OPTIMAL CONTROL OF ECOLOGICAL SYSTEMS NEAR THE EQUILIBRIUM POSITION

Abstract

The problem of the ecological system long-term keeping near the equilibrium position with minimization of control resources is considered. A mathematical formulation of this problem and a method for optimal expenditure control constructing are presented. While deviations from the equilibrium point are very small, its controlled dynamics can be described by linear differential equations with constant coefficients. The control actions are modeled by piecewise-constant unidirectional functions with a finite number of switching points. The choice of this control class is due to the characteristic features of environmental tasks: control actions on living systems are unidirectional and often have a periodic character [3], [5-11]. It can be individuals catching from the population, feeding, watering, thinning, etc. Therefore, the chosen control class takes into account both the unidirectionality and the periodicity of the control actions. Two types of regulation are discussed: adding to the system any substances promoting the population growth (substrates, fertilizers, etc.), and removal of any volumes (catching individuals from the population, thinning, etc.). As a minimizable functional, the expenditure criterion is used, that is proportional in practice to the consumption of resources used for controlling [1], [4]: pesticides, fertilizers, medicines, etc. In some cases, this criterion minimizing, in addition to the economic one, also gives a direct environmental effect. For example, the constructed optimal control allows the pest population to be destroyed with the minimum quantity of toxic chemicals that adversely affect the environment. The method for solving the problem is based on the mechanical systems optimal control problems methods of solution developed by the authors (e.g. [2]). In this paper, taking into account the ecological specifics of the problem, the system of transcendental equations is obtained. Numerical solution of this system allows us to find the control switching points that satisfy the necessary conditions of the expenditure criterion extremum. To find an adequate initial approximation for the standard numerical methods for solving the equations system an original algorithm is presented.