The project deals with the finite element approximation of the monodomain equation, which models the propagation of the electrical potential in the cardiac muscle through a coupled system of PDE’s. The goal is to recover the shape of an infarcted area inside the cardiac muscle, by measuring the electrical potential on the boundary of the domain representing a portion of tissue. An optimal control formulation of the inverse problem will be analyzed, by considering infarcted areas described by few parameters (e.g width and position). To reach the description of the state problem through a monodomain model for the transmembrane potential, coupled with a system of ODEs describing the ionic model, some steps will be performed by considering first a linear time-dependent state problem; then a nonlinear time-independent state problem; finally, a nonlinear coupled state problem.