WEAK HAMILTONIAN FINITE-ELEMENT METHOD FOR OPTIMAL-CONTROL PROBLEMS

A temporal finite element method based on a mixed form of Hamilton's weak 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 momenta and displacements do not appear therein; instead, only the virtual momenta and virtual displacements are differentiated with respect to time. Based on the similarity in structure that is observed to exist between the mixed form of Hamilton's weak principle and variational principles governing classical optimal control problems, a temporal finite element formulation of the latter can be developed in a rather straightforward manner. Very crude shape functions (so simple that element numerical quadrature is not necessary) can be used to develop an efficient procedure for obtaining candidate solutions (i.e., those which satisfy all the necessary conditions) even for highly nonlinear problems. Solutions for some well-known problems in dynamics and optimal control are illustrated. The example dynamics problem involves a nonlinear time-marching problem. As optimal control examples, trajectory optimization problems with both fixed and open final time are treated.