  The Inverse Electrical Impedance Tomography (EIT) problem on recovering
electrical conductivity tensor and potential in the body based on the
measurement of the boundary voltages on the electrodes for a given electrode
current is analyzed. A PDE constrained optimal control framework in Besov space
is pursued, where the electrical conductivity tensor and boundary voltages are
control parameters, and the cost functional is the norm declinations of the
boundary electrode current from the given current pattern and boundary
electrode voltages from the measurements. The state vector is a solution of the
second order elliptic PDE in divergence form with bounded measurable
coefficients under mixed Neumann/Robin type boundary condition. Existence of
the optimal control and Fr\'echet differentiability in the Besov space setting
is proved. The formula for the Fr\'echet gradient and optimality condition is
derived. Extensive numerical analysis is pursued in the 2D case by implementing
the projective gradient method, re-parameterization via principal component
analysis (PCA) and Tikhonov regularization.
