Search

Your search keyword '"Cots, Olivier"' showing total 94 results
Start Over Author "Cots, Olivier"
94 results on '"Cots, Olivier"'

1. Singular versus boundary arcs for aircraft trajectory optimization in climbing phase

Author

Cots, Olivier, Gergaud, Joseph, Goubinat, Damien, and Wembe, Boris

Subjects
Mathematics - Optimization and Control
Abstract

In this article, we are interested in optimal aircraft trajectories in climbing phase. We consider the cost index criterion which is a convex combination of the time-to-climb and the fuel consumption. We assume that the thrust is constant and we control the flight path angle of the aircraft. This optimization problem is modeled as a Mayer optimal control problem with a single-input affine dynamics in the control and with two pure state constraints, limiting the Calibrated AirSpeed (CAS) and the Mach speed. The candidates as minimizers are selected among a set of extremals given by the maximum principle. We first analyze the minimum time-to-climb problem with respect to the bounds of the state constraints, combining small time analysis, indirect multiple shooting and homotopy methods with monitoring. This investigation emphasizes two strategies: the common CAS/Mach procedure in aeronautics and the classical Bang-Singular-Bang policy in control theory. We then compare these two procedures for the cost index criterion.

Published
2022

3. Abnormal Geodesics in 2D-Zermelo Navigation Problems in the Case of Revolution and the Fan Shape of the Small Time Balls

Author

Bonnard, Bernard, Cots, Olivier, Gergaud, Joseph, and Wembe, Boris

Subjects
Mathematics - Differential Geometry
Abstract

In this article, based on two case studies, we discuss the role of abnormal geodesics in planar Zermelo navigation problems. Such curves are limit curves of the accessibility set, in the domain where the current is strong. The problem is set in the frame of geometric time optimal control, where the control is the heading angle of the ship and in this context, abnormal curves are shown to separate time minimal curves from time maximal curves and are both small-time minimizing and maximizing. We describe the small-time minimal balls. For bigger time, a cusp singularity can occur in the abnormal direction, which corresponds to a conjugate point along the non-smooth image. It is interpreted in terms of the regularity property of the time minimal value function., Comment: Systems and Control Letters, Elsevier, In press

Published
2022

4. Tangency property and prior-saturation points in minimal time problems in the plane

Author

Bayen, Térence and Cots, Olivier

Subjects
Mathematics - Optimization and Control, 49J15, 49N35
Abstract

In this paper, we consider minimal time problems governed by control-affine-systems in the plane, and we focus on the synthesis problem in presence of a singular locus that involves a saturation point for the singular control. After giving sufficient conditions on the data ensuring occurence of a prior-saturation point and a switching curve, we show that the bridge (i.e., the optimal bang arc issued from the singular locus at this point) is tangent to the switching curve at the prior-saturation point. This property is proved using the Pontryagin Maximum Principle that also provides a set of non-linear equations that can be used to compute the prior-saturation point. These issues are illustrated on a fed-batch model in bioprocesses and on a Magnetic Resonance Imaging (MRI) model for which minimal time syntheses for the point-to-point problem are discussed.

Published
2019
Full Text
View/download PDF

5. A Zermelo navigation problem with a vortex singularity

Author

Bonnard, Bernard, Cots, Olivier, and Wembe, Boris

Subjects
Mathematics - Optimization and Control, 49K15, 53C60, 70H05
Abstract

Helhmoltz-Kirchhoff equations of motions of vortices of an incompressible fluid in the plane define a dynamics with singularities and this leads to a Zermelo navigation problem describing the ship travel in such a field where the control is the heading angle. Considering one vortex, we define a time minimization problem which can be analyzed with the technics of geometric optimal control combined with numerical simulations, the geometric frame being the extension of Randers metrics in the punctured plane, with rotational symmetry. Candidates as minimizers are parameterized thanks to the Pontryagin Maximum Principle as extremal solutions of a Hamiltonian vector field. We analyze the time minimal solution to transfer the ship between two points where during the transfer the ship can be either in a strong current region in the vicinity of the vortex or in a weak current region. The analysis is based on a micro-local classification of the extremals using mainly the integrability properties of the dynamics due to the rotational symmetry. The discussion is complex and related to the existence of an isolated extremal (Reeb) circle due to the vortex singularity. The explicit computation of cut points where the extremal curves cease to be optimal is given and the spheres are described in the case where at the initial point the current is weak.

Published
2019

6. Geometric and numerical methods for a state constrained minimum time control problem of an electric vehicle

Author

Cots, Olivier

Subjects
Mathematics - Optimization and Control, 49K15, 49M05, 90C90, 80M50
Abstract

In this article, the minimum time control problem of an electric vehicle is modeled as a Mayer problem in optimal control, with affine dynamics with respect to the control and with state constraints. The candidates as minimizers are selected among a set of extremals, solutions of a Hamiltonian system given by the maximum principle. An analysis, with the techniques of geometric control, is used first to reduce the set of candidates and then to construct the numerical methods. This leads to a numerical investigation based on indirect methods using the HamPath software. Multiple shooting and homotopy techniques are used to build a synthesis with respect to the bounds of the boundary sets.

Published
2019
Full Text
View/download PDF

8. Minimum Energy Control of Passive Tracers Advection in Point Vortices Flow

Author

Balsa, Carlos, Cots, Olivier, Gergaud, Joseph, Wembe, Boris, Angrisani, Leopoldo, Series Editor, Arteaga, Marco, Series Editor, Panigrahi, Bijaya Ketan, Series Editor, Chakraborty, Samarjit, Series Editor, Chen, Jiming, Series Editor, Chen, Shanben, Series Editor, Chen, Tan Kay, Series Editor, Dillmann, Rüdiger, Series Editor, Duan, Haibin, Series Editor, Ferrari, Gianluigi, Series Editor, Ferre, Manuel, Series Editor, Hirche, Sandra, Series Editor, Jabbari, Faryar, Series Editor, Jia, Limin, Series Editor, Kacprzyk, Janusz, Series Editor, Khamis, Alaa, Series Editor, Kroeger, Torsten, Series Editor, Liang, Qilian, Series Editor, Martín, Ferran, Series Editor, Ming, Tan Cher, Series Editor, Minker, Wolfgang, Series Editor, Misra, Pradeep, Series Editor, Möller, Sebastian, Series Editor, Mukhopadhyay, Subhas, Series Editor, Ning, Cun-Zheng, Series Editor, Nishida, Toyoaki, Series Editor, Pascucci, Federica, Series Editor, Qin, Yong, Series Editor, Seng, Gan Woon, Series Editor, Speidel, Joachim, Series Editor, Veiga, Germano, Series Editor, Wu, Haitao, Series Editor, Zhang, Junjie James, Series Editor, Gonçalves, José Alexandre, editor, Braz-César, Manuel, editor, and Coelho, João Paulo, editor

Published
2021
Full Text
View/download PDF

10. Homotopic approach for turnpike and singularly perturbed optimal control problems

Author

Cots Olivier, Gergaud Joseph, and Wembe Boris

Subjects
Applied mathematics. Quantitative methods, T57-57.97, Mathematics, QA1-939
Abstract

The first aim of this article is to present the link between the turnpike property and the singular perturbations theory: the first one being a particular case of the second one. Then, thanks to this link, we set up a new framework based on continuation methods for the resolution of singularly perturbed optimal control problems. We consider first the turnpike case, then, we generalize the approach to general control problems with singular perturbations (that is with fast but also slow variables). We illustrate each step with an example.

Published
2021
Full Text
View/download PDF

14. Zermelo Navigation Problems on Surfaces of Revolution and Geometric Optimal Control

Author

Bonnard, Bernard, Cots, Olivier, Wembe, Boris, Mathematics for Control, Transport and Applications (McTAO), 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), Algorithmes Parallèles et Optimisation (IRIT-APO), Institut de recherche en informatique de Toulouse (IRIT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-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)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Toulouse Mind & Brain Institut (TMBI), Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT), Institut National Polytechnique (Toulouse) (Toulouse INP), and Cots, Olivier

Subjects
Zermelo navigation problems, Regularity of the value function, Abnormal geodesics, [MATH] Mathematics [math], [MATH]Mathematics [math], Conjugate and cut loci, Optimal control
Abstract

In this article, the historical study from Carathéodory-Zermelo about computing the quickest nautical path is generalized to Zermelo navigation problems on surfaces of revolution, in the frame of geometric optimal control. Using the Maximum Principle, we present two methods dedicated to analyze the geodesic flow and to compute the conjugate and cut loci. We apply these calculations to investigate case studies related to applications in hydrodynamics, space mechanics and geometry., Dans cet article, on généralise l’étude historique de Carathéodory-Zermelo sur le calcul du chemin nautique le plus rapide, aux problèmes de navigation de Zermelo sur des surfaces de révolution dans le cadre du contrôle optimal géométrique. En utilisant le Principe du Maximum, on présente deux méthodes permettant d’analyser le flot géodésique et de calculer les lieux conjugués et de coupure en lien avec l’optimalité locale et globale des trajectoires. Ces calculs sont ensuite appliqués à des cas d’études liés à des applications en hydrodynamique, en mécanique spatiale et en géométrie.

Published
2023

19. Geometric preconditioner for indirect shooting and application to hybrid vehicle

Author

Cots, Olivier, Dutto, Rémy, Jan, Sophie, and Laporte, Serge

Abstract

In this article, we are interested in the hybrid electric vehicle torque split and gear shift problem, which can be formulated as a classical Lagrange optimal control problem with fixed initial condition. The Pontryagin maximum principle gives necessary optimality conditions adjoining to the state a covector called costate. Thus, the optimal state trajectory has to be found among the projections of the lifted trajectories, called Pontryagin extremals, given by the maximum principle. The indirect simple shooting method aims to compute Pontryagin extremals reducing the resolution to the research of the initial costate. Classically, a Newtonlike solver is used to compute zeros of the so-called shooting equations. The main drawback of this method is its sensitivity to the initial guess. Indeed, a good one needs to be given to make the Newton solver converge, which is not an easy task in practice. We propose a preconditioning method of the shooting function based on a geometrical interpretation of the costate, in relation with the reachable set of the extended system and the underlying symplectic structure. We numerically show that this method reduces the number of iterations of our solver. Remarkably, in our experiments, it is better to use the preconditioner than a clever initial guess for the Newton solver.

Published
2024
Full Text
View/download PDF

21. Zermelo Navigation Problems on Surfaces of Revolution and Hamiltonian Dynamics

Author

Bonnard, Bernard, Cots, Olivier, Wembe, Boris, Mathematics for Control, Transport and Applications (McTAO), 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), Algorithmes Parallèles et Optimisation (IRIT-APO), Institut de recherche en informatique de Toulouse (IRIT), 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), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, Institut National Polytechnique (Toulouse) (Toulouse INP), Institut National de Recherche en Informatique et en Automatique (Inria), Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), and Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3)

Subjects
Integrable Hamiltonian dynamicsand Morse-Reeb classification, Hamilton-Jacobi equation and regularity of the value function, Zermelo navigation problems on surface of revolution, Abnormal geodesics, [MATH]Mathematics [math], Conjugate and cut loci
Abstract

In this article, using the historical example from Carathéodory-Zermelo and a recent work describing the evolution of a passive tracer in a vortex, we present the geometric frame to analyze Zermelo navigation problems on surfaces of revolution, assuming the current invariant by symmetry of revolution. In this context, normal (polar) coordinates distinguish parallels and meridians and one will consider the case where the current is oriented along the parallels. The problem is set in the frame of time optimal control and the Maximum Principle allows to select minimizers among geodesics, solutions of an Hamiltonian dynamics. In the strong current domain, there exist both normal and abnormal geodesics, the later representing limit curves of the cone of admissible directions. We present the concepts of conjugate points in the normal and abnormal directions, associated to the singularity analysis of the central field defined by the Lagrangian manifold formed by geodesics curves with fixed initial conditions, in relation with Hamilton-Jacobi equation. This leads to a parameterization of the conjugate locus in the general case and conjugate points along abnormal directions are cusp points of the geodesics when crossing the limit of the weak current domain and are associated to non-continuity of the value function, due to bad accessibility propertie in the weak current domain. The dynamics is Liouville integrable but this dynamics is intricated due to noncompactness of the Liouville tori related to separatrices curves and interaction between parallels geodesics causing the existence of Reeb components. We present a generalized Morse-Reeb classification associated to an extended potential and this leads to a stratification of the geodesics set. Another complementary point of view is described which goes back to the historical example, using the parameterization of the geodesics by the heading angle of the ship, corresponding to the so-called Goh transformation in optimal control. This leads to a different stratification of the set of geodesics using Lie brackets computations and integration with Clairaut relation, interpreted as computing the control given by the derivative of the heading angle. The abnormal geodesics being seen as a projection with two branches of a determinantal variety. The final problem is to compute the cut locus and we introduce the first return mappings to equator and meridian to order the geodesics and to compute the separating locus formed by intersecting time minimizing geodesics. We apply our approach to analyse the historical example and a set of cases studies related in particular to Zermelo navigation on spheres of revolution and the generalized vortex case where the different techniques are used to analyze the dynamics of the geodesics and in fine to compute the optimal syntheses in the geodesically complete case.

Published
2021

23. Geometric and numerical methods in optimal control for the time minimal saturation in Magnetic Resonance Imaging (Dynamics, Control, and Geometry, In honor of Bronislaw Jakubczyk's 70th birthday, Banach Center, Warsaw, 12/09/2018-15/09/2018)

Author

Bonnard, Bernard, Cots, Olivier, Rouot, Jérémy, Verron, Thibaut, Institut de Mathématiques de Bourgogne [Dijon] (IMB), Université de Bourgogne (UB)-Centre National de la Recherche Scientifique (CNRS), Mathematics for Control, Transport and Applications (McTAO), 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), Algorithmes Parallèles et Optimisation (IRIT-APO), Institut de recherche en informatique de Toulouse (IRIT), Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées, Ecole Nationale Supérieure d'Electrotechnique, d'Electronique, d'Informatique, d'Hydraulique et de Télécommunications (ENSEEIHT), Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées, EPF [Troyes], SEITA-ITG (SEITA-ITG), and SEITA

Subjects
[INFO]Computer Science [cs]
Published
2020

24. Abnormal Curves in a Zermelo Navigation Problem in the Plane and the Fan Shape of Small Time Balls

Author

Bonnard, Bernard, Cots, Olivier, Gergaud, Joseph, Wembe, Boris, Institut de Mathématiques de Bourgogne [Dijon] (IMB), Université de Bourgogne (UB)-Centre National de la Recherche Scientifique (CNRS), Mathematics for Control, Transport and Applications (McTAO), 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), Algorithmes Parallèles et Optimisation (IRIT-APO), Institut de recherche en informatique de Toulouse (IRIT), Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées, and Université Toulouse III - Paul Sabatier (UT3)

Subjects
[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], Time minimal value function, Zermelo navigation problem, Geometric optimal control, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Abnormal curves, Helhmoltz-Kirchhoff N vortices model
Abstract

In this note, motivated by the Zermelo navigation problem in the flat plane, where the current field is associated to a point vortex, we discuss the role of the abnormal curves in the shape of the small time balls. Abnormal curves are occurring in the strong current domain in the vicinity of the vortex and form with the hyperbolic geodesics the boundaries of the small time accessibility sets and give to the small time balls a fan shape. Implications with the regularity of the value function are discussed.

Published
2020

26. About the minimal time control of passive tracers in presence of a single point vortex

Author

Cots, Olivier, Gergaud, Joseph, Wembe, B, Algorithmes Parallèles et Optimisation (IRIT-APO), Institut de recherche en informatique de Toulouse (IRIT), 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), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, and Institut National Polytechnique (Toulouse) (Toulouse INP)

Subjects
[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
Abstract

In this work, we are interested in time-optimal displacements of passive tracers (that is point vortices with zero circulation) in presence of a single point vortex (solution of the two-dimensional incompressible Euler equation). This problem from fluid mechanics, is formulated as an optimal control problem in Mayer form that can be seen as a Zermelo-like deformation (presence of a drift) of a Riemannian situation. We first prove the controllability of the control system and the existence of optimal solutions, and then we apply the Pontryagin Maximum Principle (PMP) in order to reduce the set of candidates as minimizers. The transcription of the PMP gives us a set of nonlinear equations to solve, the so-called shooting equations. We numerically check the absence of conjugate points along the computed extremals in relation with second-order conditions of local optimality. This is a first step before computing the optimal synthesis which essentially depends on the existence of abnormal extremals and which consists, for a part, in computing the cut locus, where the extremals ceases to be optimal.

Published
2019

27. About the prior-saturation phenomenon for minimal time problems in the plane

Author

Bayen, Térence, Cots, Olivier, Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Algorithmes Parallèles et Optimisation (IRIT-APO), Institut de recherche en informatique de Toulouse (IRIT), 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), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, and Institut National Polytechnique (Toulouse) (Toulouse INP)

Subjects
[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [MATH]Mathematics [math]
Abstract

We consider minimal time problems governed by control-affine-systems in the plane. We focus on the synthesis problem in presence of a singular locus that involves a saturation point for the singular control. We show that the minimum time synthesis can exhibit a prior-saturation point at the intersection of the singular locus and a switching curve. We also provide a set of non-linear equations to compute the prior-saturation point, and, at this point, we show a tangency property involving the switching curve.

Published
2019

28. Tangency property and prior-saturation points in minimal time problems in the plane

Author

Bayen, T��rence and Cots, Olivier

Subjects
Optimization and Control (math.OC), FOS: Mathematics, 49J15, 49N35
Abstract

In this paper, we consider minimal time problems governed by control-affine-systems in the plane, and we focus on the synthesis problem in presence of a singular locus that involves a saturation point for the singular control. After giving sufficient conditions on the data ensuring occurence of a prior-saturation point and a switching curve, we show that the bridge (i.e., the optimal bang arc issued from the singular locus at this point) is tangent to the switching curve at the prior-saturation point. This property is proved using the Pontryagin Maximum Principle that also provides a set of non-linear equations that can be used to compute the prior-saturation point. These issues are illustrated on a fed-batch model in bioprocesses and on a Magnetic Resonance Imaging (MRI) model for which minimal time syntheses for the point-to-point problem are discussed.

Published
2019
Full Text
View/download PDF

29. Algebraic-geometric techniques for the feedback classification and robustness of the optimal control of a pair of Bloch equations with application to Magnetic Resonance Imaging

Author

Bonnard, Bernard, Cots, Olivier, Faugère, Jean-Charles, Jacquemard, Alain, Rouot, Jérémy, Safey El Din, Mohab, Verron, Thibaut, Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS ), Mathematics for Control, Transport and Applications ( McTAO ), 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 de recherche en informatique de Toulouse ( IRIT ), Institut National Polytechnique [Toulouse] ( INP ) -Université Toulouse 1 Capitole ( UT1 ) -Université Toulouse - Jean Jaurès ( UT2J ) -Université Paul Sabatier - Toulouse 3 ( UPS ) -Centre National de la Recherche Scientifique ( CNRS ), Polynomial Systems ( PolSys ), Laboratoire d'Informatique de Paris 6 ( LIP6 ), Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Centre National de la Recherche Scientifique ( CNRS ) -Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Centre National de la Recherche Scientifique ( CNRS ) -Inria de Paris, Équipe Méthodes et Algorithmes en Commande ( LAAS-MAC ), Laboratoire d'analyse et d'architecture des systèmes [Toulouse] ( LAAS ), Institut National Polytechnique [Toulouse] ( INP ) -Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Université Paul Sabatier - Toulouse 3 ( UPS ) -Centre National de la Recherche Scientifique ( CNRS ) -Institut National Polytechnique [Toulouse] ( INP ) -Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Université Paul Sabatier - Toulouse 3 ( UPS ) -Centre National de la Recherche Scientifique ( CNRS ), Work supported by ANR Project DFG Explosys (ANR-14-CE35-0013-01, GL203/9-1) and ERC-Advanced Grant for the TAMING project, Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), Mathematics for Control, Transport and Applications (McTAO), 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), Algorithmes Parallèles et Optimisation (IRIT-APO), Institut de recherche en informatique de Toulouse (IRIT), 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), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, Institut National Polytechnique (Toulouse) (Toulouse INP), Polynomial Systems (PolSys), Laboratoire d'Informatique de Paris 6 (LIP6), Université Pierre et Marie Curie - Paris 6 (UPMC)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, Équipe Méthodes et Algorithmes en Commande (LAAS-MAC), Laboratoire d'analyse et d'architecture des systèmes (LAAS), Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1), Université de Bourgogne (UB)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-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)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Toulouse Mind & Brain Institut (TMBI), Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT), 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), and Cots, Olivier

Subjects
[ MATH.MATH-OC ] Mathematics [math]/Optimization and Control [math.OC], optimal control, MSC. 49K15, 14Q20, 81Q93, [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], singular trajectories, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Bloch equations, Magnetic Resonance Imaging, symbolic computation
Abstract

The aim of this article is to classify the singular trajectories associated with the optimal control problems of a pair of controlled Bloch equations. The motivation is to analyze the robustness of the optimal solutions to the contrast and the time-minimal saturation problem, in magnetic resonance imaging, with respect to the parameters and B1-inhomogeneity. For this purpose, we use various computer algebra algorithms and methods to study solutions of polynomial systems of equations and inequalities which are used for classification issues: Gröbner basis, cylindrical algebraic decomposition of semi-algebraic sets, Thom's isotopy lemma., L'objectif de cet article est de classifier les trajectoires singulières associées aux problèmes de contrôle optimaux d'une paire d'équations de Bloch contrôlées. La motivation est d'analyser la robustesse de la solution optimale du problème de contraste et de multisaturation en temps minimal, en imagerie par résonance magnétique nucléaire, par rapport aux paramètres et les inhomogénéités B1. On utilise le calcul symbolique pour étudier les solutions de systèmes polynomiaux d'équations et d'inéquations dans les problèmes de classification : base de Gröbner, décomposition algébrique cylindrique des ensembles semi-algébriques et le lemme d'isotopie de Thom.

Published
2017

30. The minimum time-to-climb and fuel consumption problems and CAS/Mach procedure for aircraft

Author

Cots, Olivier, Gergaud, Joseph, Goubinat, D, Algorithmes Parallèles et Optimisation (IRIT-APO), Institut de recherche en informatique de Toulouse (IRIT), 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), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, Institut National Polytechnique (Toulouse) (Toulouse INP), and Thales (France)

Subjects
[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
Abstract

In this article, we are interested in optimal aircraft trajectories in climbing phase. We consider the cost index criterion which is a convex combination of the time-to-climb and the fuel consumption. We assume that the thrust is constant and we control the air slope of the aircraft. This optimization problem is modeled as a Mayer optimal control problem with a single-input affine dynamics in the control and with two pure state constraints, limiting the Calibrated AirSpeed (CAS) and the Mach speed. The candidates as minimizers are selected among a set of extremals given by the maximum principle. We first analyze the minimum time-to-climb problem with respect to the bounds of the state constraints, combining small time analysis, indirect multiple shooting and homotopy methods. This investigation emphasizes two strategies: the common CAS/Mach procedure in aeronautics and the classical Bang-Singular-Bang policy in control theory. We then compare these two procedures for the cost index criterion.

Published
2018

31. Geometric and numerical methods for the contrast and saturation problems in Magnetic Resonance Imaging

Author

Cots, Olivier, Bonnard, Bernard, Rouot, Jérémy, Verron, Thibaut, Algorithmes Parallèles et Optimisation (IRIT-APO), Institut de recherche en informatique de Toulouse (IRIT), 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), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, Institut National Polytechnique (Toulouse) (Toulouse INP), Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), Mathematics for Control, Transport and Applications (McTAO), 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), EPF [Troyes], SEITA-ITG (SEITA-ITG), SEITA, Johannes Kepler University Linz [Linz] (JKU), Grélaud, Françoise, Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-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)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Toulouse Mind & Brain Institut (TMBI), Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT), and Université de Bourgogne (UB)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS)

Subjects
[INFO]Computer Science [cs], [INFO] Computer Science [cs]
Abstract

Talk; International audience; In this talk, we present the time minimal control problem about the saturation of a pair of spins of the same species but with inhomogeneities on the applied RF-magnetic field, in relation with the contrast problem in MRI. We make a complete analysis based on geometric control to classify the optimal syntheses in the single spin case, to pave the road to analyze the case of two spins. This points out the phenomenon of bridge, which consists in linking two singular arcs by a bang arc to bypass some singularities of the singular extremal flow. In the case of two spins, the question about global optimality is more intricate. The Bocop software is used to determine local minimizers for physical test cases and Linear Matrix Inequalities approach is applied to estimate the global optimal value and validate the previous computations. This is complemented by numerical investigations combining shooting and continuation methods implemented in the HamPath software to analyze the structure of the time minimal solution with respect to the set of parameters of the species. This reveals new complex structures, compare to the single spin case.

Published
2018

32. Geometric and numerical methods in optimal control for the time minimal saturation in Magnetic Resonance

Author

Rouot, Jérémy, Bonnard, Bernard, Cots, Olivier, Verron, Thibaut, Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), Mathematics for Control, Transport and Applications (McTAO), 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), Algorithmes Parallèles et Optimisation (IRIT-APO), Institut de recherche en informatique de Toulouse (IRIT), 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), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, SEITA-ITG (SEITA-ITG), SEITA, Université de Bourgogne (UB)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-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)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Toulouse Mind & Brain Institut (TMBI), Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT), and Rouot, Jérémy

Subjects
[MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], [INFO]Computer Science [cs], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [INFO] Computer Science [cs], ComputingMilieux_MISCELLANEOUS
Abstract

International audience

Published
2018

33. About prior saturation points for affine control systems (Journées MODE 2018, Autrans)

Author

Bayen, Terence, Cots, Olivier, Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Algorithmes Parallèles et Optimisation (IRIT-APO), Institut de recherche en informatique de Toulouse (IRIT), 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), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, Institut National Polytechnique (Toulouse) (Toulouse INP), Société de Mathématiques Appliquées et Industrielles (SMAI), European Network for Game Theory (COST Action CA16228), and Grélaud, Françoise

Subjects
Contrast imaging, Singular arc, Saturation phenomenon, [INFO]Computer Science [cs], [INFO] Computer Science [cs], Bio-processes, Optimal control
Abstract

International audience; We consider minimal time control problems governed by an affine system w.r.t. the control. Our aim is to study properties of the optimalsynthesis in presence of a singular locus that involves a saturation point for the singular control. We show that the optimal synthesis exhibits a prior saturation point at the intersection of the singular arc and a switching curve,and we also discuss qualitative properties of this curve. We high light this phenomenon on several models arising in nuclear magnetic resonance andin bioprocesses.

Published
2018

34. Working Notes on the Time Minimal Saturation of a Pair of Spins and Application in Magnetic Resonance Imaging

Author

Bonnard , Bernard, Cots , Olivier, Rouot , Jérémy, Verron , Thibaut, Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), Mathematics for Control, Transport and Applications (McTAO), 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), Algorithmes Parallèles et Optimisation (IRIT-APO), Institut de recherche en informatique de Toulouse (IRIT), 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), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, Institut National Polytechnique (Toulouse) (Toulouse INP), EPF [Troyes], Johannes Kepler University Linz [Linz] (JKU), Rouot, Jérémy, Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS ), Mathematics for Control, Transport and Applications ( McTAO ), 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 de recherche en informatique de Toulouse ( IRIT ), Institut National Polytechnique [Toulouse] ( INP ) -Université Toulouse 1 Capitole ( UT1 ) -Université Toulouse - Jean Jaurès ( UT2J ) -Université Paul Sabatier - Toulouse 3 ( UPS ) -Centre National de la Recherche Scientifique ( CNRS ), Johannes Kepler University Linz [linz] ( JKU ), Université de Bourgogne (UB)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-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)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Toulouse Mind & Brain Institut (TMBI), Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), and Université de Toulouse (UT)

Subjects
[ MATH.MATH-OC ] Mathematics [math]/Optimization and Control [math.OC], [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
Abstract

In this article, we analyse the time minimal control for the saturation of a pair of spins of the same species but with inhomogeneities of the applied RF-magnetic field, in relation with the contrast problem in Magnetic Resonance Imaging. We make a complete analysis based on geometric control to classify the optimal syntheses in the single spin case to pave the road to analyze the case of two spins. The Bocop software is used to determine local minimizers for physical test cases and Linear Matrix Inequalities approach is applied to estimate the global optimal value and validate the previous computations. This is complemented by numerical computations combining shooting and continuation methods implemented in the HamPath software to analyze the structure of the time minimal solution with respect to the set of parameters species.

Published
2018

35. Direct and indirect methods in optimal control with state constraints and the climbing trajectory of an aircraft

Author

Cots, Olivier, Gergaud, Joseph, Goubinat, Damien, Centre National de la Recherche Scientifique - CNRS (FRANCE), Institut National Polytechnique de Toulouse - INPT (FRANCE), Thales (FRANCE), Université Toulouse III - Paul Sabatier - UT3 (FRANCE), Université Toulouse - Jean Jaurès - UT2J (FRANCE), Université Toulouse 1 Capitole - UT1 (FRANCE), Algorithmes Parallèles et Optimisation (IRIT-APO), Institut de recherche en informatique de Toulouse (IRIT), 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), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, Institut National Polytechnique (Toulouse) (Toulouse INP), Thales Avionics SAS, THALES, and Institut National Polytechnique de Toulouse - Toulouse INP (FRANCE)

Subjects
Aircraft trajectory, State constraints, Direct and indirect methods, state constraints, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Optimisation et contrôle, Geometric optimal control, aircraft trajectory, direct and indirect methods
Abstract

to appear; International audience; In this article, the minimum time and fuel consumption of an aircraft in its climbing phase is studied. The controls are the thrust and the lift coefficient and state constraints are taken into account: air slope and speed limitations. The application of the Maximum Principle leads to parameterize the optimal control and the multipliers associated to the state constraints with the state and the costate and leads to describe a Multi- Point Boundary Value Problem which is solved by multiple shooting. This indirect method is the numerical implementation of the Maximum Principle with state-constraints and it is initialized by the direct method, both to determine the optimal structure and to obtain a satisfying initial guess. The solutions of the boundary value problems we define give extremals which satisfy necessary conditions of optimality with at most two boundary arcs. Note that the aircraft dynamics has a singular perturbation but no reduction is performed.

Published
2018

36. Perturbation singulière en contrôle optimal et le problème de montée en temps minimal d'un avion (Congrès SMAI 2017, Ronce-les-bains, 05/06/17-09/06/17)

Author

Cots, Olivier, Gergaud, Joseph, Goubinat, Damien, Algorithmes Parallèles et Optimisation (IRIT-APO), Institut de recherche en informatique de Toulouse (IRIT), 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), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, Institut National Polytechnique (Toulouse) (Toulouse INP), Thales Avionics, and Grélaud, Françoise

Subjects
[INFO]Computer Science [cs], [INFO] Computer Science [cs], ComputingMilieux_MISCELLANEOUS
Abstract

International audience

Published
2017

37. HAMPATH : Suivi de chemin différentiel et conditions du second ordre en contrôle optimal (Congrès SMAI 2017)

Author

Cots, Olivier, Algorithmes Parallèles et Optimisation (IRIT-APO), Institut de recherche en informatique de Toulouse (IRIT), 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), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, Institut National Polytechnique (Toulouse) (Toulouse INP), Société de Mathématiques Appliquées et Industrielles (SMAI), Fédération de recherche ARC Mathématiques, CNRS 3399, and Grélaud, Françoise

Subjects
[INFO]Computer Science [cs], [INFO] Computer Science [cs]
Abstract

International audience; Le code HamPath [7] est un logiciel open-source développé depuis 2009 et destiné à la résolution de problèmes de contrôle optimal.

Published
2017

38. On the minimum time optimal control problem of an aircraft in its climbing phase

Author

Cots, Olivier, Delpy, Patrick, Gergaud, Joseph, Goubinat, Damien, Centre National de la Recherche Scientifique - CNRS (FRANCE), Institut National Polytechnique de Toulouse - INPT (FRANCE), Thales (FRANCE), Université Toulouse III - Paul Sabatier - UT3 (FRANCE), Université Toulouse - Jean Jaurès - UT2J (FRANCE), Université Toulouse 1 Capitole - UT1 (FRANCE), Institut National Polytechnique de Toulouse - Toulouse INP (FRANCE), Algorithmes Parallèles et Optimisation (IRIT-APO), Institut de recherche en informatique de Toulouse (IRIT), 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), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, Institut National Polytechnique (Toulouse) (Toulouse INP), and Thales Avionics

Subjects
CONTROL, AIRCRAFT, NUMERICAL SIMULATION, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Optimisation et contrôle, Aircraft climbing phase, Time optimal control problem, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analyse numérique
Abstract

International audience; In this article, the minimum time optimal control problem of an aircraft in its climbing phase is studied.First, a reduction of the initial dynamics into a three dimensional single-input system with a linear de-pendence with respect to the control is performed. This reduced system is then studied using geometriccontrol techniques. In particular, the maximum principle leads to describe a multi-point boundary valueproblem which is solved by indirect methods. These methods are the implementation of the maximumprinciple and are initialized by direct methods. We check that the extremal solution of the boundary valueproblem satisfies necessary and sufficient conditions of optimality. From this reference case and consider-ing small-time optimal trajectories, we give a local classification with respect to the initial mass and finalvelocity of BC-extremals for the climbing phase.

Published
2017

39. Time minimal saturation of a pair of spins and application in Magnetic Resonance Imaging.

Author

Bonnard, Bernard, Cots, Olivier, Rouot, Jérémy, and Verron, Thibaut

Subjects
MAGNETIC resonance imaging, MAGNETICS, SYMBOLIC computation, LINEAR matrix inequalities, CONTINUATION methods, GEOMETRIC analysis
Abstract

In this article, we analyze the time minimal control for the saturation of a pair of spins of the same species but with inhomogeneities of the applied RF-magnetic field, in relation with the contrast problem in Magnetic Resonance Imaging. We make a complete analysis based on geometric control to classify the optimal syntheses in the single spin case to pave the road to analyze the case of two spins. The BocopBocop software is used to determine local minimizers for physical test cases and Linear Matrix Inequalities approach is applied to estimate the global optimal value and validate the previous computations. This is complemented by numerical computations combining shooting and continuation methods implemented in the HamPathHamPath software to analyze the structure of the time minimal solution with respect to the set of parameters of the species. Symbolic computations techniques are used to handle the singularity analysis. [ABSTRACT FROM AUTHOR]

Published
2020
Full Text
View/download PDF

40. Skeleton-Based Multiview Reconstruction

Author

Durix, Bastien, Morin, Géraldine, Chambon, Sylvie, Roudet, Céline, Garnier, Lionel, Cots, Olivier, Centre National de la Recherche Scientifique - CNRS (FRANCE), Institut National Polytechnique de Toulouse - INPT (FRANCE), Université de Bourgogne - UB (FRANCE), Université Toulouse III - Paul Sabatier - UT3 (FRANCE), Université Toulouse - Jean Jaurès - UT2J (FRANCE), Université Toulouse 1 Capitole - UT1 (FRANCE), Real Expression Artificial Life (IRIT-REVA), Institut de recherche en informatique de Toulouse (IRIT), 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), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, Institut National Polytechnique (Toulouse) (Toulouse INP), Laboratoire Electronique, Informatique et Image [UMR6306] (Le2i), Université de Bourgogne (UB)-Centre National de la Recherche Scientifique (CNRS)-École Nationale Supérieure d'Arts et Métiers (ENSAM), Arts et Métiers Sciences et Technologies, HESAM Université (HESAM)-HESAM Université (HESAM)-Arts et Métiers Sciences et Technologies, HESAM Université (HESAM)-HESAM Université (HESAM)-AgroSup Dijon - Institut National Supérieur des Sciences Agronomiques, de l'Alimentation et de l'Environnement, Sponsor(s):Inst Elect & Elect Engineers, Inst Elect & Elect Engineers, Signal Proc Soc, IEEE, Institut de recherche en informatique de Toulouse ( IRIT ), Institut National Polytechnique [Toulouse] ( INP ) -Université Toulouse 1 Capitole ( UT1 ) -Université Toulouse - Jean Jaurès ( UT2J ) -Université Paul Sabatier - Toulouse 3 ( UPS ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire Electronique, Informatique et Image ( Le2i ), Université de Bourgogne ( UB ) -AgroSup Dijon - Institut National Supérieur des Sciences Agronomiques, de l'Alimentation et de l'Environnement-Centre National de la Recherche Scientifique ( CNRS ), Institut National Polytechnique de Toulouse - Toulouse INP (FRANCE), and université de Bourgogne, LE2I

Subjects
topology, reconstruction, [SPI] Engineering Sciences [physics], ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION, 02 engineering and technology, Iterative reconstruction, Skeleton (category theory), Network topology, Graph-edit distance, Topology, [SPI]Engineering Sciences [physics], Traitement des images, Medial axis, [ INFO.INFO-TI ] Computer Science [cs]/Image Processing, 0202 electrical engineering, electronic engineering, information engineering, [ SPI ] Engineering Sciences [physics], Traitement du signal et de l'image, Computer vision, Synthèse d'image et réalité virtuelle, Topology (chemistry), Skeleton, Mathematics, ComputingMethodologies_COMPUTERGRAPHICS, business.industry, 3D reconstruction, Perspective (graphical), 020207 software engineering, Vision par ordinateur et reconnaissance de formes, Intelligence artificielle, [SPI.TRON] Engineering Sciences [physics]/Electronics, [ SPI.TRON ] Engineering Sciences [physics]/Electronics, [SPI.TRON]Engineering Sciences [physics]/Electronics, [INFO.INFO-TI] Computer Science [cs]/Image Processing [eess.IV], Shock graphs, [INFO.INFO-TI]Computer Science [cs]/Image Processing [eess.IV], graph-edit distance, 020201 artificial intelligence & image processing, Topological skeleton, Artificial intelligence, Shapes, Reconstruction, business
Abstract

International audience; The advantage of skeleton-based 3D reconstruction is to completely generate a single 3D object from well chosen views. Having numerous views is necessary for a reliable reconstruction but projections of skeletons lead to different topologies. We reconstruct 3D objects with curved medial axis (whose topology is a tree) from the perspective skeletons on an arbitrary number of calibrated acquisitions. The main contribution is to estimate the 3D skeleton, from multiple images: its topology is chosen as the closest to those of the perspective skeletons on the set of images, which means that the number of topology changes to map the 3D skeleton topology to topologies on images is minimum. This way, the number of matched branches is maximum.

Published
2016

41. A combination of algebraic, geometric and numerical methods in the contrast problem by saturation in magnetic resonance imaging

Author

Bonnard, Bernard, Claeys, Mathieu, Cots, Olivier, Jacquemard, Alain, Martinon, Pierre, Mathematics for Control, Transport and Applications ( McTAO ), 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 de Mathématiques de Bourgogne [Dijon] ( IMB ), Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS ), Équipe Méthodes et Algorithmes en Commande ( LAAS-MAC ), Laboratoire d'analyse et d'architecture des systèmes [Toulouse] ( LAAS ), Institut National Polytechnique [Toulouse] ( INP ) -Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Université Paul Sabatier - Toulouse 3 ( UPS ) -Centre National de la Recherche Scientifique ( CNRS ) -Institut National Polytechnique [Toulouse] ( INP ) -Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Université Paul Sabatier - Toulouse 3 ( UPS ) -Centre National de la Recherche Scientifique ( CNRS ), Wolfgang Pauli Institute ( WPI ), University of Vienna [Vienna], Centre de Mathématiques Appliquées - Ecole Polytechnique ( CMAP ), École polytechnique ( X ) -Centre National de la Recherche Scientifique ( CNRS ), Control, Optimization, Models, Methods and Applications for Nonlinear Dynamical Systems ( Commands ), École polytechnique ( X ) -Centre National de la Recherche Scientifique ( CNRS ) -École polytechnique ( X ) -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 ) -Unité de Mathématiques Appliquées ( UMA ), École Nationale Supérieure de Techniques Avancées ( Univ. Paris-Saclay, ENSTA ParisTech ) -École Nationale Supérieure de Techniques Avancées ( Univ. Paris-Saclay, ENSTA ParisTech ), LAAS-CNRS, Mathematics for Control, Transport and Applications (McTAO), 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 de Mathématiques de Bourgogne [Dijon] (IMB), Université de Bourgogne (UB)-Centre National de la Recherche Scientifique (CNRS), Équipe Méthodes et Algorithmes en Commande (LAAS-MAC), Laboratoire d'analyse et d'architecture des systèmes (LAAS), 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)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT), Wolfgang Pauli Institute (WPI), Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Control, Optimization, Models, Methods and Applications for Nonlinear Dynamical Systems (Commands), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, and Cots, Olivier

Subjects
[ MATH.MATH-OC ] Mathematics [math]/Optimization and Control [math.OC], Moment optimization, [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], Direct method, Contrast imaging in NMR, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Geometric optimal control, Shooting and continuation techniques
Abstract

In this article, the contrast imaging problem by saturation in nuclear magnetic resonance is modeled as a Mayer problem in optimal control. The optimal solution can be found as an extremal solution of the Maximum Principle and analyzed with the recent advanced techniques of geometric optimal control. This leads to a numerical investigation based on shooting and continuation methods implemented in the HamPath software. The results are compared with a direct approach to the optimization problem and implemented within the Bocop toolbox. In complement lmi techniques are used to estimate a global optimum. It is completed with the analysis of the saturation problem of an ensemble of spin particles to deal with magnetic fields inhomogeneities.

Published
2014

42. Problème de contrôle optimal en temps minimal pour un avion contraint en phase de montée

Author

Goubinat, Damien, Cots, Olivier, Gergaud, Joseph, Thales Avionics, Algorithmes Parallèles et Optimisation (IRIT-APO), Institut de recherche en informatique de Toulouse (IRIT), 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), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, Institut National Polytechnique (Toulouse) (Toulouse INP), Groupe thématique MODE (Mathématiques de l’Optimisation et de la DEcision) de la Société de Mathémathiques Appliquées et Industrielles (SMAI), and Grélaud, Françoise

Subjects
[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM], [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM]
Abstract

Slides; International audience; 23-25 mars, Toulouse 2/22 Contexte Étude géométrique Résultats numériques Conclusion et Travail Futur Sommaire Contexte Position du problème Dynamique Un problème de Mayer Étude géométrique Principe du maximum Extrémales non contraintes Extrémales contraintes Résultats numériques Résumé Simulation Conclusion et Travail Futur

Published
2016

43. Reconstruction basée squelette d'un objet 3D à partir de points de vue multiples

Author

Durix , Bastien, Morin , Géraldine, Chambon , Sylvie, Roudet , Céline, Garnier , Lionel, Cots , Olivier, Centre National de la Recherche Scientifique - CNRS (FRANCE), Institut National Polytechnique de Toulouse - Toulouse INP (FRANCE), Université de Bourgogne - UB (FRANCE), Université Toulouse III - Paul Sabatier - UT3 (FRANCE), Université Toulouse - Jean Jaurès - UT2J (FRANCE), Université Toulouse 1 Capitole - UT1 (FRANCE), Institut de recherche en informatique de Toulouse ( IRIT ), Institut National Polytechnique [Toulouse] ( INP ) -Université Toulouse 1 Capitole ( UT1 ) -Université Toulouse - Jean Jaurès ( UT2J ) -Université Paul Sabatier - Toulouse 3 ( UPS ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire Electronique, Informatique et Image ( Le2i ), Université de Bourgogne ( UB ) -AgroSup Dijon - Institut National Supérieur des Sciences Agronomiques, de l'Alimentation et de l'Environnement-Centre National de la Recherche Scientifique ( CNRS ), Real Expression Artificial Life (IRIT-REVA), Institut de recherche en informatique de Toulouse (IRIT), 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), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, Institut National Polytechnique (Toulouse) (Toulouse INP), Laboratoire Electronique, Informatique et Image [UMR6306] (Le2i), Université de Bourgogne (UB)-Centre National de la Recherche Scientifique (CNRS)-École Nationale Supérieure d'Arts et Métiers (ENSAM), Arts et Métiers Sciences et Technologies, HESAM Université (HESAM)-HESAM Université (HESAM)-Arts et Métiers Sciences et Technologies, HESAM Université (HESAM)-HESAM Université (HESAM)-AgroSup Dijon - Institut National Supérieur des Sciences Agronomiques, de l'Alimentation et de l'Environnement, Algorithmes Parallèles et Optimisation (IRIT-APO), and Institut National Polytechnique de Toulouse - INPT (FRANCE)

Subjects
[ INFO.INFO-TS ] Computer Science [cs]/Signal and Image Processing, [INFO.INFO-CV]Computer Science [cs]/Computer Vision and Pattern Recognition [cs.CV], Squelette, Topologie, [ INFO.INFO-GR ] Computer Science [cs]/Graphics [cs.GR], Vision par ordinateur et reconnaissance de formes, Intelligence artificielle, [ INFO.INFO-CV ] Computer Science [cs]/Computer Vision and Pattern Recognition [cs.CV], [INFO.INFO-GR]Computer Science [cs]/Graphics [cs.GR], [INFO.INFO-AI]Computer Science [cs]/Artificial Intelligence [cs.AI], Modélisation géométrique, Traitement des images, [INFO.INFO-TS]Computer Science [cs]/Signal and Image Processing, [INFO.INFO-TI]Computer Science [cs]/Image Processing [eess.IV], [ INFO.INFO-TI ] Computer Science [cs]/Image Processing, Traitement du signal et de l'image, Reconstruction, [ INFO.INFO-AI ] Computer Science [cs]/Artificial Intelligence [cs.AI], Synthèse d'image et réalité virtuelle
Abstract

National audience; L'avantage d'une reconstruction 3D basée squelette est de générer complètement un objet 3D à partir d'un ensemble de vues calibrées. Nous reconstruisons dans cet article des objets 3D dont le squelette est curviligne, à partir d'un nombre arbitraire de vues calibrées desquelles nous extrayons un squelette perspectif. Avoir plusieurs vues est nécessaire pour une reconstruction correcte, mais la projection d'un squelette peut modifier sa topologie. La contribution principale est d'estimer un squelette curviligne 3D, à partir de multiples images : sa topologie est choisie comme étant la plus proche des squelettes sur l'ensemble des images, ce qui signifie que le nombre de changements topologiques du squelette curviligne 3D aux images est minimisé. De cette façon, le nombre de branches appariées est maximal. Ensuite, nous introduisons une méthode de mise en correspondance pour la triangulation des branches 3D associées. Enfin, nous évaluons le conditionnement induit par les points de vue choisis, ce qui permet d'estimer une nouvelle vue à ajouter pour améliorer la reconstruction du squelette 3D de façon significative.

Published
2016

46. Numerical Analysis in Optimal Control Problem for Aircraft Trajectories

Author

Goubinat, Damien, Cots, Olivier, Gergaud, Joseph, Algorithmes Parallèles et Optimisation (IRIT-APO), Institut de recherche en informatique de Toulouse (IRIT), 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), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, Thales (France), Mathematics for Control, Transport and Applications (McTAO), 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 Polytechnique (Toulouse) (Toulouse INP), Department of Computing, Imperial College London, Centre for Process Systems Engineering, Imperial College London, Imperial College Business School, Grélaud, Françoise, Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-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)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Toulouse Mind & Brain Institut (TMBI), Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT), and THALES [France]

Subjects
[INFO]Computer Science [cs], [INFO] Computer Science [cs], ComputingMilieux_MISCELLANEOUS
Abstract

International audience

Published
2015

47. Geometric optimal control : homotopic methods and applications

Author

Cots , Olivier, Cots, Olivier, Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS ), Université de Bourgogne, Jean-Baptiste Caillau, Joseph Gergaud, Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), and Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB)

Subjects
[SPI.OTHER]Engineering Sciences [physics]/Other, Méthodes de tir, Homotopie différentielle, [ SPI.OTHER ] Engineering Sciences [physics]/Other, Orbital transfer, Contrôle optimal géométrique, [SPI.OTHER] Engineering Sciences [physics]/Other, [ MATH.MATH-GM ] Mathematics [math]/General Mathematics [math.GM], Shooting methods, Differential homotopy, Automatic differentiation, Contraste en RMN, Quantum control, [MATH.MATH-GM] Mathematics [math]/General Mathematics [math.GM], Geometric optimal control, Conditions du deuxième ordre, Transfert orbital, Lieux conjugués et de coupure, Différenciation automatique, Second order conditions, [MATH.MATH-GM]Mathematics [math]/General Mathematics [math.GM], Cut and conjugate loci, Contrast imaging in NMR, Contrôle quantique
Abstract

This work is about geometric optimal control applied to celestial and quantum mechanics. We first dealt with the minimum fuel consumption problem of transfering a satellite around the Earth. This brought to the creation of the code HamPath which permits first of all to solve optimal control problem for which the command law is smooth. It is based on the Pontryagin Maximum Principle (PMP) and on the notion of conjugate point. This program combines shooting method, differential homotopic methods and tools to compute second order optimality conditions. Then we are interested in quantum control. We study first a system which consists in two different particles of spin 1/2 having two different relaxation time. Both sub-systems are driven by the same control. The problem consists in bringing to zero the magnetization of one of the two system while maximizing the magnetization of the second one. This problem comes from constrast imaging in Nuclear Magnetic Resonance and consists in maximising the contrast between two areas of the image. The use of geometrical and numerical tools has given a very precise sub-optimal synthesis for two particular cases (deoxygenated/oxygenated blood and cerebrospinal fluid/water cases). The last contribution of this thesis is about the Lindblad equations in the two-level case. The model is based upon the minimisation of the transfer energy. We restrict the study to a particular case for which the Hamiltonian given by the PMP is Liouville integrable.We describe the conjugate and cut loci for this Riemannian with drift problem, Le contexte de ce travail est le contrôle optimal géométrique appliqué à la mécanique céleste et au contrôle quantique. On s’est tout d’abord intéressé au problème de transfert orbital de satellite autour de la Terre à consommation minimale,qui amena à la réalisation du code HamPath, permettant tout d’abord la résolution de problème de contrôle optimal dont la loi de commande est lisse. Il se base sur le Principe du Maximum de Pontryagin (PMP) et sur la notion de point conjugué. Ce programme combine méthode de tir, méthodes homotopiques différentielles et calcul des conditions d’optimalité du deuxième ordre. Nous nous intéressons par la suite au contrôle quantique. On étudie tout d’abord le contrôle d’un système composé de deux types de particules de spin 1/2 ayant des temps de relaxation différents et dont la dynamique est gouvernée par les équations de Bloch. Ces deux sous-systèmes,correspondant aux deux types de particules, sont couplés par un même contrôle (un champ electromagnétique), le but étant alors d’amener la magnétisation des particules du premier type à zéro tout en maximisant celle du second (dans un système de coordonnées bien choisi). Ce modèle intervient en imagerie médicale par Résonance Magnétique Nucléaire et consiste à maximiser le contraste entre deux régions d’une même image. L’utilisation des outils géométriques et numériques aura permis de donner une très bonne synthèse sous-optimale pour deux cas particuliers (mélange sang oxygéné/désoxygéné et liquide cérébrospinal/eau). La dernière contribution de cette thèse porte sur l’étude d’un système quantique à deux niveaux d’énergie dontl a dynamique est régie par les équations de Lindblad. Le modèle est basé sur la minimisation d’énergie du transfert. On se restreint à un cas particulier pour lequelle Hamiltonien donné par le PMP est Liouville intégrable. On décrit alors les lieux conjugués et de coupure pour ce problème riemannien avec dérive

Published
2012

48. Numerical Analysis of Optimal Control Problems for Aircraft Trajectory Computation (Poster)

Author

Cots, Olivier, Gergaud, Joseph, Goubinat, Damien, Grélaud, Françoise, Mathematics for Control, Transport and Applications (McTAO), 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), Algorithmes Parallèles et Optimisation (IRIT-APO), Institut de recherche en informatique de Toulouse (IRIT), 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), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, Thales (France), Jean-Baptiste Caillau, University of Dijon, France, Thomas Haberkorn, Université d'Orléans, France, Johann Radon Institute for Computational and Applied Mathematics (RICAM) of the Austrian Academy of Sciences (ÖAW), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-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)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Toulouse Mind & Brain Institut (TMBI), Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT), and THALES [France]

Subjects
[INFO]Computer Science [cs], [INFO] Computer Science [cs], ComputingMilieux_MISCELLANEOUS
Abstract

International audience

Published
2014

49. Differential pathfollowing for regular optimal control problems

Author

Caillau , Jean-Baptiste, Cots , Olivier, Gergaud , Joseph, Caillau, Jean-Baptiste, Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de recherche en informatique de Toulouse ( IRIT ), Institut National Polytechnique [Toulouse] ( INP ) -Université Toulouse 1 Capitole ( UT1 ) -Université Toulouse - Jean Jaurès ( UT2J ) -Université Paul Sabatier - Toulouse 3 ( UPS ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), Algorithmes Parallèles et Optimisation (IRIT-APO), Institut de recherche en informatique de Toulouse (IRIT), 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), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, and Institut National Polytechnique (Toulouse) (Toulouse INP)

Subjects
[ MATH.MATH-OC ] Mathematics [math]/Optimization and Control [math.OC], [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], ComputingMilieux_MISCELLANEOUS
Abstract

International audience

Published
2012

Catalog

Books, media, physical & digital resources
See catalog results