UCLA

While the motivations behind the desire for sustainable processes and sustainable development are clear, the fundamental question of “what does it mean for a process to be sustainable?” is often met with an ambiguous answer. Clearly, the answer to this question of sustainability for any given commercial, natural, or industrial process can be a matter of perspective for the parties involved. For example, fisherman and ecologists observing the same fishing grounds may well reach wholly different conclusions as to the sustainability of commercial fishing in that ecosystem. This lack of clarity bred a variety of definitions and methodologies for assessing the sustainability of a system or process. Previous work from this research group made strides towards addressing this ambiguity by introducing the concepts of “Sustainability Over Sets” (SOS) and “Sustainizability Over Sets” (SIZOS) – mathematical frameworks which provide rigorous, quantitative metrics by which the questions of “is this system sustainable?” and “if not, can it be made sustainable?” can be objectively evaluated. This thesis builds upon the foundations laid by the introduction of these mathematical concepts and expands them to a wider variety of mathematical system descriptions, thereby increasing the utility of these tools for sustainable system synthesis. The concepts of SOS and SIZOS rely on the mathematical principle of an invariant set for a dynamic system. For a brief overview of set invariance, consider an arbitrary dynamic system and the range of attainable values, or state space, for that system. A subset of state space is deemed “invariant” if starting from any point inside of the set, the future trajectory of the system will remain inside of the set forever. Put another way, once a dynamic system attains a value inside of an invariant region of state space, the system will never escape that region. This invariance concept serves as the foundation of the sustainability criterion in the SOS framework. Namely, the question of sustainability for a region in state space translates equivalently to whether the set is invariant under the dynamic model being considered. The original SOS framework was developed for dynamic systems described by a set of ordinary differential equations. These systems are sometimes called “lumped parameter” systems because the dynamic quantity under consideration is solely a function of time and no other parameter. While ODEs capture a wide range of natural phenomena, they can be inadequate for capturing the behavior of systems with more complex behavior. This work addresses this issue by introducing further mathematical criteria which allows for the broader application of SOS concepts to distributed parameter and to systems with uncertainty. Unlike lumped parameter systems, distributed parameter systems capture dynamic behavior which is dependent on other parameters in addition to time. In engineering application, this additional parameter is often a spatial dimension, and partial differential equations are necessary to describe the system. Systems with uncertainty are those where the future behavior of the system can be random and are mathematically described by differential inclusions. This thesis first introduces analogous sustainability criteria for systems with mathematical forms of these types and illustrates the utility of these new criterion with a series of case studies analyzing chemical reactions, predator prey systems, and other ecological systems. Ultimately, this work significantly expands the capabilities of the previously introduced sustainability framework, allowing for the application of sustainability concepts to a far wider set of dynamic systems.