Dynamic Optimization: Time Scales and Nonsmooth Analysis

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This research program lies at the interface between six subjects, namely, optimization theory; nonsmooth analysis; discrete systems; continuous systems; systems over time scales; and impulsive-systems. These subjects govern a broad spectrum of applications arising from industry, engineering, economics, biology, and many others fields. In the last decade, more applications pointed toward combining both continuous-time and discrete-time systems which resulted in the birth of two different fields: Systems over time scales and impulsive hybrid systems. Nowadays they are two of the most rapidly growing areas of current research in dynamical systems theory. Results for either type of system are challenging, due to the discrepancies between the continuous and discrete settings and to their combined complexities. The objective of this investigation is to address key challenges in four topics of applied mathematics: optimal controls over time scales or over impulsive systems, nonsmooth analysis, and infinite dimensional optimization. More specifically, the aim of the proposal is four-fold. (i) To launch the field of optimal controls over time scales in which constraints are allowed. (ii) To build a bridge between optimal controls over time scales and that over impulsive systems.(iii) To introduce derivative- and Hessian- like objects for infinite dimensional "nonsmooth" functions that approximate such functions by linear and bilinear operators and that actually possess all the properties required from such sets; as was the case with Clarke's generalized Jacobian and Hessian in finite dimension. (iv) To develop in terms of those objects, optimality criteria for infinite dimensional nonsmooth optimization problems with constraints. In order execute these projects, a mixed bag of new and known techniques is needed such as nonsmooth and variational analysis techniques, optimization methods, Dubovitskii-Milyutin-type approach, dynamic programming- type techniques, time scales tools etc.Optimal control is a field whose mere existence is the product of applications arising from fields such as: aerospace, mechanical and electrical engineering, automatics, robotics, automotive electronics, economics, biology, and more. Therefore, important systematic developments of this research program have a significant impact on those fields. The outcome of this project will be a fundamental breakthrough leading to a new methodology in approaching some hybrid systems, and a new incentive to further advance optimal controls over time scales. The investigator integrates her research into education by incorporating her findings into her two graduate courses, which are well attended by engineering students, and by crafting a capstone course at the undergraduate level in which the applications of optimal control theory to the above disciplines are enhanced. Such an activity contributes greatly in training K-12 science and math teachers who graduate from MSU.