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.

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This paper presents an extension of a finite element formulation based on a weak form of the necessary conditions to solve optimal control problems. First, a general formulation for handling internal boundary conditions and discontinuities in the state equations is presented. Then, the general formulation is modified for optimal control problems subject to state-variable inequality constraints. Solutions with touch points and solutions with stateconstrained arcs are considered. After the formulations are developed, suitable shape and test functions are chosen for a finite element discretization. It is shown that all element quadrature (equivalent to one-point Gaussian quadrature over each element) may be done in closed form, yielding a set of algebraic equations. To demonstrate and analyze the accuracy of the finite element method, a simple state-constrained problem is solved. Then, for a more practical application of the use of this method, a launch vehicle ascent problem subject to a dynamic pressure constraint is solved. The paper also demonstrates that the finite element results can be used to determine switching structures and initial guesses for a shooting code.

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