Nonlinear feedback control and artificial intelligence computational methods applied to a dissipative dynamic contact problem

In this paper we consider a vibrational percussion system described by a one-dimensional hyperbolic partial differential equation with boundary dissipation at one extremity and a normal compliance contact condition at the other extremity. Firstly, we obtain the mathematical model using the Calculus of variations and we prove the existence of weak solutions. Secondly, we focus on the numerical approximation of solutions by using a neuromathematics approach - a well-structured numerical technique which combines a specific approach of Method of Lines with the paradigm of Cellular Neural Networks. Our technique ensures from the beginning the requirements for convergence and stability preservation of the initial problem and, exploiting the local connectivity of the approximating system, leads to a low computational effort. A comprehensive set of numerical simulations, considering both contact and non-contact cases, ends the contribution.