Generalized conjugacy in Hamilton-Jacobi theory for fully convex Lagrangians

Control problems with fully convex Lagrangians and convex initial costs are considered. Generalized conjugacy and envelope representation in terms of a dualizing kernel are employed to recover the initial cost from the value function at some, fixed future time, leading to a generalization of the cancellation rule for inf-convolution. Such recovery is possible subject to persistence of trajectories of a generalized Hamiltonian system, associated with the Lagrangian. Global analysis of Hamiltonian trajectories is carried out, leading to conditions on the Hamiltonian, and the corresponding Lagrangian, guaranteeing persistence of the trajectories.