Solutions of optimal feedback control problems with general boundary conditions using Hamiltonian dynamics and generating functions

Given a nonlinear system and performance index to be minimized, we present a general approach to evaluating the optimal feedback control law for this system that can be easily modified to satisfy different types of boundary conditions. Formulated in the context of Hamiltonian systems theory, this work allows us to analytically construct optimal feedback control laws from generating functions. Given our feedback control law solution, our approach enables us to obtain the feedback control for a different set of boundary conditions only using a series of algebraic manipulations, partial differentiations, and solutions of implicit algebraic equations. Furthermore, the proposed approach reveals a fundamental insight: that the optimal cost function that satisfies the HJB equation can be expressed as a generating function for a class of canonical transformations of the Hamiltonian system defined by the necessary conditions for optimality. This result is formalized as a theorem, which relates the sufficient condition to the necessary conditions for optimality. The whole procedure provides an advantage over the conventional method based on dynamic programming, which requires one to solve the HJB PDE repetitively for each type of boundary condition.