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 the biomass of population and the harvesting effort, the optimal control problem is stated as

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

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.