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Analysis, control and optimisation of PDEs, application to the biology and therapy of cancer

  • Pouchol, Camille
Publication Date
Jun 29, 2018
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This PhD originates from a joint project on chemotherapy optimisation, bringing together three advisors: Jean Clairambault, medical doctor and mathematician, Michèle Sabbah, cancer biologist, and Emmanuel Tr\'elat, mathematician specialised in optimal control. Most of the work undertaken has thus been motivated by questions from cancer biology or therapy.Answering them has required using and further developing tools from several different mathematical areas, among them the asymptotic analysis for partial differential equations, and theoretical and numerical optimal control. These developments have in turn posed new mathematical problems, interesting in their own right, with applications in the mathematical fields of adaptive dynamics, population dynamics, optimal control or numerical analysis.More precisely, we propose results of asymptotic analysis for some selection/mutation and reaction/diffusion non-local equations or systems. The Dirichlet control towards homogeneous states of 1D monostable and bistable equations is investigated in detail. A numerical and theoretical analysis for an optimal control is performed on a system representing cancer and healthy cells exposed to chemotherapy. Finally, Turing instabilities are shown to be exhibited by some Keller-Segel equations, for which we design finite-volume numerical schemes preserving positivity, energy dissipation, mass conservation and steady states.

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