Toward Efficient Global Solutions to Optimal Control Problems via Second-Order Polynomial Approximations

Optimal control problems are used for many tasks such as model-based control, state and parameter estimation, and experimental design for complex dynamic systems. The solution to these problems can be divided into two tasks, where the first corresponds to the enumeration of different arc sequences and the second is the computation of the optimal values of the decision variables for each arc sequence. For the latter task, this paper proposes a method to approximate the cost and constraints of the problem as polynomial functions of the decision variables via computation of partial derivatives up to second order and multivariate Hermite interpolation. This method allows reformulating the problem for an arc sequence as a polynomial optimization problem, which is expected to enable efficiently solving optimal control problems to global optimality. The method is illustrated by a simulation example of a reaction system.