In this project the principal investigator will explore the use of new techniques to study what are called "abnormal" cases in optimization and optimal control problems that involve nonregular operators. These techniques come from functional analysis, and they have been applied successfully to treat abnormal cases involving the Euler-Lagrange equations and the Pontryagin Maximum Principle. The principal investigator is concerned specifically with determining the limits of these techniques and with obtaining sharper results than those obtained previously for the derivation of nontrivial optimality conditions for general extremal problems. The derivation and exploitation of optimality conditions for problems in optimization and optimal control have been one of the major success stories of the decades since World War II. These results have paved the way for the development of a host of control systems that have improved the quality of life. More recent advances involving robotic systems can be traced to this work as well. In this project the principal investigator will examine optimality conditions in atypical cases where the traditional theory is inadequate. To do so requires the use of powerful methods of abstract functional analysis combined with a thorough knowledge of classical control theory.
