Analysis and Synthesis of Systems Sustainable Over Sets

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The complex nature of sustainability suggests that its assessment can benefit from the incorporation of human input into the assessment process. The novel concept of "sustainability over sets" (SOS) meets this challenge. This concept is flexible, comprehensive, and can readily incorporate human input. The SOS concept opens a new chapter in environmental and ecological sustainability. In addition, the involvement of K-12, undergraduate, and graduate students occurs through the creation of new projects, infrastructures utilized in summer research programs, environmental sustainability courses, and upper level mathematics and optimization classes. The project also provides training for involved individuals, by exposing them to high level mathematics and computer programming concepts, methods, and software. The project researchers are facilitating the placement of participants in graduate schools, and in university faculty positions. Assistance on these fronts is provided by the UCLA Center for Excellence in Engineering and Diversity (CEED) and the UCLA High School Summer Research Program (HSSRP).  This research project focuses on the analysis and synthesis of sustainable systems through the novel concept of sustainability over sets (SOS). The SOS concept is first formally defined, and then mathematically quantified for memory-less, spatially uniform systems, and demonstrated in system sustainability assessment. The research focuses on the continued conceptual development and analysis of the SOS concept; extension of its applicability to systems that are spatially distributed and/or have memory; applications and case studies on ecological, biological, and societal systems; and development of sustainable system synthesis methodologies. SOS is readily quantifiable, thus enabling definitive (yes or no) answers to the question "is a system sustainable?" The concept of invariant sets plays a key role in this quantification process, leading to simple mathematical criteria for sustainability assessment, in particular for rectangular sets for which definitive (yes or no) answers can be given to the question "does there exist a set over which the system is sustainable?"  The project also helps K-12, undergraduate, and graduate students, by creating new projects, and infrastructure to be utilized in summer research programs, environmental sustainability courses, and upper level mathematics and optimization classes.