Optimal control model for oil palm biomass incorporating felling rate and carbon absorption

Abstract

The problem of optimal harvesting in controlling the outcome has often been considered, yet the optimal felling in the control of oil palm biomass has rarely been treated. The main objective of this thesis is to mathematically model the optimal control of oil palm biomass, incorporating felling rate and carbon absorption. Optimal control of oil palm biomass is mathematically a controlled system to optimize the amount of oil palm yield while absorbing carbon, controlled by state control (felling rate) and can be influenced by state variables. The first objective considered is the felling problem. A time-invariant linear quadratic optimal control model is developed for controlling the felling and harvesting rates of the oil palm biomass using the linear quadratic regulator approach. The optimal control solution is solved using Pontryagin’s Minimum Principle. The second objective of this thesis is to develop a non-linear optimal control model of oil palm biomass incorporating carbon absorption, again assuming that the rate of felling can be controlled. The results indicate that felling rate affects the amount of biomass but not biomass growth rate. As the biomass growth rate is not affected by felling, separating the whole biomass into young and mature biomass and setting the biomass growth rate to be constant is the third objective accomplished in this thesis. Separating is considered vital since at the early age, no yield is produced and the absorption of carbon is quite low for young oil palm trees. A nonlinear system of ordinary differential equations is implemented for formulating the nonlinear optimal control models. To support the proposed models, a theoretical analysis of the positivity and boundedness of the solutions as well as their stability is carried out for each model. The equilibrium points are identified, and it is shown that a small perturbation of the felling rate near the equilibrium point decays exponentially, indicating that the model is stable with the existence of a felling rate. Model solutions are obtained using the control parametrization method and the Nelder–Mead method for both nonlinear models. The computational results show that the model is able to increase the amount of fruit yield and carbon absorption. This study has succeeded in developing models to control the oil palm biomass optimally in producing fruit whilst absorbing carbon from the atmosphere.