Consistency of Approximation of Bernstein Polynomial-Based Direct Methods for Optimal Control

Venanzio Cichella, Isaac Kaminer, Claire Walton, Naira Hovakimyan, António Pascoal

Research output: Contribution to journalArticlepeer-review


Bernstein polynomial approximation of continuous function has a slower rate of convergence compared to other approximation methods. “The fact seems to have precluded any numerical application of Bernstein polynomials from having been made. Perhaps they will find application when the properties of the approximant in the large are of more importance than the closeness of the approximation.”—remarked P.J. Davis in his 1963 book, Interpolation and Approximation. This paper presents a direct approximation method for nonlinear optimal control problems with mixed input and state constraints based on Bernstein polynomial approximation. We provide a rigorous analysis showing that the proposed method yields consistent approximations of time-continuous optimal control problems and can be used for costate estimation of the optimal control problems. This result leads to the formulation of the Covector Mapping Theorem for Bernstein polynomial approximation. Finally, we explore the numerical and geometric properties of Bernstein polynomials, and illustrate the advantages of the proposed approximation method through several numerical examples.

Original languageEnglish (US)
Article number1132
Issue number12
StatePublished - Dec 2022


  • numerical optimal control
  • Bernstein polynomials
  • Bezier curves

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Computer Science (miscellaneous)
  • Mechanical Engineering
  • Control and Optimization
  • Industrial and Manufacturing Engineering
  • Electrical and Electronic Engineering


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