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An introduction to Viability Theory

and management of renewable resources

Coupling Climate and Economic DynamicsGeneva, November 21, 2002

Jean-Pierre Aubin1

Abstract

The main purpose of viability theory is to explain the evolution of the state of a controlsystem, governed by nondeterministic dynamics and subjected to viability constraints,to reveal the concealed feedbacks which allow the system to be regulated and provideselection mechanisms for implementing them. It assumes implicitly an “opportunistic”and “conservative” behavior of the system: a behavior which enables the system to keepviable solutions as long as its potential for exploration (or its lack of determinism) —described by the availability of several evolutions — makes possible its regulation.

We shall illustrate the main concepts and results of viability theory by revisiting theVerhulst type models in population dynamics, by providing the class of all Malthusianfeedbacks (mapping states to growth rates) that guarantee the viability of the evolutions,and adapting these models to the management of renewable resources.

Other examples of viability constraints are provided by architectures of networksimposing constraints described by connectionist tensors operating on coalitions of actorslinked by the network. The question raises how to modify a given dynamical systemgoverning the evolution of the signals, the connectionist tensors and the coalitions insuch a way that the architecture remains viable.

1 Introduction

The purpose of this survey paper is to present a brief introduction of viability theorythat studies adaptive type evolution of Complex Systems under Uncertain Environ-

1The author thanks warmly Alain Haurie for inviting this contribution, and Noel Bonneuil andPatrick Saint-Pierre for their hidden collaboration.

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ment and Viability Constraints that are found in many domains involving living be-ings, from biology to cognitive sciences, from ecology to sociology and economics. Itinvolves polyscientific investigations spanning fields that have traditionally developedin isolation.

Instead of applying only known mathematical and algorithmic techniques, most ofthem motivated by physics and not necessarily adapted to such problems, viabilitytheory designs and develops mathematical and algorithmic methods for studying theevolution of such systems, organizations and networks of systems,

1. constrained to adapt to a (possibly co-evolving) environment,

2. evolving under contingent, stochastic or tychastic uncertainty,

3. using for this purpose regulation controls, and in the case of networks, connec-tionist matrices or tensors,

4. the evolution of which is governed by regulation laws 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 whenimpulses 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 whennecessary,

8. or by introducing the ”viability kernel” of a constrained set under a nonlinearcontrolled system (either continuous or hybrid), that is the set of initial statesfrom which starts at least one evolution reaching a target in finite time whileobeying state (viability) constraints.

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 evolvein a deterministic way, and may be, not even in a stochastic way as it is usuallyunderstood, but with a Darwinian flavor, where intertemporal optimality selectionmechanisms are replaced by several forms of ”viability”, a word encompassing poly-semous concepts as stability, confinement, homeostasis,tolerable windows approach 2

2see the contribution by Ferenc Toth and [44, Petschel-Held, Schellnhuber, Brucknert, Toth&Hasselmann], [26, Bruckner, Petschel-Held, Toth, Fussel, Helm, Leimbach & Schellnhuber], [54, Toth],etc.

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etc., expressing the idea that some variables must obey some constraints. Intertem-poral optimization is replaced by myopic selection mechanisms that involve presentknowledge, sometimes the knowledge of the history (or the path) of the evolution,instead of anticipations or knowledge of the future (whenever the evolution of thesesystems cannot be reproduced experimentally). Uncertainty does not necessarily obeystatistical laws, but only unforcastable rare events (tyches, or perturbations, distur-bances) that obey no statistical law, that must be avoided at all costs (precautionprinciple or robust control). These systems can be regulated by using regulation (orcybernetical) controls that have to be chosen as feedbacks for guaranteeing the viabilityof a system and/or the capturability of targets and objectives, possibly against tyches(perturbations played by Nature).

The purpose of viability theory is to attempt to answer directly the question thatsome economists or biologists ask: Complex organizations, systems and networks, yes,but for what purpose? One can propose the following answer: to adapt to the envi-ronment. This is the case in biology, since the Claude Bernard’s ”constance du milieuinterieur” and the ”homeostasis” of Walter Cannon. This is naturally the case in ecol-ogy and environmental studies. This is also the case in economics when we have toadapt to scarcity constraints, balances between supply and demand, and many otherones.

The environment is described by constraints of various kinds (representing objec-tives, physical and economic constraints, ”stability” constraints, etc.) that can neverbe violated. In the same time, the actions, the messages, the coalitions of actors andconnectionist operators do evolve, and their evolution must be consistent with the con-straints, with objectives reached at (successive) finite times (and/or must be selectedthrough intertemporal criteria).

There is no reason why collective constraints are satisfied at each instant by evolu-tions under uncertainty governed by stochastic or thychastic control dynamical systems.This leads to the study of how to correct either the dynamics, and/or the constraintsin order to reestablish this consistency. This may allow us to provide an explanation ofthe formation and the evolution of the architecture of the system and of their variables.

Presented in such an evolutionary perspective, this approach of (complex) evolutiondeparts from the main stream of modelling studying static networks with graph theoryand dynamical complex systems by ordinary or partial differential equations, a taskdifficult outside the physical sciences.

For dealing with these issues, one needs innovative concepts and formal tools, al-gorithms and even mathematical techniques motivated by complex systems evolvingunder uncertainty. For instance, and without entering into the details, we can mention

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systems sharing such common features arising in

economics, where the viability constraints are the scarcity constraints. We can re-place the fundamental Walrasian model of resource allocations by decentralizeddynamical model in which the role of the controls is played by the prices or othereconomic decentralizing messages (as well as coalitions of consumers, interestrates, and so forth). The regulation law can be interpreted as the behavior ofAdam Smith’s invisible hand choosing the prices as a function of the allocations,

dynamical connectionist networks and/or dynamical cooperative games, wherecoalitions of player may play the role of controls: each coalition acts on the en-vironment by changing it through dynamical systems. The viability constraintsare given by the architecture of the network allowed to evolve,

genetics and population genetics, where the viability constraints are the ecologi-cal constraints, the state describes the phenotype and the controls are genotypesor fitness matrices.

sociological sciences, where a society can be interpreted as a set of individuals sub-ject to viability constraints. They correspond to what is necessary to the survivalof the social organization. Laws and other cultural codes are then devised to pro-vide each individual with psychological and economical means of survival as wellas guidelines for avoiding conflicts. These cultural codes play the role of regula-tion controls.

cognitive sciences, where, at least at one level of investigation, the variables describethe sensory-motor activities of the cognitive system, while the controls translateinto what could be called a conceptual control (which is the synaptic matrix inneural networks.)

control theory and differential games, conveniently revisited, can provide manymetaphors and tools for grasping the above problems. Many problems in controldesign, stability, reachability, intertemporal optimality, viability and capturabil-ity, observability and set-valued estimation can be formulated in terms of viabilitykernels. The Viability Kernel Algorithm computes this set.

The ways of using these concepts and tools dealing with viability issues are verydifferent whether the domain of applications is socio-economic, biologic or technologi-cal.

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• In the first case, the theoretical results about the above ways to think long termviability are useful for the (re)designing of regulatory institutions: regulatedmarkets when political convention must exist for global purpose; mediation ormetamediation of all kinds, including law, social conflicts in production and inthe cities; institutions for sustainable development. And progressively, when moredata gathered by these institutions will be available, qualitative (and sometimesquantitative) prescriptions of viability theory will be useful.

• In the second case, the technological systems (robot of all kinds, from very littleones including nanorobots, to animats and to very big large ones (drones, un-derwater vehicles, etc) need embarked systems autonomous enough to regulateviability/capturability problems by adequate regulation (feedback) control laws.Modular and/or object-oriented general portable and distributed software flexi-ble enough for integrating new problems (hybrid, distributed systems, dynamicalgames) are needed and could be approached in the viability framework.

It is time to cross the interdisciplinary gap and to confront and hopefully to mergethe points of view rooted in different disciplines. Mathematics, thanks to its abstractionpower by isolating only few key features of a class of problems, can help to bridge thesebarriers as long as it proposes new methods motivated by these new problems insteadof applying the classical ones only motivated until now by physical sciences. If weaccept that physics studies much simpler phenomena than the ones investigated bysocial and biological sciences, and that for this very purpose, they motivated and useda more and more complex mathematical apparatus, we have to accept also that socialsciences require a new and dedicated mathematical arsenal which goes beyond whatis presently available. Paradoxically, the very fact that the mathematical tools usefulfor social and biological sciences are and have to be quite sophisticated impairs theiracceptance by many social scientists, economists and biologists, and the gap menacesto widen.

Outline We devote the three first sections to the description of the main conceptsand basic results of viability theory: evolutions, viability kernels and capture basinsunder evolutionary systems in the first section, characterization of viability and ofthe adaptation law in the second one, construction of static and dynamic feedbacks,including the slow and heavy ones, in the third.

We shall illustrate the main concepts and results of viability theory by revisiting theVerhulst type models in population dynamics, by providing the class of all Malthusianfeedbacks (mapping states to growth rates) that guarantee the viability of the evolu-

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tions, and adapting these models to the management of renewable resources. Morecomplex applications go beyond the format of this introduction to viability theory.

We next address the comparison between viability and intertemporal optimizationissues and indicate how the Frankowska approach to the study of Hamilton-Jacobi-Bellman equations involves viability theory.

We also mention the fact that the attractor of an evolutionary system is containedin the viability kernel of an absorbing set under the backward (negative) system andillustrate this fact by computing a superset of the Lorenz attractor (and not only itsapproximation suggested by the trajectories of the solutions to the system), providingalso an explanation of the fluctuation property.

We finally address the issues related to the restoration of the viability when theconstraints are not viable under an evolutionary system : correcting the dynamicsby viability multipliers, reinitializing the state whenever the viability is at stakes,providing hybrid of continuous and discrete time evolutions, or changing the constraintsby governing their evolutions by mutational equations.

2 The Mathematical Framework

2.1 Viability and Capturability

Let X denote the state space of the system. Evolutions describe the behavior of thestate of the system as a function of time t ∈ R+ := [0, . . . , +∞[ ranging over the set ofnonnegative real numbers or scalars t ∈ R+. We shall assume all along that

1. the state space is a finite dimensional vector space X := Rn,

2. evolutions are continuous functions x(·) : t ∈ R+ 7→ x(t) ∈ X describing theevolution of the state x(t).

We denote the space of continuous evolutions x(·) by C(0,∞; X) or, in short, C(X).

Some evolutions, mainly motivated by physics, are classical: equilibria and periodicevolutions. But these properties are not necessarily adequate for problems arisingin economics, biology, cognitive sciences and other domains involving living beings.Hence we add the concept of evolutions viable in a constrained set K ⊂ X (the theenvironment) or capturing a target C ⊂ K in finite time to the list of properties satisfiedby evolutions. Therefore, we consider mainly evolutions x(·) viable in a subset K ⊂ Xrepresenting a constrained set (an environment) in which the trajectory of the evolutionmust remain forever:

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∀ t ≥ 0, x(t) ∈ K (1)

Alternatively, a “target” C ⊂ K being given, we distinguish evolutions x(·) captur-ing the target C in the sense that they are viable in K until they reach the target C infinite time:

∃ T ≥ 0 such that

x(T ) ∈ C∀ t ∈ [0, T ], x(t) ∈ K

(2)

We devote our paper to the study of the set of evolutions viable in K outside C, i.e.that are viable in K forever or until they reach the target C in finite time.

2.2 The Evolutionary System

Next, we provide the mathematical description of one of the “engines” governing theevolution of the state. We assume that there exists a control parameter, or, better, aregulatory parameter, called a regulon, that influences the evolution of the state of thesystem. This dynamical system takes the form of a control system with (multivalued)feedbacks :

i) x′(t) = f(x(t), u(t)) (action)ii) u(t) ∈ U(x(t)) (contingent retroaction)

(3)

taking into account the a priori availability of several regulons u(t) ∈ U(x(t)) chosenin a subset U(x(t)) ⊂ Y of another finite dimensional vector-space Y subjected tostate-dependent constraints.

Once the initial state is fixed, the first equation describes how the regulon acts onthe velocities of the system whereas the second inclusion shows how the state (or anobservation on the state) can retroact through (several) regulons in a vicariant way.

We observe that there are many evolutions starting from a given initial state x0,one for each time-dependent regulon t 7→ u(t). The set-valued map U : X ; Y alsodescribes the state-dependent constraints on the regulons. In this case, system (3) canno longer be regarded as a parameterized family of differential equations, as in the casewhen U(x) ≡ U does not depend upon the state, but as a differential inclusion (see [15,Aubin & Cellina] for example). Fortunately, differential inclusions enjoy most of theproperties of differential equations.

A solution to system (3) is an evolution t → x(t) satisfying this system for some(measurable) open-loop control t → u(t) (almost everywhere).

We associate with the control system the evolutionary system x ; S(x) associatingwith any initial state x ∈ K the subset S(x) ⊂ C(0,∞; X) of solutions starting at x.

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Most of the results on viability kernels and capture basins depend upon few propertiesof this evolutionary system, that are shared by other “engine of evolutions”, such asdiffusion-reaction systems, path (or history) dependent systems, mutational equationsgoverning the evolution of states.

2.3 Viability Kernels and Capture basins

The problems we shall study are all related to the viability of a constrained subset Kand/or the capturability of a target C ⊂ K under the dynamical system modelling thedynamic behavior of the system.

1. The subset Viab(K) of initial states x0 ∈ K such that one solution x(·) to system(3)ii) starting at x0 is viable in K for all t ≥ 0 is called the viability kernel of Kunder the control system. A subset K is a repeller if its viability kernel is empty.

2. The subset Capt(K, C) of initial states x0 ∈ K such that the target C ⊂ K isreached in finite time before possibly leaving K by one solution x(·) to system(3)ii) starting at x0 is called the viable-capture basin of C in K. A subset C ⊂ Ksuch that Capt(K, C) = C is said to be isolated in K.

We say that

1. a subset K is viable under S if K = Viab(K),

2. that K is a repeller if Viab(K) = ∅.

In other words, the viability of a subset K under a control system is a consistencyproperty of the dynamics of the system confronted to the constraints it must obeyduring some length of time.

To say that a singleton c is viable amounts to saying that the state c is anequilibrium — sometime called a fixed point. The trajectory of a periodic solution isalso viable.

Contrary to the century-old tradition going back to Lyapunov, we require the systemto capture the target C in finite time, and not in an asymptotic way, as in mathematicalmodels of physical systems. However, there are close mathematical links between thevarious concepts of stability and viability. For instance, Lyapunov functions can beconstructed using tools of viability theory. Or one can prove that the attractor iscontained in the viability kernel of an absorbing set under the backward (negative)

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system. This needs much more space to be described: we refer to Chapter 8 of [1,Aubin] and Chapter 8 of [3, Aubin] for more details on this topic.

One can prove that the viability kernel Viab(K) of the subset K is the largest subsetof K viable under the control system. Hence, all interesting features such as equilibria,trajectories of periodic solutions, limit sets and attractors, if any, are all contained inthe viability kernel.

One can prove that the viability kernel is the unique subset D ⊂ K viable andisolated in K such that K\D is a repeller. If K is a repeller, the capture basinCapt(K, C) of C ⊂ K is the unique subset D between C and D such that D isisolated in K and D\C is locally viable.

The viability kernels of a subset and the capture basins of a target can thus becharacterized in diverse ways through tangential conditions thanks to the viabilitytheorems. They play a crucial role in viability theory, since many interesting conceptsare often viability kernels or capture basins.

Furthermore, algorithms designed in [50, Saint-Pierre]) allow us to compute viabilitykernels and capture basins (see also [28, Cardaliaguet, Quincampoix & Saint-Pierre]and [47, Quincampoix & Saint-Pierre]). In general, there are no explicit formulasproviding the viability kernel and capture basins.

3 Characterization of Viability and/or Capturabi-

lity

The main task is to characterize the subsets having this viability/capturability prop-erty. To be of value, this task must be done without solving the system for checkingthe existence of viable solutions for each initial state.

3.1 Tangent Directions

An immediate intuitive idea jumps to the mind: at each point on the boundary of theconstrained set outside the target, where the viability of the system is at stake, thereshould exist a velocity which is in some sense tangent to the viability domain and servesto allow the solution to bounce back and remain inside it. This is, in essence, what theviability theorem below states. Before stating it, the mathematical implementation ofthe concept of tangency must be made.

We cannot be content with viability sets that are smooth manifolds (such as spheres,which have no interior), because inequality constraints would thereby be ruled out (as

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for balls, that possess distinct boundaries). So, we need to implement” the concept ofa direction v tangent to K at x ∈ K, which should mean that starting from x in thedirection v, we do not go too far from K: The adequate definition due to G. Bouligandand F. Severi proposed in 1930 states that a direction v is tangent to K at x ∈ K ifit is a limit of a sequence of directions vn such that x + hnvn belongs to K for somesequence hn → 0+. The collection of such directions, which are in some sense inward”,constitutes a closed cone TK(x), called the tangent cone3 to K at x. Naturally, exceptif K is a smooth manifold, we lose the fact that the set of tangent vectors is a vector-space, but this discomfort in not unbearable, since advances in set-valued analysis builta calculus of these cones allowing us to compute them. See [20, Aubin & Frankowska]and [49, Rockafellar & Wets] for instance.

3.2 The Adaptive Map

We then associate with the dynamical system (described by (f, U) and with the viabilityconstraints (described by K) the (set-valued) adaptive or regulation map RK . It mapsany state x ∈ K\C to the subset RK(x) (possibly empty) consisting of regulons u ∈U(x) which are viable in the sense that f(x, u) is tangent to K at x:

RK(x) := u ∈ U(x) | f(x, u) ∈ TK(x)

We can for instance compute the adaptive map in many instances.

3.3 The Viability Theorem

The Viability Theorem states that the target C can be reached in finite time from eachinitial condition x ∈ K\C by at least one evolution of the control system viable in K ifand only if for every x ∈ K\C, there exists at least one viable control u ∈ RK(x).

This Viability Theorem holds true when both C and K are closed and for a ratherlarge class of systems, called Marchaud systems: Beyond imposing some weak technicalconditions, the only severe restriction is that, for each state x, the set of velocitiesf(x, u) when u ranges over U(x) is convex (This happens for the class of control systemsof the form

x′(t) = f(x(t)) + G(x(t))u(t)

3replacing the linear structure underlying the use of tangent spaces by the tangent cone is at theroot of Set-Valued Analysis.

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where G(x) are linear operators from the control space to the state space, when themaps f : X 7→ X and G : X 7→ L(Y,X) are continuous and when the control set U(or the images U(x)) are convex).

Curiously enough, viability implies stationarity, i.e., the existence of an equilibrium.Equilibria being specific evolutions, their existence requires stronger assumptions. TheEquilibrium Theorem states that when the constrained set is assumed to be viable,convex and compact, then there exists a (viable) equilibrium. Without convexity, wededuce only the existence of minimal viable closed subsets.

The proofs of the above Viability Theorem and the Equilibrium Theorem are diffi-cult: The Equilibrium Theorem is derived from the 1910 Brouwer Fixed Point Theorem,and the proof of the Viability Theorem uses all the theorems of functional analysis ex-cept the closed graph theorem and the Lebesgue Convergence Theorem. However, theirconsequences are much easier to obtain and can be handled with moderate mathemat-ical competence.

3.4 The Adaptation Law

Once this is done, and whenever a constrained subset is viable for a control system,the second task is to show how to govern the evolution of viable evolutions. We thusprove that viable evolution of system (3) are governed by

i) x′(t) = f(x(t), u(t))ii) u(t) ∈ RK(x(t)) (adaptation law)

(4)

until the state reaches the target C.

We observe that the initial set-valued map U involved in (3)ii) is replaced by theadaptive map RK in (4)ii). The inclusion u(t) ∈ RK(x(t)) can be regarded as anadaptation law (rather than a learning law, since there is no storage of information atthis stage of modelling).

3.5 Planning Tasks: Qualitative Dynamics

Reaching a target is not enough for studying the behavior of control systems, that haveto plan tasks in a given order. This issue has been recently revisited in [18, Aubin &Dordan] in the framework of qualitative physics (see [29, Dordan], [33, Eisenack &Petschel-Held], [48, Kuipers] and [2, Aubin] for more details on this topic). We describethe sequence of tasks or objectives by a family of subsets regarded as qualitative cells.Giving an order of visit of these cells, the problem is to find an evolution visiting thesecells in the prescribed order.

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3.6 The Metasystem

In order to bound the chattering (rapid oscillations or discontinuities) of the regulons,we set a priori constraints on the velocities of the form

∀ t ≥ 0, ‖u′(t)‖ ≤ c

Let B(0, c) ⊂ Y denote the ball of radius c centered at the origin. The bound on thevelocity of the regulons is taken into account by the metasystem associating with theinitial viability problem is the system

i) x′(t) = f(x(t), u(t))ii) u′(t) ∈ B(0, c)

(5)

subjected to the metaconstraints

∀ t ≥ 0, x(t) ∈ K & u(t) ∈ U(x(t)) (6)

Unfortunately, the above metaconstraints may no longer be viable under the metasys-tem.

Remark: The Crisis Function — If the solution is allowed to leave the con-strained set K, one can use the crisis function introduced in [31, Doyen & Saint-Pierre],which measures the minimal time spent by the evolutions x(·) outside K (see figure 4below for an illustration in the framework of management of renewable resources.).This takes into account the fact the zero damage within, infinite damage outside theemission corridor”, as it is said in [54, Toth], is not the only preoccupation of viabilitytheory, as well as the tolerable window approach.

4 Selecting Viable Feedbacks

4.1 Static Feedbacks

A (static) feedback r is a map x ∈ K 7→ r(x) ∈ X which is used to pilot evolutionsgoverned by the differential equation x′(t) = f(x(t), r(x(t))). A feedback r is said tobe viable if the solutions to the differential equation x′ = f(x, r(x)) are viable in K.The most celebrated examples of linear feedbacks in linear control theory designed tocontrol a system have no reason to be viable for an arbitrary constrained set K, and,according to the constrained set K, the viable feedbacks are not necessarily linear.

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However, the Viability Theorem implies that a feedback r is viable if and only if ris a selection of the adaptive map RK in the sense that

∀ x ∈ K\C, r(x) ∈ RK(x) (7)

Hence, the method for designing feedbacks for control systems to evolve in a con-strained subset amounts to find selections r(x). One can design a factory” for design-ing selections (see Chapter 6 of [1, Aubin], for instance). Ideally, a feedback shouldbe continuous to guarantee the existence of a solution to the differential equationx′ = f(x, r(x)). But this is not always possible. This is the case of slow selection r ofRK of minimal norm, governing the evolution of slow viable evolutions (despite its lackof continuity).

4.2 Dynamic Feedbacks

One can also look for dynamic feedbacks gK : X : K×Y 7→ Y that governs the evolutionof both the states and the regulons through the metasystem of differential equations

i) x′(t) = f(x(t), u(t))ii) u′(t) = gK(x(t), u(t))

(8)

A dynamic feedback gk is viable if the metaconstraints (6) are viable under the meta-system (8).

As for the (static) feedbacks, one can prove that all the viable dynamic feedbacksare selections of a dynamical adaptive map GK : K×Y ; Y obtained by differentiating”the adaptation law (3)ii) thanks to the differential calculus of set-valued maps see [20,Aubin & Frankowska]).

4.3 Heavy Evolutions and the Inertia Principle

Among the viable dynamic feedbacks, one can choose the heavy viable dynamic feedbackgK ∈ GK with minimal norm that governs the evolution of heavy viable solutions, i.e.,viable evolutions with minimal velocity. They are called heavy” viable evolutions4 in thesense of heavy trends in economics.

Heavy viable evolutions offer convincing metaphors of the evolution of biological,economic, social and control systems that obey the inertia principle. It states in essence

4When the regulons are the velocities, heavy solutions are the ones with minimal acceleration, i.e.,maximal inertia.

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that the regulons are kept constant as long as viability of the system is not at stake”.Heavy viable evolutions can be viewed as providing mathematical metaphors for theconcept of punctuated equilibrium introduced in paleontology by Eldredge and Gould in1972. In our opinion, this is a mode of regulation of control systems (see Chapter 8 of[2, Aubin] for further justifications).

Indeed, as long as the state of the system lies in the interior of the constrained set(i.e., away of its boundary), any regulon will do. Therefore, the system can maintainthe regulon inherited from the past. This happens if the system obeys the inertiaprinciple. Since the state of the system evolves while the regulon remains constant, itmay reach the viability boundary with an outward” velocity. This event correspondsto a period of viability crisis: To survive, the system must find other regulons such thatthe new associated velocity forces the solution back inside the viability set until thetime when a regulon can remain constant for some time.

5 Management of Renewable Resources

5.1 From Malthus to Verhulst and Beyond

A population for which there is a constant supply of resources, no predators, but limitedspace provides a simple one-dimensional example: at each instant t ≥ 0, the populationx(t) must remain confined in an interval K := [a, b] where 0 < a < b. The maximalpopulation size b is called the carrying capacity.

The dynamics are unknown, really, and several models have been proposed. Theyare all particular cases of a general dynamical systems of the form

x′(t) = u(x(t))x(t)

where u : [a, b] 7→ R is a model of the growth rate of the population feeding back onthe size of the population. Malthusian advocated in 1798 to choose a constant positivegrowth rate u(x) = u > 0, leading to an exponential evolution x(t) = xeut starting atx, which cannot be viable in any bounded interval. This the price to pay for linearity ofthe dynamic of the population:“Population, when unchecked, increases in a geometricalratio.”.

Population dynamical models providing evolutions growing fast when the popula-tion is small and declining when it becomes large where looked for to compensate forthe never ending expansion of the Malthusian model.

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The purely logistic Verhulst model with feedback of the form

u(x) := r(b− x)

proposed in 1838 by the Belgium mathematician Pierre-Francois Verhulst (rediscoveredin the 1930’s by Raymond Pearl) does the job. The solution to the logistic differentialequation x′(t) = rx(t)(b− x(t)) starting from x ∈ [a, b] are respectively equal to

x(t) =bx

x + (b− x)e−rt

They remain confined in the interval [a, b] and converge to the equilibrium b whent 7→ +∞.

The logistic model and the S-shape graph of its solution became popular in the1920’s and stood as the evolutionary model of a large manifold of growths, from thetail of rats to the size of men.

The affine feedback map r(b − x) is always positive on the interval [a, b], so thatthe velocity of the population is always nonnegative, even though the population slowsdown. In order to have negative velocities (still with positive growth rates), we shouldrequire that the feedback satisfies the following phenomenological properties: for someξ ∈ [a, b[,

(i) ∀ x ∈ [a, ξ[, u′(x) > 0(ii) ∀ x ∈]ξ, b], u′(x) < 0(iii) u(b) = 0

The increasing behavior of u(x) on the interval [a, ξ[ is called the Allee effect, statingthat at a low population size, an increase of the population size is desirable and haspositive effects on population growth, whereas the decreasing behavior of u(x) on theinterval ]ξ, b] is called the logistic effect, stating that at high population, an increase ofthe size has a negative effect on the growth of the population.

Instead of finding one feedback u satisfying the above requirements by trial anderror, we proceed systematically to design feedbacks by leaving the choice of the growthrates open, regarding them as regulons of the system

x′(t) = u(t)x(t) (9)

where the regulon is chosen to govern evolutions confined in the interval [a, b]. Imposinga bound on the velocities of the growth rates of the population further restrict theselection of such a feedback.

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The equilibrium map U∝ is defined by U∝(x) = 0 and the monotonic maps U+ andU− by

U+(x) := R+ & U−(x) := R−

The regulation map is equal to

RK(x) :=

R+ if x = aR if x ∈]a, b[R− if x = b

It is set-valued, has nonempty values, but has too poor continuity property, a sourceof mathematical difficulties. The interval [a, b] is obviously viable under such a controlsystem.

By using the concepts of viability kernel, we shall characterize all of the feedbacksgoverning evolutions viable on the interval [a, b] under inertia bound c. Among them,we will find the Verhulst feedback, to which we add two other ones, providing inertand heavy evolutions.

Inert evolutions are obtained by taking the maximal velocities allowed: u′(t) = ±c.

Heavy evolutions combine Malthusian and inert growth. They are obtained startingwith constant growth rate (Malthusian evolution) of the state (by taking u′(t) = 0 untilthe time when the inertia bound is met (an thus, implying a (weak) Allee effect duringthis phase of the evolution). This provides warning signals telling when, where and howthe regulons must evolve: The feedback becomes a specific one, the inert feedback,providing constant (negative) velocities u′(t) = −c of growth rates driving the state atits carrying capacity b.

5.2 Discrete Versus Continuous Time

Discrete evolutions are maps associating with each discrete time j ∈ N := 0, . . . , +∞ranging over the set of nonnegative integers a state x(j) =: xj ∈ X. Unfortunately,for discrete time evolutions, tradition imposes upon us to regard discrete evolutions assequences and to use the notation −→x : j ∈ N 7→ xj := x(j) ∈ X.

The choice between these two representations of time, the discrete and the continu-ous, is not easy. The natural one, that appears the simplest for the non mathematicians,is the choice of the set N of discrete times. It has drawbacks, though. On one hand, itmay be difficult to find a common time scale for the different components of the statevariables of the state space of a given type of models. On the other one, by doingso, we deprive ourselves of the concepts of velocity, acceleration and other dynamicalconcepts that are not well taken into account by discrete time systems as well as of

17

the many results of the differential and integral calculus gathered for more than fourcenturies. However, for computational purposes, we shall approximate continuous-timesystems by discrete time ones where the time scale becomes infinitesimal.

Figure 1: Viability Kernel of [0, 1] under the Quadratic Map: The interval [0, 1] hasbeen thickened for easier visibility of the viability kernel (in dark blue).

However, viability properties of the discrete analogues of continuous-time systemscan be drastically different : we shall see on the simple example of the Verhulst logisticequation that the interval [0, 1] is invariant under the continuous system

x′(t) = rx(t)(1− x(t))

whereas the viability kernel of [0, 1] under its discrete analog

xn+1 = rxn+1(1− xn+1)

is a Cantor subset of [0, 1] when r > 4. See figure 1.

5.3 Introducing Inertia Bounds

The constraint on the velocities of the growth rates imposed by the inertia propertymeans that

|u′(t)| ≤ c

This suggests to take the velocity of the regulons as “metaregulons”, required torange over the interval [−c, +c]. The state-regulon pairs (x, u) are the “metastates” ofthe “metasystem”:

(i) x′(t) = u(t)x(t)(ii) |u′(t)| ≤ c

(10)

where the metaregulon is chosen to be the velocity or the growth rate of the system.

Unfortunately, the metaconstrained set [a, b] × R+ is obviously not viable underthe above metasystem: Every solution starting from (a, u) with u < 0 leaves the set[a, b]× R+ immediately, as do evolutions starting from (b, u) with u > 0.

18

We thus define the graph of the set-valued map Uc by

Graph(Uc) := Viab(10)([a, b]× R+)

The regulation map Uc for the one-dimensional model (9) can be explicitly describedby two feedbacks maps:

We associate the inert feedbacks

r](x) :=

√2 log

(b

x

), r[(x) :=

√2 log

(x

a

)and R(x) :=

[−r[(x), +r](x)

]We can prove that for system x′(t) = u(t)x(t), the graph of the regulation map Uc islimited by the graphs of −

√c r[ below and

√c r] above: The regulation map is equal to

Uc(x) :=√

c[−r[(x), +r](x)

]=√

c R(x)

We shall now construct several feedbacks as selections of the regulation map Uc.

5.3.1 Affine Feedbacks and the Verhulst Logistic Equation

Affine feedbacks defined byu(x) := r(b− x)

are selections of the regulation map Uc when r ≤√

c 2b, so that the viability of the

interval [a, b] under the models using such affine feedbacks is guaranteed. The Ver-hulst logistic differential equation x′(t) = rx(t)(b− x(t)) corresponds to such an affinefeedback.

The solutions starting from x ∈ [a, b] are equal to

x(t) =bx

x + (b− x)e−rt

They remain confined in the interval [a, b] and converge to the equilibrium b whent 7→ +∞.

We observe that the velocity and the growth rate u(t) are respectively given by

u(t) = re−rt(b− x) and u(t) =r(b− x)e−rt

x + (b− x)e−rt

19

They start r(b−x)b

, converge to 0 when t 7→ +∞ and are always decreasing since theirderivatives

u′(t) =−r2(b− x)e−rt

(x + (b− x)e−rt)2

are negative.

5.3.2 Inert Evolutions

For a given inertia bound c, inert evolutions are governed by the feedbacks u(t) =√c r](x(t)) if u > 0 and u(t) = −

√c r[(x(t)) if u < 0.

Let us consider the case when u > 0 (the case when u < 0 being symmetrical).Then the derivative of the feedback

√c r] being negative on [a, b], it is a purely logistic

feedback with no Allee effect.

In both cases, the velocity governing the inert evolution is constant and equal tou′(t) = −c, so that u(t) = u− ct. Therefore the evolution of the inert regulon is givenby

u(t) = u

(1− ut

2 log(

bx

))and the evolution of the inert state by

x(t) = xeut− u2t2

4 log( bx)

The state reaches the equilibrium (b, 0) at time

τ(x, u) = 2log(

bx

)u

After, taking x(t) ≡ b and u(t) ≡ 0, the solution remains at equilibrium for t ≥ τ(x, u).

The derivative of the feedback u(x) :=√

c r](x) governing the evolution of the inertevolution being negative whenever a ≤ x < b, the inert evolution does not show anyAllee effect, but only a logistic one.

This is the same situation than with the Verhulst equation. For obtaining feedbacksconveying the Allee effect, we introduce heavy evolutions associated with a bound c > 0on the velocities of the regulons.

20

Figure 2: Viability Kernel and Inert Evolution. Source: Patrick Saint-Pierre

21

5.3.3 Heavy Evolutions

Heavy solutions xc are obtained when the regulon is kept constant as long as possible.Starting from (x, u), the state xc(·) of the heavy solutions evolves according to

xc(t) = xeut

and reaches b at timelog( b

x)u

(half the time needed for the inert evolution to reachequilibrium) with velocity equal to u > 0 and ub > 0, so that xc(·) leaves the interval[a, b] in finite time.

This evolution reaches the boundary of the graph of Uc atwarning state ξc(x, u) = be−

u2

2c

warning time σc(x, u) :=log( b

x)u

− u2c

Hence, once a velocity limit c is fixed, the heavy solution evolves with constantregulon u until the last instant σc(x, u) when the state reaches ξc(x, u) and the velocityof the regulon α(ξc(x, u), u) = c. This is the last time when the regulon has to changeby taking

uc(t) = u− c

(t−

log(

bx

)u

+u

2c

)so that the evolution follows the inert solution starting at (ξc(x, u), u). It reachesequilibrium (b, 0) at time t? := σc(x, u) + u

2c:

t? :=log(

bx

)u

+u

2c

with an advance equal to σc(x, u) over the inert solution.

The heavy evolution (xc(·), uc(·)) is associated with the heavy feedback uc definedby

uc(y) :=

u if x ≤ y ≤ ξc(x, u)√c r](y) if ξc(x, u) ≤ y < b]

b if y = b

The heavy feedback has an Allee effect on the interval [x, ξc(x, u)] and a logistic effecton [ξc(x, u), b]. Bounding the inertia by c, the feedback governing the heavy evolutionmaximizes the Allee effect.

Indeed, we observe that the part of the graph of any feedback u passing through(x, u) (u = u(x)) has an Allee effect only when it lies above the horizontal line passing

22

through (x, u). It necessarily intersects the graph of√

c r] before reaching (ξc(x, u), u).The Allee effect of the heavy evolution is weak in the sense that the velocity of theregulon is equal to 0 instead of being strictly positive. But it lasts longer.

In summary, the heavy evolution under bound c > α(x, u) is described by thefollowing formulas: the regulons are equal to

uc(t) =

u if t ∈[0,

log( bx)

u− u

2c

]u− c

(t− log( b

x)u

+ u2c

)if t ∈

[log( b

x)u

− u2c

,log( b

x)u

+ u2c

]and the states to xc(t) =

xeut if t ∈[0,

log( bx)

u− u

2c

]

be−u2

2c+u

(t−

log( bx)

u+ u

2c

)−

u2

t−log( b

x)u + u

2c

2

2

if t ∈[

log( bx)

u− u

2c,

log( bx)

u+ u

2c

]Remark: — Let f : [a, b] 7→ [0,∞[ be a function such that the primitives Fa(x) :=

∫ x

adx

f(x)

and Fb(x) :=∫ b

xdx

f(x) exist. In the above examples, we took f(x) = x, so that Fa(x) = log(

xa

)and

Fb(x) = log(

bx

). We can adapt the former formulas to the case of control systems of the form

x′(t) = u(t)f(x(t))

defined on the interval [a, b]. The corresponding feedbacks r] and r[ are given by

r](x) :=√

2Fb(x) & r[(x) :=√

2Fa(x)

the derivatives of which are given by

r] ′(x) :=

−1r](x)f(x)

& r[ ′(x) :=

1r[(x)f(x)

The regulation map Uc is equal to

Uc(x) :=√

c[−r[(x),+r](x)

]

23

so that the metaviability constrained set [a, b] ∈ R is viable under the control system

x′(t) = u(t)f(x(t)) where u′(t) ∈ Uc(x(t))

under bounded inflation. We may choose among the evolutions governed by this system the inertevolution, or the heavy evolution maximizing the Allee effect on the interval for a given inertia boundc as we did for x′(t) = u(t)x(t).

5.3.4 The Heavy Hysteresis Cycle

The heavy evolution (xc(·), uc(·)) starting at (x, u) where u > 0 stops when reachingthe equilibrium (b, u) at time t? := σc(x, u) + u

2c.

But as (b, u) lies on the boundary of [a, b]×R, there are (many) other possibilitiesto find evolutions starting at (b, 0) remaining viable while respecting the velocity limiton the regulons.

For instance, for t ≥ t?, we keep the evolutions xh(t) defined by

∀ t ≥ t?, xh(t) = xeut− u2t2

4 log( bx)

associated with the regulons

∀ t ≥ t?, uh(t) = u

(1− ut

2 log(

bx

))

The metastate (xh(·), uh(·)) ranges over the graph of the map −√

c r]. Therefore theregulon uh(t) and the state velocity x′h(t) become negative, so that the population sizexh(t) starts decreasing. The velocity of the negative regulon is constant.

But it is no longer viable on the interval [a, b], because with such a strictly negativevelocity, xh(·) leaves [a, b] in finite time. Hence regulons have to be switched before theevolution leaves the graph of Uc by crossing through the graph of −

√c r[.

Therefore, letting the heavy solution bypass the equilibrium by keeping its velocityequal to −c instead of switching it to 0, allows us to build a periodic evolution bytaking velocities of regulons equal successively to 0, −c, 0, +c, and so on. We obtainin this way a periodic evolution showing an hysteresis property: The evolution oscillatesbetween a and b back and forth by ranging alternatively two different trajectories on themetastate space X × U .

More precisely, we introduce the following notations: Denote by ac(u) and bc(u) theroots

24

ac(u) = aeu2

2c & bc(u) = be−u2

2c

of the equations r[(x) = u and r](x) = u. Then x? := ac(u) = bc(u) if and only ifu :=

√c u? where

x? :=√

ab & u? :=

√log

(b

a

)Therefore ac(u) ≤ bc(u) if and only if u ≤

√cu?.

We also set

τ ?(u) = 2log(

ba

)u

The periodic heavy hysteresis cycle xh(·) (of period 2τ ?(u) + 3uc) is described in the

following way:

1. The metastate (xh(·), uh(·)) starts from (ac(u), u) by taking the velocity of theregulon equal to 0. It remains viable on the time interval [0, τ ?(u) − u

2c] until it

reaches the metastate (bc(u), u).

2. The metastate (xh(·), uh(·)) starts from (bc(u), u) at time τ ?(u) − u2c

by takingthe velocity of the regulon equal to −c. It ranges over the graph of

√c r] on the

time interval [τ ?(u)− u2c

, τ ?(u) + 3u2c

] until it reaches the metastate (bc(u),−u).

3. The metastate (xh(·), uh(·)) starts from (bc(u),−u) at time τ ?(u) + 3u2c

by takingthe velocity of the regulon equal to 0. It remains viable on the time interval[τ ?(u) + 3u

2c, 2τ ?(u) + u

c] until it reaches the metastate (ac(u),−u).

4. The metastate (xh(·), uh(·)) starts from (ac(u),−u) at time 2τ ?(u) + uc

by takingthe velocity of the regulon equal to +c. It ranges over the graph of

√c r[ on the

time interval [2τ ?(u) + uc, 2τ ?(u) + 3u

c] until it reaches the metastate (ac(u), u).

For the limiting case when u :=√

c u?, it becomes the inert hysteresis cycle xh(·) (ofperiod 4u?

√c) described in the following way:

1. The metastate (xh(·), uh(·)) starts from (x?,√

c u?) at time 0 with the velocity ofthe regulon equal to −c. It ranges over the graph of

√c r] on the time interval

[0, 2u?√

c] until it reaches the metastate (x?,−

√c u?),

2. The metastate (xh(·), uh(·)) starts from (x?,−√

c u?) at time 2u?√

cby taking the

velocity of the regulon equal to +c. It ranges over the graph of√

c r[ on the timeinterval [2u?

√c, 4u?√

c] until it reaches the metastate (x?,

√c u?).

25

5.4 Verhulst and Graham

Biologists and ecologists are rather interested to long horizons and survival — viability— problems whereas economists most of the time are preoccupied with short horizons,concentrating on efficiency and substitutability of commodities (see [38, Gabay] for thisbasic point).

The simplest ecological models are based on the logistic dynamics proposed byVerhulst. This evolution is slowed down by economic activity which depletes it. Thebasic model was originated by Graham in 1935 (see [39, Graham]) and taken up bySchaeffer in 1954 (see [51, Schaeffer]), where it is assumed that the exploitation rate isproportional to the biomass and the economic activity. The equilibria of such dynamicsare called in the literature the sustainable yields. The purpose of many investigationsis to find an equilibrium yield maximizing some profit function. Beyond this staticapproach, optimal intertemporal optimization was used in economics for proposing“bang-bang” solutions, which may be not viable economically, and which bypass theinertia principle, translating many rigidities in the way of operating the capture of therenewable resource.

We present here a qualitative study by Luc Doyen and Daniel Gabay (see [30, Doyen& Gabay] for more details) dealing with the viability of both economic and ecologicalconstraints (see also [52, Scheffran]).

We denote by x ∈ R+ the biomass of the renewable resource and by v ∈ R+ theeconomic effort for exploiting it, playing the role of the regulon. The state and theregulon is subjected to viability constraints of the form

1. Ecological constraints∀ t ≥ 0, 0 ≤ x(t) ≤ b

where b is the carrying capacity of the resource.

2. Economic constraints

∀ t ≥ 0, cv(t) + C ≤ γv(t)x(t)

where C ≥ 0 is a fixed cost, c ≥ 0 the unit cost of economic activity and γ ≥ 0the price of the resource.

3. Production constraints∀ t ≥ 0, 0 ≤ v(t) ≤ v

where v is maximal exploitation effort.

26

We assume that

γb > c &C

γb− c≤ v

Therefore, setting a :=C + cv

γv, the economic constraint implies that

∀ t ≥ 0, x(t) ∈ [a, b]

The above constraints are summarized under the set-valued map U : [a, b] ; R+

defined by

∀ x ∈ [a, b], := V (x) :=

[C

γx− c, v

]The dynamics involve the Verhulst logistic dynamics and the Schaeffer proposal:

(i) x′(t) = rx(t)

(1− x(t)

b

)− v(t)x(t)

(ii) v(t) ∈ V (x(t)) :=

[C

γx(t)− c, v

] (11)

The “equilibrium curve” is here the union of the singleton 0, 0 and the graph of

the Verhulst feedback v(x) = r(1− x

b

). They are called the “sustainable yields” in

the literature. They are stable and viable if the graph of v(x) :=C

γx− cintersects

the equilibrium line: viable equilibrium belong to the interval [x−, x+] where x± arethe real roots of the equation

γrx2 − rx(γb− c) + (C + rc)b = 0

Setting v± := r(1− x±

b

), we can see that [a, b] is viable whenever v− ≥ v, i.e., if and

only if the growth rate r is large enough:

r ≥ bγv2 + rC

bγ − c

Otherwise, it easy to check that the viability kernel of the interval [a, b] undersystem (11) is equal to the interval [x−, x+].

We set a bound to the velocity of the economic effort, which translates the rigidityof the economic behavior:

∀ t ≥ 0, −d ≤ v(t) ≤ +d

27

Therefore the metasystem governing the evolution of the state and the regulon aredescribed by i) x′(t) = rx(t)

(1− x(t)

b

)− v(t)x(t)

ii) |v(t)| ≤ d(12)

and the metaconstrained set is the graph of V .

Figure 3: Source: Doyen and Gabay

The viability kernel is equal to

Viab(Graph(V )) = (x, v) ∈ Graph(V ) | x ≥ ρ](v)

where ρ] is the solution to the differential equation

−ddρ]

dv= r(1− ρ](v)

b)− uρ](v)

satisfying the initial condition ρ](v−) = x−.

In this case, danger happens when the economic effort v is larger than v−. Onecannot maintain the economic effort constant. The heavy evolution consists in keeping

28

the economic effort constant as long as the biomass is larger than ρ](v). At thislevel, the economic effort has to be drastically reduced with the velocity d, i.e., usingv(t) = −d, while the biomass decreases until it reaches the level v−.

5.5 Towards Dynamical Games

We have chosen the Verhulst feedback x 7→ r(1 − xb) to represent the growth of the

resource to be exploited for anchoring our study in history. But we could have chosen

instead the inert feedback x 7→√

c√

2 log(

bx

)and study the viability of the interval

[a, b] under the system(i) x′(t) = x(t)

(√

c

√2 log

(b

x(t)

)− v(t)

)(ii) v(t) ∈ V (x(t)) :=

[C

γx(t)− c, v

] (13)

Actually, since we do not really know what are the dynamical equations governing theevolution of the resource, we could take Malthusian feedbacks u in a given class U ofcontinuous feedbacks as parameters and study the viability kernel Viabu([a, b]) of theinterval [a, b] under the system (i) x′(t) = (u(x(t))− v(t))x(t)

(ii) v(t) ∈ V (x(t)) :=

[C

γx(t)− c, v

](14)

This suggest to introduce the conditional viability kernel⋂u∈U

Viabu([a, b])

This is the very first question with which one can study viability issues of dynam-ical games, that are dynamical systems parameterized by two parameters under thecontrol of two different players. We refer to papers by Pierre Cardaliaguet and MarcQuincampoix who studied these questions.

We can also study “metagames” by setting bounds c and d on the velocities of thegrowth rate u(t) and the exploitation effort v(t), regarded as metacontrols, whereasthe metastates of the metagame are the triples (x, u, v):

29

Figure 4: Example of crisis function, that measures the time spent outside the viabilitykernel. Source: Patrick Saint-Pierre

30

(i) x′(t) = (u(t)− v(t))x(t)(ii) u′(t) ∈ B(0, c)(iii) v′(t) ∈ B(0, d)

(15)

subjected to the viability constraints

u(t) ∈ R &C

γx(t)− c≤ v(t) ≤ v

6 Viability and Optimality

Interestingly enough, viability theory implies the dynamical programming approachfor optimal control.

Denote by S(x) the set of pairs (x(·), u(·)) solutions to the control problem (3)starting from x at time 0. We consider the minimization problem

V (T, x) = inf(x(·),u(·))∈S(x) inft∈[0,T ](c(T − t, x(t)) +

∫ t

0

l(x(τ), u(τ))dτ

)where c and l are cost functions. We can prove that the graph of the value function(T, x) 7→ V (T, x) of this optimal control problem is the capture basin of the graph ofthe cost function c under an auxiliary system involving (f, U) and the cost functionl. The regulation map of this auxiliary system provides the optimal solutions and thetangential conditions furnish Hamilton-Jacobi-Bellman equations of which the valuefunction is the solution (see for instance ([34, 35, 36, 37, Frankowska] and more re-cently, [10, Aubin]). This is a very general method covering numerous other dynamicoptimization problems.

However, contrary to optimal control theory, viability theory does not require anysingle decision-maker (or actor, or player) to guide” the system by optimizing an in-tertemporal optimality criterion5.

Furthermore, the choice (even conditional) of the controls is not made once and forall at some initial time, but they can be changed at each instant so as to take into accountpossible modifications of the environment of the system, allowing therefore for adaptationto viability constraints.

5the choice of which is open to question even in static models, even when multicriteria or severaldecision makers are involved in the model.

31

Finally, by not appealing to intertemporal criteria, viability theory does not requireany knowledge of the future6 (even of a stochastic nature.) This is of particular impor-tance when experimentation7 is not possible or when the phenomenon under study isnot periodic. For example, in biological evolution as well as in economics and in theother systems we shall investigate, the dynamics of the system disappear and cannot berecreated.

Hence, forecasting or prediction of the future are not the issues which we shall addressin viability theory.

However, the conclusions of the theorems allow us to reduce the choice of possibleevolutions, or to single out impossible future events, or to provide explanation of somebehaviors which do not fit any reasonable optimality criterion.

Therefore, instead of using intertemporal optimization8 that involves the future,viability theory provides selection procedures of viable evolutions obeying, at each in-stant, state constraints which depend upon the present or the past. (This does notexclude anticipations, which are extrapolations of past evolutions, constraining in thelast analysis the evolution of the system to be a function of its history.)

7 Attractors and Backward Viability Kernels

Viability and invariance kernels provide useful tools for analyzing the behavior of evo-lutions.

Fir instance, if there exists a neighborhood Ω of C such that C := ViabS(Ω), thenstarting from Ω\C, all evolutions leave the neighborhood Ω in finite time withoutreaching C. Such a subset cannot be reached in finite time for outside.

Also, consider the case when C1 ⊂ K and C2 ⊂ K are two closed subsets coveringK: K = C1 ∪ C2. Setting c0 := C1 ∩ C2, we assume that

C0 ∩ ViabS(C1) = C0 ∩ ViabS(C1)

6Most systems we investigate do involve myopic behavior; while they cannot take into account thefuture, they are certainly constrained by the past.

7Experimentation, by assuming that the evolution of the state of the system starting from a giveninitial state for a same period of time will be the same whatever the initial time, allows one to translatethe time interval back and forth, and, thus, to know” the future evolution of the system.

8which can be traced back to Sumerian mythology which is at the origin of Genesis: one Decision-Maker, deciding what is good and bad and choosing the best (fortunately, on an intertemporal basis,thus wisely postponing to eternity the verification of optimality), knowing the future, and havingtaken the optimal decisions, well, during one week...

32

Then all evolutions x(·) ∈ S(x) starting from x ∈ ViabS(K)\(ViabS(C1)∪ViabS(C2))viable in K fluctuate back and forth between C1 to C2 in finite times without enteringthe union ViabS(C1) ∪ C.

Observe that the above assumption is satisfied whenever the subsets C1 and C2 arerepellers, or whenever they satisfy

ViabS(C1) ∪ ViabS(Ci) ⊂ Ci\C0, i = 1, 2

One can also prove that whenever the viability kernel under a backward (or negative)system is contained in the interior of an absorbing set, then the attractor is containedin the viability kernel under the backward system.

We illustrate this fact by the example of the Lorenz system.

Studying a simplified meteorological model made of a system of three differentialequations, the American meteorologist Edward Lorenz discovered in the beginning ofthe 1960‘s that for certain parameters where the system has three repelling equilibria,the attractor” was quite strange,chaotic” in the sense that evolutions approach oneequilibrium while circling around it, then suddenly leave away toward an other equilib-rium around which it turns again, and so on. In other words, this behavior is strangein the sense that the limit set of an evolution is not a trajectory of a periodic solu-tion9. After Henri Poincare discovering the lack of predictability of evolutions of thethree-body problem, Lorenz presented in 1979 a lecture to the American Associationfor the advancement of Sciences entitled:

1 Lorenz’s Butterfly

Predictability: Does the flap of a butterfly’s wing in Brazil set off a tornado inTexas ?

Paul Henri Thiry, baron d’Holbach (1694-1778), preceded him three centuries earlierin his book Systeme de la nature (1770):

9as in the case of two-dimensional systems, thanks to the the Poincare-Bendixon Theorem.

33

2 Baron d’Holbach

Finally, if everything in nature is linked to everything, if all motions are born from eachother although they communicate secretely to each other unseen from us, we must hold forcertain that there is no cause small enough or remote enough which sometimes does not bringabout the largest and the closest effects on us. The first elements of a thunderstorm maygather in the arid plains of Lybia, then will come to us with the winds, make our weatherheavier, alter the moods and the passions of a man of influence, deciding the fate of severalnations.Enfin, si tout est lie dans la nature, si tous les mouvements y naissent les uns des autresquoique leurs communications secretes echappent souvent a notre vue, nous devons etreassures qu’il n’est point de cause si petite ou si eloignee qui ne produise quelque fois leseffets les plus grands et les plus immediats sur nous-memes. C’est peut-etre dans les plainesarides de la Lybie que s’amassent les premiers elements d’un orage qui, portes par les vents,viendra vers nous, appesantira notre atmosphere, influera sur le temperament et les passionsd’un homme que ses circonstances mettent a la portee d’influer sur beaucoup d’autres, et quidecidera, d’apres ses volontes, du sort de plusieurs nations”

Lorenz introduced the following variables

1. x, proportional to the intensity of convective motion,

2. y, proportional to the temperature difference between ascending and descendingcurrents,

3. z, proportional to the distortion (from linearity) of the vertical temperature pro-file.

Their evolution is governed by the following system of differential equations:(i) x′(t) = σy(t)− σx(t)(ii) y′(t) = rx(t)− y(t)− x(t)z(t)(iii) z′(t) = x(t)y(t)− bz(t)

(16)

where σ > b + 1.

Differential equations are continuous-time versions of discrete systems, and can beapproximated by discrete systems: With any h > 0, we associate the discretization

(i) xn+1 = xn + h(σyn − σxn)(ii) yn+1 = yn + h(rxn − yn − xnzn)(iii) z′n = zn + h(xnyn − bzn)

(17)

We also introduce the backward (or negative) discretized system system

34

(i) xn+1 = xn − h(σyn − σxn)(ii) yn+1 = yn − h(rxn − yn − xnzn)(iii) z′n = zn − h(xnyn − bzn)

(18)

The parameter r is the normalized Rayleigh number. If r ∈]0, 1[, then 0 is anasymptotically stable equilibrium. If r = 1, the equilibrium 0 is neutrally stable”.When r > 1, the equilibrium 0 becomes unstable and two more equilibria appear:

c1 :=(√

b(r − 1),√

b(r − 1), r − 1)

& c2 :=(−√

b(r − 1),−√

b(r − 1), r − 1)

They are stable when

1 < r? :=σ(σ + b + 3)

σ − b− 1

and unstable when r > r?.

We consider the backward viability kernel of the cube [−α, +α] × [−β, +β] ×[−γ, +γ]. The Lorenz attractor is contained in this backward viability kernel. Forσ = 10, b = 8

3and r = 28, the three critical points are unstable.

We observe that taking K1 := Viab←−S (K)∩R2+×R and K2 := K∩R2

−×R containingthe equilibria c1 and c2 respectively, the viability kernels of Kj are strictly containedin Kj respectively. Therefore, any solution starting from Viab←−S (K)\(Viab(K1) ∪Viab(K2)) fluctuate back and forth between K1 and K2.

8 Restoring Viability

The above example shows that there are no reasons why an arbitrary subset K shouldbe viable under a control system. Therefore, the problem of reestablishing viabilityarises. One can imagine several methods for this purpose:

1. Keep the constraints and change initial dynamics by introducing regulons thatare viability multipliers”

2. or change the initial conditions by introducing a reset map Φ mapping any stateof K to a (possibly empty) set Φ(x) ⊂ X of new initialized states” (impulsecontrol),

3. Keep the same dynamics and looking for viable constrained subsets by lettingthe set of constraints evolve according to mutational equations, as in [6, Aubin].

We shall describe succinctly these methods.

35

Figure 5: Examples of viability kernel of a cube K in R3 of the forward and backwardLorenz system where σ > b + 1. One can prove that whenever the backward viabilitykernel is contained in the interior of K, the backward viability kernel is contained in theforward viability kernel. The famous attractor is contained in the backward viabilitykernel. The color scale provides the third coordinates. Source: Patrick Saint-Pierre.

36

Figure 6: Four views of the viability kernel under the backward system and the tra-jectory of one solution approaching the viability kernel, which contains the attractor.Source: Patrick Saint-Pierre.

37

Figure 7: Usually, the Lorenz attractor is illustrated (but not computed) by computingsolutions to the systems and drawing their trajectories. By the way, these trajectoriesdo not belong to the attractor (except when they start from the attractor), just ap-proach it. In our example, we draw the trajectory of only one solution and we superposeit to the viability kernel under the backward system. Source: Patrick Saint-Pierre.

38

8.1 Designing Regulons

When an arbitrary subset is not viable under an intrinsic” system x′(t) = f(x(t)),the question arises to modify the dynamics by introducing regulons and designingfeedbacks so that the constrained subset K becomes viable under the new system.Using the above results and characterizations, one can design several mechanisms. Wejust describe three of them, that are described in more details in [3, Aubin]:

8.1.1 Viability Multipliers

If the constrained set K is of the form

K := x ∈ X such that h(x) ∈ M

where h : X 7→ Z := Rm and M ⊂ Z, we regard elements u ∈ Z as viability multi-pliers, since they play a role analogues to Lagrange multipliers in optimization underconstraints. They are candidates to the role of regulons regulating such constraints.Indeed, we can prove that K is viable under the control system

x′j(t) = fj(x(t)) +m∑

k=1

∂hk(x(t))

∂xj

uk(t)

in the same way than the minimization of a function x 7→ J(x) over a constrained setK is equivalent to the minimization without constraints of the function

x 7→ J(x) +m∑

k=1

∂hk(x)

∂xj

uk

for an adequate Lagrange multiplier u ∈ Z.

8.1.2 Connection Matrices

Instead of introducing viability multipliers, we can use a connection matrix W ∈L(X, X) as in neural networks (see [2, Aubin] for instance). We replace the intrin-sic system x′ = If(x) (where I denotes the identity) by the system

x′(t) = W (t)f(x(t))

and choose the connection matrices W (t) in such a way that the solutions of the abovesystem are viable in K.

39

The evolution of the state no longer derives from intrinsic dynamical laws validin the absence of constraints, but requires some self-organization” — described byconnection matrices — that evolves together with the state of the system in order toadapt to the viability constraints, the velocity of connection matrices describing theconcept of emergence”. The evolution law of both the state and the connection matrixresults from the confrontation of the intrinsic dynamics to the viability constraints.

One can prove that the regulation by viability multipliers u is a particular case ofthe regulation by connection matrices W : We associate with x and u the matrix Wthe matrix the entries of which are equal to

wi,j := − fi(x)

‖f(x)‖2

m∑k=1

∂hk(x(t))

∂xj

uk(t) if i 6= j

and, of j = i,

wi,i := 1− fi(x)

‖f(x)‖2

m∑k=1

∂hk(x(t))

∂xi

uk(t)

The converse is false in general. However, if we introduce the connectionist complex-ity index ‖I − W‖, one can prove that the viable evolutions governed with connectionmatrices minimizing at each instant the connectionist complexity index are actuallygoverned by the viability multipliers with minimal norms .

The concept of heavy evolution when the regulon is a connection matrix amounts tominimize the norm of the velocity W ′(t) of the connection matrix W (t) starting fromthe identity matrix, that can be used as measure of dynamical connectionist complexity.Such a velocity could encapsulate the concept of emergence” in the systems theoryliterature. The connection matrix remains constant — without emergence — as longas the viability of the system is not at stakes, and evolves as slowly as possible otherwise.

8.1.3 Hierarchical Organization

One can also design dynamic feedbacks for obeying constraints of the form

∀ t ≥ 0, Wm−1(t) · · ·W j(t) . . . W 0(t)x(t) ∈ M ⊂ Y

is satisfied at each instant. Such constraints can be regarded as describing a sequenceof m planning procedures.

Introducing at each level of such a hierarchical organization” xi(t) := W i(t)xi−1(t),on can design dynamical systems modifying the evolution of the intermediate statesxi(t) governed

x′j(t) = gj(xj(t))

40

and the entries of the matrices W i(t) by

wj′

k,l(t) = ejk,l(W

j(t))

Using viability multipliers, one can prove that dynamical systems of the form

(1)0 x′0(t) = g0(x0(t))−W 0?(t)p1(t)(j = 0)

(1)j x′j(t) = gj(xj(t)) + pj(t)−W j?(t)pj+1(t)

(j = 1, . . . ,m− 1)(1)m x′m(t) = gm(xm(t)) + pm(t) (j = m)

(2)j(k,l) wj′k,l(t) = ej

k,le(Wj(t))− xjk

(t)p(j+1)l(t)

(j = 0, . . . ,m− 1, k = 1, . . . , n, l = 1, . . . m)

govern viable solutions. Here, the viability multipliers pj are used as messages to bothmodify the dynamics of the jth level state xj(t) and to link to consecutive levels j + 1and j.

Furthermore, the connection matrices evolve in a Hebbian way, since the correctionof the velocity Wj′k,l of the entry is the product of the kth component of the j-levelintermediate state xj and the lth component of the (j + 1)-level viability multiplierpj+1.

8.1.4 Evolution of the Architecture of a Network

This hierarchical organization is a particular case of a network. Indeed, the simplestgeneral form of coordination is to require that a relation between actions of the formg(A(x1, . . . , xn)) ∈ M must be satisfied. Here A :

∏ni=1 Xi 7→ Y is a connectionist

operator relating the individual actions in a collective way. Here M ⊂ Y is the subset ofthe resource space Y and g is a map, regarded as a propagation map.

We shall study this coordination problem in a dynamic environment, by allowingactions x(t) and connectionist operators A(t) to evolve according to dynamical systemswe shall construct later. In this case, the coordination problem takes the form

∀ t ≥ 0, g(A(t)(x1(t), . . . , xn(t))) ∈ M

However, in the fields of motivation under investigation, the number n of variablesmay be very large. Even though the connectionist operators A(t) defining the architec-ture” of the network are allowed to operate a priori on all variables xi(t), they actuallyoperate at each instant t on a coalition S(t) ⊂ N := 1, . . . , n of such variables, varyingnaturally with time according to the nature of the coordination problem.

Therefore, our coordination problem in a dynamic environment involves the evolution

41

1. of actions x(t) := (x1(t), . . . , xn(t)) ∈∏n

i=1 Xi,

2. of connectionist operators AS(t)(t) :∏n

i=1 Xi 7→ Y ,

3. acting on coalitions S(t) ⊂ N := 1, . . . , n of the n actors

and requires that∀ t ≥ 0, g

(AS(t)(x(t))S⊂N

)∈ M

where g :∏

S⊂N YS 7→ Y .

The question we raise is the following. Assume that we may know the intrinsic lawsof evolution of the variables xi (independently of the constraints), of the connectionistoperator AS(t) and of the coalitions S(t), there is no reason why collective constraintsdefining the above architecture are viable under these dynamics, i.e, satisfied at eachinstant.

One may be able, with a lot of ingeniousness and the intimate knowledge of a givenproblem, and for simple constraints”, to derive dynamics under which the constraintsare viable.

However, we can investigate whether there is a kind of mathematical factory pro-viding classes of dynamics correcting” the initial (intrinsic) ones in such a way that theviability of the constraints is guaranteed. One way to achieve this aim is to use theconcept of viability multipliers” q(t) ranging over the dual Y ∗ of the resource space Ythat can be used as controls” involved for modifying the initial dynamics.

This may allow us to provide an explanation of the formation and the evolution ofthe architecture of the network and of the active coalitions as well as the evolution of theactions themselves.

In order to tackle mathematically this problem, we shall

1. restrict the connectionist operators to be multiaffine, and thus, involve tensorproducts,

2. next, allow coalitions S to become fuzzy coalitions so that they can evolve con-tinuously.

Fuzzy coalitions χ = (χ1, . . . , χn) are defined by memberships χi ∈ [0, 1] between0 and 1, instead of being equal to either 0 or 1 as in the case of usual coalitions. Themembership γS(χ) :=

∏i∈S χi is by definition the product of the memberships of the

members i ∈ S of the coalitions. Using fuzzy coalitions allows us to defined theirvelocities and study their evolution.

42

The viability multipliers q(t) ∈ Y ∗ can be regarded as regulons, i.e., regulationcontrols or parameters, or virtual prices in the language of economists. They arechosen adequately at each instant in order that the viability constraints describing thenetwork can be satisfied at each instant, and the main theorem of this paper guaranteesthat it is possible. Another one tells us how to choose at each instant such regulons(the regulation law).

For each actor i, the velocities x′i(t) of the state and the velocities χ′i(t) of itsmembership in the fuzzy coalition χ(t) are corrected by subtracting

1. the sum over all coalitions S to which he belongs of adequate functions weightedby the membership γS(χ(t)):

2. the sum over all coalitions S to which he belongs of the costs of the constraintsassociated with connectionist tensor AS of the coalition S weighted by the mem-bership γS\i(χ(t)). This type of dynamics describes a panurgean effect . The(algebraic) increase of actor i’s membership in the fuzzy coalition aggregatesover all coalitions to which he belongs the cost of their constraints weighted bythe products of memberships of the actors of the coalition other than him.

As for the correction of the velocities of the connectionist tensors AS, their correc-tion is a weighted multi-Hebbian” rule: for each component of the connectionist tensor,the correction term is the product of the membership γS(χ(t)) of the coalition S, ofthe components xik(t) and of the component qj(t) of the regulon.

In other words, the viability multipliers appear in the regulation of the multiaffineconnectionist operators under the form of tensor products, implementing the Hebbianrule for affine constraints (see [2, Aubin])), and multi-Hebbian” rules for the multiaffineones, as in [14, Aubin & Burnod].

Even though viability multipliers do not provide all the dynamics under which aconstrained set is viable, they provide classes of them exhibiting interesting structuresthat deserve to be investigated and tested in concrete situations.

Remark: Learning Laws and Supply and Demand Law — It is curiousthat both the standard supply and demand law, known as the Walrasian tatonnementprocess, in economics and the Hebbian learning law in cognitive sciences were the start-ing points of the Walras General Equilibrium Theory and Neural Networks. In boththeories, this choice of putting such adaptation laws as a prerequisite led to the samecul de sacs. Starting instead from dynamic laws of agents, viability theory providesdedicated adaptation laws”, so to speak, as the conclusion of the theory instead as the

43

primitive feature. In both cases, the point is to maintain the viability of the system,that allocation of scarce commodities satisfy the scarcity constraints in economics, thatthe viability of the neural network is maintained in the cognitive sciences. For neu-ral networks, this approach provides learning rules that possess the features meetingthe Hebbian criterion. For the general networks studied here, these features are stillsatisfied in spirit. 2

We refer to [11, Aubin] for more details on this topic.

8.2 Impulse Systems

There are many other dynamics that obey the inertia principle, among which heavyviable evolutions are the smoothest ones. At the other extreme, on can study also the(discontinuous) impulsive” variations of the regulon. Instead of waiting the system tofind a regulon that remains constant for some length of time, as in the case of heavysolutions, one can introduce another (static) system that resets a new constant regulonwhenever the viability is at stakes, in such a way that the system evolves until the nexttime when the viability is again at stakes.

This regulation mode is a particular case of what are called impulse control in con-trol theory (see for instance [7, Aubin], [23, Aubin, Lygeros, Quincampoix, Sastry &Seube]), hybrid systems in computer sciences and Integrate and Fire models in neurobi-ology, etc. Impulse systems are described by a control system governing the continuousevolution of the state between two impulsions, and a reset map resetting new initialconditions whenever the state enters the domain of the reset map.

An evolution governed by an impulse dynamical system, called a run in the controlliterature, is defined by a sequence of cadences (periods between two consecutive im-pulse times), of reinitialized states and of motives describing thecontinuous” evolutionalong a given cadence, the value of a motive at the end of a cadence being reset as thenext reinitialized state of the next cadence.

Given an impulse system, one can characterize the map providing both the nextcadence and the next reinitialized state without computing the impulse system, as a set-valued solution of a system of partial differential inclusions. It provides a summary” ofthe behavior of the impulse system from which one can then reconstitute the evolutionsof the continuous part of the run by solving the motives of the run that are the solutionsto the dynamical system starting at a given reinitialized state.

A cadenced run is defined by constant cadence, initial state and motive, where thevalue at the end of the cadence is reset at the same reinitialized state. It plays the roleofdiscontinuous” periodic solutions of a control system.

44

We prove in [21, Aubin & Haddad] that if the sequence of reinitialized states of arun converges to some state, then the run converges to a cadenced run starting fromthis state, and that, under convexity assumptions, that a cadenced run does exist.

8.3 Mutational Equations Governing the Evolution of the Con-strained Sets

Alternatively, if the viability constraints can evolve, another way to resolve a viabilitycrisis is to relax the constraints so that the state of the system remains inside the newviability set. For that purpose, kind of differential equation governing the evolutionof subsets, called mutational equations, have been designed. This requires an adequate

definition of the velocityK (t) of a tube” t ; K(t), called mutation, that makes

sense and allows us to prove results analogous to the ones obtained in the domain ofdifferential equations. This can be done, but cannot be described in few lines.

Hence the viability problem amounts to find evolutions of both the state x(t) andthe subset K(t) to the system

i) x′(t) = f(x(t), K(t)) (differential equation)

ii)K (t) 3 m(x(t), K(t)) (mutational equation)

(19)

viable in the sense that for every t, x(t) ∈ K(t). For more details, see [6, Aubin].

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Contents

1 Introduction 1

2 The Mathematical Framework 6

2.1 Viability and Capturability . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 The Evolutionary System . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Viability Kernels and Capture basins . . . . . . . . . . . . . . . . . . . 8

3 Characterization of Viability and/or Capturability 9

3.1 Tangent Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2 The Adaptive Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.3 The Viability Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.4 The Adaptation Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.5 Planning Tasks: Qualitative Dynamics . . . . . . . . . . . . . . . . . . 11

3.6 The Metasystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Selecting Viable Feedbacks 12

4.1 Static Feedbacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.2 Dynamic Feedbacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.3 Heavy Evolutions and the Inertia Principle . . . . . . . . . . . . . . . . 13

5 Management of Renewable Resources 14

5.1 From Malthus to Verhulst and Beyond . . . . . . . . . . . . . . . . . . 14

5.2 Discrete Versus Continuous Time . . . . . . . . . . . . . . . . . . . . . 16

5.3 Introducing Inertia Bounds . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.3.1 Affine Feedbacks and the Verhulst Logistic Equation . . . . . . 18

5.3.2 Inert Evolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.3.3 Heavy Evolutions . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5.3.4 The Heavy Hysteresis Cycle . . . . . . . . . . . . . . . . . . . . 23

5.4 Verhulst and Graham . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.5 Towards Dynamical Games . . . . . . . . . . . . . . . . . . . . . . . . . 28

6 Viability and Optimality 29

50

7 Attractors and Backward Viability Kernels 30

8 Restoring Viability 33

8.1 Designing Regulons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

8.1.1 Viability Multipliers . . . . . . . . . . . . . . . . . . . . . . . . 37

8.1.2 Connection Matrices . . . . . . . . . . . . . . . . . . . . . . . . 37

8.1.3 Hierarchical Organization . . . . . . . . . . . . . . . . . . . . . 38

8.1.4 Evolution of the Architecture of a Network . . . . . . . . . . . . 39

8.2 Impulse Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

8.3 Mutational Equations Governing the Evolution of the Constrained Sets 43