 |
(B.1) |
You can load the pdf version of this web page.
Code: C07E03
Title: Viable control of discrete time systems and applications
Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech, Université Paris-Est, Marne-la-Vallée, France
Director: Jean-François Delmas
Departamento de Matemática, Universidad Técnica Federico Santa María (UTFSM), Valparaíso, Chile
Director: Verónica Gruenberg
Michel De Lara, Ingénieur en chef des ponts et forêts
CERMICS, École des Ponts ParisTech, 6 et 8 avenue Blaise Pascal, Cité Descartes, 77455 Marne la Vallée Cedex 2, France
Phone +33 (0)1 64 15 36 21 Fax +33 (0)1 64 15 35 86 email: delara cermics.enpc.fr
Pedro Gajardo, Professor
Departamento de Matemática, Universidad Técnica Federico Santa María, Avda. España 1680, Valparaíso, Chile
Phone +56 32 265 44 90 ax +56 32 265 45 63 email: pedro.gajardo usm.cl
Michel De Lara, CERMICS, École des Ponts ParisTech, Université Paris-Est, Marne la Vallée
Vincent Martinet, UMR Économie publique, INRA Paris Grignon
Pedro Gajardo, Departamento de Matemática, Universidad Técnica Federico Santa María, Valparaíso, Chile
Hector Ramirez, Centro de Modelamiento Matemático (CMM), UMI CNRS, Universidad de Chile, Santiago de Chile.
Gestión de Recursos Pesqueros at Centro de Modelamiento Matemático (CMM)
Are also attached to the action the marine biologists Alejandro Zuleta and Pedro Rubilar (Centro de Estudios Pesqueros CEPES) and the economist Julio Pena Torres (Universidad Jesuita Alberto Hurtado).
M. De Lara, P. Gajardo, H. Ramirez.
Viable states for monotone harvest models.
In
Systems and Control Letters,
in press, 2010.
Vincent Martinet, Julio Pena Torres, Hector Ramirez and Michel De Lara, Evaluation of Management Procedures: Application to Chilean Jack Mackerel Fishery, IIFET 2010 Montpellier Proceedings.
Our group had the opportunity to present its researches at conferences in mathematics, in biology and in economics. See details in Appendix A.
See details in Appendix A.
The aim of this project was to provide mathematical tools to deal with dynamical systems under state and control constraints and to shed new lights on some applied viability problems especially in the economics and environmental fields.
Our first contribution has been to obtain lower and upper approximations of the viability kernel for a class of bio-economic models, by exploiting monotonicity properties. Numerical applications are given for two Chilean fisheries. We provided upper bounds for production which are interesting for managers in that they only depend on the model parameters, and not on the current stocks.
Our second contribution has been to introduce a new class of dynamic bargaining problems, that we coin ``bargaining problems with intertemporal maximin payoffs.'' Such problems may reflect sustainability issues, with conflicting objectives and long term issues.
Our third contribution has been to develop a theoretical framework to assess resources management procedures from a sustainability perspective, when resource dynamics is marked by uncertainty. Using stochastic viability, management procedures are ranked according to their probability to achieve economic and ecological constraints over time. This framework is applied to a fishery case-study, facing El Niño uncertainty. We study the viability of constant effort and constant quota strategies, when a minimal catch level and a minimal biomass are required. Conditions on the sustainability objectives are derived for the superiority of each of the two management methods.
We join the original version of the project in Appendix. Our objective of exploring the consequences of monotonicity properties in bio-economic models has been attained. We published our main results in the paper Viable harvest of monotone bioeconomic models in the journal Systems & Control Letters in 2010.
We also opened a new path with the analysis of bargaining problems with intertemporal maximin payoffs, and we used stochastic viability to develop a theoretical framework to assess resources management procedures.
As a consequence of the project, let us mention that Pedro Gajardo was invited to the Comité Científico de la Merluza de cola 2010, committee that depends from the Subsecretaría de Pesca - Chile (SUBPESCA), and where the assessment of the resource and the annual quotas are discussed. This invitation was made after we presented some of our results to SUBPESCA.
As mentioned in the Agenda of scientific activities in Appendix, the ECOS program has permitted many research meetings, the supervision of four students, the participation to at least six international conferences. One paper has been accepted for publication, while two other papers are currently written, involving all the members of the Chilean-French team.
We have as perspectives to continue the research initiated during the project, to apply conjointly to other research contracts, together with the training of students. During 2010 we will have at least one undergraduate student that will be co-trained (with CEPES: industrial partnership) in the topics of this project. On the other hand, we have as perspective for our research to implement actions in order to transfer the obtained knowledge to regulatory organism and industrial sectors (see Section 2.5.3).
Following the observations above, the ECOS program has deepened the relations between the teams. Our immediate objective is to finish our joint papers.
The following paper, in a peer-reviewed international journal, is co-signed by the two teams and explicitely thanks ECOS for its support.
M. De Lara, P. Gajardo, H. Ramirez.
Viable states for monotone harvest models.
In
Systems and Control Letters,
in press, 2010.
Michel De Lara visits Pedro Gajardo and Hector Ramirez at Centro de Modelamiento Matemático (CMM), Universidad de Chile, Santiago de Chile.
A new paper, Viable harvest of monotone bioeconomic models, between Michel De Lara, Pedro Gajardo and Hector Ramirez is initiated.
Two joint meetings with marine biologists Alejandro Zuleta, Pedro Rubilar (Centro de Estudios Pesqueros CEPES) and with economist Julio Pena Torres (Universidad Jesuita Alberto Hurtado) are organized. The visit at CEPES provides new applied themes; this will help prepare the work of two ENSTA students in 2008.
The mission costs (plane ticket and per diem) were not paid by French ECOS budget.
Vincent Martinet and Michel De Lara visit Pedro Gajardo and Hector Ramirez at CMM, Santiago de Chile, and at Universidad Técnica Federico Santa María, Valparaíso.
Two joint meetings with marine biologists Alejandro Zuleta, Pedro Rubilar (Centro de Estudios Pesqueros CEPES) and with economist Julio Pena Torres (Universidad Jesuita Alberto Hurtado) are organized.
The meetings provide two directions of research: direct applications of results on viable management to different fisheries models; formalization of management strategy evaluation. This gives two subjects for the ENSTA students in May 2008
A new paper Risk and Sustainability. Assessing Resource Management Procedures between Vincent Martinet, Julio Pena Torres, Hector Ramirez and Michel De Lara is initiated.
Michel De Lara visits Laurent Bonneau and Ingrid Chanefo from Délégation régionale pour le cône sud et le Brésil, Ambassade de France au Chili.
The plane ticket of Michel De Lara was paid by French INRIA budget, as well as daily expenses (accomodation, etc.).
The plane ticket of Vincent Martinet was paid by French ECOS budget. Daily expenses (accomodation, etc.) were supported by Chilean CONICYT budget and by Universidad Técnica Federico Santa María-Valparaíso
Cluster Sustainable Natural Resources Management within XIV Congreso Latino Ibero Americano de Investigación de Operaciones (CLAIO 2008)
Michel De Lara, Pedro Gajardo and Hector Ramirez give conferences.
Mission of Pedro Gajardo.
We continue working on our paper Viable harvest of monotone bioeconomic models.
The plane ticket of Pedro Gajardo was paid by Chilean CONICYT budget. Daily expenses (accomodation, etc.) were supported partly by French ECOS budget and partly by Universidad Técnica Federico Santa María, Valparaíso, Chile.
Mission of Hector Ramirez.
We continue working on our paper Viable harvest of monotone bioeconomic models.
The plane ticket of Hector Ramirez was paid by Chilean CONICYT budget. Daily expenses (accomodation, etc.) were supported partly by French ECOS budget and partly by the Centro de Modelamiento Matemático (CMM), Universidad de Chile, Santiago de Chile.
Mission of Vincent Martinet and Michel De Lara.
Thursday, April 16, meeting at CMM with Hector Ramirez and French students Pauline Dochez and Aurore Lavigne; meeting with Hector Ramirez, Pedro Gajardo, Julio Peña Torres, Alejandro Zuleta and Pedro Rubilar.
Friday, April 17, meeting at CEPES with Alejandro Zuleta, Pedro Rubilar, Hector Ramirez, Pedro Gajardo and French students Arthur Henriot, Pauline Dochez and Aurore Lavigne.
Tuesday, April 21, meeting at Depto de Economía, Universidad Alberto Hurtado with Julio Peña Torres: we work on our joint paper Risk and Sustainability. Assessing Resource Management Procedures.
Wednesday, April 22, meeting at CMM with Hector Ramirez and French students Pauline Dochez and Aurore Lavigne: ongoing work presentation; meeting at CMM with Julio Peña Torres.
Thursday, April 23, meeting at Universidad Técnica Federico Santa María-Valparaíso with French student Arthur Henriot; Vincent Martinet gives a conference Multi-criteria dynamic decision under uncertainty: A stochastic viability analysis and an application to sustainable fishery management at the seminar.
A new paper Sustainable thresholds, between Vincent Martinet, Pedro Gajardo, Hector Ramirez and Michel De Lara is initiated.
Friday, April 24, meeting with Laurent Bonneau, Consejero Regional de Cooperacion a la Delegacion Regional de Cooperacion para el cono sur y Brasil, Embajada de Francia en Chile.
The plane ticket of Michel De Lara was paid by French ECOS budget. Daily expenses (accomodation, etc.) were supported by Chilean CONICYT budget and by Universidad Técnica Federico Santa María-Valparaíso
The plane ticket of Vincent Martinet was paid by French ECOS budget. Daily expenses (accomodation, etc.) were supported by Chilean CONICYT budget and by Universidad Técnica Federico Santa María-Valparaíso
XXIII Jornada de Matemática de la Zona Sur, Universidad de Magallanes, Punta Arenas, Chile.
Michel De Lara gives a plenary conference Discrete-Time Viability Methods for Sustainable Management of Natural Resources
The ECOS group is invited to give conferences within the session on Teoría de Control y Optimización: Vincent Martinet gives a conference Multi-criteria dynamic decision under uncertainty: A stochastic viability analysis and an application to sustainable fishery management; Michel De Lara gives a conference Sustainable quotas and viable management of ecosystems; Hector Ramirez gives a conference Harvesting technology and the risk of fishing collapse: the case of small pelagic fish.
XXIX Congreso de Ciencias del Mar, Hualpén - Talcahuano, 25-28 May 2009.
Our group is offered an invited session Modelos bioeconomicos en pequeria on Wednesday 27 May 2009.
Hector Ramirez gives a conference Evaluation of Management Procedures: Application to Chilean Jack Mackerel Fishery
Julio Peña Torres gives a conference ITQ's in Chile: Measuring the Economic Benefits of Reform
Pedro Gajardo gives a conference Sustainable Reference Points
Alejandro Zuleta a conference Recuperando pesquerías sustentables
Mission of Hector Ramirez.
Our paper Viable harvest of monotone bioeconomic models is submitted to a journal, and a preprint is put on HAL and on arXiv.
Michel De Lara, Pedro Gajardo, Hector Ramirez Cabrera,
Viable harvest of monotone bioeconomic models,
arXiv, 16 pages
The plane ticket of Hector Ramirez was paid by Chilean CONICYT budget. Daily expenses (accomodation, etc.) were supported partly by French ECOS budget and partly by the Optmisation and System team of CERMICS, École des Ponts ParisTech, Université Paris-Est, Marne-la-Vallée.
Mission of Michel De Lara.
Concerning the paper Risk and Sustainability: Assessing Resource Management Procedures, we devoted our time with Hector Ramirez to check and modify the simulation Scilab code for the Jack-Mackerel Chilean fishery facing El Niño uncertainty. Julio Peña Torres joined to discuss the economic interpretation.
Mission of Pedro Gajardo.
We worked on our paper Viable harvest of monotone bioeconomic models in order to submit it to the journal Systems & Control Letters.
The plane ticket of Pedro Gajardo was paid by Chilean CONICYT budget. Daily expenses (accomodation, etc.) were supported partly by French ECOS budget and partly by the Optmisation and System team of CERMICS, École des Ponts ParisTech, Université Paris-Est, Marne-la-Vallée.
IIFET 2010, Montpellier, 13-16 July 2010, ``Economics of fish resources and aquatic ecosystems: balancing uses, balancing costs''
Hector Ramirez gives a conference Evaluation of Management Procedures: Application to Chilean Jack Mackerel Fishery
Mission of Michel De Lara.
Participation to the International Conference on Continous Optimization (ICCOPT) 2010 in Santiago: special course (Viability methods: application to fishery management) at the Winter School and conference.
With Julio Peña Torres, we specified biological and environmental data in the Scilab code for the Jack-Mackerel Chilean fishery facing El Niño uncertainty.
Our paper Viable harvest of monotone bioeconomic models is accepted for publication in the journal Systems & Control Letters.
Viability means the ability of survival, namely the capacity for a system to maintain, during time, condition of existence, good health, safety, or by extension, effectiveness (in the sense of cost-effectiveness in economics).
Viability issues arise obviously in engineering sciences and automatic control. Keeping a vehicle on a road, maintaining safety conditions in a industrial plant may constitute illustrations of viability problems.
The question of survival obviously arise in life sciences. The analysis of adaptative mechanisms adopted by a population with respect to its environment, the study of coexistence conditions for several species, the maintenance of biodiversity are preoccupations met by ecological and biological scientists, which are close to viability issues.
Also, viability issues are fundamental in the economic field through environmental and ecological economics. Harvesting a resource without exhausting it, preserving biodiversity, avoiding or mitigating climate change are examples of sustainability concerns. More generally, the main question raised by sustainable development is how to reconcile economic growth and ecological conditions. The concepts of precaution, irreversibility, and intergenerational equity are closely related to viability issues. This implies to study the compatibility between biological, physical and economical dynamics, to mix considerations of ecologists, biologists, demographs and economists and, in particular, to handle long term horizons.
The main common features of these problems are:
These two point being stated, an important question arises quite naturally: Which compatibility, if any, exists between the dynamics of the system and these constraints?
In other words, we have to study dynamical systems under constraints. In applied mathematics and system theory, this question has mostly been neglected to concentrate rather on steady state equilibria or optimization concepts.
The aim of this project is to provide mathematical tools to deal with dynamical systems under state and control constraints and to shed new lights on some applied viability problems especially in the economics and environmental fields.
The scientific project is organized as follows. In the next section we present some discrete time viability issues and we establish the main subjects that we will approach within the framework of this project. In the last section we present a field of applicability where viability issues arise naturally.
Let us consider a nonlinear control system described in discrete time by the difference equation
belongs to the finite
dimensional state space
, the control
variable 
is an element of the control set
while the dynamics 
maps
into
.
A decision maker describes ``desirable configurations of the
system'' through a set
termed the
desirable set
includes both system states and controls
constraints. Typical instances of such a desirable set are given
by inequalities requirements:
.
The state constraints set associated with
is
obtained by projecting the desirable set
onto the state
space
:
Viability is defined as the ability to choose, at each time step
, a control
such that the system
configuration remains desirable. More precisely, the system is
viable if the following feasible set is not empty:
is called the viability
kernel associated with the dynamics and the desirable
set
. By definition, we have
but, in general, the inclusion is
strict. For a decision maker, knowing the viability kernel has
practical interest since it describes the states from which
controls can be found that maintain the system in a desirable
configuration forever. However, computing this kernel is not an
easy task in general.
A subset
is said to be weakly invariant for the
dynamics in the desirable set
, or a
viability domain of in
, if
 such that and  |
(B.5) |
, a suitable control may transfer
the state in
and the system into a desirable configuration.
In particular, it is worth pointing out that any desirable
equilibrium is a viability domain of in
.
A desirable equilibrium is an equilibrium of
the system that belongs to
, that is a pair
such that
.
Moreover, according to viability theory [1], the
viability kernel
turns out to be the union
of all viability domains, that is the largest set such that
   |
(B.6) |
In this section we present problems that we will study during the project. We expect to obtain lower and upper approximations of the viability kernel in particular contexts together with to determinate some structure about it (convexity, polyhedral, increasing, etc). Our work is based, among others, in the paper [7] where some estimations of the kernel are deduced under monotonicity assumptions in the dynamics and desirable set. This estimations allow to compute or approximate the viability kernel through weakly invariant domains under suitable assumptions. On the other hand, we want to establish some rules in order to determinate when a particular set (e.g. polyhedral) of the state space is a weakly invariant domain.
At the beginning of the project, we will work with a generic form for dynamics and desirable sets corresponding to what we shall call bioeconomics viability issues. This relies upon monotonicity properties.
The first basic assumptions, as in [7], are the
monotonicity properties. At the beginning of the project we will
work with, and afterwards, we expect to relax some of these
hypothesis. The monotonicity properties deal with the state space
and the control space
supplied with
the componentwise order: 
if and only if each component
of 
is greater than or equal to the corresponding component of
:
of 
as follows:
We now define the monotonicity of constraint sets.
is
increasing if it satisfies the following property:
is increasing
if it satisfies the following property:
in the first
case, and by
in the second case (where state and control do not play
the same role).
Similarly, we define monotonicity for the dynamics as follows.
is increasing
with respect to the state if it satisfies
We say that
is a
bioeconomics dynamics if is increasing w.r.t
the state and decreasing with respect to the control.
We say that
is a
bioeconomics quasi-linear dynamics if
, where 
is a
matrix and
is a vector in
for all
.
A desirable set
is said to be a production desirable
set if
is
increasing with respect both to the state and to the control, that is
is increasing with respect to both variables (state and control).
A desirable set
is said to be a preservation
desirable set if
is
increasing with respect to the state and
decreasing with respect to the control, that is
is increasing
with respect to the state but decreasing with respect to the
control.
The first objectives during the project is to study the following subject in the context of bioeconomics viability issues defined above:
and
;
and
,
assuming some structure (convexity, polyhedral, etc) on desirable
sets
and
;
and
.
The following section presents one field of application where viability issues arises naturally and on which our results could be an important impact.
Fisheries are under extreme pressure worldwide despite the endeavors for better regulations in terms of economic instruments and measures of stocks and catches. Regulating agencies have to drive the resource management on more sustainable paths which could conciliate both ecological and economic goals within an ecosystem perspective.
Nowadays, sustainability is a major goal of international agreements and guidelines to fisheries management [13,10]. However, fisheries management objectives are often stated in terms of ecology, aiming at resource sustainability, in the restricted sense of a stock. For example the International Council for the Exploration of the Sea (ICES) precautionary framework aims at conserving stocks, but considerations of economics and society are excluded from the advice and left to the managers. In the framework of an ecosystem approach to fisheries management, the multiplicity of objectives is more explicitly recognized [11,14], even if this implies a more comprehensive account of ecological sustainability. In this perspective, objectives can also pertain to economics and consider sustainability of harvesting and related activities, and to society with concern of the well-being of the community.
Indicators and their associated reference points are key elements
of current fisheries management advice, as well as of the
developing ecosystem approach. For a given implicit or explicit
management objective, one or several indicators have to be
selected to monitor the progress toward this objective. Reference
points are used as benchmarks for the indicator values
[13]. In many cases, there is some confusion between
objective and reference point. For example, in the ICES
precautionary approach, the objectives are to maintain spawning
stock biomass above a limit reference point
, while
keeping fishing mortality below a limit
[13].
To achieve this in a context of high uncertainty on both the
current value of the indicator and the reference point,
operational precautionary reference points
(
) and
(
) are used
to trigger management action before reaching the limits
[13]. However, as an exploited stock or a fishery or an
ecosystem evolves, the dynamics of their components are tied, bear
a shared inertia and many uncertainties occurs. In other words,
systems are generally not at equilibrium, contrary to what is
often assumed while defining long-term reference points. For this
reason, it might not always be sufficient to keep an indicator
above a reference point to be able to keep it again at subsequent
time steps.
Optimal control theory has been extensively used to define fisheries management strategies [19,12]. Although it offers a more dynamic perspective, undesirable outcomes of such an optimality approach reduces its applicability in terms of sustainability. One problem is resource extinction: it could be optimal to "liquidate" the resource; Another difficulty displayed by optimal control relies on intergenerational equity issue: optimal catches may be zero at the beginning or the end of the period at stake depending on the preferences for future or present induced by the discount rate. Moreover, optimal control is not easy to apply in a multi-criteria context.
All these remarks show the interest to address the problem from a different perspective, attempting in particular at reconciling ecological and economic issues. This is part of an ecosystem based approach which aims at taking into account overall ecosystem complexity for the sustainability of ecosystem use. In particular, it seems relevant to adopt a dynamic framework and not restrict the analysis to stationary and steady state viewpoint while not focusing on optimal control framework. Viability approach do not strive to determine optimal paths for the co-evolution of resources, but rather for desirable corridors. These corridors are constrained by normative measures representing our knowledge of what should at least be avoided in order to achieve sustainability or prevent catastrophic developments. The viable control approach [1] or weak invariance approach [5] enables us to formally define the corridor borders. Such an analytical strategy allows to provide knowledge for decision-making, although constraints are normative and an outcome of public discussions. In addition, it shifts attention from the whole set of options to a constrained set of options, leaving space for adjustments acceptable in political frameworks. Basically, the viable control approach focuses on inter-temporal feasible paths. It first requires the identification of a set of constraints that represents the "good health", the safety or by extension the effectiveness of the system. Then the approach relies on the study of conditions which allow these constraints to be fulfilled at any time, including both present and future. We refer for instance to [15] for applied work in the exhaustible resource context and [3,2,8,9,18] for renewable resources management. The paper [6] advocates for the use of viability approach to integrate ecosystem considerations. From the ecological viewpoint, the so-called population viability analysis (PVA) [16] and conservation biology has concerns close to viable control by focusing to extinction process, situations and regulations in an uncertain (stochastic) framework. Connections with safe standard of conservation invoked in [17] are worth to be pointed out. The Tolerable Windows Approach (TWA) proposes a similar framework on climatic change issues [4]. More specifically, in the environmental context, viability may allow for the satisfaction of both economic and environmental constraints. In this sense, it is a multi-criteria approach. Moreover, since the viability constraints are the same at any moment and the term horizon is infinite, an intergenerational equity feature is naturally integrated within this framework. As emphasized in [15], this approach is deeply close to the maximin approach exposed by Rawl's or Solow. When system dimension and/or non-linearity increase, technical and numerical difficulties to apply viability concepts generally arise. However, it is shown in [7] how viability tools may enjoy some nice monotonicity properties which can be applied to age structured problems. We propose to use viable control approach to make explicit the relationships between management objectives and reference points for the ICES model.
As an example of time discrete systems with some viabilities concepts, we present an age structured abundance population model with a possibly non linear stock-recruitment relationship, derived from fish stock management [19].
Time is measured in years, and the time index
represents the beginning of year 
and of yearly period
. Let
denote a maximum age, and
an age class index, all expressed in years. The
state variable is
, the abundances at age: for
,
is the number of individuals of age between
and 
at the beginning of yearly period ;
is the number of individuals of age greater
than 
.
For
, we have
is the natural mortality rate of
individuals of age ;
is the mortality rate of individuals
of age
due to harvesting between and 
, supposed to remain
constant during period ;
the vector
is termed the
exploitation pattern;
is
the fishing effort multiplier,
supposed to be applied in the middle of period .
Since 
is the number of individuals of age
greater than , an additional term appears in the
dynamical relation
is related to the existence of a
so called plus-group: if we neglect the survivors older
than age 
then 
, else 
and the last age class is
a plus group.
Recruitment involves complex biological and environmental
processes that fluctuate in time, and are difficult to integrate
into a population model. The recruits
are supposed to be a function of the
spawning stock biomass
defined by
are the
proportions of mature individuals (some may be zero) at age
and
are the weights at
age (all positive). We write
describes a stock-recruitment
relationship, of which typical examples are
Taking for state vector
is defined for any
and
by
The exploitation is described by catch-at-age 
and
yield 
, respectively defined for a given vector of abundance
and a given control 
by the so called
Baranov catch equations
[19, p. 255-256]. The catches are the number
of individuals captured over the period 
:
is
 |
(B.19) |
We focus our analysis on the desirable set
is given by (B.18).
According to (B.7) in Section 2, we observe that
is a production desirable set. So, the question
is: does an abundance at age in
satisfy some preservation requirement? That
is, if the viability kernel for the production set
is not empty, that imposes some preservation requirement as the
spawning stock biomass
has to be greater than some
reference point?. This kind of questions, also in the context of
more general systems (e.g. no monotonicity assumptions), are the
principals subjects on which we focus this project.
given,
