## ABSTRACT

In this chapter, a spectral formulation is introducedfor a fractional optimal control problem (FOCP) defined in spherical coordinates in the cases of half and complete axial symmetry. The dynamics of optimally controlled system are described by space––time fractional diffusion equation in terms of the Caputo and fractional Laplacian differentiation operators. The first step in the numerical methodology is to represent the state and control functions of the system as eigenfunction expansion series. This is clearly obtained by the discretization of space fractional Laplacian operator term. In the next step, the necessary optimality conditions are determined by using fractional Euler––Lagrange equations. Therefore, we also reduce the space––time fractional differential equation (FDE) into time FDEs in terms of forward and backward fractional Caputo derivatives. In the last step, the time domain is discretized into a number of subintervals by using Gr\"{u}nwald––Letnikov approach. An illustrative example is considered for various orders of fractional derivatives and different spatial and temporal discretizations. As a result, a limited number of grid points are sufficient to obtain good results. In addition, the numerical solutions converge as the size of the time step is reduced. Note that solution techniques in the sense of analytical or numerical for space––time FDEs defined by fractional Laplacian, Riesz or Riesz––Feller spatial derivatives arequite complicated. In reality, these equations correspond to systems that show the behaviour of anomalous diffusion. For this complexity, the spectral approaches depend on the spatial domain, on which the main problem is constructed, are more clear and so extensive applicability than the alternative ones.