Dans une première partie, nous étudions la stabilité des solutions d'une inégalité variationnelle de la forme cône normal perturbé par une fonction. Pour ce faire, nous généralisons la méthode de S. Robinson, basée sur le degré topologique, aux espaces de Hilbert et à une classe de multi-applications non nécessairement convexes, appelées multi-applications prox-régulières.  Dans une deuxième partie, nous étudions des problèmes de contrôle optimal liés à la modélisation de problèmes de bio-procédés, et l'on s'intéresse à des contraintes périodiques sur l'état. Ainsi, nous étendons les résultats d'existence de solutions périodiques des EDOs de Kolmogorov au cadre du contrôle en rajoutant un paramètre contrôlé à ces équations. Ceci nous permet d'étudier par la suite un problème de commande optimale d'un chemostat sous forçage périodique, et d'en déduire la synthèse optimale pour ce problème. / In the first part, we study stability of solutions of a variational inequality of the form normal cone perturbed by a mapping. To do so, we generalize the method introduced by S. Robinson, based on the topological degree, to the general Hilbert setting on the class of non-necessarily convex set-valued mapping, called prox-regular set-valued mapping. In the second part, we study optimal control problems connected to the modelization of bio-processes and we consider periodic constraints on the state variable. We first extend the existence result of periodic solutions of Kolmogorov ODEs to the setting of control by adding a controlled parameter to those ODEs. This allows us to study an optimal control problem modeling a chemostat under a periodic forcing for which we give the optimal synthesis.

We present a model of coupled transcriptional-translational ultradian oscillators (TTOs) as a possible mechanism for the circadian rhythm observed at the cellular level. It includes nonstationary Poisson interactions between the transcriptional proteins and their affined gene sites. The associated reaction-rate equations are nonlinear ordinary differential equations of stochastic switching type. We compute the deterministic limit of this system, or the limit as the number of gene-proteins interactions per unit of time becomes large. In this limit, the random variables of the model are simply replaced by their limiting expected value. We derive the Kolmogorov equations — a set of partial differential equations —, and we obtain the associated moment equations for a simple instance of the model. In the stationary case, the Kolmogorov equations are linear and the moment equations are a closed set of equations. In the nonstationary case, the Kolmogorov equations are nonlinear and the moment equations are an open-ended set of equations. In both cases, the deterministic limit of the moment equations is in agreement with the deterministic state equations.

We present a model of coupled transcriptional-translational ultradian oscillators (TTOs) as a possible mechanism for the circadian rhythm observed at the cellular level. It includes nonstationary Poisson interactions between the transcriptional proteins and their affined gene sites. The associated reaction-rate equations are nonlinear ordinary differential equations of stochastic switching type. We compute the deterministic limit of this system, or the limit as the number of gene-proteins interactions per unit of time becomes large. In this limit, the random variables of the model are simply replaced by their limiting expected value. We derive the Kolmogorov equations — a set of partial differential equations —, and we obtain the associated moment equations for a simple instance of the model. In the stationary case, the Kolmogorov equations are linear and the moment equations are a closed set of equations. In the nonstationary case, the Kolmogorov equations are nonlinear and the moment equations are an open-ended set of equations. In both cases, the deterministic limit of the moment equations is in agreement with the deterministic state equations.

My thesis addresses a systematic approach to stochastic models in population genetics; in particular, the Wright-Fisher models affected only by the random genetic drift. I used various mathematical methods such as Probability, PDE, and Geometry to answer an important question: \"How do genetic change factors (random genetic drift, selection, mutation, migration, random environment, etc.) affect the behavior of gene frequencies or genotype frequencies in generations?”. In a Hardy-Weinberg model, the Mendelian population model of a very large number of individuals without genetic change factors, the answer is simple by the Hardy-Weinberg principle: gene frequencies remain unchanged from generation to generation, and genotype frequencies from the second generation onward remain also unchanged from generation to generation. With directional genetic change factors (selection, mutation, migration), we will have a deterministic dynamics of gene frequencies, which has been studied rather in detail. With non-directional genetic change factors (random genetic drift, random environment), we will have a stochastic dynamics of gene frequencies, which has been studied with much more interests. A combination of these factors has also been considered. We consider a monoecious diploid population of fixed size N with n + 1 possible alleles at a given locus A, and assume that the evolution of population was only affected by the random genetic drift. The question is that what the behavior of the distribution of relative frequencies of alleles in time and its stochastic quantities are. When N is large enough, we can approximate this discrete Markov chain to a continuous Markov with the same characteristics. In 1931, Kolmogorov first introduced a nice relation between a continuous Markov process and diffusion equations. These equations called the (backward/forward) Kolmogorov equations which have been first applied in population genetics in 1945 by Wright. Note that these equations are singular parabolic equations (diffusion coefficients vanish on boundary). To solve them, we use generalized hypergeometric functions. To know more about what will happen after the first exit time, or more general, the behavior of whole process, in joint work with J. Hofrichter, we define the global solution by moment conditions; calculate the component solutions by boundary flux method and combinatorics method. One interesting property is that some statistical quantities of interest are solutions of a singular elliptic second order linear equation with discontinuous (or incomplete) boundary values. A lot of papers, textbooks have used this property to find those quantities. However, the uniqueness of these problems has not been proved. Littler, in his PhD thesis in 1975, took up the uniqueness problem but his proof, in my view, is not rigorous. In joint work with J. Hofrichter, we showed two different ways to prove the uniqueness rigorously. The first way is the approximation method. The second way is the blow-up method which is conducted by J. Hofrichter. By applying the Information Geometry, which was first introduced by Amari in 1985, we see that the local state space is an Einstein space, and also a dually flat manifold with the Fisher metric; the differential operator of the Kolmogorov equation is the affine Laplacian which can be represented in various coordinates and on various spaces. Dynamics on the whole state space explains some biological phenomena.

My thesis addresses a systematic approach to stochastic models in population genetics; in particular, the Wright-Fisher models affected only by the random genetic drift. I used various mathematical methods such as Probability, PDE, and Geometry to answer an important question: \"How do genetic change factors (random genetic drift, selection, mutation, migration, random environment, etc.) affect the behavior of gene frequencies or genotype frequencies in generations?”. In a Hardy-Weinberg model, the Mendelian population model of a very large number of individuals without genetic change factors, the answer is simple by the Hardy-Weinberg principle: gene frequencies remain unchanged from generation to generation, and genotype frequencies from the second generation onward remain also unchanged from generation to generation. With directional genetic change factors (selection, mutation, migration), we will have a deterministic dynamics of gene frequencies, which has been studied rather in detail. With non-directional genetic change factors (random genetic drift, random environment), we will have a stochastic dynamics of gene frequencies, which has been studied with much more interests. A combination of these factors has also been considered. We consider a monoecious diploid population of fixed size N with n + 1 possible alleles at a given locus A, and assume that the evolution of population was only affected by the random genetic drift. The question is that what the behavior of the distribution of relative frequencies of alleles in time and its stochastic quantities are. When N is large enough, we can approximate this discrete Markov chain to a continuous Markov with the same characteristics. In 1931, Kolmogorov first introduced a nice relation between a continuous Markov process and diffusion equations. These equations called the (backward/forward) Kolmogorov equations which have been first applied in population genetics in 1945 by Wright. Note that these equations are singular parabolic equations (diffusion coefficients vanish on boundary). To solve them, we use generalized hypergeometric functions. To know more about what will happen after the first exit time, or more general, the behavior of whole process, in joint work with J. Hofrichter, we define the global solution by moment conditions; calculate the component solutions by boundary flux method and combinatorics method. One interesting property is that some statistical quantities of interest are solutions of a singular elliptic second order linear equation with discontinuous (or incomplete) boundary values. A lot of papers, textbooks have used this property to find those quantities. However, the uniqueness of these problems has not been proved. Littler, in his PhD thesis in 1975, took up the uniqueness problem but his proof, in my view, is not rigorous. In joint work with J. Hofrichter, we showed two different ways to prove the uniqueness rigorously. The first way is the approximation method. The second way is the blow-up method which is conducted by J. Hofrichter. By applying the Information Geometry, which was first introduced by Amari in 1985, we see that the local state space is an Einstein space, and also a dually flat manifold with the Fisher metric; the differential operator of the Kolmogorov equation is the affine Laplacian which can be represented in various coordinates and on various spaces. Dynamics on the whole state space explains some biological phenomena.