Sufficient and necessary conditions for stochastic near-optimal controls: A stochastic chemostat model with non-zero cost inhibiting

Abstract

Near-optimal controls are as important as optimal controls for both theory and applications. Meanwhile, using inhibitor to control harmful microorganisms and ensure maximum growth of beneficial microorganisms (target microorganisms) is a very interesting topic in the chemostat. Thus, in this paper, we consider a stochastic chemostat model with non-zero cost inhibiting in finite time. The near-optimal control problem was constructed by minimizing the number of harmful microorganisms and minimizing the cost of inhibitor. We find that the Hamiltonian function is key to estimate objective function, and according to the adjoint equation, we obtain some error estimations of the near-optimality. Finally, we establish sufficient and necessary conditions for stochastic near-optimal controls of this model and numerical simulations and some conclusions are given.