Weak Hamiltonian finite element method for optimal control problems

Abstract

A temporal finite element method based on a mised form of the Hamiltonian n-eal; principle is developed for dynamics and optimal control problems. The mixed form of Hamilton's weak principle contains both displacements and momenta as primary variables that are expanded in terms of nodal values and simple polynomial shape functions. Unlike other forms of Hamilton's principle, however, time derivatives of the nionient a and displacements do not appear therein; instead, only the virtual momenta and virtual displaceinents are differentiated with respect to time. Based on the duality that is obser\-cd to esist between the mixed form of Hamilton's weak principle and variational principles governing classical optiinal control problems, a temporal finite element forinulation of the latter can be developed in a rather straightforward manner. Several n-ell-l;nowl problelxs in dynamics and optimal control are illustrated. The esample dynamics problem inr-olws a time-marching problem. As optimal control esaniples, elementary trajectory optimizat ion problems are treated.

Global solution of multi-objective optimal control problems with multi agent collaborative search and direct finite elements transcription

“…DFET was initially proposed in [8] and uses finite elements in time on spectral bases to transcribe the differential equations into a set of algebraic equations. Finite Elements in Time for the indirect solution of optimal control problems were initially proposed by Hodges et al in [9], and during the late 1990s evolved to the discontinuous version. As pointed out by Bottasso et al in [10], FET for the forward integration of ordinary differential equations are equivalent to some classes of implicit RungeKutta integration schemes, can be extended to arbitrary highorder, are very robust and allow full h-p adaptivity.…”

Dynamic Simulation of Multibody Systems Using a New State-Time Methodology

“…Dynamics community have also considered this type of approach in regard to various applications such as structural dynamics, nonlinear vibrations, inverse dynamics of flexible robots, gate analysis, and optimal control problem for converting problems from infinite dimension to finite dimension. Some of the related works can be found in [23]- [29]. These prior time-parameterization methods often produced highly nonlinear and coupled sets of equations, thus limiting their tractability with complex systems.…”

Simulation of a Moving, Elastic Beam Using Hamilton's Weak Principle

“…Riff and Baruch [9] [15,29] demonstrated that a mixed finite element formulation (where generalized coordinates and their momenta appear as independent unknowns) yields solutions with unconditional stability without having to use reduced element quadrature. The current research is also based on a mixed finite element formulation.…”

“…Because there are no derivatives, only simple shape functions are necessary to satisfy C 0 continuity between elements. Higher order elements (p elements) could also be developed; however, for general nonlinear problems the use of crude shape functions is more efficient in that it allows element quadrature to be performed by inspection [29]. Independent variables need only be piecewise continuous, while variational quantities must be continuous and piecewise differentiable.…”

“…A time marching procedure was used in [45] to demonstrate accuracy for some simple problems. Hodges [29] applies HWP in mixed variational form to optimal control problems and nonlinear initial value problems. He also uses the technique to develop exact intrinsic equations for a moving beam [26], a derivation which is used extensively in this research.…”