An approximation method for the stabilizing solution of the Hamilton-Jacobi equation for integrable systems using Hamiltonian perturbation theory

Abstract

An approximation method for the stabilizing solution of the Hamilton-Jacobi equation for integrable systems using Hamiltonian perturbation theory Sakamoto, Noboru; Schaft, Arjan J. van der Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Abstract-In this report, a method for approximating the stabilizing solution of the Hamilton-Jacobi equation for integrable systems is proposed using symplectic geometry and a Hamiltonian perturbation technique. Using the fact that the Hamiltonian lifted system of an integrable system is also integrable, the Hamiltonian system (canonical equation) that is derived from the theory of 1-st order partial differential equations is considered as an integrable Hamiltonian system with a perturbation caused by control. Assuming that the approximating Riccati equation from the Hamilton-Jacobi equation at the origin has a stabilizing solution, we construct approximating behaviors of the Hamiltonian flows on a stable Lagrangian submanifold, from which an approximation to the stabilizing solution is obtained.

An analytical approximation method for the stabilizing solution of the Hamilton-Jacobi equation based on stable manifold theory

“…4, the calculation result by the Hamiltonian perturbation approach in [26] is shown. Since the integrable nonlinearity is fully taken into account in this approach, the feedback function is better approximated in the region further from the origin.…”

“…See also [30] for the treatment of the Hamilton-Jacobi equation as well as recently developed techniques in nonlinear control theory such as the theory of port-Hamiltonian systems. Recently, the authors proposed a Hamiltonian perturbation approach to obtain an approximation of the stabilizing solution when the uncontrolled part of the system is integrable [26].…”