Solving two-point boundary value problems using generating functions: Theory and Applications to optimal control and the study of Hamiltonian dynamical systems

A methodology for solving two-point boundary value problems in phase space for Hamiltonian systems is presented. Using Hamilton-Jacobi theory in conjunction with the canonical transformation induced by the phase flow, we show that the generating functions for this transformation solve any two-point boundary value problem in phase space. Properties of the generating functions are exposed, we especially emphasize multiple solutions, singularities, relations with the state transition matrix and symmetries. Then, we show that using Hamilton’s principal function we are also able to solve two-point boundary value problems, nevertheless both methodologies have fundamental differences that we explore. Finally, we present some applications of this theory. Using the generating functions for the phase flow canonical transformation we are able to solve the optimal control problem (without an initial guess), to study phase space structures in Hamiltonian dynamical systems (periodic orbits, equilibrium points) and classical targeting problems (this last topic finds its applications in the design of spacecraft formation trajectories, reconfiguration, formation keeping, etc...).