9500798 Sussmann Research will be carried out on nonlinear control theory, continuing the principal investigator s previous work in this area on a broad class of problems in optimal control, applications to robotics, feedback stabilization, and nonholonomic motion planning. The methods used will be those of differential geometric control theory, nonsmooth analysis, and the theory of real analytic maps and their associated stratifications. Necessary mathematical tools will be developed to solve a number of open problems in the areas of optimal control, controllability, and realization theory. In particular, recent results of the principal investigator on the Pontrayagin Maximum Principle for systems of vector fields will be extended, with the goal of proving a general Maximum Principle for systems of differential inclusions under minimal hypotheses. The question of the regularity of optimal trajectories will also be addressed. The proposed research fits within the general framework of control theory, which is concerned with design problems in which some components of a dynamical system are accessible to a controlling agent that can modify them so as to alter the system s behavior. The object is to control the system so as to achieve a desired behavior, or meet some specifications. Situations of this kind occur everywhere in engineering as well as in all of the physical and biological sciences, and many apparently different problems turn out to have common structural features from the mathematical point of view. The purpose of the mathematical theory of control systems is to study these features in a systematic way and to develop techniques for solving particular design problems, such as causing a robot arm to carry out a particular task, or keeping an antenna focused. The issues considered in this project deal with (a) optimal control, in which the goal is to determine how to control a system so as to optimize some criterion; (b) stabiliz ation, where the object is to drive the system towards some desired operating point and keep it there; and (c) motion planning, where it is sought to find a control strategy that will cause a system to move along some prescribed path. ***
