Stochastic Systems and Optimization: Proceedings of the 6th IFIP WG 7.1 Working Conference Warsaw, Poland, September 12–16, 1988

376 Pages·1989·5.075 MB·English

by  C. Bruni, G. Koch (auth.), Jerzy Zabczyk (eds.)

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Stochastic Systems
and Optimization
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gnikroW Conference
,wasraW ,dnaloP September 12-16, 1988
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Series Editors
M. Thoma • A. Wyner
Advisory Board
L. D. Davisson • A. G. .J MacFarlane • H. Kwakernaak
.J L. Massey • Ya Z. Tsypkin A • .J Viterbi
Editor
Jerzy Zabczyk
Institute of Mathematics
Polish Academy of Sciences
Sniadeckich 8
00- 950 Warsaw
Poland
ISBN 3-540-51619-0 Spdnger-Vedag Berlin Heidelberg New York
ISBN 0-387-51619-0 Springer-Verlag New York Berlin Heidelberg
Library of Congress Cataloging-in-Publication Data
IFIP GW 1.7 Working Conference (6th 1988 : : Warsaw, Poland)
Stochastic systems and optimization :
proceedings of the 6th WG IFIP 1.7 Working Conference, Warsaw, Poland, September 12-16,
1988 / J. .W Zabczyk, editor.
(Lecture notes in control and information sciences 136) ;
ISBN 0-387-51619-0 (U.S.)
.1 Control theory--Congresses. 2. Stochastic processes--Congresses.
3. Mathematical optimization--Congresses.
.I Zabczyk, Jerzy. .1I Title. .1II Series.
QA402.3.~4539 1988
629.8'312--dc20 89-21791
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© Spdnger-Verlag Berlin, Heidelberg 1989
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PREFACE
This volume presents the scripts of most of the lectures given at the Sixth
IFIP Working Conference on Stochastic Systems and Optimization which took place in
Warsaw, Poland, September 12-16, 1988. The conference was held under the auspices
of IFIP WG 7.1 and was organized by the Institute of Mathematics of the Polish
Academy of Sciences in cooperation with the Institute of Mathematics of Warsaw
University.
eliT programme of the conference was prepared with the help of the International
Programme Comm~ittee: A.Bensoussan, A.V.Balakrishnan (Committee Chairman), M.H.A.
Davis~ }[.Engelbert, W.Fleming, B.Grigelionis, A.Halanay, K.Helmes, L.le±Ksnys and
J.Zabczyk. The Organizing Committee consisted of T.Bielecki, T.Bojdeckl, ~.Goldys,
K.Malanowski, W.Smoledskl, L.Stettner (Secretary), J.Zabczyk (Conference Chairman)
and P.Zaremba.
The meeting was a continuation of the foregoing conferences in Kyoto (1976),
VilnJus (197g), Visegrad (1980), Marseille-L~nlny ~1984) and Eisenach (1986) and
focused on topics of current interest in the field of stochastic systems and opti-
mization. Particular emphasis was placed on stochastic differential equations both
in finite and infinite dimensional spaces, stochastic control and estimation, asym-
ptotic methods and periodic systems. During a special evening session Professor
T.Hida and Professor G.Kallianpur shared with the participants their views on the
white noise theory.
On behalf of the organizers I want to thank all the participants for making
the conference an interesting and a memorable event.
J. Zabczyk
CONTENTS
1. STOCHASTIC FINITE-DIMENSIONAL SYSTEMS
C. BltUNI m~d G. KOCII:
Some Results about Two-Mode Stoclnmtic Compartmcutal Models ....................... 3
It. BUCKDAIIN:
Anticipating Lincar Stochastic Diffcrcntinl Equations .................................... 18
O. ENCIIEV:
Nonlinc~ Filtcring for Signal Correlated with the Noise ................................. 24
II. J. ENGELBEILT and W. SCIIMIDT:
Continuous Local Martingales: Strong Markov Property, Solutions
of Stocha.~tic Equatious, mid the Interplay between Them ................................ 48
A. IIALANAY and T. MOILOZAN:
Tracking Almost Periodic Signals under Whitc Noisc Perturbations ...................... 75
T. IIIDA aald SI SI:
Variational Calculus for Gaussian Rmldom Ficlds ........................................ 86
J. KISYI(ISKI:
Local Uniqueness of Fcllcr Processes with Intcgrodiffcrcutlal Gem-tarots .................. 98
A. KOWALSKI and D. SZYNAL:
On GcncrM ARMA Models and Rcgul~ity Conditions ................................... 112
A. ILOZKOSZ aud L. SLOMINSKI:
On Limit Points of a Sequence of Weak Solutions of 0nc-DimcnsionM
Stochastic Diffcrcntial Equations ........................................................ 125
K. SOBCZYK:
Modelling of Rml(lom Fatigue Accumulation ............................................. 134
V
K. UItBANIK:
Functionals on Stoch~tic Processes ..................................................... 142
C. VARSAN:
Asymptotic Ahnost Periodic Solutions for S~ochastic Differential Equations .............. 152
2. STOCHASTIC INFINITE-DIMENSIONAL SYSTEMS
T. BOJDECKI and J. JAKUBOWSKI:
Stochastic Integral with Respect to a Generalized Wiener Process
ill a Coauelcar St)ace .................................................................... 161
A. CHOJN OWSKA-MICtIALIK:
Periodic Linear Equations with General Additive Noise in Hilbcrt Spaces ................. 169
P. KOTELENEZ:
Low and High Density Reaction-Diffusion Models ........................................ 185
Z. KOTULSKI:
EquaLions for the Characteristic Fullctiolml aald Moments of the Complex
Stochastic Evolutiolls--Motiw~tion and Results .......................................... 191
It. MANTIlEY:
On a Class of Semiliurar Stochast, ic Partial Diffcrcnti~d Equations ........................ 201
B, MASLOWSKI:
Strong Feller Property for Scmilinc~u" Stochaztic Evolution Equations
and Applications ........................................................................ 210
E. PLATEN:
On the Macroscopic Noncquilibl5um Dynmnics of ml Exclusion Process .................. 225
J. ZABCZYK:
On L,'u'gc Deviations for Stochastic Evolution Equations ................................. 240
3. STOCHASTIC CONTltOL AND ESTIMATION
A. BENSOUSSAN and K. GLOWII~SKI:
Al)l)roximation of Zakai Equation by the Splitting up Method ............................ 257
VI
A. IIENSOUSSAN and NAGAI:
An Ergodic Cont1"ol Problem on Whole Euclidcaal Space ................................. 266
T. BIELECKI and L. STETTNEIt:
On Limit Control Principle for Singulm'ly Perturbed Markov Processes ................... 274
T. DUNCAN:
Some Solvable Stochastic Control Problems in Symmctrlc Spaces of Type IV ............. 284
D. GATAItEK:
Impulsive Control of Picccwisc-Dctcrministic Processes .................................. 298
U. IIAUSSMAN:
Synthcsis of Optimal Controls ........................................................... 309
P, KAZIMIEItCZYK:
Consistent ML Estimator for Drift Pm'amctcrs of both
Ergodic mid Noncrgodic Diffusions ...................................................... 318
B. PASIK-DUNCAN:
On Adaptive Control of Contimlous Time Linear Stochastic Systcms ..................... 328
Z, POILOSIi</SKI ~uld K. SZAJOWSKI:
A Minim;~x Control of Lineal" Systems ................................................... 344
A. SIEItOCII{/SKI and J. ZABCZYK:
On a Packing Problem .................................................................. 356
C. TUDOR:
Quadratic Control for Linear Stc~chastic Equations with Pathwisc Cost ................... 360
ADDRESSES OF CONTRIBUTORS .............................................. 371
.i STOCHASTIC FINITE DIMENSIONAL SYSTEMS

SOME RESULTS ABOUT TW0-MODE STOCHASTIC COMPARTMENTAL MODELS
C. Bruni i, G. Koch 2
I) Dipartimento di Informatica e Sistemistica, Universita di Roma "La
Sapienza"
via Eudossiana 18, 00184 Roma
2) Dipartimento di Matematica, Universita di Roma "La Sapienza"
piazzale A. Moro 5, 00186 Roma
I. Models rof two-mode compartmental systems.
In this paper we consider a class fo population models, which will be
named "two-mode" and si suitable ot represent phenomena characterized
by the interaction fo two compartment groups. The two groups, both
containing the same number fo compartments, exhibit some difference ni
their behaviour ("mode"), and ni the following we conventionally denote
them as "active" and "inactive" compartments.
a i
Let 2n be the total number fo compartments, and tel Xkt , Xkt ,
,n...,2,l=k denote the number fo individuals at time ,t respectively ni the
a i
active and inactive k-th compartment. By X t and Xt we denote the n-
a i
dimensional vectors respectively with components Xkt , Xkt , and by t X we
denote the 2n-dimensional aggregate vector :
= t X )l.i(
\×t)
4
We assume that within both active and inactive subsystems the
exchanges of individuals may occur in all possible ways, and we denote by
a i
vkj' ~kj the exchange rates from k-th to j-th compartment, k,j=l,2,...n,
respectively in the active and inactive subsystem. We also assume that
between the active and inactive subsystems exchanges only occur in a
pairwise fashion, that is each active compartment communicates in both
direction with just one inactive compartment (by a suitable ordering, any
two communicating compartments will be given the same index).
Moreover a special feature of our model is that these last exchanges occur
much faster than those ones within each subsystem. For this reason, the
exchange rate from j-th active (inactive) compartment to j-th inactive
ai ia
(active) one will be denoted by c.~j (c.t)j ), where c is a gain factor, c>>1.
In order to account for possible birth/reproduction and death
phenomena, each exchange between communicating compartments is
attached a reproduction factor ,¢o which is a non negative integer
expressing the number of individuals appearing into the arrival
compartment whenever one individual disappears from the departure one.
This factor will be indexed in the same way as the corresponding exchange
a i
rate. By assigning to a reproduction factor of the type ~Xkk (CXkk) a value
greater than 1, one can model reproduction phenomena occurring within
the k-th active (inactive) compartment itself. No reproduction is allowed in
the exchanges between an active (inactive) compartment and the inactive
ai ia
(active) corresponding one, thus C~k = °~k = 1. This is motivated by the
fact that such exchanges have indeed to be considered as changes of
"status" (of mode) for the individual, rather than real compartmental
transitions; besides, they will be supposed to occur at a very high rate,
which cannot be matched by necessarily slower reproduction processes.
Deaths can be accounted for by simply giving one compartment (let us say
the n-th one) in each subsystem the role of death compartment, defined
as the arrival compartment for all (and only those) exchanges with a zero
a i
reproduction factor. Assuming for the initial condition Xn0, Xn0, the zero
value., the death compartments stay permanently empty and death
corresponds indeed to a disappearing of individuals.