International audienc

In this paper, we consider a class of optimal control problems governed by a differential system. We analyse the sensitivity relations satisfied by the co-state arc of the Pontryagin maximum principle and the value function that associates the optimal value of the control problem to the initial time and state. Such a relationship has been already investigated for state-constrained problems under some controllability assumptions to guarantee          Lipschitz regularity property of the value function. Here, we consider the case with intermediate and final state constraints,  without any controllability assumption on the system, and without  Lipschitz  regularity of the value function. Because of this lack of regularity, the sensitivity relations cannot be expressed with the sub-differentials of the value function. This work shows that the  constrained problem can be reformulated with an auxiliary value function which is more regular and suitable to express the    sensitivity of the adjoint arc of the original state-constrained control problem along an optimal trajectory. Furthermore, our analysis covers the case  of normal optimal solutions, and abnormal solutions as well.

In this article, we use the potential of computational biology to highlight the key role of cell apoptosis for studying some tissue's properties through in silico experiments of morphogenesis. Our morphogenesis model is a new approach focusing on the deterministic program within cells that controls their placement and their differentiation at the beginning of the embryogenesis. Indeed, when the tissue is made by just a few pair of cells, we consider that cellular mechanisms are related neither to the influence of mechanical forces nor to the spread of chemicals. Dynamics are based on spatial and logical choices, the other factors being involved when the tissue contains a large number of cells. We had established a mathematical formulation of such a model and had enlightened the link between phenotype (cell placement and cell differentiation) and genotype (cell program) at the early embryogenesis. Indeed, that work allowed for generating any early tissue and the associated program that designs it. We propose now to study and assess some properties of these tissues for further selection and classification purposes. More precisely, we present in this article novel methods to measure tissue robustness based on the backward morphogenesis of our model. We also show some implementations of their self-maintenance properties, on the one hand to deal with environment disturbances through autopoiesis and on the other hand to achieve a dynamical steady state which ensures tissue renewal.

The resolution of the launcher ascent trajectory problem by the so-called Hamilton–Jacobi–Bellman (HJB) approach, relying on the Dynamic Programming Principle, has been investigated. The method gives a global optimum and does not need any initialization procedure. Despite these advantages, this approach is seldom used because of the dicculties of computing the solution of the HJB equation for high dimension problems. The present study shows that an eccient resolution is found. An illustration of the method is proposed on a heavy class launcher, for a typical GEO (Geostationary Earth Orbit) mission. This study has been performed in the frame of the Centre National d’Etudes Spatiales (CNES) Launchers Research & Technology Program.

We present a few applications of the viability theory to the solution to the Hamilton-Jacobi-Moskowitz problems when the Hamiltonian (fundamental diagram) depends on time, position and/or some regulation parameters. We study such a problem in its equivalent variational formulation. In this case, the corresponding lagrangian depends on the state of the characteristic dynamical system. As the Lax-Hopf formulae that give the solution in a semiexplicit form for an homogeneous lagrangian do not hold, we use a capture basin algorithm to compute the Moskowitz function as a viability solution of the Hamilton-Jacobi-Moskowitz problem with general conditions (including initial, boundary and internal conditions). We present two examples of applications.In the first one we introduce the variable speed limit as a regulation parameter. Our approach allows to compute the Moscowitz function for all values of the variable speed limit in a selected range and then to analyze its influence on the traffic flow. In particular, we study the case when the variation of the speed limit is applied locally, in space and time.Our second example deals with the local load capacity variations on the road. Such a variation can be a permanent property of the road (road narrowing) or it can be due to a temporary change of number of lanes (an accident or roadworks, for example). One can also use it as a regulation parameter by variable assignment of a supplementary lane.