Controllability on infinite-dimensional manifolds

Following the unified approach of A. Kriegl and P.W. Michor (1997) for a treatment of global analysis on a class of locally convex spaces known as convenient, we give a generalization of Rashevsky-Chow's theorem for control systems in regular connected manifolds modelled on convenient (infinite-dimensional) locally convex spaces which are not necessarily normable.Comment: 19 pages, 1 figur

Moving constraints as stabilizing controls in classical mechanics

The paper analyzes a Lagrangian system which is controlled by directly assigning some of the coordinates as functions of time, by means of frictionless constraints. In a natural system of coordinates, the equations of motions contain terms which are linear or quadratic w.r.t.time derivatives of the control functions. After reviewing the basic equations, we explain the significance of the quadratic terms, related to geodesics orthogonal to a given foliation. We then study the problem of stabilization of the system to a given point, by means of oscillating controls. This problem is first reduced to the weak stability for a related convex-valued differential inclusion, then studied by Lyapunov functions methods. In the last sections, we illustrate the results by means of various mechanical examples.Comment: 52 pages, 4 figure

The rolling problem: overview and challenges

In the present paper we give a historical account -ranging from classical to modern results- of the problem of rolling two Riemannian manifolds one on the other, with the restrictions that they cannot instantaneously slip or spin one with respect to the other. On the way we show how this problem has profited from the development of intrinsic Riemannian geometry, from geometric control theory and sub-Riemannian geometry. We also mention how other areas -such as robotics and interpolation theory- have employed the rolling model.Comment: 20 page

The Greenhouse Gas Climate Change Initiative (GHG-CCI): comparative validation of GHG-CCI SCIAMACHY/ENVISAT and TANSO-FTS/GOSAT CO₂ and CH₄ retrieval algorithm products with measurements from the TCCON

Column-averaged dry-air mole fractions of carbon dioxide and methane have been retrieved from spectra acquired by the TANSO-FTS (Thermal And Near-infrared Sensor for carbon Observations-Fourier Transform Spectrometer) and SCIAMACHY (Scanning Imaging Absorption Spectrometer for Atmospheric Cartography) instruments on board GOSAT (Greenhouse gases Observing SATellite) and ENVISAT (ENVIronmental SATellite), respectively, using a range of European retrieval algorithms. These retrievals have been compared with data from ground-based high-resolution Fourier transform spectrometers (FTSs) from the Total Carbon Column Observing Network (TCCON). The participating algorithms are the weighting function modified differential optical absorption spectroscopy (DOAS) algorithm (WFMD, University of Bremen), the Bremen optimal estimation DOAS algorithm (BESD, University of Bremen), the iterative maximum a posteriori DOAS (IMAP, Jet Propulsion Laboratory (JPL) and Netherlands Institute for Space Research algorithm (SRON)), the proxy and full-physics versions of SRON's RemoTeC algorithm (SRPR and SRFP, respectively) and the proxy and full-physics versions of the University of Leicester's adaptation of the OCO (Orbiting Carbon Observatory) algorithm (OCPR and OCFP, respectively). The goal of this algorithm inter-comparison was to identify strengths and weaknesses of the various so-called round- robin data sets generated with the various algorithms so as to determine which of the competing algorithms would proceed to the next round of the European Space Agency's (ESA) Greenhouse Gas Climate Change Initiative (GHG-CCI) project, which is the generation of the so-called Climate Research Data Package (CRDP), which is the first version of the Essential Climate Variable (ECV) "greenhouse gases" (GHGs). For XCO₂, all algorithms reach the precision requirements for inverse modelling (< 8 ppm), with only WFMD having a lower precision (4.7 ppm) than the other algorithm products (2.4–2.5 ppm). When looking at the seasonal relative accuracy (SRA, variability of the bias in space and time), none of the algorithms have reached the demanding < 0.5 ppm threshold. For XCH₄, the precision for both SCIAMACHY products (50.2 ppb for IMAP and 76.4 ppb for WFMD) fails to meet the < 34 ppb threshold for inverse modelling, but note that this work focusses on the period after the 2005 SCIAMACHY detector degradation. The GOSAT XCH₄ precision ranges between 18.1 and 14.0 ppb. Looking at the SRA, all GOSAT algorithm products reach the < 10 ppm threshold (values ranging between 5.4 and 6.2 ppb). For SCIAMACHY, IMAP and WFMD have a SRA of 17.2 and 10.5 ppb, respectively

Cohomological tautness for Riemannian foliations

In this paper we present some new results on the tautness of Riemannian foliations in their historical context. The first part of the paper gives a short history of the problem. For a closed manifold, the tautness of a Riemannian foliation can be characterized cohomologically. We extend this cohomological characterization to a class of foliations which includes the foliated strata of any singular Riemannian foliation of a closed manifold

On local linearization of control systems

We consider the problem of topological linearization of smooth (C infinity or real analytic) control systems, i.e. of their local equivalence to a linear controllable system via point-wise transformations on the state and the control (static feedback transformations) that are topological but not necessarily differentiable. We prove that local topological linearization implies local smooth linearization, at generic points. At arbitrary points, it implies local conjugation to a linear system via a homeomorphism that induces a smooth diffeomorphism on the state variables, and, except at "strongly" singular points, this homeomorphism can be chosen to be a smooth mapping (the inverse map needs not be smooth). Deciding whether the same is true at "strongly" singular points is tantamount to solve an intriguing open question in differential topology

Tautness for riemannian foliations on non-compact manifolds

For a riemannian foliation $\mathcal{F}$ on a closed manifold $M$, it is known that $\mathcal{F}$ is taut (i.e. the leaves are minimal submanifolds) if and only if the (tautness) class defined by the mean curvature form $\kappa_\mu$ (relatively to a suitable riemannian metric $\mu$) is zero. In the transversally orientable case, tautness is equivalent to the non-vanishing of the top basic cohomology group $H^{^{n}}(M/\mathcal{F})$, where n = \codim \mathcal{F}. By the Poincar\'e Duality, this last condition is equivalent to the non-vanishing of the basic twisted cohomology group $H^{^{0}}_{_{\kappa_\mu}}(M/\mathcal{F})$, when $M$ is oriented. When $M$ is not compact, the tautness class is not even defined in general. In this work, we recover the previous study and results for a particular case of riemannian foliations on non compact manifolds: the regular part of a singular riemannian foliation on a compact manifold (CERF).Comment: 18 page