Observability analysis by Poincaré normal forms

International audienceThis paper deals with quadratic equivalence, normal forms of observability, characteristic matrices and normal quadratic numbers for nonlinear Single-Input Single-Output (SISO) systems. We investigated both cases; nonlinear systems linearly observable and nonlinear systems with one linear unobservable mode. Particularly, the effect of the normal quadratic numbers on the observer design is pointed out. Finally, a faster observability analysis is proposed using characteristic matrices and normal quadratic numbers. Throughout the paper, academic examples as well as bio-reactor example highlight our purpose

No entailing laws, but enablement in the evolution of the biosphere

Biological evolution is a complex blend of ever changing structural stability, variability and emergence of new phenotypes, niches, ecosystems. We wish to argue that the evolution of life marks the end of a physics world view of law entailed dynamics. Our considerations depend upon discussing the variability of the very "contexts of life": the interactions between organisms, biological niches and ecosystems. These are ever changing, intrinsically indeterminate and even unprestatable: we do not know ahead of time the "niches" which constitute the boundary conditions on selection. More generally, by the mathematical unprestatability of the "phase space" (space of possibilities), no laws of motion can be formulated for evolution. We call this radical emergence, from life to life. The purpose of this paper is the integration of variation and diversity in a sound conceptual frame and situate unpredictability at a novel theoretical level, that of the very phase space. Our argument will be carried on in close comparisons with physics and the mathematical constructions of phase spaces in that discipline. The role of (theoretical) symmetries as invariant preserving transformations will allow us to understand the nature of physical phase spaces and to stress the differences required for a sound biological theoretizing. In this frame, we discuss the novel notion of "enablement". This will restrict causal analyses to differential cases (a difference that causes a difference). Mutations or other causal differences will allow us to stress that "non conservation principles" are at the core of evolution, in contrast to physical dynamics, largely based on conservation principles as symmetries. Critical transitions, the main locus of symmetry changes in physics, will be discussed, and lead to "extended criticality" as a conceptual frame for a better understanding of the living state of matter

Null controllability of Grushin-type operators in dimension two

International audienceWe study the null controllability of the parabolic equation associated with the Grushin-type operator in the rectangle, under an additive control supported in an open subset of the rectangle. We prove that the equation is null controllable in any positive time when the degeneracy is not too strong, and that there is no time for which it is null controllable when the degeneracy is too strong. In the transition regime and when the control support is a strip, a positive minimal time is required for null controllability. Our approach is based on the fact that, thanks to the particular geometric con guration, null controllability is closely linked to the one-dimensional observability of the Fourier components of the solution of the adjoint system, uniformly with respect to the Fourier frequency

A cancer model for the angiogenic switch

International audienceThe occurrence of metastasis is an important feature in cancer development. In order to have a one-site model taking into account the interactions between host, effector immune and tumor cells which is not only valid for the early stages of tumor growth, we developed in this paper a new model where are incorporated interactions of these three cell populations with endothelial cells. These latter cells are responsible for the neo-vascularization of the tumor site which allows the migration of tumor cells to distant sites. It is then shown that, for some parameter values, the resulting model for the four cell populations reproduces the angiogenic switch, that is, the transition from avascular to vascular tumor

Some elements for a history of the dynamical systems theory

Leon Glass would like to thank the Natural Sciences and Engineering Research Council (Canada) for its continuous support of curiosity-driven research for over 40 years starting with the events recounted here. He also thanks his colleagues and collaborators including Stuart Kauffman, Rafael Perez, Ronald Shymko, Michael Mackey for their wonderful insights and collaborations during the times recounted here. R.G. is endebted to the following friends and colleagues, listed in the order encountered on the road described: F. T. Arecchi, L. M. Narducci, J. R. Tredicce, H. G. Solari, E. Eschenazi, G. B. Mindlin, J. L. Birman, J. S. Birman, P. Glorieux, M. Lefranc, C. Letellier, V. Messager, O. E. Rössler, R. Williams. U.P. would like to thank the following friends and colleagues who accompanied his first steps into the world of nonlinear phenomena: U. Dressler, I. Eick, V. Englisch, K. Geist, J. Holzfuss, T. Klinker, W. Knop, A. Kramer, T. Kurz, W. Lauterborn, W. Meyer-Ilse, C. Scheffczyk, E. Suchla and M. Wisenfeldt. The work by L. Pecora and T. Carroll was supported directly by the Office of Naval Research (ONR) and by ONR through the Naval Research Laboratory’s Basic Research Program. C.L. would like to thank Jürgen Kurths for his support to this project.Peer reviewedPostprintPublisher PD

ADI schemes for the time integration of Maxwell equations

This thesis is concerned with the analysis and construction of alternating direction implicit (ADI) splitting schemes for the time integration of linear isotropic Maxwell equations on cuboids. The work is organized in two major parts. The first part deals with time discrete approximations to exponentially stable Maxwell equations. By means of a divergence cleaning technique and artificial damping, we obtain an ADI scheme with approximations that also decay exponentially in time. The decay rate is here uniform with respect to the time discretization. One of the main ingredients in the proof of the decay behavior is an observability estimate for the numerical approximations. This inequality is obtained by means of a discrete multiplier technique. We also provide a rigorous error analysis for the uniformly exponentially stable ADI scheme, yielding convergence of order one in a space similar to $H^{-1}$. The error result makes only assumptions on the initial data and the model parameters. In the second part, we analyze time discrete approximations to linear isotropic Maxwell equations on a heterogeneous cuboid. In this setting, the domain consists of two different homogeneous subcuboids. The Maxwell equations are here integrated in time by means of the Peaceman-Rachford ADI splitting scheme which is well-known in literature. The main result provides a rigorous error bound of order 3/2 in $L^2$ for the numerical approximations. It is significant that the final error statement involves conditions only on the initial data and model parameters, but not on the solution. To achieve this result, we establish a detailed regularity analysis for the considered Maxwell system