Optimality and sustainability of hybrid limit cycles in the pollution control problem with regime shifts

In this paper, we consider the problem of pollution control in a system that undergoes regular regime shifts. We first show that the optimal policy of pollution abatement is periodic and is described by the unique hybrid limit cycle. We next introduce the notion of an environmentally-sustainable solution, and demonstrate that this policy is the only one that yields the best possible trade-off between steadily achieving profit and preserving the environment. In contrast, the policy that is not environmentally-sustainable eventually enters stagnation. To further illustrate our findings, we compare the optimal periodic solution with a myopic solution. Interestingly, the myopic solution yields higher overall payoff in the short-run but completely fails in the long-run, while the environmentally-sustainable policy yields maximal payoff and preserves the environment over the infinite time interval.