The main objectives of this project are to develop nonsmooth analysis for semicontinuous functions and multivalued mappings on smooth manifolds and to apply these new analytical techniques to a variety of optimization and control problems. Examples of nonsmooth (nondifferentiable) functions on manifolds are provided by the maximum eigenvalue of a symmetric matrix on the manifold of symmetric matrices, the Riemannian metric and Riemannian distance functions on Riemannian manifold, control Lyapunov functions for stabilization of control systems on manifolds and optimal value functions for optimal control problems. The major directions for research are the following: (i) subdifferential(infinitesimal) calculus for nonsmooth functions on smooth manifolds; (ii) applications of subdifferential calculus to optimization problems on smooth manifolds; (iii) applications of subdifferential calculus to the study of invariance of closed sets and monotonicity of semicontinuous functions with respect to solutions of differential inclusions on manifolds; (iv) applications of these results to the study of generalized solutions to Hamilton-Jacobi equations on manifolds; (v) applications to the derivation of optimality conditions for general nonsmooth control problems on manifolds; (vi) applications of subdifferential calculus for constructing discontinuous optimal and stabilizing feedback controls using nonsmooth optimal value and control Lyapunov functions; (vii) the study of interior point methods in semistable and semidefinite programmings via discontinuous feedback techniques and the study of generalized gradient flows on manifolds.
Design of high-performance feedback controls for nonlinear systems whose mathematical models include manifolds is an active research area oriented to automotive, aerospace, and naval applications. It has been recognized that, in general, control tasks (such as stabilization or robust optimal control) for these systems cannot be performed using traditional continuous feedback and require discontinuous feedback. Since existing mathematical tools (differential inclusion theory) have proved inadequate for analysis of discontinuous feedback performance, this research project is aimed at developing new analytical tools and techniques for such analysis. But feedback control is not the only one research field which will benefit from these new analytical tools. Other fields of application for these research results include numerical and theoretical optimization, in particular, a development of robust numerical methods for new semistable and semidefinite optimization problems, and generalized solutions of partial differential equations on manifolds .