A p_nh-Adaptive Refinement Procedure for Numerical Optimal Control Problems

This paper presents an automatic procedure to enhance the accuracy of the numerical solution of an optimal control problem (OCP) discretized via direct collocation at Gauss–Legendre points. First, a numerical solution is obtained by solving a nonlinear program (NLP). Then, the method evaluates its accuracy and adaptively changes both the degree of the approximating polynomial within each mesh interval and the number of mesh intervals until a prescribed accuracy is met. The number of mesh intervals is increased for all state vector components alike, in a classical fashion. Instead, improving on state-of-the-art procedures, the degrees of the polynomials approximating the different components of the state vector are allowed to assume, in each finite element, distinct values. This explains the $pnh$ definition, where n is the state dimension. With respect to the approaches found in the literature, where the degree is always raised to the highest order for all the state components, our methods allow a sensible reduction of the overall number of variables of the resulting NLP, with a corresponding reduction of the computational burden. Numerical tests on three OCP problems highlight that, under the same maximum allowable error, by independently selecting the degree of the polynomial for each state, our method effectively picks lower degrees for some of the states, thus reducing the overall number of variables in the NLP. Accordingly, various advantages are brought about, the most remarkable being: (i) an increased computational efficiency for the final enhanced mesh with solution accuracy still within the prescribed tolerance, (ii) a reduced risk of being trapped by local minima due to the reduced NLP size, and (iii) a gain of the robustness of the convergence process due to the better-behaved solution landscapes.