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

Sakamoto, N. & van der Schaft, A. J., Nov-2008, In : IEEE Transactions on Automatic Control. 53, 10, p. 2335-2350 16 p.

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  • Analytical Approximation Methods for the StabilizingSolution of the Hamilton–Jacobi Equation

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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.

Original languageEnglish
Pages (from-to)2335-2350
Number of pages16
JournalIEEE Transactions on Automatic Control
Issue number10
Publication statusPublished - Nov-2008


  • Hamilton-Jacobi equation, Hamiltonian systems, nonlinear control theory, perturbation method, stable manifold theory, symplectic geometry, H-INFINITY-CONTROL, INNER-OUTER FACTORIZATION, NONLINEAR-SYSTEMS, VISCOSITY SOLUTIONS, DYNAMICAL-SYSTEMS, FEEDBACK, GEOMETRY, DESIGN

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