# Harvest

We study here the delay problem studied in [1] and [2], originating from [4]. The aim is to find the optimal harvesting of a renewable ressource whose growth follows a logistic function. Denoting $Latex formula$ the biomass of population and $Latex formula$ the harvesting effort, the optimal control problem is stated as

$Latex formula$

with the harvesting cost $Latex formula$, the growth rates $Latex formula$, the discount rate $Latex formula$ and market price $Latex formula$, and the growth delay $Latex formula$. Bocop can handle the delayed term $Latex formula$ without having to perform the classical Guinn transformation ([3]), but for a fixed final time only. Therefore we perform a batch of optimizations for $Latex formula$, and iterate the process for $Latex formula$ to find a better estimate of the optimal time. Batch optimizations indicate an optimal final time $Latex formula$ with an objective $Latex formula$.

Fig1: Delay problem. Cost function with respect to final time T.

Fig2: Delay problem. Optimal state and control for T = 12.24.

References
[1] A. Boccia, P. Falugi, H. Maurer, and R. Vinter. Free time optimal control problems with time delays. pages 520–525. IEEE, 2013.
[2] L. Goellmann, D. Kern, and H. Maurer. Optimal control problems with delays in state and control variables subject to mixed control-state constraints. Optimal Control Applications and Methods, 30(4):341–365, 2009.
[3] T. Guinn. Reduction of delayed optimal control problems to non-delayed problems. Journal of Optimization Theory and Applications, 18(3):371–377, 1976.
[4] R. May. Stability and Complexity in Model Ecosystems. 1975.