VIABILITY THEORY
By Jean-Pierre Aubin
Birkhauser, 1991
Viability theory is a mathematical theory that offers mathematical
metaphors of evolution of macrosystems arising in biology, economics,
cognitive sciences, games, and similar areas, as well as in nonlinear
systems of control theory.
The author is specifically concerned with three main common features:
-- A nondeterministic (or contingent) engine of evolution, providing
several (and even many) opportunities to explore the environment,
-- Viability constraints that the state of the system must obey at each
instant under ``death penalty'',
-- An inertia principle stating that the ``controls'' of the system are
changed only when viability is at stake.
Viability theorems yield selection procedures of viable evolutions, i.e.,
characterize the connections between the dynamics and the constraints for
guaranteeing the existence of at least one viable solution starting from
any initial state. These theorems also provide the regulation processes
(feedbacks) that maintain viability, or, even as time goes by, improve the
state according to some preference relation.
Contrary to optimal control theory, viability theory does not require any
single decision-maker (or actor, or player) to ``guide'' the system by
optimizing an intertemporal optimality criterion.
Furthermore, the choice (even conditional) of the controls is not made once
and for all at some initial time, but they can be changed at each instant
so as to take into account possible modifications of the environment
of the system, allowing therefore for adaptation to viability
constraints.
Finally, by not appealing to intertemporal criteria, viability theory
does not require any knowledge of the future.
In a nutshell, the main purpose of viability theory is to explain the
evolution of a system, determined by given nondeterministic dynamics and
viability constraints, to reveal the concealed feedbacks which allow the
system to be regulated and provide selection mechanisms for implementing
them.
On the mathematical side, viability theory contributed to vigorous renewed
interest in the field of ``differential inclusions'', as well as an engine
for the development of a differential calculus of set-valued maps. Only the
results needed in this book are presented.
An exposition of Set-Valued Analysis can be found in the companion
monograph ``Set-Valued Analysis'' by H. Frankowska and the author.
These techniques have already found applications to other domains, for
instance, to nonlinear systems theory, control theory (tracking, zero
dynamics) and differential games.
CONTENTS:
1 Viability Theorems for Ordinary and Stochastic Differential Equations
(Replicator Systems)
2 Set-Valued Maps
3 Viability Theorems for Differential Inclusions
(Stability of Viability Domains, Limit Sets and Equilibria, Cesaro means of
the velocities, Viability implies Stationarity, Chaotic Solutions to
Differential Inclusions)
4 Viability Kernels and Exit Tubes
(Permanence and Fluctuation, Viability Envelopes, Anatomy of a Sets,
Boundary of Viability Kernels, Viability Domain Algorithms,
Finite-Difference Approximation of Viability Kernels)
5 Invariance Theorems for Differential Inclusions
(Graphical Lower Limits of Solution Maps, Accessibility Map, Stability of
Invariance Domains, Semipermeability of the Boundary of the Viability
Kernel, Defeat and Victory domains of a Target and its Barrier)
6 Regulation of Control Systems 199
(Regulation Map, Selection Procedures, Closed-Loop Controls and Slow
Solutions, Continuous Closed Loop Controls)
7 Smooth and Heavy Viable Solutions
(Contingent Derivatives, Smooth Viable Solutions, Regularity Theorem,
Subregulation and Metaregulation Maps, Punctuated Equilibria, Ramp Controls
and Polynomial Open-Loop Controls, Heavy Viable Solutions, Dynamical Closed
Loops)
8 Partial Differential Inclusions of Tracking Problems
(The Tracking Property, Decentralization of a control system, Hierarchical
Decomposition Property, Partial Differential Inclusions, The Variational
Principle, Feedback Controls Regulating Smooth Evolutions)
9 Lyapunov Functions 315
(Contingent Epiderivatives, Stability Theorems, Attractors, Optimal
Lyapunov Functions, Lyapunov Preorders, Asymptotic Observability of
Differential Inclusions)
10 Miscellaneous Viability Issues
10.1 Variational Differential Inequalities
10.2 Fuzzy Viability
10.3 Finite-Difference Schemes
10.4 Newton's Method
11 Viability Tubes
(Cauchy Problem for Viability Tubes, Asymptotic Target)
12 Functional Viability
(History-dependent Viability Constraints, Viability constraints with
delays, Volterra Viability constraints)
13 Viability Theorems for Partial Differential Inclusions
(Elliptic & Parabolic Inclusions)
14 Differential Games
Bibliographical Comments