Piecewise Polynomial Control Parameterization in the Direct Solution of Optimal Control Problems

This paper describes an algorithm for the direct solution of a class of optimal control problems. The algorithm is based on approximating the unknown control inputs via a finite dimensional parameterization. Specific control approximations that are implemented include (i) piecewise constant, (ii) piecewise linear, or (iii) piecewise cubic polynomials. The cubic approximation presented here is believed to be new. Another novel feature of the algorithm is that the state equations are approximated using single step Runge–Kutta methods on a fixed mesh. The parameterized optimal control problem is solved using a sequential quadratic programming technique. The paper presents examples to illustrate the convergence behavior of the various control parameterization schemes.