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Optimal Control Theory for Sustainable Environmental Management

Yogendra Shastri, Department of Bioegineering, University of Illinois at Chicago, Vishwamitra Research Institute, Center for Uncertain Systems: Tools for Optimization and Management, 34, N. Cass Avenue, Westmont, IL 60559, Urmila Diwekar, Center for Uncertain Systems: Tools for Optimization and Management (CUSTOM), Vishwamitra Research Institute, 368 56th Street, Clarendon Hills, IL 60514, and Heriberto Cabezas, National Risk Management Research Laboratory, U.S. Environmental Protection Agency, 26 West Martin Luther King Drive, Cincinnati, OH 45268.

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.  





[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).