An approximation method for exact controls of vibrating systems with numerical viscosity

¹ Laboratoire de Mathématiques Blaise Pascal, Université Clermont Auvergne Campus des Cézeaux – 3 place Vasarely, 63178 Aubière, France
² Department of Mathematics, University of Craiova, Craiova, 200585 and “Gheorghe Mihoc-Caius Iacob” Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy 70700, Romania
³ Department of Mathematics, University of Craiova Craiova 200585, Romania

^* Corresponding author: ionelroventa@yahoo.com

Received: 22 June 2023
Accepted: 16 March 2024

Abstract

We analyze a method for the approximation of exact controls of a second order infinite dimensional system with bounded input operator. The algorithm combines Russell’s “stabilizability implies controllability” principle and a finite elements method of order θ with vanishing numerical viscosity. We show that the algorithm is convergent for any initial data in the energy space and that the error is of order θ for sufficiently smooth initial data. Both results are consequences of the uniform exponential decay of the discrete solutions guaranteed by the added viscosity and improve previous estimates obtained in the literature. Several numerical examples for the wave and the beam equations are presented to illustrate the method analyzed in this article.