Theory and Applications of Finite-Dimensional Nonlinear Control

9803411

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

theoretical and applied control theory problems. The methods used will be

those of differential-geometric control theory, nonsmooth analysis, and the

theory of real analytic maps and their associated stratifications.

Specifically, efforts will be made to solve a number of open problems in

the areas of optimal control, controllability, and realization theory, while

pursuing the development of the necessary mathematical tools. In

particular, work will continue on a major project that began in 1992, and

has evolved since then and has led to a strong, general, unified version of

the necessary conditions for optimality usually known as the ``Pontryagin

Maximum Principle.'' This new version of the Maximum Principle

requires the use of a new theory of generalized differentials, called

``multidifferentials,'' and part of the work will involve the systematic

development of this theory and its applications.

Recent developments in nonlinear control have led to many applications to

various issues in robotics and nonholonomic motion planning. The

general question here is that of finding a path for a given system that takes

it from a given state to another desired state, satisfying some constraints or

optimizing some cost functional. (For example, steer a vehicle from a

given position to another desired position while avoiding certain obstacles,

possibly with the extra requirement that this be done in minimum time or

with minimum expenditure of energy.) Thanks to the extraordinary recent

advances in computing power, it has now become realistic to expect to

solve many of these problems in real time, and we intend to develop

methods that will contribute to this endeavor, by providing an a priori

understanding of the structure of the solutions, so as to reduce the search

needed to find it. A special effort will be made to improve communication

between specialists in the field of nonlinear control and the general

mathematical and engineering community, by bringing to the attention of

the community examples of applications that show the advantages and the

power of the techniques of nonlinear control.