The main thrust of this research is to study problem formulations in optimal control theory for which the extant mathematical tools are not adequately developed. The approach relies on methods of nonsmooth analysis, which is an extension of classical calculus that systematically handles derivative-like properties of functions that may not be differentiable in the usual sense. Its development was largely motivated by problems in optimization, where inequality constraints and min/max operations are ubiquitous, but which do not preserve classical differentiability. This project investigates four related problem areas. (1) Fully convex control, which is on the one hand quite special since it requires convexity assumptions in the state and velocity. But on the other hand, it is a natural generalization of the linear-quadratic regulator, which is the workhorse of optimal control in applications. Problems with state constraints, impulse trajectories, and infinite horizon will be studied from this viewpoint. (2) One-sided Lipschitz dynamics, where the data may have non-Lipschitz behavior but only in a dissipative manner. Such structure in the dynamics is present in the modeling of dry friction. (3) Time-delay problems. (4) Impulsive systems, in which the states evolve according to two time scales and can jump over short time intervals.
To reflect the desirability of certain behaviors of engineered systems in preference to others, aspects of optimization are often included in mathematical models, and dynamic optimization problems arise. Optimal control theory offers deep insight into the nature of these problems. Because the world appears to have many more nonsmooth characteristics than previously imagined, new theoretical challenges in optimal control have emerged. The development of nonsmooth analysis was largely motivated by these considerations, and now consists of a substantial body of results that is being increasingly utilized by engineers. This research project will broaden the range of application of nonsmooth analytic tools, and the results will allow control engineers to employ mathematical models that are more accurate and realistic than those in current use.
