This thesis is about the use of analytical mathematics within the framework of inverse methods applied to industry. These works are devoted to the development of sensors using real-time inverse methods adapted for rolling process. Steel producer ArcelorMittal leads a European project that aims to demonstrate the feasibility of sensors measuring fields (temperature, stress) in the contact between the product and the tool without altering physical conditions of this contact. The thesis has been funded by l'école des Ponts ParisTech. However, a three-year contract signed with ArcelorMittal has enabled the author to be part of the European project, and thus his research has been motivated by a real industrial demand. The aim of this thesis is twofold, academic and industrial; academic in the sense that these works seek to identify and use efficiently existing analytical methods for inverse problems occurring in thermo-elasticity in the context of real-time metrology. Indeed, analytical solutions are often exploited as purely theoretical cases, finding in many ways some success in engineering (for example in linear elasticity with stress intensity factors), but are most of the time relegated to the validation of numerical codes under simple assumptions. As a matter of fact, there is no research any more in pure mathematics concerning these solutions and analytical methods. However, the specific complications related to the inverse nature of problems under consideration, penalize numerical algorithms because inverse problems are ill-posed and stabilization is needed. But it remains often difficult if we consider the extreme loads (very sharp gradients) applied to industrial tools during rolling. Moreover, the real-time metrology excludes the use of numerical codes too costly in terms of computation time (iterative methods etc...). Both aspects contribute to renew widely interest for analytical solutions. It is then necessary to collect most effective and efficient (in terms of computation time and precision) methods and emphasis the most suitable for metrology. We will see various series expansions, not only to give a sequence of measured points an analytical form, but also to simplify the partial differential equations to solve. On the other hand, the goal of this thesis is also industrial, as these works are part of a process of development of sensors adapted for steel rolling industry. Thus, the robustness to measurement noise, technological constraints related to the local measurement systems (such as limitations in terms of frequency of acquisition) and calibration issues are central in the developments. Thus, the whole work can be a kind of rehabilitation of analytical methods. Their superiority over numerical methods (in terms of computation time and sometime accuracy) is highlighted, in the specific context of metrology in real-time on simple geometries. Three inverse methods in two-dimensions suitable for rolling process were developed successfully (isothermal elastic, thermal and thermoelastic coupling) and a series of experimental tests were made on the laboratory mill of ArcelorMittal. In addition, three-dimensional extensions of elastic and thermal inverse methods are also detailed
