Analytical Approximation Methods for the Stabilizing Solution of the Hamilton–Jacobi Equation

Abstract

In this paper, two methods for approximating the stabilizing solution of the Hamilton-Jacobi equation are proposed using symplectic geometry and a Hamiltonian perturbation technique as well as stable manifold theory. The first method uses the fact that the Hamiltonian lifted system of an integrable system is also integrable and regards the corresponding Hamiltonian system of the Hamilton-Jacobi equation as an integrable Hamiltonian system with a perturbation caused by control. The second method directly approximates the stable flow of the Hamiltonian systems using a modification of stable manifold theory. Both methods provide analytical approximations of the stable Lagrangian submanifold from which the stabilizing solution is derived. Two examples illustrate the effectiveness of the methods.

Approximate solutions to the Hamilton-Jacobi equations for generating functions: The general cost function case

“…The generating functions S(x, λ 0 ,t) is well-defined at initial time t = t 0 for (9) as follows due to (8):…”

“…Then it is easy to obtain the costate λ (x) by the first equation in the canonical transformations (8). Finally, we can obtain the optimal control u * (x) by (6).…”

“…An approximate stabilizing solution of the HJE is obtained by using symplectic geometry and the Hamiltonian perturbation technique for feedback optimal control problems [8]. However, all of these methods are for the value function.…”

“…However, all of these methods are for the value function. In particular, papers [4], [5], [7], [8] are for the infinite time optimal control problems. For a finite time nonlinear optimal control problem with fixed initial and terminal state values, it is reduced to a Two-Point Boundary-Value Problem (TPBVP) for an ordinary differential equation (ODE) with respect to a Hamiltonian system [9].…”

Nonlinear Optimal Servo Control Design for PMSM with Inverter Voltage Norm Constraints

“…Anti-windup [1] (PMSM : Permanent Magnet Synchronous Motor) Hamilton-Jacobi [5] [6,7] Anti-windup [8,9] [10] . [11] Hamilton-Jacobi HamiltonJacobi PMSM …”