10 research outputs found
Computation of order conditions for symplectic partitioned Runge-Kutta schemes with application to optimal control
We discuss the derivation of order conditions for the discretization of (unconstrained) optimal control problems, when the scheme for the state equation is of Runge-Kutta type. This problem appears to be essentially the one of checking order conditions for symplectic partitioned Runge-Kutta schemes. We show that the the computations using bi-coloured trees are naturally expressed in this case in terms of oriented free tree. This gives a way to compute them by an appropriate computer program. Our software is able to compute conditions up to order 7 (we display them up to order 6). The results are in accordance with those of Hager (where they were computed for order up to 4) as well as those of Murua where the number of conditions up to order 7 is stated
Asymptotic Preserving time-discretization of optimal control problems for the Goldstein-Taylor model
We consider the development of implicit-explicit time integration schemes for
optimal control problems governed by the Goldstein-Taylor model. In the
diffusive scaling this model is a hyperbolic approximation to the heat
equation. We investigate the relation of time integration schemes and the
formal Chapman-Enskog type limiting procedure. For the class of stiffly
accurate implicit-explicit Runge-Kutta methods (IMEX) the discrete optimality
system also provides a stable numerical method for optimal control problems
governed by the heat equation. Numerical examples illustrate the expected
behavior
Implicit-Explicit Runge-Kutta schemes for numerical discretization of optimal control problems
Implicit-explicit (IMEX) Runge-Kutta methods play a major rule in the
numerical treatment of differential systems governed by stiff and non-stiff
terms. This paper discusses order conditions and symplecticity properties of a
class of IMEX Runge-Kutta methods in the context of optimal control problems.
The analysis of the schemes is based on the continuous optimality system. Using
suitable transformations of the adjoint equation, order conditions up to order
three are proven as well as the relation between adjoint schemes obtained
through different transformations is investigated. Conditions for the IMEX
Runge-Kutta methods to be symplectic are also derived. A numerical example
illustrating the theoretical properties is presented
High order variational integrators in the optimal control of mechanical systems
In recent years, much effort in designing numerical methods for the
simulation and optimization of mechanical systems has been put into schemes
which are structure preserving. One particular class are variational
integrators which are momentum preserving and symplectic. In this article, we
develop two high order variational integrators which distinguish themselves in
the dimension of the underling space of approximation and we investigate their
application to finite-dimensional optimal control problems posed with
mechanical systems. The convergence of state and control variables of the
approximated problem is shown. Furthermore, by analyzing the adjoint systems of
the optimal control problem and its discretized counterpart, we prove that, for
these particular integrators, dualization and discretization commute.Comment: 25 pages, 9 figures, 1 table, submitted to DCDS-
Discrete mechanics and optimal control: An analysis
The optimal control of a mechanical system is of crucial importance in many application areas. Typical examples are the determination of a time-minimal path in vehicle dynamics, a minimal energy trajectory in space mission design, or optimal motion sequences in robotics and biomechanics. In most cases, some sort of discretization of the original, infinite-dimensional optimization problem has to be performed in order to make the problem amenable to computations. The approach proposed in this paper is to directly discretize the variational description of the system's motion. The resulting optimization algorithm lets the discrete solution directly inherit characteristic structural properties from the continuous one like symmetries and integrals of the motion. We show that the DMOC (Discrete Mechanics and Optimal Control) approach is equivalent to a finite difference discretization of Hamilton's equations by a symplectic partitioned Runge-Kutta scheme and employ this fact in order to give a proof of convergence. The numerical performance of DMOC and its relationship to other existing optimal control methods are investigated