RUI: Integral Equations with Delay; Optimal Control and Applications

Prairie View A&M Research Foundation, Prairie View TX

Investigators

Abstract

The project will develop a new innovative approach to model complex dynamics processes in technology, biology, ecology, economics, and the environmental sciences. While many models have arisen in these disciplines, mathematical theory is far behind fast growing application needs. The distinguishing feature of the problems under study is the optimal control of endogenous delays. These delays reflect the age of individuals to be harvested or the lifetime of obsolete equipment to be replaced due to technological innovations. While such models have immense potential for applied sciences, prior use has primarily been restricted due to the assumption of constant delays. The use of variable endogenous delays will provide greater modeling flexibility and open new horizons in qualitative analysis of the models. Optimal control with endogenous delays will be reduced to the analysis of integral equations with unknown upper limits of integration. Such equations represent a completely new mathematical object and will be part of the study. Thus, the project outcomes will enhance both mathematics with new results in the optimal control and integral equations and applications with new investigation techniques relevant to applications. Models based on equations with delay are currently employed by various interdisciplinary groups of scientists in economic growth theory, operations research, management sciences, environmental protection, life sciences, population biology, and related areas. The developed methods will be beneficial to these fields and lead to deeper analysis of many phenomena. As a consequence, the research will foster collaboration among scientists from different countries and disciplines and stimulate new contacts. Societal benefits include the ability to allocate usage of novel technological inventions and optimal management of natural resources under environmental and climate changes. The project also will attract attention of mathematical community to new applications and increase participation of African-American undergraduates in research. These students will be involved in all research activities and will present at conferences.