)z(t) s =x(t),
! " ! t1$ 5
""& "$ =!
& $ *& XX
K
2
KL
L
2 KM
M
2
LM
K
L M
2u2 u4 u2
2u4 2u3
u5
u3 2u5 u3
u5
u2 u4
u2
u4 u3
u5 u1
5" 1"',
u1(t) = 0$
"' D% ) E* "
! !& XZ "Y ZV$
K L
M
%
O
t1
t2
t3
t4
5
KM LM
KO
LO
M%
u1
u2 u1
u4 u3
LMO KO
2
LO
2
M%O
%M
2
LM
2
u2 KMO
u1
2u4 u3 u4 u3
2u2 u1
5 " , ""
1"'
u4(t)z(t)2w(t) =u2(t)v(t)2x(t).
$ 5""DE&.
$
"( =! " " !
"!" "
"$ *#
""$
( ! "" !
8"$
!
& % 34
& 5 + $
& 5&" + $ 5!
#$
K L M %
t1
t2 t3
" " " "
D"'E " "
& $
5 # ! " "
K L M
"
t1 t2 t3
! t1 " t2 !Y " Z .
$ 5 1"' , #
!"$ *# #?$
! C<
• • •
t2 t4
C" & XZ$
&! 5#
! $ ) #" !
D&" E<
K
L
M
t1
t2
> & X t1 !($ !
! ,<