Nonsmooth and Geometric Methods in Nonlinear Control

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The main purpose of the research is to derive new necessary conditions for a minimum in optimal control theory, using the approach developed by the PI in recent years, based on the systematic use of generalized differentials instead of classical differentials, of flows instead of vector fields, and of abstract variations instead of the standard needle variations. This will achieve a unification of the various existing versions of the finite-dimensional Pontryagin Maximum Principle, incorporating them all into a single result, which, in addition, will be more general in scope and will also apply to hybrid problems. Several other related lines of research will be pursued: sufficient conditions for an optimum for families of trajectories, using extensions ---due to the PI in collaboration with B. Piccoli---of V. Boltyainskii's idea of a "regular synthesis," ergodic properties of skew-product flows (in collaboration with M. Nerurkar), viscosity solutions of first-order Bellman equations corresponding to problems with degenerate Lagrangians, and subanalyticity of value functions.This work is motivated by the need for powerful new and usable tools for studying curve optimization problems. Such problems occur in many areas of science, ranging from the more traditional automatic control questions that arise naturally in engineering (e.g. control of power plants, aircraft, or various mechanical devices) to the more recent applications in biological and medical problems (e.g. the search for optimal ways to administer combinations of several medications). The common aspects of all these problems are (a) that they involve the search for strategies for influencing (i.e., "controlling") the behavior of a system so as to get it to achieve a desired goal in the best possible way ("optimal control") or, at least, to come as close as possible to that goal, and (b) that they have a definite dynamical structure (for example, one needs to know not only how much of each medication to administer, but also the specific time sequence in which this is to be done). The search for solutions to optimal control problems has been made more difficult by the fact that the existing mathematical methods involve a collection of different techniques that cannot be combined into a single theory. This means that most real-life problems do not fit within the framework of existing techniques, especially when these problems are "hybrid," in the sense that they combine continuous aspects with discrete ones. The ultimate goal of the PI's proposed research is to produce tools that will make it possible to attack large classes of such problems by means of a single, systematic, user-friendly approach.