New trends in fractional optimal control problems

In this thesis, I study the basics of some fractional derivatives (e. g. Riemann-Liouville, Caputo) with the corresponding approximation based on Gru ̈nwald-Letnikov definitions. Later, the fundamentals of fractional optimal control problem are presented via mentioned fractional derivatives which are used in the definition of constraints and optimality conditions given through the formulation. Some new aspects are studied for the numerical solutions of fractional optimal control problems in the sense of integrating new orthogonal polynomials to approximate the considered fractional derivatives. In this respect, Bernstein polynomials, shifted Chebyshev polynomials and shifted Legendre orthonormal polynomials are newly used within the Legendre-Gauss quadrature method in order to approximate and solve numerically the Caputo based fractional partial differential equations coming from the formulation of fractional optimal control problem. Two dynamical systems are considered as illustrative examples based on the given control functions and the corresponding responses of the systems are presented under fractional derivatives. Then the comparison with the classical derivative is discussed. It is observed from numerical results and presented simulations that the system response increases as the fractional order of the derivative decreases for the same point of the variable t.