Spectral methods to approximate control and inverse problems for the wave equation and elasticity system

• Autores: Somia Boumimez
• Directores de la Tesis: Carlos Manuel Castro Barbero (dir. tes.)
• Lectura: En la Universidad Politécnica de Madrid ( España ) en 2024
• Idioma: inglés
• Tribunal Calificador de la Tesis: María Luisa Rapún (presid.) , Fabricio Maciá Lang (secret.) , María López Fernández (voc.) , Francisco Periago Esparza (voc.) , Anna Doubova Krasotchenko (voc.)
• Programa de doctorado: Programa de Doctorado en Ingeniería Matemática, Estadística e Investigación Operativa por la Universidad Complutense de Madrid y la Universidad Politécnica de Madrid
• Calificación: Sobresaliente Cum Laude
• Enlaces
  • Tesis en acceso abierto en: Archivo Digital UPM
• Resumen

  • In this Thesis, we explore the use of spectral methods to approximate numerically some control and inverse problems for the wave equation and the elasticity system. Roughly speaking, in a control problem one is interested in acting on the system (though a control) in order to drive an initial state into a desired final state after some prescribed time interval. Of course the main interest (and difficulty) arises when the control acts in a small part of the domain or the boundary. Here we focus on boundary controllability, i.e. the particular case where the control acts in one part of the boundary, since it is the case with more practical applications. After the seminal work by \cite{lions3} it is well-known that boundary controllability of the wave equation can be reduced to a suitable observability inequality for the solutions of the adjoint uncontrolled wave equation. Roughly speaking, such observability inequalities give an estimate of global information, typically the energy of the solution, from local one obtained from the solution on the part of the boundary where the control acts, and for sufficiently large time. They are obtained by specific techniques such as the multipliers method, Carleman inequalities, moment problems, etc. To obtain a numerical approximation of controls a natural procedure is as follows: first, replace the control problem by a discrete one where the controlled system is approximated by a suitable discretization of the continous one. This discrete approximation will depend on a small discretization parameter $h$. Then, solve the discrete control problem and, finally, prove the convergence of the discrete control to a control of the continuous system, as the discretization parameter $h$ goes to zero. It is important to note that a convergent numerical scheme for the continuous system does not guarantee the convergence of the discrete controls. In fact, the existence of discrete controls requires a discrete observability inequality for the discrete adjoint system, while the convergence will require the uniformity of this constant with respect to the discretization parameter $h$. It turns out that this uniformity is not true for the classical discretization of the wave equation (finite difference or finite elements) and in fact, the discrete controls obtained by this natural procedure constitute an unbounded sequence, as $h$ goes to zero. This difficulty has given rise to an active research topic in the last 30 years and a number of different techniques have been studied to overcome this difficulty: Tychonoff regularization, bigrid techniques, space-time finite elements, perturbation, etc. Here we focus on the spectral collocation method to discretize the wave equation. In contrast with other methods, the main advantage here is the high order accuracy that gives good approximation with a not-too-fine discretization, when the data are smooth. This is important in higher dimensions or vector problems like the elasticity system. On the other hand, the main drawback is that the method is mainly restricted to the rectangle domain due to the difficulty of imposing boundary conditions when considering polynomial approximation in separated variables. In this Thesis, we find a method to obtain a discrete version of the boundary observability inequality which is uniform with respect to the discretization parameter. The method is applied to the wave model and the elasticity system. This allows us to find convergent numerical approximations of the associated control problems and some related inverse source problems. This is particularly significant as it overcomes the limitations encountered in classical discretization techniques which fail to achieve such uniformity. By harnessing the high-order accuracy inherent in spectral collocation methods, our approach enables efficient approximation of controls and solutions to inverse source problems. Furthermore, we demonstrate the applicability of our method across various dimensions and vectorial systems, exemplifying its versatility and effectiveness in practical settings.