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Consistent approximation of optimal control problems using Bernstein polynomials

We present a direct method for the solution of nonlinear optimal control problems based on Bernstein polynomial approximations. We show, using a rigorous setting, that the proposed method yields consistent approximations of time continuous optimal control problems. We demonstrate that the method can also be used for the estimation of optimal control problems costates. This result leads to the formulation of the Covector Mapping Theorem for Bernstein polynomial approximation. Finally, we exploit the numerical and geometric properties of Bernstein polynomials, and illustrate the advantages of the method through numerical examples.