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Non-smooth dynamics driven by stochastic disturbance arise in a wide varietyof engineering problems. Impulsive interventions are often employed to controlstochastic systems; however, the modeling and analysis subject to executiondelay have been less explored. In addition, continuously receiving informationof the dynamics is not always possible. In this paper, with an application toan environmental restoration problem, a continuous-time stochastic impulsecontrol problem subject to execution delay under discrete and randomobservations is newly formulated and analyzed. The dynamics have a non-smoothcoefficient modulated by a Markov chain, and eventually attain an undesirablestate like a depletion due to the non-smoothness. The goal of the controlproblem is to find the most cost-efficient policy that can prevent the dynamicsfrom attaining the undesirable state. We demonstrate that finding the optimalpolicy reduces to solving a non-standard system of degenerate ellipticequations, the optimality equation, which is rigorously and analyticallyverified in a simplified case. The associated Fokker-Planck equation for thecontrolled dynamics is derived and solved explicitly as well. The model isfinally applied to numerical computation of a recent river environmentalrestoration problem. The optimality and Fokker-Planck equations aresuccessfully computed, and the optimal policy and the probability densityfunctions are numerically obtained. The impacts of execution delay arediscussed to deeper analyze the model.

Stochastic impulse control of non-smooth dynamics with partial observation and execution delay: application to an environmental restoration problem