212 research outputs found
Analytical Approximation Methods for the Stabilizing Solution of the Hamilton–Jacobi Equation
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.
Optimal Stabilization of Periodic Orbits
In this contribution the optimal stabilization problem of periodic orbits is
studied in a symplectic geometry setting. For this, the stable manifold theory
for the point stabilization case is generalized to the case of periodic orbit
stabilization. Sufficient conditions for the existence of a \gls{nhim} of the
Hamiltonian system are derived. It is shown that the optimal control problem
has a solution if the related periodic Riccati equation has a unique periodic
solution. For the analysis of the stable and unstable manifold a coordinate
transformation is used which is moving along the orbit. As an example, an
optimal control problem is considered for a spring mass oscillator system,
which should be stabilized at a certain energy level.Comment: Submitted for IFAC World Congress 202
Hamilton-Jacobi-Bellman equation of nonlinear optimal control problems with fractional discount rate
This paper derives the Hamilton-Jacobi-Bellman equation of nonlinear optimal
control problems for cost functions with fractional discount rate from the
Bellman's principle of optimality. The fractional discount rate is described by
Mittag-Leffler function that can be considered as a generalized exponential
function
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