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)
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. (2011) Viability Theory.
Regulation of Uncertain Systems, 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 (présentation.pdf)
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.
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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 Articles.pdf
et
Habilitations à Diriger
des Recherches
|
Thèses de Doctorat d'Etat |
|
|
|
|
|
|
|
|
|
DI GUGLIELMO |
Francis |
1972 |
Approximations par éléments finis |
|
MOULIN |
Hervé |
1977 |
|
|
CORNET |
Bernard |
1979 |
Economie
mathématique |
|
HADDAD |
Georges |
1982 |
Théorie de la viabilité |
|
FRANKOWSKA |
Hélène |
1984 |
Analyse multivoque et contrôle optimal |
|
|
|
|
|
|
Habilitations
à diriger des recherches |
|
|
|
|
|
|
|
|
|
SCHMITT |
Michel |
1991 |
Morphologie Mathématique |
|
SAINT-PIERRE |
Patrick |
1992 |
Méthodes Numériques Multivoques |
|
QUINCAMPOIX |
Marc |
1996 |
Jeux Différentiels et contrôle |
|
CARDALIAGUET |
Pierre |
1999 |
Propagation de fronts |
|
DOYEN |
Luc |
2001 |
Développement Viable |
|
SEUBE |
Nicolas |
2003 |
Contrôle et Robotique |
|
|
|
|
|
|
Thèses
de Doctorat |
|
|
|
|
|
|
|
|
|
DORDAN |
Olivier |
1990 |
Analyse Qualitative |
|
QUINCAMPOIX |
Marc |
1991 |
Jeux Différentiels et contrôle |
|
SEUBE |
Nicolas |
1992 |
Réseaux de Neurones en Automatique |
|
MATTIOLI |
Juliette |
1993 |
Morphologie Mathématique |
|
DOYEN |
Luc |
1993 |
Asservissement Visuel |
|
CARDALIAGUET |
Pierre |
1994 |
Jeux Différentiels |
|
NAJMAN |
Laurent |
1994 |
Morphologie Mathématique |
|
MONROCQ |
Christophe |
1994 |
Réseaux de Neurones |
|
MULLERS |
Katharina |
1995 |
Jeux inertiels et oscillateurs |
|
GORRE |
Anne |
1996 |
Tubes Opérables et Equations Mutationnelles |
|
ROSSI |
Fabrice |
1996 |
Réseaux de Neurones |
|
LACOUDE |
Philippe |
1998 |
Fiscalité |
|
PUJAL |
Dominique |
2000 |
Valuation de portefeuilles |
|
MARTIN |
Sophie |
2005 |
Résilience et Viabilité |
|
|
|
|
|
|
Thèses
de Doctorat d'Universités Etrangères |
|
|
|
|
|
|
|
|
|
GUIDY |
Joséphine |
1980 |
Economie quadratique (Un. de Côte d'Ivoire) |
|
MADERNER |
Nina |
1992 |
Viabilité (Un. de Vienne) |
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