Comparison of two methods for the calculation of the gradient and of the Hessian of cost functions associated with differential systems

Identification and control problems associated with lumped and distributed systems are studied. The problems are set in terms of minimization of a cost function and two methods for the calculation of the gradient and of the Hessian of the cost function are considered: the first method is based on the classical Lagrangian approach using an adjoint state and an adjoint equation, while the second method involves a direct differentiation of the state equation. The paper is devoted to a comparison between the number of elementary calculations required by each method. The Lagrangian method turns out to be more advantageous.