Abstract
The approximate solution for solving a class of composition fractional order optimal control problems (FOCPs) is suggested and studied in detail. However, the properties of Caputo and Riemann-Liouville derivatives are also given with complete details on Chebyshev approximation function to approximate the solution of fractional differential equation with different approach. Also, the relation between Caputo and Riemann-Liouville of fractional derivative took a big role for simplifying the fractional differential equation that represents the constraints of optimal control problems. The approximate solutions are defined on interval [0,1] and are compared with the exact solution of order one which is an important condition to support the working method. Finally, illustrative examples are included to confirm the efficiency and accuracy of the proposed method.
Highlights
The idea of fractional derivative dates back to a conversation between two mathematicians: Leibniz and L’Hopital
A fractional optimal control problem (FOCPs) is an optimal control problem focused on the performance index or the fractional differential equations governing the dynamics of the system or both contain at least one fractional order derivative term
The formulation and solution of state and variables fractional order optimal control problems (FOCPs) were first established by Agrawal, where the applied fractional variational calculus (FVC) presented a general formulation and solution scheme for FOCPs in the RiemannLiouville (RL) sense; it was based on variational virtual work coupled with the Lagrange multiplier technique
Summary
The idea of fractional derivative dates back to a conversation between two mathematicians: Leibniz and L’Hopital. Agrawal and Baleanu [4] obtained necessary conditions for FOCPs with Riemann-Liouville derivative and were able to solve the problem numerically. In [6], Khader and Hendy studied an efficient numerical scheme for solving fractional optimal control problems. In [7], Akbarian and Keyanpour studied a new approach to the numerical solution of fractional order optimal control Problems. The multi-fractional differential equations corresponding to optimal control problems and basic theorems have been given with algorithm for multi-composite fractional order optimal control problems.
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