Direct Transcription of Optimal Control Problems with Finite Elements on Bernstein Basis

Abstract

The paper introduces the use of Bernstein polynomials as a basis for the direct transcription of optimal control problems with finite elements in time. Two key properties of this new transcription approach are demonstrated in this paper: Bernstein bases return smooth control profiles with no oscillations near discontinuities or abrupt changes of the control law, and for convex feasible sets, the polynomial representation of both states and controls remains within the feasible set for all times. The latter property is demonstrated theoretically and experimentally. A simple but representative example is used to illustrate these properties and compare the new scheme against a more common way to transcribe optimal control problems with finite elements in time.