Applications of variational analysis to optimal trajectories and nonsmooth Hamilton-Jacobi theory

The ability to analyze set-valued mappings through tools of variational analysis has allowed for a greater understanding of problems in optimization and optimal control. In this thesis, new results for necessary conditions on optimal trajectories and regularity of these arcs are presented. Employing these tools in Hamilton-Jacobi theory, we show that the value function of a generalized Bolza problem is the unique (nonsmooth) solution to the Cauchy form of the Hamilton-Jacobi equation. This greatly extends the class of Hamiltonians for which uniqueness can be established. Crucial to these developments are set-valued mappings that arise from epigraphs of extended real-valued functions. We overcome the difficulty confronting us of the unbounded images by introducing cosmically Lipschitz set-valued mappings.