Geometric reduction in optimal control theory with symmetries

A general study of symmetries in optimal control theory is given, starting from the presymplectic description of this kind of system. Then, Noether's theorem, as well as the corresponding reduction procedure (based on the application of the Marsden-Weinstein theorem adapted to the presymplectic case) are stated both in the regular and singular cases, which are previously described.Comment: 24 pages. LaTeX file. The paper has been reorganized. Additional comments have been included in Section 3. The example in Section 5.2 has been revisited. Some references have been adde

Stratospheric and tropospheric NO₂ variability on the diurnal and annual scale: a combined retrieval from ENVISAT/SCIAMACHY and solar FTIR at the Permanent Ground-Truthing Facility Zugspitze/Garmisch

International audienceColumnar NO2 retrievals from solar FTIR measurements at the Zugspitze (47.42° N, 10.98° E, 2964 m a.s.l.), Germany were investigated synergistically with columnar NO2 retrieved from SCIAMACHY data by the University of Bremen scientific algorithm UB1.5 for the time span July 2002-October 2004. A new concept to match FTIR data to the time of satellite overpass makes use of the NO2 daytime increasing rate retrieved from the FTIR data set itself [+1.02(6)E+14 cm-2/h]. This measured increasing rate shows no significant seasonal variation. SCIAMACHY data within a 200-km radius around Zugspitze were considered, and a pollution-clearing scheme was developed to select only pixels corresponding to clean background (free) tropospheric conditions, and exclude local pollution hot spots. The resulting difference between SCIAMACHY and FTIR columns (without correcting for the different sensitivities of the instruments) varies between 0.60-1.24E+15 cm-2 with an average of 0.83E+15 cm-2. A day-to-day scatter of daily means of ?7-10% could be retrieved in mutual agreement from FTIR and SCIAMACHY. Both data sets are showing sufficient precisions to make this assessment. Analysis of the averaging kernels gives proof that at high-mountain-site FTIR is a highly accurate measure for the pure stratospheric column, while SCIAMACHY shows significant tropospheric sensitivity. Based on this finding, we set up a combined a posteriori FTIR-SCIAMACHY retrieval for tropospheric NO2, based upon the averaging kernels. It yields an annual cycle of the clean background (free) tropospheric column (-2, an average of 1.09E+15 cm-2, and an intermediate phase between that of the well known boundary layer and stratospheric annual cycles. The outcome is a concept for an integrated global observing system for tropospheric NO2 that comprises DOAS nadir satellite measurements and a set of latitudinally distributed mountain-site or clean-air FTIR stations

BFV-complex and higher homotopy structures

We present a connection between the BFV-complex (abbreviation for Batalin-Fradkin-Vilkovisky complex) and the so-called strong homotopy Lie algebroid associated to a coisotropic submanifold of a Poisson manifold. We prove that the latter structure can be derived from the BFV-complex by means of homotopy transfer along contractions. Consequently the BFV-complex and the strong homotopy Lie algebroid structure are $L_{\infty}$ quasi-isomorphic and control the same formal deformation problem. However there is a gap between the non-formal information encoded in the BFV-complex and in the strong homotopy Lie algebroid respectively. We prove that there is a one-to-one correspondence between coisotropic submanifolds given by graphs of sections and equivalence classes of normalized Maurer-Cartan elemens of the BFV-complex. This does not hold if one uses the strong homotopy Lie algebroid instead.Comment: 50 pages, 6 figures; version 4 is heavily revised and extende

Weak Liouville-Arnold Theorems & Their Implications

This paper studies the existence of invariant smooth Lagrangian graphs for Tonelli Hamiltonian systems with symmetries. In particular, we consider Tonelli Hamiltonians with n independent but not necessarily involutive constants of motion and obtain two theorems reminiscent of the Liouville-Arnold theorem. Moreover, we also obtain results on the structure of the configuration spaces of such systems that are reminiscent of results on the configuration space of completely integrable Tonelli Hamiltonians.Comment: 24 pages, 1 figure; v2 corrects typo in online abstract; v3 includes new title (was: A Weak Liouville-Arnold Theorem), re-arrangement of introduction, re-numbering of main theorems; v4 updates the authors' email and physical addresses, clarifies notation in section 4. Final versio

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

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

Conservation laws for invariant functionals containing compositions

The study of problems of the calculus of variations with compositions is a quite recent subject with origin in dynamical systems governed by chaotic maps. Available results are reduced to a generalized Euler-Lagrange equation that contains a new term involving inverse images of the minimizing trajectories. In this work we prove a generalization of the necessary optimality condition of DuBois-Reymond for variational problems with compositions. With the help of the new obtained condition, a Noether-type theorem is proved. An application of our main result is given to a problem appearing in the chaotic setting when one consider maps that are ergodic.Comment: Accepted for an oral presentation at the 7th IFAC Symposium on Nonlinear Control Systems (NOLCOS 2007), to be held in Pretoria, South Africa, 22-24 August, 200

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