Controllability and observability of coupled PDE Systems: boussinesq and thermoelasticity

Abstract

In this thesis, we examine two distinct problems in the context of distributed parameter systems: one concerns system control, more precisely, the issue of local null controllability, and the other concerns observability and inverse problems. First, we establish the local null controllability of the 3D Boussinesq system using a single distributed scalar control acting solely on the temperature equation. To achieve this, we first linearize the system around a suitable trajectory. Inspired by the classical work of M. Gromov and the recent contributions of J. M. Coron and P. Lissy on the Navier- Stokes equation, we employ an algebraic method to systematically reduce the number of required controls, leveraging known controllability results for systems with more controls. Finally, using an inverse function theorem, we prove the local null controllability of the nonlinear system. These results demonstrate that the Boussinesq system is locally nullcontrollable without direct action on the velocity field, highlighting the strength of the coupling between the thermal and fluid dynamic equations. Second, we investigate the optimal sensor placement problem for a linear thermoelasticity system on a bounded domain. Adopting a randomized approach to the initial data, we reduce the problem to the maximization of a spectral functional defined by the eigenfunctions of the Laplacian and a sequence of weights corresponding to the inverse of the maximum eigenvalues of a particular sequence of matrices. From a theoretical perspective, we analyze the existence of optimal sets by studying the relaxation gap between characteristic functions and density functions. We prove that if the spectral weights converge to their limit from below, the existence of a classical optimal set is guaranteed under Quantum Ergodicity (QUE) assumptions or weaker asymptotic separation conditions. We also address cases where QUE fails, such as the n-dimensional orthotope, proving existence results under specific geometric constraints. Applying this framework to the thermoelasticity system, we perform a detailed highfrequency asymptotic analysis of the spectral weights. We identify two distinct physical regimes depending on the wave speed and coupling parameters. We derive explicit algebraic sufficient conditions on the physical parameters (coupling strength, viscosity, wave speed, and time horizon) that ensure the spectral weights converge from below, thereby guaranteeing the existence of an optimal sensor domain.