Quasi-linear partial differential equations for the optimal control of nonlinear systems over an infinite horizon

The infinite-horizon optimal control problem is studied in the linear setting with the objective of revisiting the role of the underlying Algebraic Riccati Equation. It is shown that this may be interpreted in terms of a triangularizing change of coordinates for the Hamiltonian dynamics associated to the optimal control problem. Such a perspective leads to a few implications, including the observation that duality between the Riccati equations arising in various contexts is simply related to a choice of coordinates for the Hamiltonian dynamics. Similar arguments are then extended to the nonlinear setting. It is shown that, by relying on such alternative view point, the computational complexity associated to the solution of nonlinear optimal control problems over an infinite horizon tantamounts to the solution of an Algebraic Riccati Equation, for the linearized problem, and a quasi-linear partial differential equation.