(106g) Optimal Control Theory for Sustainable Environmental Management

1.
Introduction

The available
resources being rapidly diminishing, and the harmful effects of various
pollutants being realized, research priorities in the field of ecology and
environment have evolved. The focus has shifted from achieving short term goals
to envisioning long term sustenance, also known as sustainability.
Sustainable development is defined as ?the development that meets the needs of
the present without compromising the ability of the future generations to meet
their own needs? [1]. In its simplest terms, it calls for the consideration of
long terms effects, benefits and drawbacks in all decisions relevant to the
society as a whole. As a consequence, management strategies targeting
sustainability are sought.

The goal of a sustainable management strategy is to promote the structure
and operation of the human component of a system (society, economy, technology,
etc.) in such a manner as to reinforce the persistence of the structures and
operation of the natural component (i.e., the ecosystem) [2]. The proposed work
is aimed to accomplish this task for a model system representing an ecosystem
that includes humans, a very rudimentary industrial process, and a very simple
agricultural system [2].

It is evident that this will be a highly interdisciplinary approach, involving
interactions of systems on multiple temporal and spatial scales. A systematic
approach, based on sound mathematical techniques, is essential to communicate
between such diverse systems. Two important aspects of this approach are:
formulation of sustainability based objectives and development of the
management strategies. Fisher information based sustainability hypotheses,
proposed in [3] and based on information theory, allow the formulation of
mathematical objectives relevant to disparate systems. This work therefore formulates
objectives based on these hypotheses. Once the objectives are developed, systems
theory based methods and tools provide the means to derive the appropriate management
strategies. This work proposes to use optimal control theory to derive time
dependent management strategies. To formulate the control problem, it is
important to identify the right control variables, which is done in this work
through partial correlation coefficient analysis of the model.

2. Theory

2.1. Model

The food web
model considered in this work is presented in [2]. The model system represents
an ecosystem that includes humans, a very rudimentary industrial process, and a
very simple agricultural system. The model system tracks the flow of mass
resources (biomass, nutrients, water, etc.) within a closed system (i.e. the
cumulative sum of masses of all the system compartments is constant). The
system includes a resource pool (RP) generically representing all biological
resources (water and nutrients), four plant (P1, P2, P3, and P4), three
herbivore (H1, H2, and H3), two carnivore (C1 and C2), a human population (HH),
and an inaccessible resource pool (IRP) representing resources that are biologically
unavailable as a result of human activity. The model is divided into two
characteristic branches: domesticated (representing agricultural and livestock
activities) comprising compartments P1, P2, and H1, and non-domesticated
(representing species hunted, gathered, and species not consumed by humans) consisting
of compartments P3, P4, H2, H3, C1, and C2. Humans (HH) rely on the
non-domesticated branch for both resources and for the recycling of mass from the
inaccessible resource pool back into the rest of the system. The industrial
process is meant to represent at a very elementary level a generic human industrial
activity that offers a benefit to the human population. The industrial process simply
takes mass from three compartments (P1, RP, and H3) in different proportions
and combines it to form a product. More information about the model is given in
[3].

2.2.
Fisher Information and Sustainability Objective

Cabezas and Fath
[3] have recently proposed the sustainability hypotheses for natural systems
using Fisher information as the sustainability index. It states that: the
time-averaged Fisher information of a system in a persistent regime does not
change with time. Any change in the regime will manifest itself through a
corresponding change in Fisher information value. Accordingly, two possible
objectives for the control problem are: Maximization of time averaged Fisher
information and minimization of Fisher information variance over time.

            Shastri
and Diwekar [4, 5] have compared these objectives for a three-species
predator-prey model (deterministic as well as stochastic), The results indicate
that the objective of minimization of Fisher information variance is guaranteed
to give a stable response. Consequently, this work considers the objective of
Fisher information variance minimization for the food web model.

2.3. PRCC Analysis

To formulate any
control, it is important to first identify the model parameters that can be used
as the control variables. Equally important is the identification of model variables
that are indicators of the model sustainability i.e. used in the computation of
the Fisher information. This is achieved in this work through the partial
correlation coefficient (PCC) and partial rank correlation coefficient (PRCC) analysis.
PCC and PRCC values indicate a major or unique or unshared contribution of each
variable and explain the unique relationship between two variables that cannot
be explained in terms of the relations of these variables with any other
variable. For a nonlinear model, such as the one in this work, PRCC values are
more important. The PCC and PRCC results are used to identify the appropriate
control variables for the problem, along with the right model sustainability
indicators. Based on the results, the control variables identified for this study
are: coefficient of mass transfer from RP to P1 and P2, constant for the waste
term associated with human consumption, and constant reflecting the
effectiveness of the industrial process in reducing the human mortality rate

2.4
Optimal Control Problem Formulation and Solution

To account for the complex nature
of the model and the objectives, this work proposes to use optimal control
theory to derive time dependent liming strategy. Optimal control theory is
designed to optimize a time dependent performance objective of the system. It
has been extensively used in control applications for natural systems,
primarily because of its generality and its ability to handle any type of
system (including nonlinear) [4, 5]. Literature proposes different methods to
formulate and solve the optimal control problem. This work uses the Pontryagin's
maximum principle for the same. This method of formulating the control problem
leads to a set of algebraic and ordinary differential equations that need to be
solved as a boundary value problem. Due to the complex nature of the resulting
equations, the work will use the numerical method of steepest ascent of
Hamiltonian.

The control problem will be used to solve various cases of the model
behavior. The simulated cases are meant to represent situations when uncontrolled
model is showing undesirable dynamic behavior and external intervention is
essential. The control problem will be solved using different control variables
and different combinations of the objectives (different variables in FI
calculation). The results should provide a guideline as to which combination is
most effective to exercise external control on the considered food web mode.

3. Summary

Sustainable
management of the human and natural systems, taking into account their
interactions, has become paramount. To achieve this complex multidisciplinary
objective, systems theory based techniques prove useful. The proposed work is a
step in that direction. Taking a food web model incorporating the essential aspects
of the complete spectrum, it uses Fisher information based sustainability
objectives and optimal control theory to derive sustainable management
strategies. For the considered model, the results will highlight the important
aspects parameters and variables. However, on broader perspectives, the results
should be viewed as the proof of concept for the application of systems theory
based techniques in sustainability. The important conclusions and results from
this study should form the basis for the use of such approaches for more
complicated models.

References:

[1] C.
Tomlinson (1987). Our Common Future. World Commision on Environment and Development,
Oxford University Press, Oxford.

[2] H. Cabezas,
C. Pawlowski, A. Mayer, & N.T. Hoagland (2005). Simulated experiments with
complex sustainable systems: Ecology and technology. Resources Conservation
and Recycling, 44:279?291, 2005.

[3] H. Cabezas
& B. Fath (2002). Towards a theory of sustainable systems. Fluid Phase Equilibria,
2, 184?197.

[4] Y. Shastri
& U. Diwekar (2006a). Sustainable ecosystem management using optimal
control theory: Part 1 (Deterministic systems). Journal of Theoretical
Biology (Accepted for publication).

[5] Y. Shastri
& U. Diwekar (2006b). Sustainable ecosystem management using optimal
control theory: Part 2 (Stochastic systems). Journal of Theoretical Biology
(Accepted for publication).