Synthesis of Optimal Control in the Problem of Regulating the Degree of Lake Pollution
Abstract
This paper studies the problem of optimal control of the degree of lake pollution by the pollutants of an industrial plant. A mathematical model is considered that represents the process of lake pollution, taking into account the change in its self-cleaning rate as a function of time. This function is assumed to be piecewise constant and switches between its two possible values periodically, that reflects seasonal fluctuations in the rate of self-cleaning of the lake. An optimal control problem is formulated with respect to a given quality functional characterizing the profit of the plant.
To solve this problem, the Pontryagin's maximum principle is applied. Analytical expressions for the optimal trajectories of the state variable and the adjoint variable are obtained, a qualitative analysis of the obtained solutions is carried out. The concept of environmentally sustainable solution is introduced, and it is shown that such a solution gives the best possible trade-off between the profit of the plant and penalties for lake pollution. The obtained optimal solution is compared with the so-called "myopic" solution, that is, a solution greedily pursuing instantaneous profit.
It is shown that the “myopic” solution can increase short-term profits due to environmental degradation, but in the long term it not only brings worse results compared to the environmentally sustainable solution, but also causes irreparable harm to the environment, which can only be neutralized at the cost of complete shutdown of production for an extended period of time. As a result of the work, a new mixed solution is proposed that combines the advantages of both strategies and is more effective for practical use. The results obtained in this paper are illustrated by examples of numerical simulation for various values of the system parameters.
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