Título: Solution of Fractional Optimal Control Problems with Specified Final State by using Orthogonal Collocation and Differential Evolution
Autores: LIMA, J. V. C. F.; LOBATO, F. S.; STEFFEN Jr, V.
Revista: Latin-American Journal of Physics Education, 16(4), 4312-4329, 2023.
Resumo: In the last years, the study of Fractional Optimal Control Problem (FOCP) configures a very interesting challenge due to numerical difficulties inherent in this type of investigation. Traditionally, this problem has been solved by considering three different approaches, namely, Direct and Indirect strategies and Hamilton–Jacobi–Bellman (HJB) equation. The first consists in solving the FOCP through the discretization of state and/or control variables. The resulting nonlinear optimization problem is solved by considering either classical or heuristic methods. On the other hand, the indirect approach consists in obtaining the necessary conditions, i.e., the original FOCP is converted into a two-point boundary value problem. The third strategy considers an extension of the well-known HJB equation for fractional order dynamic systems. In the present contribution, the solution of FOCP with specified final state variable is addressed by using the direct approach. For this purpose, the association involving the Orthogonal Collocation Method (OCM) and the Differential Evolution (DE) algorithm is investigated. In order to evaluate the proposed methodology, a classical mathematical problem and a two degree-of-freedom given by a spring-mass-damper system are considered. As expected, the results indicate that the variation of the fractional order implies different values for the original objective function. Furthermore, depending on the fractional order value, it may not be possible to find a solution that satisfies the boundary conditions for a given application. Finally, it is pointed out that the proposed methodology is considered as a promising strategy to solve FOCPs.
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