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- Creators: Arizona State University
- Member of: Theses and Dissertations
As part of this dissertation, a yield approximation method is developed and integrated with a mixed-integer program to estimate a region’s potential to produce non-perennial, vegetable items. This integration offers practical approximations that help decision-makers identify technologies needed to protect agricultural production, alter harvesting patterns to better match market behavior, and provide an analytical framework through which external investment entities can assess different production options.
Lithium ion batteries are quintessential components of modern life. They are used to power smart devices — phones, tablets, laptops, and are rapidly becoming major elements in the automotive industry. Demand projections for lithium are skyrocketing with production struggling to keep up pace. This drive is due mostly to the rapid adoption of electric vehicles; sales of electric vehicles in 2020 are more than double what they were only a year prior. With such staggering growth it is important to understand how lithium is sourced and what that means for the environment. Will production even be capable of meeting the demand as more industries make use of this valuable element? How will the environmental impact of lithium affect growth? This thesis attempts to answer these questions as the world looks to a decade of rapid growth for lithium ion batteries.
The hyperbolic conservation law considered here is a well-established model for a highly re-entrant semiconductor manufacturing system. Prior work established well-posedness for $L^1$-controls and states, and existence of optimal solutions for $L^2$-controls, states, and control objectives. The results on measure-valued solutions presented here reduce to the existing literature in the case of initial state and in-flux being absolutely continuous measures. The surprising well-posedness (in the face of measures containing nonzero pure-point part and discontinuous velocities) is directly related to characteristic features of the model that capture the highly re-entrant nature of the semiconductor manufacturing system.
More specifically, the optimal control problem is to minimize an $L^1$-functional that measures the mismatch between actual and desired accumulated out-flux. The focus is on the transition between equilibria with eventually zero backlog. In the case of a step up to a larger equilibrium, the in-flux not only needs to increase to match the higher desired out-flux, but also needs to increase the mass in the factory and to make up for the backlog caused by an inverse response of the system. The optimality results obtained confirm the heuristic inference that the optimal solution should be an impulsive in-flux, but this is no longer in the space of $L^1$-controls.
The need for impulsive controls motivates the change of the setting from $L^1$-controls and states to controls and states that are Borel measures. The key strategy is to temporarily abandon the Eulerian point of view and first construct Lagrangian solutions. The final section proposes a notion of weak measure-valued solutions and proves existence and uniqueness of such.
In the case of the in-flux containing nonzero pure-point part, the weak solution cannot depend continuously on the time with respect to any norm. However, using semi-norms that are related to the flat norm, a weaker form of continuity of solutions with respect to time is proven. It is conjectured that also a similar weak continuous dependence on initial data holds with respect to a variant of the flat norm.