Investigation of sliding mode dynamics and near-optimal controls for a reaction–diffusion population model in a polluted environment

In this paper, we present a reaction–diffusion population model developed for a polluted environment, and investigate the impact of different control measures on population-toxicant dynamics. Toxicant concentrations is taken as the reference to determine whether to implement control measures, and sliding mode dynamics are studied. The dynamical behaviors of each subsystem are discussed, and results show that model solutions ultimately approach either two endemic equilibria or the sliding equilibrium on a surface of discontinuity. Furthermore, we formulate the near-optimal control problem by minimizing the control cost while reducing the toxicant concentration. The sufficient and necessary conditions for the near-optimality are established using Ekeland’s variational principle and Hamiltonian function. The theoretical results are illustrated by numerical simulations.