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1. Finite-Time Observers for Parameters and State Variables of Nonlinear Systems

Author: Mazenc, Frederic, Malisoff, Michael, Laboratoire des signaux et systèmes (L2S), CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Dynamical Interconnected Systems in COmplex Environments (DISCO), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire des signaux et systèmes (L2S), CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics [Baton Rouge] (LSU Mathematics), and Louisiana State University (LSU)
Subjects: estimation, Control and Systems Engineering, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, nonlinear, Observers
Abstract: International audience; We provide new classes of finite-time and fastconverging observers for continuous-time nonlinear systems that contain unknown constant parameters and measurement delays. When the dynamics are affine in the unmeasured state, we obtain fixed time observers for states and for the model parameters, i.e., the finite convergence time is independent of the initial state of the dynamics. When the dynamics contain a nonlinearity that depends on the unmeasured state, we instead obtain asymptotic convergence of our observers, whose rate of convergence converges to infinity as the growth rate of this nonlinearity converges to zero, so in this case we call the observers almost finite-time observers. Our examples illustrate the ease with which our assumptions are verified in practice.
Published: 2022

2. Almost Finite-Time Observers for a Family of Nonlin Continuous-Time Systems

Author: Mazenc, Frederic, Malisoff, Michael, Laboratoire des signaux et systèmes (L2S), CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Dynamical Interconnected Systems in COmplex Environments (DISCO), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire des signaux et systèmes (L2S), CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics [Baton Rouge] (LSU Mathematics), and Louisiana State University (LSU)
Subjects: [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering
Abstract: International audience; We provide a new class of observers for a class of nonlinear systems which are not required to be affine in the unmeasured states. The observers ensure exponential convergence of the observation errors to zero, under linear output measurements. The rate of exponential convergence converges to infinity, as the growth rate of the nonlinear statedependent part of the dynamics converges to zero, so we call the observers almost finite-time. Under global Lipschitz conditions on the state-dependent part of the dynamics, our global result ensures convergence of the observers, for all initial states. For cases where the nonlinearity is of order two at the origin, we provide local results ensuring exponential convergence of the observation errors to zero, when the initial state is small enough. We apply the results to a model of a pendulum with friction, and to dynamics with Lotka-Volterra nonlinearities.
Published: 2022

3. The minimum time function around the origin

Author: Colombo, Giovanni, Khai, Nguyen T., Marigonda, Antonio, Wolenski, Peter, Dipartimento di Matematica Pura e Applicata [Padova], Universita degli Studi di Padova, Department of Pure and Applied Mathematics ( UNIPD ), Department of Mathematics [Baton Rouge] ( LSU Mathematics ), Louisiana State University [Baton Rouge] ( LSU ), European Project : 264735,EC:FP7:PEOPLE,FP7-PEOPLE-2010-ITN,SADCO ( 2011 ), Università degli Studi di Padova = University of Padua (Unipd), Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), and European Project: 264735,EC:FP7:PEOPLE,FP7-PEOPLE-2010-ITN,SADCO(2011)
Subjects: [ MATH.MATH-OC ] Mathematics [math]/Optimization and Control [math.OC], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
Abstract: International audience; The minimum time function around the origin
Published: 2011

4. A geometric model for blocks of Frobenius kernels

Author: Achar, Pramod, Riche, Simon, Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), Laboratoire de Mathématiques Blaise Pascal (LMBP), and Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS)
Subjects: Mathematics::Algebraic Geometry, [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], Mathematics::Category Theory, FOS: Mathematics, Representation Theory (math.RT), Mathematics::Representation Theory, Mathematics - Representation Theory
Abstract: Building on a geometric counterpart of Steinberg's tensor product formula for simple representations of a connected reductive algebraic group $G$ over a field of positive characteristic, and following an idea of Arkhipov--Bezrukavnikov--Braverman--Gaitsgory--Mirkovi\'c, we define and initiate the study of some categories of perverse sheaves on the affine Grassmannian of the Langlands dual group to $G$ that should provide geometric models for blocks of representations of the Frobenius kernel $G_1$ of $G$. In particular, we show that these categories admit enough projective and injective objects, which are closely related to some tilting perverse sheaves, and that they are highest weight categories in an appropriate generalized sense., Comment: 88 pages
Published: 2022

5. Event-Triggered Control for Linear Time-Varying Systems Using a Positive Systems Approach

Author: Frederic Mazenc, Michael Malisoff, Corina Barbalata, Zhong-Ping Jiang, Laboratoire des signaux et systèmes (L2S), CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Dynamical Interconnected Systems in COmplex Environments (DISCO), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire des signaux et systèmes (L2S), CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), Department of Mechanical and Industrial Engineering [Baton Rouge], Department of Electrical and Computer Engineering [New York], New York University [New York] (NYU), NYU System (NYU)-NYU System (NYU), and Supported by NSF Grants 2009659 (Malisoff and Barbalata) and 2009644 (Jiang)
Subjects: positive, General Computer Science, Control and Systems Engineering, Mechanical Engineering, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, observer, event-triggered, Electrical and Electronic Engineering, Stability
Abstract: International audience; We provide a new event-triggered output feedback control method for time-varying linear systems, using the notions of interval observers and positive systems. Our methods ensure uniform global exponential stability of the closed-loop system that is robust with respect to uncertainty in the coefficient matrices. Our illustrations include a curve tracking system from marine robotics.
Published: 2022

6. Event-Triggered Prediction-Based Delay Compensation Approach

Author: Frederic Mazenc, Michael Malisoff, Corina Barbalata, Laboratoire des signaux et systèmes (L2S), CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Dynamical Interconnected Systems in COmplex Environments (DISCO), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire des signaux et systèmes (L2S), CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), Department of Mechanical and Industrial Engineering [Baton Rouge], and Supported by US National Science Foundation Applied Mathematics Program Grant 2009659.
Subjects: Control and Optimization, Control and Systems Engineering, delay, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, prediction, Event-triggered
Abstract: International audience; We provide a new event-triggered delay compensation approach for linear systems with arbitrarily long constant input delays. Our prediction map is expressible as a solution of a discrete time system. Our method ensures input-to-state stability. We also provide an analog under measurement delays, where the prediction map is expressible as a solution of a continuous-discrete system. Significant novel features are our combined use of matrices of absolute values and our prediction based event triggers, instead of Euclidean norms, and the fact that the predictor dynamics always has the same dimension as that of the original system. Our marine robotic example illustrates an advantage of using our new methods.
Published: 2022

7. Integral exotic sheaves and the modular Lusztig-Vogan bijection

Author: Achar, Pramod N, Hardesty, William, Riche, Simon N., Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS), Université Clermont Auvergne [2017-2020] (UCA [2017-2020]), and European Project: 677147,H2020,ERC-2015-STG,ModRed(2016)
Subjects: [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], FOS: Mathematics, Representation Theory (math.RT), Mathematics::Representation Theory, Mathematics - Representation Theory
Abstract: Let G be a reductive group over an algebraically closed field k of very good characteristic. The Lusztig-Vogan bijection is a bijection between the set of dominant weights for G and the set of irreducible G-equivariant vector bundles on nilpotent orbits, conjectured by Lusztig and Vogan independently, and constructed in full generality by Bezrukavnikov. In characteristic 0, this bijection is related to the theory of 2-sided cells in the affine Weyl group, and plays a key role in the proof of the Humphreys conjecture on support varieties of tilting modules for quantum groups at a root of unity. In this paper, we prove that the Lusztig-Vogan bijection is (in a way made precise in the body of the paper) independent of the characteristic of k. This allows us to extend all of its known properties from the characteristic-0 setting to the general case. We also expect this result to be a step towards a proof of the Humphreys conjecture on support varieties of tilting modules for reductive groups in positive characteristic., Comment: 49 pages. v2: weakened the assumptions on G throughout the paper
Published: 2022

8. Feedback Stabilization and Robustness Analysis using Bounds on Fundamental Matrices

Author: Frederic Mazenc, Michael Malisoff, Laboratoire des signaux et systèmes (L2S), CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Dynamical Interconnected Systems in COmplex Environments (DISCO), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire des signaux et systèmes (L2S), CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics [Baton Rouge] (LSU Mathematics), and Louisiana State University (LSU)
Subjects: sampling, General Computer Science, Control and Systems Engineering, Mechanical Engineering, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, positive system, Electrical and Electronic Engineering, Stabilization
Abstract: International audience; We prove a new robust stabilization theorem for time-varying systems that have sampled measurements and time-varying disturbances, and we provide new observer designs, using novel bounds for fundamental matrices for systems with disturbances. Our main tools use properties of Metzler matrices and positive systems.
Published: 2022

9. New Fixed Time and Fast Converging Reduced Order Observers

Author: Mazenc, Frederic, Malisoff, Michael, CentraleSupélec, Dynamical Interconnected Systems in COmplex Environments (DISCO), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire des signaux et systèmes (L2S), CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), and Supported by NSF Grants 1711299 and 2009659 (Malisoff)
Subjects: Finite time, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, Discrete measurements, Reduced order observer
Abstract: International audience; For nonlinear continuous-time systems with continuous measurements of the output, we provide new reduced order observers that converge in finite time. The convergence time is independent of the initial state. For cases where the measurements are discrete, we provide asymptotically converging observers, whose rate of convergence is proportional to the negative of the logarithm of the size of the sampling interval. Our observers are based on the observability Gramian.
Published: 2021

10. Stability Analysis using New Variant of Halanay's Inequality

Author: Mazenc, Frederic, Malisoff, Michael, Krstic, Miroslav, CentraleSupélec, Dynamical Interconnected Systems in COmplex Environments (DISCO), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire des signaux et systèmes (L2S), CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), Department of Mechanical and Aerospace Engineering [La Jolla] (UCSD), University of California [San Diego] (UC San Diego), University of California-University of California, Supported by US National Science Grants 1711299 (Maliso) and 1711373 (Krstic)., Department of Mechanical and Aerospace Engineering [Univ California San Diego] (MAE - UC San Diego), and University of California (UC)-University of California (UC)
Subjects: [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, Switching, Stability, Delay systems
Abstract: International audience; We present a generalized form of Halanay's inequality, having a time varying gain multiplying the delayed term and a constant decay rate. Unlike the usual Halanay's conditions where the decay rate is required to exceed an upper bound on the gain multiplying the delayed term, we provide less restrictive conditions that allow times when the decay rate can be strictly less than the gain. We include an application to continuous time systems with switched delay values. This illustrates the utility of our generalized Halanay's inequality conditions for proving asymptotic stability in significant cases that violate the contraction condition that was needed to prove asymptotic stability in previous trajectory based results, and which are also not amenable to previous Lyapunov function constructions.
Published: 2021

11. Sampled-data estimator for nonlinear systems with uncertainties and arbitrarily fast rate of convergence

Author: Mazenc, Frédéric, Malisoff, Michael, Niculescu, Silviu-Iulian, Laboratoire des signaux et systèmes (L2S), CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Dynamical Interconnected Systems in COmplex Environments (DISCO), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire des signaux et systèmes (L2S), CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), and The work of M. Malisoff was supported by US National Science Foundation Grant 1711299.
Subjects: delay, Control and Systems Engineering, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, nonlinear systems, stability, Electrical and Electronic Engineering, Estimation
Abstract: International audience; We study a class of continuous-time nonlinear systems with discrete measurements, model uncertainty, and sensor noise. We provide an estimator of the state for which the observation error enjoys a variant of the exponential input-to-state stability property with respect to the model uncertainty and sensor noise. A valuable novel feature is that the overshoot term in this stability estimate only involves a recent history of uncertainty values. Also, the rate of exponential convergence can be made arbitrarily large by reducing the supremum of the sampling intervals. Our proof uses a recently developed trajectory based approach. We illustrate our work using a model for a pendulum whose suspension point is subjected to an unknown time-varying bounded horizontal oscillation.
Published: 2022

12. ON THE HUMPHREYS CONJECTURE ON SUPPORT VARIETIES OF TILTING MODULES

Author: Simon Riche, William Hardesty, Pramod N. Achar, Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), Institut National des Sciences Mathématiques et de leurs Interactions (INSMI), Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS), ANR-13-BS01-0001,Vargen,Variétés de caractères et généralisations(2013), and European Project: 677147,H2020,ERC-2015-STG,ModRed(2016)
Subjects: Pure mathematics, Nilpotent cone, Algebra and Number Theory, Conjecture, Subvariety, 010102 general mathematics, 01 natural sciences, Cohomology, Semisimple algebraic group, 0103 physical sciences, FOS: Mathematics, 010307 mathematical physics, Geometry and Topology, Representation Theory (math.RT), [MATH]Mathematics [math], 0101 mathematics, Variety (universal algebra), Algebraically closed field, Coxeter element, Mathematics - Representation Theory, Mathematics
Abstract: Let $G$ be a simply-connected semisimple algebraic group over an algebraically closed field of characteristic $p$, assumed to be larger than the Coxeter number. The "support variety" of a $G$-module $M$ is a certain closed subvariety of the nilpotent cone of $G$, defined in terms of cohomology for the first Frobenius kernel $G_1$. In the 1990s, Humphreys proposed a conjectural description of the support varieties of tilting modules; this conjecture has been proved for $G = \mathrm{SL}_n$ in earlier work of the second author. In this paper, we show that for any $G$, the support variety of a tilting module always contains the variety predicted by Humphreys, and that they coincide (i.e., the Humphreys conjecture is true) when $p$ is sufficiently large. We also prove variants of these statements involving "relative support varieties.", 54 pages, 2 color figures. v3: minor corrections and additions
Published: 2019

13. Stability Analysis using Generalized Sup-Delay Inequalities*

Author: Michael Malisoff, Miroslav Krstic, Frédéric Mazenc, Dynamical Interconnected Systems in COmplex Environments (DISCO), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire des signaux et systèmes (L2S), CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), CentraleSupélec, Louisiana State University (LSU), University of California, Laboratoire des signaux et systèmes (L2S), CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics [Baton Rouge] (LSU Mathematics), Department of Mechanical and Aerospace Engineering [Univ California San Diego] (MAE - UC San Diego), University of California [San Diego] (UC San Diego), University of California (UC)-University of California (UC), Supported in part by NSF Grants 1711299 (Malisoff) and 1711373 (Krstic), Department of Mechanical and Aerospace Engineering [La Jolla] (UCSD), University of California-University of California, and University of California (UC)
Subjects: 0209 industrial biotechnology, Work (thermodynamics), Control and Optimization, Inequality, Generalization, media_common.quotation_subject, 020208 electrical & electronic engineering, 0206 medical engineering, Stability (learning theory), 02 engineering and technology, 020601 biomedical engineering, Data modeling, 020901 industrial engineering & automation, Exponential stability, Control and Systems Engineering, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, 0202 electrical engineering, electronic engineering, information engineering, Trajectory, Applied mathematics, media_common, Mathematics
Abstract: International audience; We provide a new generalization of a recent trajectory-based approach for proving global asymptotic stability properties, using new sup-delay inequalities. Our generalization eliminates the usual contractivity requirement from previous sup-delay inequality approaches. We show how the requirements of our generalization hold under less restrictive Halanay's conditions than those that were previously reported. While the usual Halanay's inequality requires the decay rate to exceed the gain, our less restrictive conditions allow the gain to exceed the decay rate. We apply our work to systems with switching delays and to a continuous-discrete system.
Published: 2021

14. Feedback Stabilization with Discrete Measurements using Bounds on Fundamental Matrices

Author: Frederic Mazenc, Michael Malisoff, Mazenc, Frederic, CentraleSupélec, Dynamical Interconnected Systems in COmplex Environments (DISCO), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire des signaux et systèmes (L2S), CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), Supported by NSF Grants1711299 and 2009659 (Malisoff), Laboratoire des signaux et systèmes (L2S), and CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)
Subjects: sampling, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, [INFO.INFO-AU] Computer Science [cs]/Automatic Control Engineering, stabilization
Abstract: International audience; We prove a new robust stabilization theorem for systems with time-varying disturbances and sampled measurements, using novel bounds on fundamental matrices for systems with disturbances. Our main tools use properties of positive systems and Metzler matrices. Our numerical example illustrates an advantage of using our method instead of previous methods.
Published: 2021

15. Conjectures on tilting modules and antispherical p-cells

Author: Achar, Pramod N., Hardesty, William, Riche, Simon, Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS), and European Project: 677147,H2020,ERC-2015-STG,ModRed(2016)
Subjects: [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], Mathematics::Quantum Algebra, FOS: Mathematics, Representation Theory (math.RT), Mathematics::Representation Theory, Mathematics - Representation Theory
Abstract: For quantum groups at a root of unity, there is a web of theorems (due to Bezrukavnikov and Ostrik, and relying on work of Lusztig) connecting the following topics: (i) tilting modules; (ii) vector bundles on nilpotent orbits; and (iii) Kazhdan-Lusztig cells in the affine Weyl group. In this paper, we propose a (partly conjectural) analogous picture for reductive algebraic groups over fields of positive characteristic, inspired by a conjecture of Humphreys., Comment: 16 pages. v2: minor corrections
Published: 2019

16. Modular generalized Springer correspondence: an overview

Author: Achar, Pramod, Henderson, Anthony, Juteau, Daniel, Riche, Simon, Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), School of Mathematics and statistics [Sydney], The University of Sydney, Laboratoire de Mathématiques Nicolas Oresme (LMNO), Université de Caen Normandie (UNICAEN), Normandie Université (NU)-Normandie Université (NU)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Mathématiques et de leurs Interactions (INSMI), Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université de Caen Normandie (UNICAEN), and Normandie Université (NU)-Normandie Université (NU)
Subjects: [MATH]Mathematics [math]
Abstract: International audience; This is an overview of our series of papers on the modular generalized Springer correspondence. It is an expansion of a lecture given by the second author in the Fifth Conference of the Tsinghua Sanya International Mathematics Forum, Sanya, December 2014, as part of the Master Lecture 'Algebraic Groups and their Representations' Workshop honouring G. Lusztig. The material that has not appeared in print before includes some discussion of the motivating idea of modular character sheaves, and heuristic remarks about geometric functors of parabolic induction and restriction.
Published: 2019

17. Representation theory of disconnected reductive groups

Author: Achar, Pramod, Hardesty, William, Riche, Simon, Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS), Université Clermont Auvergne [2017-2020] (UCA [2017-2020]), and European Project: 677147,H2020,ERC-2015-STG,ModRed(2016)
Subjects: [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], General Mathematics, FOS: Mathematics, Representation Theory (math.RT), 16. Peace & justice, Mathematics - Representation Theory
Abstract: We study three fundamental topics in the representation theory of disconnected algebraic groups whose identity component is reductive: (i) the classification of irreducible representations; (ii) the existence and properties of Weyl and dual Weyl modules; and (iii) the decomposition map relating representations in characteristic $0$ and those in characteristic $p$ (for groups defined over discrete valuation rings of mixed characteristic). For each of these topics, we obtain natural generalizations of the well-known results for connected reductive groups., DOCUMENTA MATHEMATICA, Vol 25 (2020), p. 2149-2177
Published: 2019

18. Dualité de Koszul formelle et théorie des représentations des groupes algébriques réductifs en caractéristique positive

Author: Achar, Pramod N, Riche, Simon, Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS), Université Clermont Auvergne [2017-2020] (UCA [2017-2020]), E. Breuillard, and European Project: 677147,H2020,ERC-2015-STG,ModRed(2016)
Subjects: [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT]
Abstract: International audience; Dans cet article nous présentons les grandes lignes de la preuve d'une formule de caractères pour les représentations basculantes des groupes algébriques réductifs sur un corps de caractéristique positive, obtenue partiellement en collaboration avec plusieurs auteurs. Nous unissons les différentes étapes de cette preuve dans la notion de "dualité de Koszul formelle", et en présentons quelques applications.
Published: 2019

19. The parabolic exotic t-structure

Author: Nicholas Cooney, Pramod N. Achar, Simon Riche, Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS), Université Clermont Auvergne [2017-2020] (UCA [2017-2020]), and European Project: 677147,H2020,ERC-2015-STG,ModRed(2016)
Subjects: Pure mathematics, exceptional collection, parity complexes, Commutator subgroup, 01 natural sciences, Representation theory, Flag varieties, 010104 statistics & probability, Simply connected space, FOS: Mathematics, Representation Theory (math.RT), 0101 mathematics, Algebraically closed field, Mathematics::Representation Theory, Mathematics, Algebra and Number Theory, Conjecture, [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], 010102 general mathematics, 16. Peace & justice, t-structure, Algebraic group, Cotangent bundle, Geometry and Topology, derived category of coherent sheaves, Coxeter element, Mathematics - Representation Theory
Abstract: Let G be a connected reductive algebraic group over an algebraically closed field k, with simply connected derived subgroup. The exotic t-structure on the cotangent bundle of its flag variety T^*(G/B), originally introduced by Bezrukavnikov, has been a key tool for a number of major results in geometric representation theory, including the proof of the graded Finkelberg-Mirkovic conjecture. In this paper, we study (under mild technical assumptions) an analogous t-structure on the cotangent bundle of a partial flag variety T^*(G/P). As an application, we prove a parabolic analogue of the Arkhipov-Bezrukavnikov-Ginzburg equivalence. When the characteristic of k is larger than the Coxeter number, we deduce an analogue of the graded Finkelberg-Mirkovic conjecture for some singular blocks., Soit G un groupe algébrique réductif connexe sur un corps k algébriquement clos. La t-structure exotique sur le fibré cotangent de sa variété de drapeaux T^*(G/B), introduite à l'origine par Bezrukavnikov, a été un outil clé pour de nombreux résultats majeurs en théorie géométrique des représentations, en particulier la démonstration de la conjecture de Finkelberg-Mirkovic graduée. Dans cet article, nous étudions (sous de légères hypothèses techniques) une t-structure analogue sur le fibré cotangent de la variété de drapeaux partiels T^*(G/P). Comme application, nous prouvons un analogue parabolique de l'équivalence de Arkhipov-Bezrukavnikov-Ginzburg. Lorsque la caractéristique de k est supérieure au nombre de Coxeter, nous déduisons un analogue de la conjecture de Finkelberg-Mirkovic graduée pour certains blocs singuliers.
Published: 2018

20. Modular generalized Springer correspondence I: the general linear group

Author: Anthony Henderson, Daniel Juteau, Pramod N. Achar, Simon Riche, Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), School of Mathematics and statistics [Sydney], The University of Sydney, Laboratoire de Mathématiques Nicolas Oresme (LMNO), Centre National de la Recherche Scientifique (CNRS)-Université de Caen Normandie (UNICAEN), Normandie Université (NU)-Normandie Université (NU), Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS), Riche, Simon, Université de Caen Normandie (UNICAEN), and Normandie Université (NU)-Normandie Université (NU)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Pure mathematics, General Mathematics, General linear group, Mathematics::Algebraic Topology, 01 natural sciences, Stratification (mathematics), symbols.namesake, Symmetric group, Mathematics::Category Theory, 0103 physical sciences, FOS: Mathematics, Representation Theory (math.RT), [MATH.MATH-RT] Mathematics [math]/Representation Theory [math.RT], 0101 mathematics, Mathematics::Representation Theory, Mathematics, Nilpotent cone, [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], business.industry, Applied Mathematics, 010102 general mathematics, Modular design, 16. Peace & justice, Algebra, Fourier transform, symbols, Equivariant map, 010307 mathematical physics, business, Springer correspondence, Mathematics - Representation Theory
Abstract: We define a generalized Springer correspondence for the group GL(n) over any field. We also determine the cuspidal pairs, and compute the correspondence explicitly. Finally we define a stratification of the category of equivariant perverse sheaves on the nilpotent cone of GL(n) satisfying the `recollement' properties, and with subquotients equivalent to categories of representations of a product of symmetric groups., v1: 27 pages; v2: 29 pages, minor corrections; v3: 29 pages, final version, to appear in JEMS
Published: 2016

21. Dualit\'e de Koszul formelle et th\'eorie des repr\'esentations des groupes alg\'ebriques r\'eductifs en caract\'eristique positive

Author: Achar, Pramod N, Riche, Simon, Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS), Université Clermont Auvergne [2017-2020] (UCA [2017-2020]), and European Project: 677147,H2020,ERC-2015-STG,ModRed(2016)
Subjects: [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], Mathematics - Representation Theory
Abstract: In this survey paper, we present the broad outlines of the proof of a character formula for tilting representations of reductive algebraic groups in positive characteristic, obtained partly in collaboration with several other authors. A unifying theme for a number of steps of this proof is the notion of "formal Koszul duality." We explain this notion and discuss some applications. ----- Dans cet article nous pr\'esentons les grandes lignes de la preuve d'une formule de caract\`eres pour les repr\'esentations basculantes des groupes alg\'ebriques r\'eductifs sur un corps de caract\'eristique positive, obtenue partiellement en collaboration avec plusieurs auteurs. Nous unissons les diff\'erentes \'etapes de cette preuve dans la notion de "dualit\'e de Koszul formelle", et en pr\'esentons quelques applications., Comment: 58 pages, in French
Published: 2018

22. Mixed perverse sheaves on flag varieties of Coxeter groups

Author: Achar, Pramod N, Riche, Simon, Vay, Cristian, Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS), Université Clermont Auvergne [2017-2020] (UCA [2017-2020]), Consejo Nacional de Investigaciones Científicas y Técnicas [Buenos Aires] (CONICET), and Riche, Simon
Subjects: [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], 20F55, 20C08, 14F05, Mathematics::Category Theory, FOS: Mathematics, [MATH.MATH-RT] Mathematics [math]/Representation Theory [math.RT], Representation Theory (math.RT), Mathematics::Representation Theory, Mathematics - Representation Theory
Abstract: In this paper we construct an abelian category of "mixed perverse sheaves" attached to any realization of a Coxeter group, in terms of the associated Elias-Williamson diagrammatic category. This construction extends previous work of the first two authors, where we worked with parity complexes instead of diagrams, and we extend most of the properties known in this case to the general setting. As an application we prove that the split Grothendieck group of the Elias-Williamson diagrammatic category is isomorphic to the corresponding Hecke algebra, for any choice of realization., Comment: 53 pages. v2: minor corrections
Published: 2018
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23. Reductive groups, the loop Grassmannian, and the Springer resolution

Author: Achar, Pramod, Riche, Simon, Riche, Simon, Blanc 2013 - Variétés de caractères et généralisations - - Vargen2013 - ANR-13-BS01-0001 - Blanc 2013 - VALID, The geometry of modular representations of reductive algebraic groups - ModRed - - H20202016-09-01 - 2021-08-31 - 677147 - VALID, Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS), Université Clermont Auvergne [2017-2020] (UCA [2017-2020]), ANR-13-BS01-0001,Vargen,Variétés de caractères et généralisations(2013), and European Project: 677147,H2020,ERC-2015-STG,ModRed(2016)
Subjects: Mathematics::Category Theory, [MATH] Mathematics [math], [MATH]Mathematics [math], Mathematics::Representation Theory
Abstract: v3 : 112 pages, corrections mineures; International audience; In this paper we prove equivalences of categories relating the derived category of a block of the category of representations of a connected reductive algebraic group over an algebraically closed field of characteristic p bigger than the Coxeter number and a derived category of equivariant coherent sheaves on the Springer resolution (or a parabolic counterpart). In the case of the principal block, combined with previous results, this provides a modular version of celebrated constructions due to Arkhipov-Bezrukavnikov-Ginzburg for Lusztig's quantum groups at a root of unity. As an application, we prove a "graded version" of a conjecture of Finkelberg-Mirkovic describing the principal block in terms of mixed perverse sheaves on the dual affine Grassmannian, and deduce a new proof of Lusztig's conjecture in large characteristic.
Published: 2018

24. Geometric Satake, Springer correspondence, and small representations II

Author: Simon Riche, Anthony Henderson, Pramod N. Achar, Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), School of Mathematics and statistics [Sydney], The University of Sydney, Laboratoire de Mathématiques Blaise Pascal (LMBP), and Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS)
Subjects: groupe de Weyl, Pure mathematics, représentations petites, Commutative ring, Langlands dual group, groupes algébriques réductifs, 01 natural sciences, Representation theory, équivalence de Satake, Mathematics (miscellaneous), Mathematics::Category Theory, 0103 physical sciences, FOS: Mathematics, Mathematics education, Representation Theory (math.RT), 0101 mathematics, Mathematics::Representation Theory, Mathematics, Nilpotent cone, Functor, [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], 010102 general mathematics, 17B08, 20G05 (Primary), 14M15 (Secondary), Reductive group, correspondance de Springer, 010307 mathematical physics, Springer correspondence, Mathematics - Representation Theory, Affine Grassmannian
Abstract: For a split reductive group scheme $G$ over a commutative ring $k$ with Weyl group $W$, there is an important functor $Rep(G,k) \to Rep(W,k)$ defined by taking the zero weight space. We prove that the restriction of this functor to the subcategory of small representations has an alternative geometric description, in terms of the affine Grassmannian and the nilpotent cone of the Langlands dual group to $G$. The translation from representation theory to geometry is via the Satake equivalence and the Springer correspondence. This generalizes the result for the $k=\mathbb{C}$ case proved by the first two authors, and also provides a better explanation than in that earlier paper, since the current proof is uniform across all types., Comment: Version 4: minor revisions; 73 pages
Published: 2015

25. Koszul duality for Kac-Moody groups and characters of tilting modules

Author: Achar, Pramod N, Makisumi, Shotaro, Riche, Simon, Williamson, Geordie, Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), Department of Mathematics [Stanford], Stanford University, Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Mathématiques et de leurs Interactions (INSMI), The University of Sydney, and European Project: 677147,H2020,ERC-2015-STG,ModRed(2016)
Subjects: FOS: Mathematics, [MATH]Mathematics [math], Representation Theory (math.RT), Mathematics::Representation Theory, Mathematics - Representation Theory
Abstract: We establish a character formula for indecomposable tilting modules for connected reductive groups in characteristic p in terms of p-Kazhdan-Lusztig polynomials, for p>h the Coxeter number. Using results of Andersen, one may deduce a character formula for simple modules if p>2h-3. Our results are a consequence of an extension to modular coefficients of a monoidal Koszul duality equivalence established by Bezrukavnikov and Yun., 50 pages
Published: 2017

26. Stabilization in a Chemostat with Sampled and Delayed Measurements and Uncertain Growth Functions

Author: Michael Malisoff, Frdric Mazenc, Jrome Harmand, Dynamical Interconnected Systems in COmplex Environments (DISCO), Laboratoire des signaux et systèmes (L2S), Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Laboratoire de Biotechnologie de l'Environnement [Narbonne] (LBE), Institut National de la Recherche Agronomique (INRA)-Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro), Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro)-Institut National de la Recherche Agronomique (INRA), Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro), and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut National de la Recherche Agronomique (INRA)
Subjects: 0301 basic medicine, 0209 industrial biotechnology, sampling, fonction de croissance, contrôle de performance, delay, [SDV]Life Sciences [q-bio], 02 engineering and technology, Chemostat, concentration en substrat, 03 medical and health sciences, 020901 industrial engineering & automation, Control theory, taux de dilution, output feedback, stabilization, Electrical and Electronic Engineering, méthode d'échantillonnage, Sampling interval, Mathematics, chemostat, Substrate concentration, Substrate (marine biology), Dilution, 030104 developmental biology, Monotone polygon, performance testing, Control and Systems Engineering, Piecewise, Constant (mathematics)
Abstract: International audience; We provide a new control design for chemostats, under constant substrate input concentrations, using piecewise constant delayed measurements of the substrate concentration. Our growth functions can be uncertain and are not necessarily monotone. The dilution rate is the control. We use a new Lyapunov approach to derive conditions on the largest sampling interval and on the delay length to ensure asymptotic stabilization properties of a componentwise positive equilibrium point.
Published: 2017

27. Free-monodromic mixed tilting sheaves on flag varieties

Author: Achar, Pramod N, Makisumi, Shotaro, Riche, Simon, Williamson, Geordie, Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), Department of Mathematics [Stanford], Stanford University, Institut National des Sciences Mathématiques et de leurs Interactions (INSMI), Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS), The University of Sydney, Max Planck Institute for Mathematics (MPIM), Max-Planck-Gesellschaft, ANR-13-BS01-0001,Vargen,Variétés de caractères et généralisations(2013), and European Project: 677147,H2020,ERC-2015-STG,ModRed(2016)
Subjects: Mathematics::Category Theory, FOS: Mathematics, Representation Theory (math.RT), [MATH]Mathematics [math], Mathematics::Representation Theory, Mathematics - Representation Theory
Abstract: In this paper we propose a construction of a monoidal category of "free-monodromic" tilting perverse sheaves on (Kac-Moody) flag varieties in the setting of the "mixed modular derived category" introduced by the first and third authors. This category shares most of the properties of their counterpart in characteristic 0, defined by Bezrukavnikov-Yun using certain pro-objects in triangulated categories. This construction is the main new ingredient in the construction of a "modular Koszul duality" equivalence for constructible sheaves on flag varieties, see [P. Achar, S. Makisumi, S. Riche, and G. Williamson, "Koszul duality for Kac-Moody groups and characters of tilting modules", J. Amer. Math. Soc. 32 (2019)]., Comment: 140 pages. v2: some corrections and updates
Published: 2017

28. Stability and Control Design for Time-Varying Systems with Time-Varying Delays using a Trajectory-Based Approach

Author: Silviu-Iulian Niculescu, Michael Malisoff, Frédéric Mazenc, Dynamical Interconnected Systems in COmplex Environments (DISCO), Laboratoire des signaux et systèmes (L2S), Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), and Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)
Subjects: Lyapunov function, 0209 industrial biotechnology, Mathematical optimization, Control and Optimization, delay, stability AMS subject classifications 93Cxx, 02 engineering and technology, 01 natural sciences, Stability (probability), Reduction (complexity), symbols.namesake, 020901 industrial engineering & automation, Control theory, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, stability I Introduction, reduction approach, 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Mathematics, Pointwise, time-varying, Applied Mathematics, 010102 general mathematics, Linear system, 93Dxx, Bounded function, Dynamic Extension, Trajectory, symbols
Abstract: A recent work by Mazenc and Malisoff provides a trajectory-based approach for proving stability of time-varying systems with time-varying delays. Here, we provide several significant applications of their approach. In two results, we use a Lyapunov function for a corresponding un-delayed system to provide a new method for proving stability of linear continuous-time time-varying systems with bounded time-varying delays. We allow uncertainties in the coefficient matrices of the systems. Our main results use upper bounds on an integral average involving the delay. The results establish input-to-state stability with respect to disturbances. We also provide a novel reduction model approach that ensures global exponential stabilization of linear systems with a time-varying pointwise delay in the input, which allows the delay to be discontinuous and uncertain. Finally, we provide an alternative to the reduction model method, based on a different dynamic extension. Our examples demonstrate the usefulness of our findings in several settings.
Published: 2017

29. The Mayer and Minimum Time Problems with Stratified State Constraints *

Author: Peter R. Wolenski, Cristopher Hermosilla, Hasnaa Zidani, Departamento de Matematicas, Univesidad Tecnica Federico Santa-Maria (Valparaiso), Univesidad Federico Santa-Maria, Valparaiso, Chile, Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), Optimisation et commande (OC), Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris), ANR-12-BS01-0008,HJnet,Equations de Hamilton-Jacobi sur des structures hétérogènes et des réseaux(2012), and European Project: 264735,EC:FP7:PEOPLE,FP7-PEOPLE-2010-ITN,SADCO(2011)
Subjects: Statistics and Probability, 0209 industrial biotechnology, Mathematical optimization, State constraints, stratified structure, Constraint domain, 02 engineering and technology, 01 natural sciences, 020901 industrial engineering & automation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, 0101 mathematics, Mathematics, Numerical Analysis, Applied Mathematics, 010102 general mathematics, Minimum time, minimal time function, Optimal control, Controllability, Geometry and Topology, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Hamilton-Jacobi equations, Mayer problem, Analysis, Hamiltonian (control theory)
Abstract: International audience; This paper studies optimal control problems with state constraints by imposing structural assumptions on the constraint domain coupled with a tangential restriction with the dynamics. These assumptions replace pointing or controllability assumptions that are common in the literature, and provide a framework under which feasible boundary trajectories can be analyzed directly. The value functions associated with the state constrained Mayer and minimal time problems are characterized as solutions to a pair of Hamilton-Jacobi inequalities with appropriate boundary conditions. The novel feature of these inequalities lies in the choice of the Hamiltonian.
Published: 2017

30. Approximate roots of a valuation and the Pierce-Birkhoff conjecture

Author: François Lucas, Daniel Schaub, James J. Madden, Mark Spivakovsky, Laboratoire Angevin de Recherche en Mathématiques (LAREMA), Université d'Angers (UA)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), European Project, Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)
Subjects: Pure mathematics, Social connectedness, Polynomial ring, 01 natural sciences, Mathematics - Algebraic Geometry, Pierce-Birkhoff, piecewise polynomial, 0103 physical sciences, FOS: Mathematics, approximate root, 0101 mathematics, blowing up, Algebraic Geometry (math.AG), Mathematics, Ring (mathematics), Conjecture, 010102 general mathematics, real spectrum, General Medicine, Regular local ring, Real closed field, complete ideal, Pierce–Birkhoff conjecture, 14P10, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], 010307 mathematical physics, Valuation (measure theory), valuation
Abstract: International audience; This paper is a step in our program for proving the Piece-Birkhoff Conjecture for regular rings of any dimension (this would contain, in particular, the classical Pierce-Birkhoff conjecture which deals with polynomial rings over a real closed field). We first recall the Connectedness and the Definable Connectedness conjectures, both of which imply the Pierce - Birkhoff conjecture. Then we introduce the notion of a system of approximate roots of a valuation v on a ring A (that is, a collection Q of elements of A such that every v-ideal is generated by products of elements of Q). We use approximate roots to give explicit formulae for sets in the real spectrum of A which we strongly believe to satisfy the conclusion of the Definable Connectedness conjecture. We prove this claim in the special case of dimension 2. This proves the Pierce-Birkhoff conjecture for arbitrary regular 2-dimensional rings.
Published: 2014

31. Modular perverse sheaves on flag varieties I: tilting and parity sheaves

Author: Achar, Pramod, Riche, Simon, Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), Institut National des Sciences Mathématiques et de leurs Interactions (INSMI), Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS), and Riche, Simon
Subjects: Mathematics::Algebraic Geometry, [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], Mathematics::Category Theory, [MATH] Mathematics [math], [MATH.MATH-RT] Mathematics [math]/Representation Theory [math.RT], [MATH]Mathematics [math], Mathematics::Representation Theory
Abstract: v1 : 40 pages, avec un appendice avec Geordie Williamson ; v2 : version finale, à paraître aux Annales de l'ENS; International audience; In this paper we prove that the category of parity complexes on the flag variety of a complex connected reductive group is a "graded version" of the category of tilting perverse sheaves on the flag variety of the dual group, for any field of coefficients whose characteristic is good. We derive some consequences on Soergel's modular category O, and on multiplicities and decomposition numbers in the category of perverse sheaves.
Published: 2016

32. Constructible sheaves on nilpotent cones in rather good characteristic

Author: Simon Riche, Anthony Henderson, Daniel Juteau, Pramod N. Achar, Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), School of Mathematics and statistics [Sydney], The University of Sydney, Laboratoire de Mathématiques Nicolas Oresme (LMNO), Centre National de la Recherche Scientifique (CNRS)-Université de Caen Normandie (UNICAEN), Normandie Université (NU)-Normandie Université (NU), Institut National des Sciences Mathématiques et de leurs Interactions (INSMI), Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS), Université de Caen Normandie (UNICAEN), and Normandie Université (NU)-Normandie Université (NU)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Pure mathematics, Derived category, Nilpotent cone, General Mathematics, 010102 general mathematics, General Physics and Astronomy, Order (ring theory), Field (mathematics), 010103 numerical & computational mathematics, Reductive group, 01 natural sciences, 20G05 (Primary), 17B08 (Secondary), Nilpotent, FOS: Mathematics, Equivariant map, 0101 mathematics, [MATH]Mathematics [math], Representation Theory (math.RT), Mathematics::Representation Theory, Springer correspondence, Mathematics - Representation Theory, Mathematics
Abstract: We study some aspects of modular generalized Springer theory for a complex reductive group $G$ with coefficients in a field $\mathbb k$ under the assumption that the characteristic $\ell$ of $\mathbb k$ is rather good for $G$, i.e., $\ell$ is good and does not divide the order of the component group of the centre of $G$. We prove a comparison theorem relating the characteristic-$\ell$ generalized Springer correspondence to the characteristic-$0$ version. We also consider Mautner's characteristic-$\ell$ `cleanness conjecture'; we prove it in some cases; and we deduce several consequences, including a classification of supercuspidal sheaves and an orthogonal decomposition of the equivariant derived category of the nilpotent cone., Comment: 26 pages
Published: 2015
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33. Constructible sheaves on affine Grassmannians and geometry of the dual nilpotent cone

Author: Achar, Pramod, Riche, Simon, Riche, Simon, Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), Laboratoire de Mathématiques Blaise Pascal (LMBP), and Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Mathematics::Algebraic Geometry, [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], Mathematics::Category Theory, [MATH.MATH-RT] Mathematics [math]/Representation Theory [math.RT], Mathematics::Representation Theory
Abstract: International audience; In this paper we study the derived category of sheaves on the affine Grassmannian of a complex reductive group G, contructible with respect to the stratification by G[[x]]-orbits. Following ideas of Ginzburg and Arkhipov-Bezrukavnikov-Ginzburg, we describe this category (and a mixed version) in terms of coherent sheaves on the nilpotent cone of the Langlands dual reductive group. We also show, in the mixed case, that restriction to the nilpotent cone of a Levi subgroup corresponds to hyperbolic localization on affine Grassmannians.
Published: 2015

34. On the Stability of a Periodic Solution of Distributed Parameters Biochemical System

Author: Frédéric Mazenc, Peter R. Wolenski, Abdou K. Drame, City College of New York [CUNY] (CCNY), City University of New York [New York] (CUNY), Dynamical Interconnected Systems in COmplex Environments (DISCO), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire des signaux et systèmes (L2S), Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS), Laboratoire des signaux et systèmes (L2S), Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)
Subjects: Partial differential equation, Applied Mathematics, Mathematical analysis, State (functional analysis), Periodic input, Lyapunov functional, Stability (probability), biochemical system, Control theory, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, partial differential equations, Stability, Analysis, Mathematics
Abstract: International audience; This paper studies the stability of periodic solutions of distributed parameters biochemical system with periodic input Sin(t). We prove that if Sin(t) is periodic then the system has a periodic solution that is input to state stable when small perturbations are acting on the input concentration Sin(t).
Published: 2015

35. Modular generalized Springer correspondence III: exceptional groups

Author: Daniel Juteau, Simon Riche, Anthony Henderson, Pramod N. Achar, Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), School of Mathematics and statistics [Sydney], The University of Sydney, Laboratoire de Mathématiques Nicolas Oresme (LMNO), Université de Caen Normandie (UNICAEN), Normandie Université (NU)-Normandie Université (NU)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Mathématiques et de leurs Interactions (INSMI), Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université de Caen Normandie (UNICAEN), and Normandie Université (NU)-Normandie Université (NU)
Subjects: Classical group, Weyl group, Pure mathematics, Series (mathematics), General Mathematics, 010102 general mathematics, Sylow theorems, Nilpotent orbit, Primary 17B08, 20G05, Reductive group, 01 natural sciences, symbols.namesake, Character (mathematics), 0103 physical sciences, symbols, FOS: Mathematics, 010307 mathematical physics, 0101 mathematics, [MATH]Mathematics [math], Representation Theory (math.RT), Mathematics::Representation Theory, Springer correspondence, Mathematics - Representation Theory, Mathematics
Abstract: We complete the construction of the modular generalized Springer correspondence for an arbitrary connected reductive group, with a uniform proof of the disjointness of induction series that avoids the case-by-case arguments for classical groups used in previous papers in the series. We show that the induction series containing the trivial local system on the regular nilpotent orbit is determined by the Sylow subgroups of the Weyl group. Under some assumptions, we give an algorithm for determining the induction series associated to the minimal cuspidal datum with a given central character. We also provide tables and other information on the modular generalized Springer correspondence for quasi-simple groups of exceptional type, including a complete classification of cuspidal pairs in the case of good characteristic, and a full determination of the correspondence in type $G_2$., Comment: 40 pages. Version 2: added section 7.5, modified Table 5.2 to match current conventions of GAP3. Version 3 has minor edits suggested by the referee, including a slight strengthening of Proposition 3.2; final version, to appear in Math. Annalen
Published: 2015
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36. Modular generalized Springer correspondence II: classical groups

Author: Simon Riche, Daniel Juteau, Anthony Henderson, Pramod N. Achar, Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), School of Mathematics and statistics [Sydney], The University of Sydney, Laboratoire de Mathématiques Nicolas Oresme (LMNO), Centre National de la Recherche Scientifique (CNRS)-Université de Caen Normandie (UNICAEN), Normandie Université (NU)-Normandie Université (NU), Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS), Université de Caen Normandie (UNICAEN), Normandie Université (NU)-Normandie Université (NU)-Centre National de la Recherche Scientifique (CNRS), and Riche, Simon
Subjects: Classical group, Nilpotent cone, Pure mathematics, Primary 17B08, 20G05, Secondary 14M15, [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], business.industry, Applied Mathematics, General Mathematics, 010102 general mathematics, [MATH] Mathematics [math], Modular design, 01 natural sciences, 0103 physical sciences, FOS: Mathematics, 010307 mathematical physics, Representation Theory (math.RT), 0101 mathematics, [MATH.MATH-RT] Mathematics [math]/Representation Theory [math.RT], [MATH]Mathematics [math], business, Mathematics::Representation Theory, Springer correspondence, Mathematics - Representation Theory, Mathematics
Abstract: We construct a modular generalized Springer correspondence for any classical group, by generalizing to the modular setting various results of Lusztig in the case of characteristic-$0$ coefficients. We determine the cuspidal pairs in all classical types, and compute the correspondence explicitly for $\mathrm{SL}(n)$ with coefficients of arbitrary characteristic and for $\mathrm{SO}(n)$ and $\mathrm{Sp}(2n)$ with characteristic-$2$ coefficients., 52 pages. Version 2 corrects a minor mistake in the combinatorics of the type D case; no numbered statements are affected. Version 3 has minor additions, mostly in Section 2; final version, to appear in J. Eur. Math. Soc
Published: 2014

37. Weyl group actions on the Springer sheaf

Author: Daniel Juteau, Simon Riche, Anthony Henderson, Pramod N. Achar, Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), School of Mathematics and statistics [Sydney], The University of Sydney, Laboratoire de Mathématiques Nicolas Oresme (LMNO), Université de Caen Normandie (UNICAEN), Normandie Université (NU)-Normandie Université (NU)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université de Caen Normandie (UNICAEN), and Normandie Université (NU)-Normandie Université (NU)
Subjects: Modular representation theory, Pure mathematics, GeneralLiterature_INTRODUCTORYANDSURVEY, General Mathematics, 17B08, 20G05 (Primary) 14M15 (Secondary), 01 natural sciences, symbols.namesake, 0103 physical sciences, ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION, FOS: Mathematics, 0101 mathematics, Representation Theory (math.RT), Mathematics::Representation Theory, Mathematics, Weyl group, [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], 010102 general mathematics, Zero (complex analysis), 16. Peace & justice, Cohomology, Character (mathematics), symbols, Sheaf, 010307 mathematical physics, Springer correspondence, Mathematics - Representation Theory, Sign (mathematics)
Abstract: We show that two Weyl group actions on the Springer sheaf with arbitrary coefficients, one defined by Fourier transform and one by restriction, agree up to a twist by the sign character. This generalizes a familiar result from the setting of l-adic cohomology, making it applicable to modular representation theory. We use the Weyl group actions to define a Springer correspondence in this generality, and identify the zero weight spaces of small representations in terms of this Springer correspondence., Comment: 27 pages; version 2: accepted version, with minor changes recommended by referee
Published: 2014

38. Koszul duality and semisimplicity of Frobenius

Author: Achar, Pramod, Riche, Simon, Riche, Simon, Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), Laboratoire de Mathématiques Blaise Pascal (LMBP), and Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS)
Subjects: [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], Mathematics::K-Theory and Homology, Mathematics::Category Theory, [MATH.MATH-RT] Mathematics [math]/Representation Theory [math.RT], Mathematics::Representation Theory
Abstract: International audience; A fundamental result of Beilinson-Ginzburg-Soergel states that on flag varieties and related spaces, a certain modified version of the category of l-adic perverse sheaves exhibits a phenomenon known as Koszul duality. The modification essentially consists of discarding objects whose stalks carry a nonsemisimple action of Frobenius. In this paper, we prove that a number of common sheaf functors (various pull-backs and push-forwards) induce corresponding functors on the modified category or its triangulated analogue. In particular, we show that these functors preserve semisimplicity of the Frobenius action.
Published: 2013

39. The dual arc inclusion with differential inclusions

Author: Piermarco Cannarsa, Francesco Marino, Peter R. Wolenski, Dipartimento di Matematica [Roma II] (DIPMAT), Università degli Studi di Roma Tor Vergata [Roma], Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), and European Project: 264735,EC:FP7:PEOPLE,FP7-PEOPLE-2010-ITN,SADCO(2011)
Subjects: 0209 industrial biotechnology, Mathematical optimization, Dual arc inclusion, 02 engineering and technology, Type (model theory), Differential inclusion, Dynamic optimization, Minimum time function, Semiconcavity, Optimality conditions, 01 natural sciences, Arc (geometry), 020901 industrial engineering & automation, Settore MAT/05 - Analisi Matematica, Bellman equation, Applied mathematics, 0101 mathematics, Mathematics, Applied Mathematics, 010102 general mathematics, Minimum time, Dual (category theory), [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Inclusion (mineral), Analysis, Hamiltonian (control theory)
Abstract: International audience; This paper continues our study of the minimum time problem with dynamics driven by a particular type of differential inclusion. We prove under certain assumptions that the dual arc associated to an optimal solution and the supergradient of the value function along the solution are intertwined in this new setting. We also derive new optimality conditions that contain nonsmooth features with an additional constancy condition on the Hamiltonian.
Published: 2013

40. Bounded Tracking Controllers and Robustness Analysis for UAVs

Author: Aleksandra Gruszka, Michael Malisoff, Frédéric Mazenc, Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), Department of Mathematics, Dynamical Interconnected Systems in COmplex Environments (DISCO), Laboratoire des signaux et systèmes (L2S), Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)
Subjects: Lyapunov function, 0209 industrial biotechnology, Engineering, Robot kinematics, business.industry, 020208 electrical & electronic engineering, Control engineering, 02 engineering and technology, Kinematics, Computer Science Applications, symbols.namesake, 020901 industrial engineering & automation, Control and Systems Engineering, Robustness (computer science), Control theory, Bounded function, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, 0202 electrical engineering, electronic engineering, information engineering, symbols, Motion planning, Electrical and Electronic Engineering, Robust control, business, Actuator
Abstract: International audience; We study a kinematic model that is suitable for control design for high level formation flight of UAVs. We design controllers that give robust global tracking for a wide class of reference trajectories in the sense of input-to-state stability while satisfying key amplitude and rate constraints on the inputs. We illustrate our work in simulations.
Published: 2013

41. Semiconcavity of the Minimum Time Function for Differential Inclusions

Author: Cannarsa, Piermarco, Marino, Francesco, Wolenski, Peter, Dipartimento di Matematica [Roma II] (DIPMAT), Università degli Studi di Roma Tor Vergata [Roma], Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), European Project: 264735,EC:FP7:PEOPLE,FP7-PEOPLE-2010-ITN,SADCO(2011), Bouzat, Estelle, and Sensitivity Analysis for Deterministic Controller Design - SADCO - - EC:FP7:PEOPLE2011-01-01 - 2014-12-31 - 264735 - VALID
Subjects: Settore MAT/05 - Analisi Matematica, [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], sadco, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
Abstract: To appear in Dynamics of Continuous, Discrete and Impulsive Systems; International audience; In this paper we consider the Minimum Time Problem with dynamics given by a differential inclusion. We prove that the minimum time function is semiconcave under suitable hypotheses on the multifunction F.
Published: 2012

42. Further Results on Robust Output Feedback Control for the Chemostat Dynamics

Author: Frédéric Mazenc, Michael Malisoff, Dynamical Interconnected Systems in COmplex Environments (DISCO), Laboratoire des signaux et systèmes (L2S), Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics [Baton Rouge] (LSU Mathematics), and Louisiana State University (LSU)
Subjects: 0209 industrial biotechnology, education.field_of_study, Systems biology, Control (management), Population, Novelty, 02 engineering and technology, Chemostat, 01 natural sciences, Stability (probability), [SPI.AUTO]Engineering Sciences [physics]/Automatic, 010101 applied mathematics, 020901 industrial engineering & automation, Robustness (computer science), Control theory, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, Quantitative Biology::Populations and Evolution, 0101 mathematics, Robust control, education, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract: We study feedback stabilization problems for chemostats with two species and one limiting substrate, which play an important role in systems biology and population dynamics. We provide a survey of the literature on feedback stabilization for chemostats, and then we construct new dilution rate output feedbacks that stabilize a componentwise positive equilibrium, and only depend on the sum of the species levels. We prove that the feedbacks are robust to model uncertainty. The novelty and importance of our new contribution is in our dropping the usual assumption on the relative sizes of the growth yield constants, and our allowing uncertain uptake functions that are not necessarily concave.
Published: 2010

43. Model-Based Nonlinear Control of the Human Heart Rate During Treadmill Exercising

Author: Michael Malisoff, Marcio de Queiroz, Frédéric Mazenc, Dynamical Interconnected Systems in COmplex Environments (DISCO), Laboratoire des signaux et systèmes (L2S), Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), and Department of Mechanical and Industrial Engineering [Baton Rouge]
Subjects: Lyapunov function, Engineering, business.industry, 010102 general mathematics, 0206 medical engineering, 02 engineering and technology, Nonlinear control, 020601 biomedical engineering, 01 natural sciences, [SPI.AUTO]Engineering Sciences [physics]/Automatic, symbols.namesake, Exponential stability, Control theory, Robustness (computer science), [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, symbols, Trajectory, 0101 mathematics, Treadmill, Actuator, business, Simulation, ComputingMilieux_MISCELLANEOUS
Abstract: We study a recently-proposed nonlinear model for human heart rate response that describes the central and peripheral local responses during and after treadmill exercise. The treadmill speed is the control input, and the control objective is to make the heart rate track a prescribed reference trajectory. Using a strict Lyapunov function analysis, we design new state and output feedback tracking controllers that render the error dynamics globally exponentially stable to the origin. This allows us to show that the feedback stabilized error dynamics enjoys input-to-state stable performance under actuator errors. This robustness condition quantifies the effects of deviations of the treadmill speed from the controller values. We illustrate our control design through simulations.
Published: 2010

44. Stabilization of a chemostat model with Haldane growth functions and a delay in the measurements

Author: Frédéric Mazenc, Michael Malisoff, Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), Dynamical Interconnected Systems in COmplex Environments (DISCO), Laboratoire des signaux et systèmes (L2S), Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)
Subjects: Lyapunov function, 0209 industrial biotechnology, education.field_of_study, Population, 02 engineering and technology, Chemostat, 01 natural sciences, Upper and lower bounds, [SPI.AUTO]Engineering Sciences [physics]/Automatic, 010101 applied mathematics, symbols.namesake, Arbitrarily large, 020901 industrial engineering & automation, Exponential stability, Control and Systems Engineering, Control theory, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, symbols, 0101 mathematics, Electrical and Electronic Engineering, education, Linear combination, Constant (mathematics), ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract: International audience; The stabilization of equilibria in chemostats with measurement delays is a complex and challenging problem, and is of significant ongoing interest in bioengineering and population dynamics. In this paper, we solve an output feedback stabilization problem for chemostat models having two species, one limiting substrate, and either Haldane or Monod growth functions. Our feedback stabilizers depend on a given linear combination of the species concentrations, which are both measured with a constant time delay. The values of the delays are unknown. Instead, one only knows an upper bound on the delays, and we allow the upper bound to be arbitrarily large. The stabilizing feedback depends on the known upper bound for the delays as well. Our work is based on a Lyapunov-Krasovskii argument.
Published: 2010

45. Strict Lyapunov function constructions under LaSalle conditions with an application to Lotka-Volterra systems

Author: Frédéric Mazenc, Michael Malisoff, Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), Dynamical Interconnected Systems in COmplex Environments (DISCO), Laboratoire des signaux et systèmes (L2S), Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS), Mathématiques, Informatique et STatistique pour l'Environnement et l'Agronomie (MISTEA), Institut National de la Recherche Agronomique (INRA)-Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro), Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Institut National de Recherche en Informatique et en Automatique (Inria), and Department of mathematics
Subjects: Lyapunov function, Periodic system, 0209 industrial biotechnology, 02 engineering and technology, SYSTEME A PARAMETRE VARIABLE, 01 natural sciences, MODELE LOTKA VOLTERRA, [SPI.AUTO]Engineering Sciences [physics]/Automatic, symbols.namesake, 020901 industrial engineering & automation, ROBUSTNESS, Robustness (computer science), Control theory, TIME VARYING SYSTEM, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, POPULATION DYNAMICS, LOTKA VOLTERRA MODEL, SYSTEME A PARAMETRE VARIABLE, MODELE LOTKA VOLTERRA, Quantitative Biology::Populations and Evolution, PREDATOR PREY RELATION, LYAPUNOV FUNCTION, ROBUSTNESS, TIME VARYING SYSTEM, BIOLOGICAL SYSTEMS, LASALLE CONDITIONS, Lyapunov equation, 0101 mathematics, Electrical and Electronic Engineering, Lyapunov redesign, LOTKA VOLTERRA MODEL, Mathematics, 010102 general mathematics, LYAPUNOV FUNCTION, LASALLE CONDITIONS, Computer Science Applications, POPULATION DYNAMICS, Nonlinear system, Control and Systems Engineering, symbols, BIOLOGICAL SYSTEMS, PREDATOR PREY RELATION
Abstract: International audience; We provide new techniques for building explicit global strict Lyapunov functions for broad classes of periodic time varying nonlinear systems satisfying LaSalle conditions. We illustrate our work using the Lotka-Volterra model, which plays a fundamental role in bioengineering. We use our strict Lyapunov constructions to prove robustness of the Lotka-Volterra tracking dynamics to uncertainty in the death rates.
Published: 2010

46. Remarks on Output Feedback Stabilization of Two-Species Chemostats Models

Author: Mazenc, Frédéric, Malisoff, Michael, Dynamical Interconnected Systems in COmplex Environments (DISCO), Laboratoire des signaux et systèmes (L2S), Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics [Baton Rouge] (LSU Mathematics), and Louisiana State University (LSU)
Subjects: [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, ComputingMilieux_MISCELLANEOUS, [SPI.AUTO]Engineering Sciences [physics]/Automatic
Abstract: International audience
Published: 2010

47. Stabilization of Two-Species Chemostats with Delayed Measurements and Haldane Growth Functions

Author: Frédéric Mazenc, Michael Malisoff, Dynamical Interconnected Systems in COmplex Environments (DISCO), Laboratoire des signaux et systèmes (L2S), Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics [Baton Rouge] (LSU Mathematics), and Louisiana State University (LSU)
Subjects: Output feedback, 0209 industrial biotechnology, 02 engineering and technology, Limiting, Chemostat, 01 natural sciences, Stability (probability), Upper and lower bounds, [SPI.AUTO]Engineering Sciences [physics]/Automatic, 010101 applied mathematics, Arbitrarily large, 020901 industrial engineering & automation, Control theory, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, Quantitative Biology::Populations and Evolution, Applied mathematics, 0101 mathematics, Linear combination, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract: The stabilization of chemostat models with delayed measurements is a challenging problem that is of significant interest in bioengineering and population biology. Here we solve a class of output feedback stabilization problems for chemostat models with one limiting substrate and two species, having Haldane or Monod growth functions. Our stabilizing feedbacks depend only on (i) a given linear combination of the species concentrations, which are measured with an unknown time delay and (ii) a known upper bound for the delay. The delay bound can be arbitrarily large. Our proof is based on the construction of a Lyapunov-Krasovskii functional.
Published: 2010

48. Stabilization in a two-species chemostat with monod growth functions

Author: Michael Malisoff, Jérôme Harmand, Frédéric Mazenc, Water Resource Modeling (MERE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de la Recherche Agronomique (INRA), Analyse des Systèmes et Biométrie (ASB), Institut National de la Recherche Agronomique (INRA), Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), Laboratoire de Biotechnologie de l'Environnement [Narbonne] (LBE), Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro)-Institut National de la Recherche Agronomique (INRA), Institut National de la Recherche Agronomique (INRA)-Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro), and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)
Subjects: Lyapunov function, 0209 industrial biotechnology, Adaptive control, [INFO.INFO-OH]Computer Science [cs]/Other [cs.OH], ARTIFICIAL INTELLIGENCE LYAPUNOV FONCTION SPECIES COEXISTENCE BIOREACTORS INTEGRAL INPUT-TO-STATE STABILITY UNCERTAIN STATES INTELLIGENCE ARTIFICIELLE ETAT INCERTAIN COEXISTANCE DES ESPÈCES COMPETITION, 02 engineering and technology, Chemostat, 01 natural sciences, Stability (probability), Computer Science Applications, 010101 applied mathematics, symbols.namesake, 020901 industrial engineering & automation, Exponential stability, Control and Systems Engineering, Control theory, Stability theory, symbols, Quantitative Biology::Populations and Evolution, 0101 mathematics, Electrical and Electronic Engineering, Linear combination, Velocity measurement, Mathematics
Abstract: aeres : ACL; International audience; We design feedback controllers for two species chemostats so that an equilibrium with arbitrary prescribed positive species concentrations becomes globally asymptotically stable. We use a new global explicit strict Lyapunov function construction, which allows us to quantify the effects of disturbances using the input-to-state stability paradigm. We assume that only a linear combination of the species concentrations is known. We illustrate our approach using a numerical example.
Published: 2009

49. A simplified design for strict Lyapunov functions under Matrosov conditions

Author: Michael Malisoff, Olivier Bernard, Frédéric Mazenc, Water Resource Modeling (MERE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de la Recherche Agronomique (INRA), Analyse des Systèmes et Biométrie (ASB), Institut National de la Recherche Agronomique (INRA), Department of Mathematics [Baton Rouge] (LSU Mathematics), Louisiana State University (LSU), Modeling and control of renewable resources (COMORE), Laboratoire d'océanographie de Villefranche (LOV), Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Observatoire océanologique de Villefranche-sur-mer (OOVM), Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Observatoire océanologique de Villefranche-sur-mer (OOVM), Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS)-Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Observatoire océanologique de Villefranche-sur-mer (OOVM), Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS)-Observatoire océanologique de Villefranche-sur-mer (OOVM), and Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS)-Inria Sophia Antipolis - Méditerranée (CRISAM)
Subjects: Lyapunov function, NONLINEAR SYSTEM BIOLOGICAL SYSTEMS LYAPUNOV FUNCTION MATROSOV THEOREM THEOREME MATROSOV, 0209 industrial biotechnology, Differential equation, 010102 general mathematics, [INFO.INFO-OH]Computer Science [cs]/Other [cs.OH], 02 engineering and technology, Type (model theory), 01 natural sciences, Computer Science Applications, Nonlinear system, symbols.namesake, 020901 industrial engineering & automation, Exponential stability, Control and Systems Engineering, Control theory, Stability theory, symbols, 0101 mathematics, Electrical and Electronic Engineering, Bifurcation, Numerical stability, Mathematics
Abstract: aeres : ACL; International audience; We construct strict Lyapunov functions for broad classes of nonlinear systems satisfying Matrosov type conditions. Our new constructions are simpler than the designs available in the literature. We illustrate the practical interest of our designs using a globally asymptotically stable biological model.
Published: 2009

50. Uniform global asymptotic stability of a class of adaptively controlled nonlinear systems

Author: M.S. de Queiroz, Michael Malisoff, Frédéric Mazenc, Water Resource Modeling (MERE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de la Recherche Agronomique (INRA), Analyse des Systèmes et Biométrie (ASB), Institut National de la Recherche Agronomique (INRA), Department of Mechanical and Industrial Engineering [Baton Rouge], Louisiana State University (LSU), and Department of Mathematics [Baton Rouge] (LSU Mathematics)
Subjects: Lyapunov function, 0209 industrial biotechnology, Adaptive control, Estimation theory, 010102 general mathematics, [INFO.INFO-OH]Computer Science [cs]/Other [cs.OH], System identification, 02 engineering and technology, Nonlinear control, 01 natural sciences, Computer Science Applications, Nonlinear system, symbols.namesake, 020901 industrial engineering & automation, Exponential stability, Control and Systems Engineering, Control theory, symbols, STABILITY OF NONLINEAR SYSTEMS ADAPTIVE CONTROL INPUT-TO-STATE STABILITY LYAPUNOV FUNCTIONS UNIFORM ASYMPTOTIC STABILITY STABILITE ASYMPTOTIQUE UNIFORME STABILITE DES SYSTEMES NON LINEAIRES COMMANDE ADAPTATIVE, Affine transformation, 0101 mathematics, Electrical and Electronic Engineering, Mathematics
Abstract: aeres : ACL; International audience; We give a new explicit, global, strict Lyapunov function construction for the error dynamics for adaptive tracking control problems, under an appropriate persistency of excitation condition. We then allow time-varying uncertainty in the unknown parameters. In this case, we construct input-to-state stable Lyapunov functions under suitable bounds on the uncertainty, provided the regressor also satisfies an affine growth condition. This lets us quantify the effects of uncertainties on both the tracking and the parameter estimation. We illustrate our results using Rossler systems.
Published: 2009

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