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Deforestation is a major threat to global environmental wellness, with illegal logging as one of the major causes. Recently, there has been increased effort to model environmental crime, with the goal of assisting law enforcement agencies in deterring these activities. We present a continuous model for illegal logging applicable to arbitrary domains. We model the practice of criminals under influence of law enforcement agencies using tools from multiobjective optimal control theory and consider non-instantaneous logging events and load-dependent travel velocity. We calibrate our model using real deforestation data from the Brazilian rainforest and demonstrate the importance of geographically targeted patrol strategies.

We present a method for optimal path planning of human walking paths inmountainous terrain, using a control theoretic formulation and aHamilton-Jacobi-Bellman equation. Previous models for human navigation wereentirely deterministic, assuming perfect knowledge of the ambient elevationdata and human walking velocity as a function of local slope of the terrain.Our model includes a stochastic component which can account for uncertainty inthe problem, and thus includes a Hamilton-Jacobi-Bellman equation withviscosity. We discuss the model in the presence and absence of stochasticeffects, and suggest numerical methods for simulating the model. We discuss twodifferent notions of an optimal path when there is uncertainty in the problem.Finally, we compare the optimal paths suggested by the model at differentlevels of uncertainty, and observe that as the size of the uncertainty tends tozero (and thus the viscosity in the equation tends to zero), the optimal pathtends toward the deterministic optimal path.

We present a method for determining optimal walking paths in steep terrainusing the level set method and an optimal control formulation. By viewing thewalking direction as a control variable, we can determine the optimal controlby solving a Hamilton-Jacobi-Bellman equation. We then calculate the optimalwalking path by solving an ordinary differential equation. We demonstrate theeffectiveness of our method by computing optimal paths which travel throughoutmountainous regions of Yosemite National Park. We include details regarding thenumerical implementation of our model and address a specific application of alaw enforcement agency patrolling a nationally protected area.