JCJC - Jeunes chercheuses & jeunes chercheurs

Control and Numerical Methods. Applications to biology. – CoNuM

Submission summary

The control of systems of partial differential equations has been the scene of important developments in the 90s, in particular in controllability to the trajectories. The derivation of so-called Carleman estimates and their use in this framework has led to new results for the controllability of semi-linear parabolic equations, parabolic equations with discontinuous coefficients, and reaction-diffusion systems. These very estimates also led to proofs of new stability results for inverse problems. The modelling of many phenomena enters this setting, for instance in biology or medical imaging. It is thus essential to address such controllability issues non only with a theoretical point of view but also with a practical point of view by designing and analysing numerical methods. The analysis of approximation methods for these control and identification problems currently remains largely open. These research topics are at the center of our project. There exist very few stability and/or convergence results of numerical schemes for the controllability to the trajectories. We plan on first studying the simple case of one-dimensional parabolic equations with regular coefficients with inner or boundary controls. We hope to obtain discrete controllable problems, for which we shall be able to derive uniform observability inequalities through explicit methods (for example discrete-type Carleman estimates for elliptic and parabolic operators). We shall then be able to construct uniformly bounded discrete controls and prove convergence towards a control of the limit continuous problem. We believe that such explicit methods will be adaptable to more general cases, in particular to the multi-dimensional case, to the non-smooth coefficient case and also to systems. This approach is one of the original points of this proposal. In a second time, we plan to focus on important open questions such as the strong convergence of the discrete control to a control function of the continuous problem, the optimality of the limit control, etc. In this framework, we shall also study different numerical schemes for the spacial discretization (finite differences, finite elements, finite volumes...). In addition, the questions of the influence of a choice of a time discretization, which is little studied in this context, and of its convergence properties are also of relevance. In parallel, reliable numerical simulations will be used to make further progress in our understanding of open problems such as the influence of the choice of the location of the support of the control function (in particular in the case of discontinuous coefficients), the controllability of semi-linear parabolic equations in critical cases, and controllability for systems. We also plan to extend this expertise to inverse problems such as parameter identifications and to their applications, in particular for the study of cancer treatments in collaboration with a pharmacology laboratory in Marseille. Goals as described above require skills in both scientific computing and numerical analysis (F. Boyer, F. Hubert and A. Münch) and control theory and inverse problem (A. Benabdallah, C. Dupaix and J. Le Rousseau) which are shared by the mathematics research laboratories in Marseille (LATP) and Besançon. In conclusion, the main aims of this project are, on the one hand, to provide new tools for the analysis of the numerical approximation of controllability problems and, on the other hand, to develop reliable numerical methods to treat these problems. The use of numerical simulations will also be a first step towards the understanding of some theoretical questions. The strength of this project is to bring together a diversity of complementary skills and experiences from applied analysis: control theory of PDEs and numerical analysis.

Project coordination

Jérôme LE ROUSSEAU (Organisme de recherche)

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.

Partner

Help of the ANR 90,000 euros
Beginning and duration of the scientific project: - 36 Months

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