Solutions of the optimal feedback control problem using Hamiltonian dynamics and generating functions

We show that the optimal cost function that satisfies the Hamilton-Jacobi-Bellman (HJB) equation is a generating function for a class of canonical transformations for the Hamiltonian dynamical system defined by the necessary conditions for optimality. This result allows us to circumvent the final time singularity in the HJB equation for a finite time problem, and allows us to analytically construct a nonlinear optimal feedback control and cost function that satisfies the HJB equation for a large class of dynamical systems. It also establishes that the optimal cost function can be computed from a large class of solutions to the Hamilton-Jacobi (HJ) equation, many of which do not have singular boundary conditions at the terminal state.

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