Camille Pouchol

Analyse, contrôle et optimisation d'EDP, application à la biologie et la thérapie du cancer: Cette thèse a pour origine un projet sur l'optimisation de la chimiothérapie, rassemblant trois directeurs: Jean Clairambault, médecin et mathématicien, Michèle Sabbah, biologiste du cancer, et Emmanuel Trélat, mathématicien spécialisé en contrôle optimal. Ainsi, l'essentiel du travail a été motivé par des questions provenant de la biologie ou la thérapie du cancer. Y répondre a nécessité l'utilisation et le développement d'outils empruntés à diverses disciplines mathématiques, parmi lesquelles l'analyse asymptotique d'équations aux dérivées partielles, leur contrôle optimal théorique et numérique. Ces développements ont posé de nouveaux problèmes mathématiques intéressants en eux-mêmes, avec des applications en dynamique adaptative, dynamique des populations, contrôle optimal ou encore analyse numérique. Plus précisément, nous proposons des résultats d'analyse asymptotique pour certaines équations ou systèmes de sélection/mutation et réaction/diffusion non-locaux. Le contrôle Dirichlet des équations monostable et bistable 1D est étudié dans le détail. On considère l'étude numérique et théorique d'un problème de contrôle optimal pour un système représentant des cellules saines et cancéreuses soumises à de la chimiothérapie. Enfin, l'existence d'instabilités de Turing pour un système de Keller-Segel est prouvée. Pour ces équations, nous développons des schémas numériques aux volumes finis qui préservent la positivité, la dissipation de l'énergie, la conservation de la masse et les états stationnaires.

Analysing reachability associated to a control system is a subtle issue, especially for infinitedimensional dynamics, and when controls are subject to bounded constraints. We develop a computer-assisted framework for establishing non-reachability in linear parabolic PDEs governed by strongly elliptic operators, extending recent finite-dimensional techniques introduced in to the PDE setting. The non-reachability of a given target is shown to be equivalent to proving that a properly defined dual functional takes negative values. Our approach combines rigorous numerics with explicit convergence estimates for discretisations of the adjoint equation, ensuring mathematically certified results with tight error bounds. We demonstrate the wide applicability of our framework on Laplacian-driven control systems, showcasing its accuracy and reliability under various types of control constraints.

Phenotypic heterogeneity along the epithelial-mesenchymal (E-M) axis contributes to cancer metastasis and drug resistance. Recent experimental efforts have collated detailed time-course data on the emergence and dynamics of E-M heterogeneity in a cell population. However, it remains unclear how different intra- and inter-cellular processes shape the dynamics of E-M heterogeneity. Here, using Cell Population Balance model, we capture the dynamics of cell density along E-M phenotypic axis resulting from interplay between—(a) intracellular regulatory interaction among biomolecules, (b) cell division and death and (c) stochastic cell-state transition. We find that while the existence of E-M heterogeneity depends on intracellular regulation, heterogeneity gets enhanced with stochastic cell-state transitions and diminished by growth rate differences. Further, resource competition among E-M cells can lead to both bi-phasic growth of the total population and/or bi-stability in the phenotypic composition. Overall, our model highlights complex interplay between cellular processes shaping dynamic patterns of E-M heterogeneity.

Motivated by applications requiring sparse or nonnegative controls, we investigate reachability properties of linear infinite-dimensional control problems under conic constraints. Relaxing the problem to convex constraints if the initial cone is not already convex, we provide a constructive approach based on minimising a properly defined dual functional, which covers both the approximate and exact reachability problems. Our main results heavily rely on convex analysis, Fenchel duality and the Fenchel-Rockafellar theorem. As a byproduct, we uncover new sufficient conditions for approximate and exact reachability under convex conic constraints. We also prove that these conditions are in fact necessary. When the constraints are nonconvex, our method leads to sufficient conditions ensuring that the constructed controls fulfill the original constraints, which is in the flavour of bang-bang type properties. We show that our approach encompasses and generalises several works, and we obtain new results for different types of conic constraints and control systems.

Motivated by applications requiring sparse or nonnegative controls, we investigate reachability properties of linear infinite-dimensional control problems under conic constraints. Relaxing the problem to convex constraints if the initial cone is not already convex, we provide a constructive approach based on minimising a properly defined dual functional, which covers both the approximate and exact reachability problems. Our main results heavily rely on convex analysis, Fenchel duality and the Fenchel-Rockafellar theorem. As a byproduct, we uncover new sufficient conditions for approximate and exact reachability under convex conic constraints. We also prove that these conditions are in fact necessary. When the constraints are nonconvex, our method leads to sufficient conditions ensuring that the constructed controls fulfill the original constraints, which is in the flavour of bang-bang type properties. We show that our approach encompasses and generalises several works, and we obtain new results for different types of conic constraints and control systems.

It is customary to design a controlled system in such a way that, whatever the chosen control satisfying the constraints, the system does not enter so-called unsafe regions. This work introduces a general computer-assisted methodology to prove that a given linear finite-dimensional control systems with compact constraints avoids a chosen unsafe set. Relying on support hyperplanes, we devise a functional such that the property of interest is equivalent to finding a point at which the functional is negative. Actually evaluating the functional first requires time-discretisation. We thus provide explicit, fine discretisation estimates for various types of matrices underlying the controlled problem. Second, computations lead to roundoff errors, which are dealt with by means of interval arithmetic. The control of both error types then lead to rigorous, computer-assisted proofs of non-reachability of the unsafe set. We illustrate the applicability and flexibility of our method in different contexts featuring various control constraints, unsafe sets, types of matrices and problem dimensions.

We consider the internal control of linear parabolic equations through on-off shape controls, i.e., controls of the form M(t) \chi_{\omega(t)} with M(t) \geq 0 and \omega(t) with a prescribed maximal measure. We establish small-time approximate controllability towards all possible final states allowed by the comparison principle with nonnegative controls. We manage to build controls with constant amplitude M(t) \equiv \bar{M} . In contrast, if the moving control set \omega(t) is confined to evolve in some region of the whole domain, we prove that approximate controllability fails to hold for small times. The method of proof is constructive. Using Fenchel–Rockafellarduality and the bathtub principle, the on-off shape control is obtained as the bang-bang solution of an optimal control problem, which we design by relaxing the constraints. Our optimal control approach is outlined in a rather general form for linear constrained control problems, paving the way for generalisations and applications to other PDEs and constraints.