  Non-smooth dynamics driven by stochastic disturbance arise in a wide variety
of engineering problems. Impulsive interventions are often employed to control
stochastic systems; however, the modeling and analysis subject to execution
delay have been less explored. In addition, continuously receiving information
of the dynamics is not always possible. In this paper, with an application to
an environmental restoration problem, a continuous-time stochastic impulse
control problem subject to execution delay under discrete and random
observations is newly formulated and analyzed. The dynamics have a non-smooth
coefficient modulated by a Markov chain, and eventually attain an undesirable
state like a depletion due to the non-smoothness. The goal of the control
problem is to find the most cost-efficient policy that can prevent the dynamics
from attaining the undesirable state. We demonstrate that finding the optimal
policy reduces to solving a non-standard system of degenerate elliptic
equations, the optimality equation, which is rigorously and analytically
verified in a simplified case. The associated Fokker-Planck equation for the
controlled dynamics is derived and solved explicitly as well. The model is
finally applied to numerical computation of a recent river environmental
restoration problem. The optimality and Fokker-Planck equations are
successfully computed, and the optimal policy and the probability density
functions are numerically obtained. The impacts of execution delay are
discussed to deeper analyze the model.
