Estimating the Average Temporal Benefit in Probabilistic Environmental–Economic Models

• Published in: Automatic control and computer sciences Vol. 53; no. 7; pp. 730 - 737
• Main Authors: Rodina, L. I., Tyuteev, I. I.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01-12-2019; Springer Nature B.V
• Subjects: Computer Science; Computer simulation; Control Structures and Microprogramming; Differential equations; Economic analysis; Economic models; Environment models; Parameters; Population; Population control; Random variables; Statistical analysis; optimal exploitation; average temporal benefit; model of a population subject to harvesting
• ISSN: 0146-4116; 1558-108X
• DOI: 10.3103/S0146411619070174

This paper considers environmental–economic models of optimal harvesting, given by impulsive differential equations, which depend on random parameters. We assume that lengths of intervals
between the moments of impulses
are random variables and the sizes of impulse influence depend on random parameters
One example of such objects is an impulsive equation that simulates dynamics of the population subject to harvesting. In the absence of harvesting, the population development is described by the differential equation
and in time points
some random share of resource
is taken from the population. We can control the gathering process so that to stop harvesting when its share will appear big enough to save the biggest possible remaining resource to increase the size of the next gathering. Let the equation
have an asymptotic stable solution
whose attraction region is the interval
where
We construct the control
which limits a share of the harvested resource at each moment of time
so that the quantity of the remaining resource, since some moment
would be not less than the given value
For any
the estimates of the average temporal benefit, valid with the probability of one, are obtained. It is shown that there is a unique
at which the lower estimate reaches the greatest value. Thus, we described the way of population control at which the value of the average time profit can be lower estimated with the probability of one by the greatest number whenever possible.