Jean-Pierre Aubin
born February 19, 1939, Abidjan, Ivory Coast
Nationalité Française/ French Citizenship
14, rue Domat,
75005 Paris France
Viabilité,
Marchés, Automatique, Décisions
Viability, Markets, Automatics, Decisions
S.A.R.L., siège social : 1, ruelle des chaudronniers, 89420 Bierry
les Belles Fontaines
Code APE-NAP
Numéro SIRET : 490 174 604 00010
http://vimades.com
LASTRE :
Laboratoire d'Applications des Systèmes Tychastiques Régulés
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Centre de Recherche
en Épistémologie Appliquée
CREA (CNRS/ÉcolePolytechnique) (UMR 7656)
Conte de Copenhague
Le
19 décembre 2009
Il
était une fois, il y a plus de trois milliards d’années, des bactéries
chenues et non moins expertes s'étaient associées en GIEC (Groupe d'experts
Interbactérial sur l'Évolution du Climat) pour tirer un signal d’alarme
sur la pollution de l'atmosphère de
méthane et d'ammoniaque qui entourait notre planète à cette époque. Inquiets,
car de jeunes algues bleu-vert avaient inventé la photosynthèse pour profiter
de l’énergie solaire, sans se soucier de rejeter un oxygène toxique pour
les bactéries de cette époque.
Heureux
de leur condition, confortés dans leur certitudes, incapables d’imaginer
l’avenir, ils invoquèrent le « principe de précaution » pour
préserver leur situation, l’enregistrèrent dans leur constitution, mais
en vain. Les jeunes bactéries n’allaient pas se priver d’une
énergie gratuite et inépuisable au bénéfice des générations futures. Opportunistes, elles explorèrent leur
environnement pour se saisir d’une
solution si par chance elles la rencontraient, chance qu’elles allaient
pour cela aider. Et elles trouvèrent une
parade : une protéine fixant
l'oxygène pour le neutraliser à la manière d’un contrepoison. Puis, parmi leurs descendantes, la verte chlorophylle, la rouge hémoglobine et leurs nombreuses
cousines transformèrent en énergie l’oxygène qu’elles
transportaient. Bel exemple de
récupération opportuniste d'un mécanisme conçu pour des fins qui n'étaient plus
les siennes, profitant de l’organe
pour créer la fonction plutôt que l'inverse.
Un
professeur de l’université de Copenhague réveilla
il y a deux ans une bactérie du
sommeil d’un demi-million d’années dans le nord-ouest canadien. Après avoir vu ce qu’était devenu le
monde, elle s’aperçut que tout
organisme vivant meurt pour qu’il survive dans sa lignée. Les organismes doivent s'adapter « à court terme »
à l’évolution de l’ensemble qu’ils forment même si, en retour, leur évolution retentit
sur cet ensemble, mais « à long terme » et sans pouvoir le piloter. L’évolution de ces ensembles est
myope, conservatrice et paresseuse, même si certains des organismes qui les
composent sont explorateurs et opportunistes.
C’est ainsi que les
bactéries règnent sur le monde, dix fois plus nombreuses que les cellules des
végétaux et animaux qui les ignorent.
L’idée que certaines
d’entre elles vinrent se loger dans chacune de leurs cellules pour leur
fournir leur principale source d'énergie en guise d’écot de leur loyer
l’égaya. Sans elles, il n’y aurait pas de vie sur terre.
Et
de conclure en haussant ses flasques
épaules : le principe de précaution n’a pas suffi, il n’est
pas possible de freiner ou de guider l’évolution d’un ensemble
d’organismes suivant leurs propres buts. Ce sont eux qui ont opposé au principe de
précaution un « principe d’anticipation » : il ne suffit
pas seulement de léguer à notre descendance un environnement durable,
c’est-à-dire stationnaire, il est vital
de lui fournir les moyens
d’anticiper son inéluctable et
implacable évolution afin de s’y adapter.
Sans
être sûre d’être écoutée, elle
suggéra aux êtres humains d’utiliser leurs cerveaux pour les imaginer et
les mettre en œuvre.
Jean-Pierre
Aubin
1 Main
Academic Positions
1.1 Permanent positions
2006-
Sociétaire
of the company VIMADES (Viability,
Markets, Automatics and Decisions)
2004-
Investigator
at LASTRE (Laboratoire d'Applications
des Systèmes Tychastiques Régulés),
1969-2004
Professeur (Full Professor), Université de Paris-Dauphine, (classe
exceptionnelle, 1979) . Since
2004, Professor Emeritus
2000-2003
Délégué au CNRS, CREA, École Polytechnique
1969-1986
Maître de Conférences (half-time Associate Professor), Ecole Polytechnique,
1967-1969
Associate Professor, Purdue University,
1965-1967
Maître de Conférences (Associate Professor), Faculté des Sciences de Lyon,
1961-1966
Ingénieur-chercheur (engineer/investigator), Électricité de France,
1.2 Main Visiting positions:
- Summers 2008,2009
and 2010,
- 1-month (2002) at
- 16 one-month (1989-2004) visits at Scuola
- 1-month (1998) and 3-month (1999) visits to
- 3 summers (1996, 1997, 1998) at SISSA (
- 15 summers (1981 -1995) at IIASA
(International Institute for Applied Systems Analysis),
- 9 summers at the
- One
semester (1987) and 3 summers (1977, 1978, 1980) at U.S.C. (
- One academic year (1976-1977) and a semester
(1986) at Université de Montréal.
2 Main Administrative Positions
- Co-founder
with Philippe Boutry and Patrick
Saint-Pierre of VIMADES (Viability, Markets, Automatics and Decisions)
and LASTRE (Laboratoire d'Applications des Systèmes Tychastiques Régulés)
- Founder and director of the Centre
de Recherche Viabilité, Jeux, Contrôle (1996-2000), Université
Paris-Dauphine
- Founder and director of the graduate school EDOMADE (École Doctorale de Mathématiques de
- Founder and director of the CEREMADE, Centre de Recherche de
Mathématiques de
- Founder
and director of the department (UFR) of Mathématiques de
- Président of Institut Henri Poincaré (1982-1985)
- Founder of the journal Analyse Non linéaire, Annales de l'Institut Henri-Poincaré
(1983) PDF
- Contractor
of the European programme MATARI (Mathematical
Toolkit for Artificial Intelligence and Regulation of Macro-Systems)
of COMETT (1990-1993)
- Contractor
of the Scientific Network Dynamics of
Complex Systems in Biosciences of the European Science
Foundation (1991-1994)
3 Books Published
Monographs
[1] AUBIN J.-P., BAYEN A. & SAINT-PIERRE
P. (2010) Applied Viability Theory. Regulation of Viable and Optimal Evolutions, Springer-Verlag
[2] AUBIN J.-P. (2000) Mutational and morphological analysis: tools for shape
regulation and morphogenesis, Birkhäuser
[3] AUBIN J.-P. (1997) Dynamic economic theory: a viability approach,
Springer-Verlag
[4] AUBIN J.-P. (1996) Neural networks and qualitative physics: a viability
approach, Cambridge University Press
[5] AUBIN
J.-P. (1991) Viability theory,
Birkhäuser
[6] AUBIN
J.-P. & FRANKOWSKA H. (1990) Set-valued
analysis, Birkhäuser
[7] AUBIN
J.-P. &
[8] AUBIN
J.-P. & EKELAND I. (1984) Applied
nonlinear analysis, Wiley-Interscience
[9] AUBIN
J.-P. (1982, 1979) Mathematical methods of
game and economic theory, North-Holland (Studies in Mathematics and
its applications, Vol. 7, 1-619)
[10] AUBIN
J.-P. (1972) Approximation of elliptic
boundary-value problems, Wiley-Interscience
Textbooks
[11] AUBIN
J.-P. (2000, 1979) Applied functional
analysis (second edition), Wiley Interscience. French Version: (1987) Analyse
fonctionnelle appliquée, tomes 1 & 2, Presses Universitaires de France
[12] AUBIN
J.-P. (1998, 1993) Optima and equilibria,
Springer-Verlag. French Version: (1984) L'analyse
non linéaire et ses motivations économiques, Masson
[13] AUBIN J.-P. (1994) Initiation à
l'analyse appliquée , Masson
[14] AUBIN J.-P. (1987) Exercices
d'analyse non linéaire, Masson
[15] AUBIN
J.-P. (1985) Explicit methods of optimization,
Dunod. French Version: (1982) Méthodes explicites de l'optimisation,
Dunod
[16] AUBIN
J.-P. (1977) Applied abstract analysis,
Wiley-Interscience
Vulgarization Books
[17] AUBIN J.-P. (2010) La mort du
devin, l'émergence du démiurge. Essai sur la contingence et la viabilité des
systèmes, Beauchesne
Lecture Notes
[18] AUBIN
J.-P. &
[19] AUBIN
J.-P. (20002) Viability Kernels and Capture
Basins, Lecture Notes, Universidad Politécnica de Cartagena
[20] AUBIN
J.-P. (2001) A Concise introduction to
viability theory, optimal control and robotics, cours DEA MVA, Ecole
Normale Supérieure de Cachan
[21] AUBIN
J.-P. (2002) Impulse differential inclusions
and hybrid systems: a viability approach, Lecture Notes, Université
Paris-Dauphine
Books Edited
[22] ATTOUCH
H., AUBIN J.-P., CLARKE F. H. & EKELAND I., Editors (1989) Analyse non linéaire, Gauthier-Villars
& C.R.M. Université de Montréal
[23] AUBIN
J.-P.,
[24] AUBIN
J.-P.,
[25] AUBIN
J.-P. & VINTER R. B. Editors (1982) Convex
analysis and optimization, Pitman, Research Notes in Math. # 57
[26] AUBIN
J.-P.,
[27] AUBIN J.-P. (1974) Analyse
convexe et ses applications, Lecture notes in Economics and
Mathematical Systems, Vol 102, Springer Verlag
4 Description of Research Activities
I have been very lucky to have
been initiated to mathematical research and guided
by Jacques-Louis Lions(May 2, 1928-May 17, 2001) 
and later, by 
to whom I owe everything, professionally and intellectually.
Among all the lessons I learned from them, both of human, moral and
scientific nature, trying to look for motivations for mathematics and find
applications of mathematics in other fields of knowledge was the most important
one, and doubtless, the most difficult to learn, the most arduous to implement.
I tried as much as I could to follow their examples, but these models were by
far too inaccessible.
My research activities began in the sixties in numerical analysis of
partial differential equations. Jacques-Louis Lions knew my deep interests in
biological, human and social sciences. Hence, while I was in the United States
of America after 1967, he told me that Université Paris-Dauphine, devoted to
the science of organizations, was opened after the ``events of 1968'' in France
and that this could interest me. He was right, as usual, and I took this
opportunity to return to France in 1969 to create the department of
Mathématiques de
Consequently, I changed drastically my former research programme for
investigating mathematical economics and game theory, concentrating on optima
and equilibria, while I was more and more involved in studying biological
evolution and cognitive sciences.
However, I rapidly became dissatisfied by the obvious shortcomings of
static game theory and mathematical economics: Economic evolution - as
biological evolution - is never at equilibrium, neither does it converge to it,
whereas it is constrained by the legacy of its history. I thought that the
reduction of rationality to a maximizing behavior is far to be in accordance
with what cognitive sciences are teaching, where adaptation and learning to
adapt should instead be the key features of the behavior of even
Homo-oeconomicus. This is the reason why I advocated the use of cognitive
sciences in decision sciences, that led me to the investigation of neural
networks and cognitive systems.
·
Even when time and dynamics were used to
tackle problems arising in these fields, it was mostly in the framework of intertemporal
optimization that requires a teleological point of view demanding
·
that decision makers act on the controls,
·
the knowledge of decision criteria (the
choice of which is open to question even in static models, even when
multicriteria or several Decision-makers are involved in the model)
·
anticipation of the future that only
experimentation provides,
·
that decision are taken once and for all
at the initial time.
Observing the evolution of economic and biological
systems led me to conclude that none of these requirements were present in the
evolution of such systems, where the
dynamics of the system disappears and cannot be recreated. Instead
of looking for ``optimal decisions'', I felt more important to choose decisions
taken at the right moment. Neither was I convinced that stochastic evolution -
requiring too much statistical regularity at certain levels of economic and
biological evolution - was the only way to capture uncertainty involved in the
evolution of such systems.
When I was looking for mathematical approaches to tackle such issues,
the available mathematical machinery available in 1975 seemed inadequate: I
thus decided to return to the root of the problems and became convinced that
evolution in some kind of Darwinian sense offered a more appropriate framework
in which to answer the main questions arising in this field. Hence, strongly
motivated by economics, history and Darwinian evolution, and since 1981 by
neurobiology, I tried several different mathematical approaches to capture
those new points of view. This is how I started to learn differential
inclusions from
It was in an exciting atmosphere that with my students, students of my
students and collaborators of Centre de Recherches Viabilité, Jeux, Contrôle
and, until 1986, of CEREMADE to whom I am very grateful (among whom
Instead of applying only known
mathematical and algorithmic techniques, most of them motivated by physics and
not necessarily adapted to adaptation problems to environmental or viability
constraints, viability theory designs and develops mathematical and algorithmic
methods for studying the evolution of such systems, organizations and networks
of systems (or organizations, organisms),
1.
constrained to adapt to a (possibly
co-evolving) environment,
2.
evolving under contingent, stochastic or
tychastic uncertainty,
3.
using for this purpose regulons
(regulation controls), and in the case of networks, connectionist matrices or
tensors,
4.
the evolution of which is regulated by
feedback laws (static or dynamic) that are then "computed" according
to given principles, such as the inertia principle,
5.
the evolution being either continuous,
discrete, or an "hybrid" of the two when impulses are involved,
6.
the evolution concerning both the
variables and the environmental constraints (mutational viability),
7.
the nonviable dynamics being corrected by
introducing adequate controls (viability multipliers) when necessary,
8.
or by introducing the viability kernel
with target under a nonlinear controlled system (either continuous or hybrid): This is the subset of initial states from
which starts at least one evolution that
a.
remains in the constrained set (i.e., is
viable) forever
b.
or reaches (i.e., captures) the target in finite
time before possibly violating the constraints (and not only asymptotically, as
it is usually studied with concepts of attractors since the pioneering works of
Alexander Lyapunov and Henri Poincaré going back to 1892).
When the target is
empty, only the first condition matters, and one says that it is simply the viability kernel of the constrained set. The set of initial
states satisfying only the second condition is called the capture basin of the target viable in the constrained
subset.
When these evolutions depend upon a parameter, such parameter can be
regarded as a control when actors, agents, decision makers, etc. can act on
them (pilot, decide, choose, etc.). When no clearly identified agent can act on
them, these parameters are regarded as regulatory parameters, in short, regulons, as genotypes in biology, fiduciary goods
in economics, cultural codes in sociology. They range over a state-dependent
cybernetic map, providing the system opportunities to adapt to viability
constraints (often, as slowly as possible) and/or to regulate intertemporal
optimal evolutions.
We also introduce the ``dual'' concept of invariance kernel
with target, which is the subset of initial states from which all evolutions
c.
remain in the constrained set (i.e., are
viable) forever
d.
or reach the target in finite time before
possibly violating the constraints. The set of initial states satisfying only
the second condition is called the absorption basin of the target invariant in the constrained
set.
This concept plays a
role whenever the evolutions are governed by evolutions depending upon
parameters on which actors, agents, decision makers, etc. These parameters are
often perturbations, disturbances (as in ``robust control'') or more generally,
tyches
ranging
over a state-dependent tychastic map. They could be called ``random variables''
if this vocabulary was not already confiscated by probabilists. This is why we
borrow to Charles Peirce who introduced the concept of tychastic evolution in [,Peirce] the term
of tyche , one of the three words
of classical Greek meaning ``chance'', and to call them in this case tychastic
systems analogous to (and actually, more general than) stochastic systems (In
this paper, Peirce associates with the Greek concept of necessity, ananke , the concept of anancastic evolution , anticipating the
``chance and necessity'' framework that has motivated viability theory in the
first place).
Tychastic control systems (or dynamical games) involve both regulons and
tyches in the dynamics, tyches describing uncertainties played by an
indifferent, may be hostile, Nature, regulons being available and chosen by the
system in order to adapt its evolutions whatever the tyches, we introduce the
concept of tychastic (or guaranteed) viability kernel, which is the subset of initial states from
which there
exist a regulon such that, for all tyches, the associated evolutions
e.
remain in the constrained set (i.e., are
viable) forever
f.
or reach the target in finite time before
possibly violating the constraints. The set of initial states satisfying only
the second condition is called the absorption basin of the target invariant in the constrained
set.
It is by now a consensus that the evolution of many
variables describing systems, organizations, networks arising in biology and
human and social sciences do not evolve in a deterministic way, and may be, not
even in a stochastic way as it is usually understood, but with a Darwinian
flavor, where intertemporal optimality selection mechanisms are replaced by
several forms of "viability", a word encompassing polysemous concepts
as stability, confinement, homeostasis, etc., expressing the idea that some
variables must obey some constraints. Intertemporal optimization is replaced by
myopic selection mechanisms that involve present knowledge, sometimes the
knowledge of the history (or the path) of the evolution, instead of
anticipations or knowledge of the future (whenever the evolution of these
systems cannot be reproduced experimentally). Uncertainty does not necessarily
obey statistical laws, but only unforcastable rare events (tyches, or
perturbations, disturbances) that obey no statistical law, that must be avoided
at all costs (precaution principle or robust control). These systems can be
regulated by using regulation (or cybernetical) controls that have to be chosen
as feedbacks for guaranteeing the viability of a system and/or the
capturability of targets and objectives, possibly against tyches (perturbations
played by Nature).
In a nutshell, the evolution under contingent and/or tychastic
(nonstochastic) uncertainty is governed by dynamical systems parameterized by
regulons (controls) and tyches (perturbations, disturbances) and confronted to
viability constraints (or scarcity constraints in economics).
The main purpose of viability theory is to
characterize and compute the viability kernel, that is the set of initial
states from which, for at least an adequate feedback regulon (control), and
whatever the tyches, if any, the evolution is viable (in the sense that it
satisfies the viability constraints for ever). The second objective is then to
reveal the concealed feedbacks, which allow the system to regulate viable
evolutions and provide selection mechanisms for implementing them. The third
one is to find ways of restoring viability when it is at stakes.
The search of evolutions that are optimal under an intertemporal
optimization is replaced by the quest of heavy solutions that, at each instant,
minimize a criterion on the state and its velocity, and satisfy the inertia
principle: the controls or regulons are changed only when viability is at
stakes. It assumes implicitly an ``opportunistic'' and ``conservative''
behavior of the system: a behavior that enables the system to keep viable
solutions as long as its potential for exploration (or its lack of determinism)
- described by the availability of several regulons - makes possible its
regulation, whatever the tyches.
Among fields where these issues are at their heart, I can mention
systems sharing such common features arising in
- economics, where the viability constraints are the scarcity constraints. We
can replace the fundamental Walrasian model of resource allocations by
decentralized dynamical model in which the role of the controls is played
by the prices or other economic decentralizing messages (as well as
coalitions of consumers, interest rates, and so forth). The regulation law
can be interpreted as the behavior of Adam Smith's invisible hand choosing
the prices as a function of the allocations,
- dynamical connectionist networks
and/or dynamical cooperative games, where coalitions of player may play
the role of controls: each coalition acts on the environment by changing
it through dynamical systems. The viability constraints are given by the
architecture of the network allowed to evolve,
- genetics and population genetics,where the viability constraints are the ecological constraints, the
state describes the phenotype and the controls are genotypes or fitness
matrices,
- sociological sciences, where a society can be interpreted as a set of individuals subject
to viability constraints. They correspond to what is necessary to the
survival of the social organization. Laws and other cultural codes are
then devised to provide each individual with psychological and economical
means of survival as well as guidelines for avoiding conflicts. These cultural codes play the role of regulation controls,
- cognitive sciences, where, at least at one level of investigation, the variables
describe the sensory-motor activities of the cognitive system, while the
controls translate into what could be called a conceptual control (which
is the synaptic matrix in neural networks),
- control theory and differential
games, conveniently revisited, can provide
many metaphors and tools for grasping the above problems. Many problems in
control design, stability, reachability, intertemporal optimality,
viability and capturability, observability and set-valued estimation can
be formulated in terms of viability kernels and capture basins. The
After years of study of various problems of different
kinds, motivated from robotics (and animat theory), population dynamics, game
theory, economics, neuro-sciences, biological evolution and, unexpectedly, from
financial mathematics, after sorting out and isolating the relevant features of
a class of problems at a level sufficiently high to be useful for all of them,
after noticing the common features of the proofs and algorithms, it became
possible to design a mathematical theory universal enough to be efficient in
many apparently different problems:
- Connectionist Complexity and Dynamical Networks
- Viability Theory
- Neural Networks
- Cognitive Systems
- Shape Regulation and Morphogenesis
- Chaotic Systems
- Impulse and Hybrid Systems
- Qualitative Physics
- Tychastic Uncertainty
- Dynamic
Management and Evaluation of Portfolios
- Stochastic Viability
- Mathematical Economics
- Nonlinear and Set-Valued Analysis
- Boundary-Value
Problems Systems of Hamilton-Jacobi-Bellman Partial Differential
Inclusions under Viability Constraints
- Intergenerational
and intertemporal transfers in social security problems
- Numerical Analysis
that I am presenting in reverse chronological order instead of a logical
one.
4.1 Viability Theory
Viability theory and set-valued analysis, using
methods that are rooted neither in linear system theory nor in differential
geometry, provides results that
- hold
true for nonlinear systems,
- are
global instead of being local,
- and
allow an algorithmic treatment.
On this route to abstraction, I regarded control
systems as differential inclusions, i.e., differential equations with
set-valued right hand sides. As long as we do not need to implicate explicitly
the regulons or the controls in our study, it is advantageous to replace
control problems by differential inclusions (and dynamical games by controlled
differential inclusions). Furthermore, by doing so, we are allowed to impose
state-dependent constraints on the controls, task imperative to deal with
realistic control problems and impossible to achieve without differential
inclusions. Actually, a sizable part of the results only depend upon few
properties of the set-valued map associating with any initial state the set of
solutions to the differential. Hence the idea to regard the solution map as an
``evolutionary system'', since many results are valid for any evolutionary
system described by a set-valued map associating with an initial state a subset
of evolutions starting from this initial state. It can be the solution map
associated with differential inclusion with memory or with a mutational on
metric spaces, etc.
Given a constrained set and a target, the key concepts for evolutionary
systems are
- the
viability kernel introduced in 1985: This is the subset of initial states
from which starts at least one evolution that remains in the constrained
set (i.e., is viable) forever
- the
viable-capture basin of the target, introduced in 1998: This is the subset
of initial states from which starts at least one evolution that reaches
(i.e., captures) the target in finite time(and not only asymptotically, as
it is usually studied with concepts of attractors since the pioneering
works of Lyapunov and Poincaré going back to 1892).
etV-(K).jpg)
Viability kernels of the Forward and Backward Lorenz
System.
The famous attractor is contained in
the Viability Kernel of the Backward system, itself contained in
the Viability Kernel of the forward System. Source:
- I
also introduced the ``dual'' concept of invariance kernel, which is the
subset of initial states from which all evolutions remain in the
constrained set (i.e., are viable) Forever and the concept of absorption
basin of the target invariant in the constrained set, which is the subset
of initial states from which all evolutions reach the target in finite
time.
Not only the
viability/capturability problem is important in itself, but it happens that
many other important concepts of control theory, mathematical economics and
finance and mathematics are viability kernels or capture basins under auxiliary
systems:
- For
economic (control) problems with bounded or minimal inflation
(chattering), the graph of the regulation map (synthesis) is a viability
kernel under the associated ``metasystem'', the regulons (controls) of
which are the velocities of the regulons (controls) of the original
system. The ``derivative'' of
this regulation map provides the dynamical feedbacks
- The
epigraph of optimal exponential Lyapunov functions is a viability kernel.
This further provides the ``Montagnes Russes Algorithm'' for finding (with
L. Najman) global minima of functions
- The
epigraph of the value function of intertemporal optimal control problems,
of stopping-time problems and of optimal controls with intertemporal
inequality constraints are capture basins in finite horizon, viability
kernels in infinite horizon (Frankowska's epigraphical approach) - see
further comments below
- The
graph of the solution to a boundary-value problem for a system of
first-order Hamilton-Jacobi-Bellman partial differential equations is the
viability kernel of an auxiliary system, allowing us in particular to
study age-structured problems (with N. Bonneuil) as well as other problems
structured by ``co-variable'' that are tracked by ``exosystems''
- The
graph of the detector (nonlinear filter) that detects solutions of a
control problems from measurements gathered along time is a viable-capture
basin (with
- The
set of equilibria of a controlled system is a viability kernel (
- Concepts
of Lyapunov stability, of attractors, of fluctuation around the boundary
of a target, of permanence are closely related to the concept of viability
kernel
- The
graph of the largest l-Lipschitz (set-valued) map contained in the graph
of a given set-valued map is the (discriminating) viability kernel of this
graph under an auxiliary dynamical game (see L. Doyen)
After gathering the properties and characterizations
of these viability kernels with targets, the next task is the characterization
of the regulation (or feedback, synthesis) law governing evolutions that remain
viable until they (possibly) reach the target. These are obtained from the
Viability and Invariance Theorems proved at the end of the seventies by many
authors (
What happens when a given set is not viable under a control system? As
in optimization under constraints, where Lagrange or Kuhn-Tucker multipliers
are used to correct the initial cost function by adding to it a linear
functional involving such a multiplier, I showed that we can correct the initial
dynamics by using ``viability multipliers" as regulons (controls) in such
a way that the constrained set is viable under this corrected system. For
instance, correcting a given dynamics by projecting it onto the tangent cone to
the constrained set is equivalent to correct the dynamics by viability
multipliers.
Therefore, starting from a given constrained set and given dynamics, we
can construct a family of control systems under which the set becomes viable.
This provides a first way to reestablish the viability of a set that is
not viable under a given control system. The second one is to restrict the
constrained set to its viability kernel, as we saw. A third one is to let the
constrained set to evolve according to morphological equations, as indicated
below. A fourth one if to reinitialize the state of the system whenever the
viability is at stakes through a ``reset map", as in impulse systems. This
list of methods for reestablishing the viability of a constrained set under a
control system is not exhaustive, and research should be active to find other
ones.
The first collection of results on differential inclusions and viability
theory was presented in
The motivations coming from Darwinian evolution in biology, from
evolutionary economics, from evolution of societies, from learning processes in
cognitive systems are the themes of the essay [16,Aubin]. This essay provides
epistemological thoughts on mathematical modeling of systems, a vernacular
description of the present state of viability theory, the connections with
history and evolution of social groups, with economics, with genetics and
biological evolution and with cognitive sciences, domains that motivated many
of his mathematical activities.
4.2 Minimal Congestion in Transportation
Networks
VIMADES contributes to some problems of traffic regulation, both for
highway and aerial traffic. The problem investigated is the computation and/or optimization of
travel times of a given vehicle between any points of nodes of a physical
network, made of highways (Real-time traffic information on (CMS) changeable
message signs) or Air Traffic Management (ATM)
Gate to Gate'' or `` Enroute to Enroute" 4D routes (tubes and funnels) for dynamic
allocation of ``available arrival
routes'' and computation of ``4D contract flights''
- For highway transportation,
we use the currently used
macroscopic Moskowitz/Lighthill-Whitham-Richards theory for
Lagrangian conditions provided by pieces of trajectories equipped with GPS
- For a microscopic approach, VIMADES investigates the regulation of
one or several prototypical vehicles between any two nodes of the system.
4.3
Robotics Management: Rallying a target in an environment
strewn with obstacles by an experimental drone
This project has emerged in the research laboratories of the French
Direction Générale de l'Armement under a research contract with investigators
and engineers VIMADES.
The problem was to drive a robot in an environment strewn with obstacles
to reach a target. The dynamics of the robot are determined by its velocity and
direction of the wheels. Viability algorithms provide the feedback to reach a
target in finite time (and even, in minimum time) while avoiding obstacles.
The embedded PILOT software of VIMADES, combined with digital maps of
the target and obstacles obtained by sensors, computes:
- The set of all the positions from which it is possible to reach the
target in finite time while avoiding obstacles;
- The feedback that can drive the vehicle to reach the target knowing
only the position and velocity of the vehicle at all times.
Reaching a target without obstacles on the course and avoiding obstacles
without a guarantee to reach this target are problems for which various
solutions have already been made. The PILOT software of VIMADES however is the
only autonomous feedback driving a vehicle in an environment strewn with
obstacles to achieve a target in finite time and a guarantee.
4.4
Asset-Liability Management : the VPPI (Viabilist Portfolio
Performance and Insurance) Trader Robot
The
VPPI Trader Robot of VIMADES manages of a portfolio hedging a liability ("variable annuities",
for instance), and transaction costs
(brokerage fees) is a software
suite ensuring both that the value of the portfolio is
always greater than or equal to the liabilities and maximize its performance
taking opportunity of the underlying price.
For
eradicating any risk of violation of constraints floor, it is necessary to
define the “minimum guaranteed investment” (MGI) by the two
following properties:
- Where the investment exceeds the minimum
guaranteed investment: whatever the realization of a series of underlying
prices, the value of the portfolio managed by the VPPI management rule is
always higher than the floor.
Where the investment is strictly less than the minimum guaranteed investment: whatever the management rule chosen, the floor constraint is violated by the completion of at least one set of underlying prices.
The initial value of
the minimum guaranteed investment may be considered a
"guaranteed/performance price," a warranty cost to eradicate ruin. It
corresponds, according to the context, to an "economic capital”.
4.1 Connectionist Complexity and Dynamical
Networks
Reading the literature on complexity, and quoting
George Cowan, the founder of the Santa Fe Institute, ``in the universe,
everything is connected with every thing'' seems to be
the consensual agreement of the members of this Institute. However, Seth Loyd
had found 31 different definitions of complexity at the beginning of the 90's.
This number increased a lot since. Complexity is indeed a polysemous word that
tries to embrace too many distinct phenomenon of interest in social and
biological sciences. One thus need to focus the concept, and confine to
connectionist complexity arising in networks. The coordination of activities of
the agents of the network operates by communicating messages that are
propagated by coalitions of agents through the ``connectionist operators".
Hence complexity cannot be studied by proposing only that dynamical systems
given a priori govern the evolution of messages, coalitions of agents and
connectionist operators, but rather by attempting to answer directly the
question that some economists or biologists ask: Complex
organizations, yes, but for what purpose? One can propose to investigate the
following answer: to adapt to the environment. This is the case in biology,
since the Claude Bernard's ``
Hence I suggest to
treat the evolution of the architecture of networks as well as connectionist
and structural complexity in the framework of viability theory.
4.5 Neural Networks
One way of classifying most of the neural networks is
to consider them as discrete or continuous control systems controlled by synaptic matrices.
In this perspective, they are examples of
interconnected systems, which include also replicator
systems which appear in biochemistry (Eigen's hypercycles), in
population dynamics, in ecology and in ethnology.
This also allows us to present in a unified way many examples of neural
networks and to use several results on control of linear and nonlinear systems
to obtain learning algorithm of pattern classification problems (including time
series in forecasting), such as the back-propagation formula, as well as
learning algorithms of feedback regulation laws of solutions to control systems
subject to state constraints (inverse dynamics.)
For these connectionist systems, we have to use the specific structure
of the space of synaptic matrices as a tensor product to justify mathematically
the connectionist features of neural networks. Tensor products explain the
Hebbian nature of many learning algorithms. This is due to the fact that
derivatives of a wide class of nonlinear maps defined on spaces of synaptic
matrices are tensor products and also, to the fact that the pseudo-inverse of a
tensor product of linear operators is the tensor product of their
pseudo-inverses.
Neural networks can also be used for learning viable solutions of
control systems, i.e., solutions to control systems satisfying given viability
(or state) constraints. One can derive learning processes of the regulation
feedbacks of control problems through neural networks. Three classes of
learning rules are presented. The first one, called the class of external
learning rules, is based on the gradient method (of optimization problems
involving nonsmooth functions). The second one deals with uniform algorithms.
The last one provides adaptive regulation
rules. Applications of these algorithms to the control of Autonomous Underwater
Vehicles (AUV) tracking the trajectory of an exosystem has been carried out by
Nicolas Seube.
These results are presented in the book [3,Aubin].
4.4 Cognitive Systems
Viability techniques may also be efficient to
contribute to the various attempts to devise mathematical metaphors of
cognitive processes. I presented in
The main (speculative) assumption is that
the informations are not coded in ``synaptic weights'', but on cyclic or
periodic evolutions of neurotransmitters through the synapses (together with
the slower and less precise circulation of hormones in the organism). Given the
mechanism of recognition of the state of the environment by conceptual
controls, perception and action laws and viability constraints, the viability
theorems allow to construct learning rules which describe how conceptual
controls evolve in terms of sensory-motor states to adapt to viability
constraints.
These results are presented in the eight chapter of the book [3,Aubin] and the eight chapter of [16,Aubin].
These ideas are now investigated in collaboration with Y. Burnod and K.
Pakdaman.
4.5 Shape Regulation and Morphogenesis
One purpose of mutational and morphological analysis
was to deal with some aspects of morphogenesis appearing in several biological
problems, with ``visual control'' or target problems
in control problems and differential games and other ``viability issues'', and
with the evolution of shapes and images, that are basically sets, not even
smooth. Therefore, their analysis, their processing, their evolution, their
optimization and/or their regulation and control require naturally an intrinsic
analysis, for which the tools of set-valued
and morphological analysis have been designed.
Set-valued analysis, including the contributions of Minkowski and
Steiner among others, underlies the original approach proposed under the name
of mathematical morphology
pioneered by Georges Matheron for image processing. Another original approach
was called shape optimization by
Jean Céa and Jean-Paul Zolésio, who introduced the basic concept of shape
derivatives of set-defined maps as well as the concept of velocities of tubes
as early as 1976. After having introduced the concept of graphical derivatives of set-valued maps
in the beginning of the 80's, I suggested ten years later to use mutations for defining velocities of evolving sets that provides
a structure that embraces and integrates the underlying frameworks of these
competing - yet complementary - concepts. In collaboration with Anne Bru-Gorre,
The results are presented in [1,Aubin], the first book presenting
mutational and morphological analysis and their application to an evolution
theory of sets.
4.6 Chaotic Systems
Some issues on chaotic evolution are related to viability
kernels of subsets under continuous time systems (attractor of the Lorenz and
Rossler systems, for instance) or discrete time
systems (Julia and Mandelbrot sets, Cantor nature of subsets under quadratic
maps or inverses of Hutchinson maps, for instance) See [17,Aubin & Saint-Pierre].
It happens that viability kernels of subsets, capture basins of targets
and the combination of the twos provide tools for analysis the local behavior
around equilibria (local stable and unstable manifolds), the asymptotic
behavior (localizing the attractor in the intersection of the forward and
backward viability kernels), the fluctuation between two areas of a domain,
etc.
Usually, the Lorenz attractor is
illustrated (but not computed) by computing solutions to the systems and
drawing their trajectories. By the way, these trajectories do not belong to the
attractor (except when they start from the attractor), just approach it. In our
example, we draw the trajectory of only one solution and we superpose it to the
viability kernel under the backward system. One can prove that small balls
around the two nonzero equilibria are contained in the complement of the
attractor. This explains the ``eyes'' around those equilibria, Source: Patrick
Saint-Pierre.
Since algorithms and softwares do exist for computing the viability
kernels and the capture basins, as well as evolutions viable in the viability
kernel until they converge to a target in finite time, we are able to localize
attractors going beyond the mere simulation of trajectories of evolutions. We
can ``trap the tamed'' few evolutions of these chaotic systems that are viable
in these kernels or basins. For systems with high sensitivity to initial
states, this is quite important, because, even starting from an initial state
in the attractor, approximations provided with very precise schemes of
solutions which should be viable in the attractor may actually leave it (and
converges to it, but from the outside).
These tools are meant to enrich the panoply of those set out by the
study of dynamical systems. These diverse and ingenious techniques born out the
pioneering works of Lyapunov and Poincaré more than one century ago provide
deep insights in the behavior of complex evolution dynamics that even simple
dynamics provide. The viability techniques are also geometric in nature, global
and bypass linearization or the dynamics around equilibria, for instance. They
bring other lights to the decipheration of complex and strange behaviors by
providing other type of algorithms.
.jpg)
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Example of a filled-in Julia Set and the corresponding Julia Set.
4.7 Impulse and Hybrid Systems
Hybrid systems of control theory, and, more generally,
impulse differential inclusions, motivate one of the methods allowing to
maintain the solution to a differential equation or a control system inside a
closed viability subset.
When the change of controls or the evolution of
the closed viability subset are not possible, the idea is to associate with the
(continuous) system a discrete system resetting initial conditions whenever the
solution of the (continuous) control system is doomed to violate the viability
constraints.
The problem is thus to find the impulse times at which this resetting is
done and the sequence of reinitialized states.
Optimal impulse control systems and hybrid systems are particular cases
of this problem. I collaborate on these issues with
4.8 Qualitative Physics
Another direction where the techniques of set-valued
analysis, differential inclusions, viability theory, etc... can play an
important role is a domain of Artificial Intelligence known under the name of
``qualitative simulation'' or ``qualitative physics''.
Instead of measuring all the inputs and applying them to the system to
observe the produced outputs, qualitative reasoning involves a first step which
consists in translating numerical date into qualitative values which can then
be reasoned about. This translation between numerical data and qualitative data
provides an important means of data reduction.
In evolution theory, the problem amounts to map a continuous dynamical
or control system into a discrete system which tells how we can go through a qualitative
cell (a cluster of states sharing a same qualitative behavior) to another.
These results are presented in the book [3,Aubin]. Recent developments are being
conducted in collaboration with O. Dordan.
4.9 Tychastic Uncertainty
The theory of tychastic control (or ``robust
control'') can be studied in the framework of dynamical games, when one player
plays the role of Nature that chooses - plays - perturbations. These
perturbations, disturbances, parameters that are not
under the control of the controller or the decision-maker, could be called
``random variables'' if this vocabulary was not already confiscated by
probabilists. I suggested to borrow to Charles Peirce the concept of tyche , one of the three words of
classical Greek meaning ``chance'', and to call in this case the control system
a tychastic system.
These tyches are allowed to range over a set depending on the state of
the system (the tychastic map). We stress the fact that tyches play the role of
event in stochastic systems. This analogy is further motivated by the Doss
Equivalence Theorem between stochastic viability and tychastic Viability
derived from the Strook & Varadhan Support Theorem.
The underlying idea is that we are looking for guaranteed (or robust)
choices under tychastic uncertainty, and not only on average situations, as in
stochastic control. This is linked by the way with the point of view proposed
by the specialists of max-plus algebras and/or exotic algebras, by advocating
the use of Maslov measures instead of the Kolmogorov ones, as well as making a
bridge towards the theory of fuzzy sets (actually, toll sets) via the Cramér
transform introduced in the theory of large deviations. The theory of dynamical
games is thus also involved in the mathematical treatment of uncertainty. Here
too, an effort of clarifying abstraction of ideas arising in many fields should
be the topic of active research.
4.10 Dynamic Management and Evaluation of
Portfolios
Curiously enough, this is in the context of tychastic
uncertainty that new and recent impetus to explore further new directions in
viability theory came from some financial problems (dynamic portfolio valuation
and management, including replicating portfolios for
evaluating options), that I investigated with 
Graph of the valuation function of an
European option. Source:
Translating these inequalities as membership conditions to epigraphs of
functions, these problems are guaranteed viability and capture problems under
dynamical games. Characterizing the capture basin of a target, one discovers
that the valuation function of the portfolio is the value function of a
concealed underlying dynamical game.
4.11 Stochastic Viability
In collaboration with
Thanks to the Strook & Varadhan ``Support Theorem'' and under
convenient regularity assumptions, stochastic viability problems are equivalent
to invariance problems for control systems, as it has been singled out by H.
Doss in 1977 for instance. By the way, this is in this framework of invariance
under control systems that problems of stochastic viability in mathematical
finance are studied.
The Invariance Theorem for control systems characterizes invariance
through first-order tangential and/or normal conditions whereas the stochastic
invariance theorem characterizes invariance under second-order tangential
conditions. Doss's Theorem states that these first-order normal conditions are
equivalent to second-order normal conditions that we expect for invariance
under stochastic differential equations for smooth subsets.
These results have been extended in collaboration with H. Doss to any
subset by defining in an adequate way the concept of contingent curvature of a
set and contingent epi-Hessian of a function, related to the contingent
curvature of its epigraph. This allows us to go one step further by
characterizing functions the epigraphs of which are invariant under systems of
stochastic differential equations and to show that they are generalized solutions
to either a system of first-order Hamilton-Jacobi equations or to an equivalent
system of second-order Hamilton-Jacobi equations.
4.12 Mathematical Economics
My contributions in this domain at the end of the
seventies are gathered in [8,Aubin] and its 1984 pedagogical companion
[11,Aubin]. They tentatively present at the
research and the pedagogical levels respectively a unified treatment in the
framework of Nonlinear Functional Analysis of (convex) optimization, game
theory and Walras equilibria in models of resource allocation.
Important simplifications of known results and new achievements are
provided, thanks, in particular, to the ``
However, despite my strong attraction to the many challenges of
non-linear analysis, I became extremely dissatisfied by the obvious
shortcomings of static game theory and mathematical economics and with what I
felt deficiencies in the main stream Arrow-Debreu framework. I thus started to
challenge the concepts of general equilibrium in mathematical economics, of
rationality equated with maximization of preferences, of (intertemporal)
optimality, of exclusive representation of uncertainty by averaging processes
(probabilities) and stochastic processes, etc. These frustrations were at the
origin of the development of viability theory; Its specific applications to
dynamic economic theory are presented in [2,Aubin].
4.13 Nonlinear and Set-Valued Analysis
The development of viability theory required advances
in both nonlinear analysis and set-valued analysis. For instance, it was
discovered that viable systems have equilibria when they are time-independent
(and periodic trajectories when they are periodic).
Viability theory led to the introduction of a differential calculus of
set-valued maps (including the extension of the Inverse Function Theorem for
set-valued maps), which I introduced at the beginning of the eighties and
developed together with H. Frankowska. She has since extended these concepts to
higher order inverse function theorems and has successfully used such results
in control theory of nonlinear systems and differential inclusions. A
differential calculus and an approximation theory of set-valued maps Based on
the concept of graphical convergence are now available, and are related to the
concept of epi-convergence (or G-convergence) and epidifferential calculus of
extended functions.
Some of the results obtained in this area are presented in [7,Aubin & Ekeland] and above all, in [5,Aubin & Frankowska], that are the
first books presenting the differential calculus if set-valued maps.
4.14 Boundary-Value Problems for Systems of
Hamilton-Jacobi-Bellman Partial Differential Inclusions under Viability
Constraints
``Viability techniques'' happen to be efficient not
only for regulating evolutions governed by a control system or a dynamical game
``viable'' in a constrained set until they reach a
target, 
but also for regulating optimal solutions
of optimal control problems in the Hamilton-Jacobi tradition, according to the
``epigraphical method'' devised by Hélène Frankowska. They allow us to revisit
also the methods of characteristics for solving boundary-value problems (with
viability constraints on the solutions) for systems of first-order
Hamilton-Jacobi-Bellman equations that arise in several domains of control
theory, such as intertemporal Paretian optimization for multicriteria control
problems, detectability of solutions satisfying measurements and their
regulation, etc. The discovery that such techniques motivated by a dissident
mathematical treatment of economic evolution could be used for investigated
optimal control, mainly due to
In collaboration with 
We stated a variational principle and an
existence theorem of a (single-valued contingent) solution to such an
inclusion. The existence of contingent single-valued
and set-valued solutions is proved for several classes of first-order systems
of partial differential inclusions.
Several comparison and localization results (which replace uniqueness
results in the case of hyperbolic systems of partial differential equations)
allow to derive useful informations on the solutions of these systems.
Concerning the specific applications to the Hamilton-Jacobi approached to
optimal control theory using the derivatives of the value function for
designing the synthesis of the (feedback) control, I am even tempted to suggest
- for practical purposes, not for mathematical ones - bypassing the revered
approach of Hamilton, Jacobi, Carathéodory, Bellman, Isaacs for devising
efficient algorithms by using the viability kernel and capture basin
algorithms. I am conscious of the fact that these dissident statements can lead
me to severe critical commentaries and intellectual ostracism, or worse ...
4.15 Intergenerational and intertemporal
transfers in social security problems
With N. Bonneuil, we used the results on
boundary-value problems (with viability constraints on the solutions) for
systems of first-order partial differential equations and inclusions to study problems with age structure or, more generally,
problems with co-variables ``structuring" the variables of the system
under investigation. This led us to investigate intergenerational and
intertemporal transfers in social security problems. We are also studying the
concept of anticipations, by introducing two time variables, once for the past
and the present, the other one for anticipations.
4.16 Numerical Analysis
Between 1962 and 1970, I worked on numerical analysis
of linear and non-linear partial differential equations. The main results I
obtained are presented in [9,Aubin], the first book on ``functional
numerical analysis''. Together with J.-L. Lions and J. Céa, I introduced the
``functional analysis'' approach to the theory of approximation of PDE's by
finite-dimensional problems. I proposed to approximate Sobolev spaces by using
what became known as the finite element method (for regular grids only), estimated
the ``error'' of these approximations and proved that the speed of convergence
is optimal. By using duality, I could estimate both the a priori and a
posteriori approximation errors for smooth and non smooth data. On the way, I
proved a ``compactness lemma'' which is a constant us in nonlinear PDE's and
the abstract ``Green formula'', which lies at the root of boundary conditions
in PDE's and ``transversality conditions'' in the calculus of variations.
5 Pedagogical Activities
The creation of the department of Mathématiques de
6 List of Papers
Articles
AUBIN J.-P.& LESNE
A. (2005) Constructing and exploring wells
of energy landscapes , Journal of Mathematical Physics, 46, 043508
AUBIN J.-P. &
AUBIN
J.-P. & SEUBE N. (2004) Conditional
Viability for Impulse Differential Games, Annals of Operations
Research, 137, 269 - 297
AUBIN
J.-P. & DOSS H. (2003) Characterization
of Stochastic Viability of any Nonsmooth Set Involving its Generalized
Contingent Curvature, Stochastic Analysis and Applications, 25, 951-981
AUBIN
J.-P. (2003) Regulation of the Evolution of
the Architecture of a Network by Connectionist Tensors Operating
on Coalitions of Actors,J. Evolutionary Economics, 13, 95-124
AUBIN
J.-P. (2003) Boundary-Value Problems for
Systems of First-Order Partial Differential Inclusions with
Constraints, Progress
in nonlinear differential equations and their applications 55, 25-60
AUBIN
J.-P. &
AUBIN
J.-P. (2002) Detectability of solutions to
control systems through informations gathered along time, Transactions of the South-African Institute of Electrical
Engineers, 93, 85-90
AUBIN
J.-P. (2002) Boundary-Value Problems for
Systems of Hamilton-Jacobi-Bellman Inclusions with Constraints, SIAM J. Control, 41, 425-456
AUBIN
J.-P. (2004) Dynamic Core of Fuzzy Dynamical
Cooperative Games, Annals of Dynamic Games, Ninth International Symposium on Dynamical Games and Applications, Adelaide, 2000,
Chapter 7, 128-162
AUBIN
J.-P. &
AUBIN
J.-P. & CATTE F. (2002) Bilateral
Fixed-Point and Algebraic Properties of Viability Kernels and Capture Basins of Sets,
Set-Valued Analysis, 10, 379-416
AUBIN J.-P., LYGEROS J.,
QUINCAMPOIX. M.,
AUBIN
J.-P., BONNEUIL N., MAURIN F. &
AUBIN
J.-P. (2001) Viability Kernels and Capture
Basins of Sets under Differential Inclusions, SIAM J. Control, 40, 853-881
AUBIN J.-P. &
AUBIN
J.-P.,
AUBIN J.-P., BONNEUIL N. & MAURIN F. (2000) Non-linear Structured Population Dynamics with
Co-Variables, Mathematical Population Studies, 91, 1-31
AUBIN J.-P. (2000) Boundary-Value
Problems for Systems of First-Order Partial Differential Inclusions,
NoDEA, 7, 61-84
AUBIN
J.-P. &
AUBIN J.-P. &
AUBIN
J.-P. & NAJMAN L. (1998) The Russian
Mountain Algorithm for global optimization, Math. Methods of
Operational Research
AUBIN
J.-P. & FRANKOWSKA H. (1997) Set-valued
Solutions to the Cauchy Problem for Hyperbolic Systems of Partial Differential
Inclusions,
NoDEA, 4, 149-168
AUBIN
J.-P. (1997) L'autonomie dans le cadre de la
théorie de la viabilité, Revue Internationale de
Systémique, 11, 535-550
AUBIN
J.-P. & FRANKOWSKA H. (1996) The
Viability Kernel Algorithm for Computing Value Functions of Infinite Horizon
Optimal Control Problems, J.Math. Anal. Appl., 201, 555-576
AUBIN J.-P. (1996) Une Métaphore Mathématique du principe de précaution, Natures, Sciences, Sociétés, 4, 146-154
AUBIN J.-P. &
AUBIN J.-P. &
AUBIN J.-P. & FRANKOWSKA H.
(1995) Partial differential inclusions governing feedback controls, J.
Convex Analysis, 2, 19-40
AUBIN
J.-P. & NAJMAN L. (1994) L'algorithme des montagnes russes pour l'optimisation
globale, Comptes-Rendus de l'Académie des Sciences,
Paris, 319, 631-636
AUBIN J.-P. (1993) Mutational
Equations in Metric Spaces, Set-Valued Analysis, 1, 3-46
AUBIN
J.-P. &
AUBIN J.-P. & FRANKOWSKA H. (1992) Hyperbolic systems of partial differential inclusions, Annali della
Scuola Normale di Pisa, 18, 541-562
AUBIN
J.-P. & SEUBE N. (1992) Apprentissage
adaptatif de lois de rétroaction de systèmes contrôlés par réseaux de neurones,
Comptes-Rendus de l'Académie des Sciences, Paris, 314, 957-963
AUBIN J.-P. (1992) A
Note on Differential Calculus in Metric Spaces and its
Applications to the Evolution of Tubes, Bulletin of the Polish
Academy of Sciences, 40, 151-162
AUBIN J.-P., FRANKOWSKA H. &
AUBIN
J.-P. & FRANKOWSKA H. (1991) Systèmes
hyperboliques d'inclusions aux dérivées partielles, Comptes-Rendus de l'Académie des Sciences, Paris, 312, 271-276
AUBIN J.-P. (1991) Evolution
of viable allocations, J. Economic Behavior and Organizations, 16, 183-215
AUBIN J.-P. &
AUBIN
J.-P. &
AUBIN J.-P. & FRANKOWSKA H. (1990) Controllability and observability of control systems
under uncertainty, Volume dedicated to Opial, P. W. N., Annales Polonici Mathematici, 37-67
AUBIN J.-P. & FRANKOWSKA H. (1990) Inclusions aux dérivées partielles gouvernant des contrôles de rétroaction, Comptes-Rendus de l'Académie des Sciences,
Paris, 311, 851-856
AUBIN
J.-P. (1990) A survey of viability theory,
SIAM J. on Control and Optimization, 28, 749-788
AUBIN J.-P. (1990) Differential
games: a viability approach, SIAM J. on Control and Optimization,
28, 1294-1320
AUBIN J.-P. (1990) Fuzzy differential inclusions, Problems On Control and Information
Theory, 19, 55-67
AUBIN J.-P. & FRANKOWSKA H. (1989) Observability
of systems under uncertainty, SIAM J. on Control and Optimization,
Special Issue dedicated to Mac Shain, 27,
AUBIN J.-P. (1989) Qualitative
simulation of differential equations, J. Differential and Integral
Equations, 2, 183-192
AUBIN J.-P. (1989) Règles d'apprentissage de systèmes cognitifs, Comptes-Rendus
de l'Académie des Sciences, Paris, 308, 147-150
AUBIN J.-P.
(1989) Smallest Lyapunov functions of
differential inclusions, J. Differential and Integral Equations,
2,333-343
AUBIN
J.-P. & SIGMUND K. (1988) Permanence and
viability, J. of Computational and Applied Mathematics, 22, 203-209
AUBIN J.-P. & WETS R. (1988) Stable approximations of set-valued maps,
Ann.Inst. Henri Poincaré, Analyse non linéaire, 5, 519-535 PDF
AUBIN J.-P. (1988) Equations qualitatives aux confluences, Comptes-Rendus
de l'Académie des Sciences, Paris, 307, 679-682
AUBIN J.-P. (1988) L'évolution des systèmes historiques vu - travers la théorie de la
viabilité, Stratégique, 1, 129-142
AUBIN J.-P. (1988) Problèmes mathématiques posés par l'algorithme QSIM (qualitative
simulation), Comptes-Rendus de l'Académie des Sciences,
Paris, 307, 731-734
AUBIN
J.-P. & FRANKOWSKA H. (1987) Observabilité
des systèmes sous incertitude, Comptes-Rendus de
l'Académie des Sciences, PARIS, 306, 413-416
AUBIN J.-P. & FRANKOWSKA H. (1987) On the inverse function theorem for set-valued maps,
J. Math. Pures Appliquées, 66, 71-89
AUBIN
J.-P., FRANKOWSKA H. & OLECH C. (1986) Controllability
of convex processes, SIAM J. of Control and Optimization, 24, 1192-1211
AUBIN J.-P., FRANKOWSKA H. & OLECH C.
(1986) Contr-labilité des processus convexes,
Comptes-Rendus de l'Académie des Sciences, Paris, 301, 153-156
AUBIN
J.-P. & FRANKOWSKA H. (1985) Heavy
viable trajectories of controlled systems, Annales de l'Institut
Heanri-Poincaré, Analyse Non Linéaire, 2, 371-395 PDF
AUBIN J.-P. (1985) Motivated
mathematics , SIAM
News, Vol.18,# 1, 2 & 3.
AUBIN
J.-P. & FRANKOWSKA H. (1984) Trajectoires
lourdes de systèmes contrôlés, Comptes-Rendus de l'Académie des
Sciences, PARIS,298, 521-524
AUBIN J.-P. (1984) Lipschitz behavior of solutions to convex minimization problems,
Mathematics of Operations Research, 8, 87-111
AUBIN J.-P. (1983) Exemples de mécanismes décentralisés de régulation par des prix dans un
contexte de déséquilibre, Economie Appliquée, 36,
333-347
AUBIN
J.-P. (1982) Comportement lipschitzien des
solutions de problèmes de minimisation convexes, Comptes-Rendus de l'Académie des Sciences, PARIS, 295, 235-238
AUBIN J.-P. (1981) A
dynamical, pure exchange economy with feedback pricing, J. Economic
Behavior and Organizations, 2, 95-127
AUBIN
J.-P. (1981) Contingent derivatives of
set-valued maps and existence of solutions to nonlinear inclusions and
differential inclusions, Advances in Mathematics 7a, Mathematical
Analysis and Applications, Ed. Nachbin L., 159-229
AUBIN
J.-P. (1981) Cooperative
fuzzy games, Math. Op. Res.,
6, 1-13
AUBIN
J.-P. (1981) Locally lipchitz cooperative
games, J. Math. Economics, 8, 241-262
AUBIN
J.-P. & DAY R.H. (1980) Homeostatic
trajectories for a class of adaptative economic systems, J. Econ. Dyn. Control., 2, 185-203
AUBIN J.-P. & EKELAND (1980) Second order evolution equation with convex hamiltonian, Can. Bull. Math, 23, 81-94
AUBIN
J.-P. & SIEGEL (1980) Fixed points and
stationary points of dissipative multivalued maps, Proceedings of
Am. Math. Soc., 78, 391-398
AUBIN
J.-P. (1980) Formation of coalitions in a
dynamical model where agents act on the environment, Economie et Société, 1583-1594
AUBIN J.-P. (1980) Further
propertiesoflagrange multipliers in nonsmooth optimization, Applied
Mathematics and Optimization, 6, 79-90
AUBIN
J.-P. (1980) Monotone trajectories of
differential inclusions : a darwinian approach, Methods of
Operations research, 37, 19-40
AUBIN
J.-P., LOUIS-GUERIN & ZAVALONNI (1979) Comptabilité
entre conduites sociales réelles dans les groupes et les représentations symboliques de ces groupes : un essai de
formalisation mathématique, Math. Sci. Hum., 68, 27-61
AUBIN J.-P. & CLARKE (1979) Shadow prices and duality for a class of optimal control problems, SIAM J. Opt.
and Control, 17, 567-586
AUBIN
J.-P. (1979) C-nes tangents - un
sous-ensemble convexe fermé, Ann. Sc. Math. Québec, 3, 63-80
AUBIN J.-P. (1979) Monotone
evolution of ressource allocation, J. Math. Economics, 6, 43-62
AUBIN
J.-P. (1978) Analyse
fonctionnelle non linéaire et applications - l'équilibre économique,
Ann. Sc. Math. Quebec, 2, 5-47
AUBIN J.-P. (1978) Gradients généralisés de Clarke, Ann. Sc. Math. Québec, 2, 197-252
AUBIN J.-P. (1978) Propriété de
perron-frobenius pour des
correspondances , Comptes-Rendus de l'Académie des Sciences, 286,
911-914
AUBIN
J.-P., CELLINA & NOHEL (1977) Monotone trajectories of
multivalued dynamical systems, Annali di Matematica pura e Applicata, 115, 99-117
AUBIN
J.-P. & CLARKE (1977) Monotone invariant
solutions to differential inclusions, J. London Math. Soc., 16, 357-366
AUBIN
J.-P. & CLARKE (1977) Multiplicateurs de
lagrange en optimisation non convexe et applications, Comptes-Rendus
de l'Académie des Sciences, 285, 451-454
AUBIN J.-P. (1977) Evolution monotone d'allocations de biens disponibles,
Comptes-Rendus de l'Académie des Sciences, 285, 293-296
AUBIN
J.-P. (1977) Représentation linéaire
canonique de systèmes dynamiques non linéaires, Comptes-Rendus de
l'Académie des Sciences, 285, 411-414
AUBIN J.-P. & CORNET (1976) Règles de décision en théorie des jeux et théorèmes
de points fixes, Comptes-Rendus de l'Académie des
Sciences, 282, 11-14
AUBIN
J.-P. & EKELAND I. (1976) Estimates of
the duality gap in non convex optimization, Math. Op. Res., 1, 11-14
AUBIN
J.-P. & NOHEL (1976) Existence de
trajectoires monotones de systèmes dynamiques discrets,
Comptes-Rendus de l'Académie des Sciences, 282, 267-270
AUBIN J.-P. & EKELAND I. (1975) Minimisation de critères intégraux,
Comptes-Rendus de l'Académie des Sciences, 281, 285-288
AUBIN
J.-P. (1974) Coeur et valeur des jeux flous
- paiements latéraux, Comptes-Rendus de l'Académie des Sciences,
279, 963-966
AUBIN
J.-P. (1974) Coeur et équilibres des jeux
flous sans paiement latéraux, Comptes-Rendus de l'Académie des
Sciences, 279, 963-966
AUBIN J.-P. (1974) Règles de décision optimales en théorie des jeux - 2 personnes,
Comptes-Rendus de l'Académie des Sciences, 279, 173-176
AUBIN J.-P. (1973) Multigames
and decentralization in management, Multiple Criteria Decision
Making, 313-326
AUBIN
J.-P. (1973) Multiplicateurs de Kuhn et
Tucker pour des jeux non coopératifs contraints, Ann. Scuola Normale
Pisa, 27, 561-589
AUBIN J.-P.
(1973) Représentation d'un c-ne convexe de
fonctions par un c-ne de fonctions convexes, Boll. dell'Unione matematica italiana, 7, 212-228
AUBIN J.-P. & MOULIN H. (1972) Condition nécessaire et suffisante d'existence d'une
solution du problème dual d'un problème d'optimisation, Comptes-Rendus de l'Académie des
Sciences, 274, 547-549
AUBIN J.-P. (1972) Théorème du minimax pour une classe de fonctions, Comptes-Rendus de l'Académie des
Sciences, 274, 455-458
AUBIN
J.-P. (1971) Remarks about the construction
of optimal subspaces of approximants of Hilbert spaces, J. Theory of
Approximation, 4, 21-36
AUBIN J.-P. (1970) A
Pareto minimum principle, Differential games, 147-175
AUBIN
J.-P. (1970) Abstract boundary value operators and
their adjoint, Rend. Sem. Padova, 43, 1-33
AUBIN
J.-P. (1970) Approximation des problèmes aux limites non homogènes pour des opérateurs non
linéaires, J. Math. Anal. Appl., 30, 510-521
AUBIN J.-P. (1970) Characterization
of the sets of constraints for which the necessary conditions for optimization problems hold,
SIAM J. of Control, 8, 148-162
AUBIN J.-P. (1970) Optimal
approximation and characterization of the error and stability functions in Banach spaces, Journal of Approximation Theory, 3, 430-444
AUBIN
J.-P. (1969) Approximation dans Lp
des problèmes aux limites non homogènes par des schémas aux
différences finies,
Comptes-Rendus de l'Académie des Sciences, 268, 861-863
AUBIN
J.-P. (1969) Approximation des problèmes aux
limites non homogènes et régularité de la convergence, Calcolo, 6, 117-140
AUBIN
J.-P. (1969) Approximation des problèmes aux
limites non homogènes pour l'opérateur åDi(|u|p-2Diu)
, Comptes-Rendus de l'Académie des Sciences, 268,
950-953
AUBIN J.-P. (1968) Approximation et interpolation optimales et ßplines functions",
J. Math. Appl., 24, 1-24
AUBIN J.-P. (1968) Best
approximation of linear operators in Hilbert spaces, SIAM J. Num. Anal., 5, 518-521
AUBIN J.-P. (1968) Evaluation des erreurs de troncature des approximations des
espaces de sobolev, J. Math. Anal. Appl., 24, 1-24
AUBIN
J.-P. (1967) Approximation des espaces de
distributions et des opérateurs différentiels, Bull. Soc. Math. Fr. Mémoire, 12,
1-139 PDF
AUBIN
J.-P. (1967) Behavior of the error of the
approximate solutions of boundary-value problems for
linear elliptic operators by Galerkin's and finite difference methods,
Ann. Sc. Nor. Pisa, 21, 599-637
AUBIN
J.-P. (1966) Approximation of variational inequations ,
Functional Analysis and Optimization, 7-14
AUBIN J.-P. (1963) Un théorème de compacité, Comptes-Rendus de l'Académie des
Sciences, 265, 5042-5045
AUBIN J.-P. (1963) Etude d'une
certaine classe d'équations
aux dérivées partielles non linéaires, Comptes-Rendus de l'Académie
des Sciences, 256, 350-353
Proceedings
AUBIN
J.-P.& LESNE A. (2005) Analyse
morphologique et mutationnelle: des outils pour la
morphogenèse, in Morphogenèse, Belin
AUBIN J.-P.,
AUBIN J.-P. (2005) Dynamical
Connectionist Network and Cooperative Games, in Dynamic Games: Theory and Applications ,
AUBIN J.-P. &
AUBIN J.-P., BERNARDO T. &
AUBIN J.-P. &
AUBIN
J.-P. & MURILLO-HERNANDEZ J. A.. (2004) Morphological Equations and Sweeping
Processes, Nonsmooth Mechanics and Analysis:
Theoretical and Numerical Advances, ch. 21, 249-259, Klüwer
AUBIN J.-P. (2004) Elements
of Viability Theory for the Regulation of the Evolution of the
Architecture of Networks, in COGNITIVE
ECONOMICS - An Interdisciplinary Approach,
AUBIN J.-P. (2004) Evolution of Complex Economic Systems and Uncertain Informations,
in Connectionist Approaches in Economics and
Management Sciences, Cédric Lesage ,
AUBIN J.-P. (2003) Adaptive
evolution of complex systems under uncertain environmental
constraints: A viability approach in Formal descritpions of developing systems, Eds. NATION J. et
al., 165-184, Klüver
AUBIN J.-P. (2002) Systems
of First-Order Partial Differential Inclusions with Constraints,
in Proceedings of Evolution Equations 2000