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In this paper, we investigate an optimal harvesting problem of a spatially explicit fishery model that was previously analyzed. On the surface, this problem looks innocent, but if parameters are set to where a singular arc occurs, two complex questions arise. The first question pertains to Fuller's phenomenon (or chattering), a phenomenon in which the optimal control possesses a singular arc that cannot be concatenated with the bang-bang arcs without prompting infinite oscillations over a finite region. 1) How do we numerically assess whether or not a problem chatters in cases when we cannot analytically prove such a phenomenon? The second question focuses on implementation of an optimal control. 2) When an optimal control has regions that are difficult to implement, how can we find alternative strategies that are both suboptimal and realistic to use? Although the former question does not apply to all optimal harvesting problems, most fishery managers should be concerned about the latter. Interestingly, for this specific problem, our techniques for answering the first question results in an answer to the the second. Our methods involve using an extended version of the switch point algorithm (SPA), which handles control problems having initial and terminal conditions on the states. In our numerical experiments, we obtain strong empirical evidence that the harvesting problem chatters, and we find three alternative harvesting strategies with fewer switches that are realistic to implement and near optimal. 

In this chapter, we demonstrate how to use a nonlinear polyhedral con- strained optimization solver called the Polyhedral Active Set Algorithm (PASA) for solving a general singular control problem. We present a method for discretizing a general optimal control problem involving the use of the gradient of the Lagrangian for computing the gradient of the cost functional so that PASA can be applied. When a numerical solu- tion contains artifacts that resemble “chattering,” a phenomenon where the control oscillates wildly along the singular region, we recommend a method of regularizing the singular control problem by adding a term to the cost functional that measures a scalar multiple of the total variation of the control, where the scalar is viewed as a tuning parameter. We then demonstrate PASA’s performance on three singular control problems that give rise to different applications of mathematical biology. We also provide some exposition on the heuristics that we use in determining an appropriate size for the tuning parameter.