Geometric Optimal Control of the Generalized Lotka-Volterra Model with Applications Controlled Stability of Microbiota

International audience; In this talk we present the Generalized Lotka–Volterra dynamics associated to themodel of C-difficile infection of the intestine microbiote and aiming to transfer the systemfrom an infected state to an healthy state. The control inputs are of two types : fecalinjection or bactericides which act as Dirac pulses and prebiotics or antibiotics which act ascontinuous controls. An uniform frame can be introduced using the tools from geometriccontrol to analyze the accessibility set as the orbit of a pseudo-semi group. Optimalcontrol can be considered in the frame of permanent control or sampled-data control. Thelater being adapted to the practical constraints of a finite s…

Geodesic flow of the averaged controlled Kepler equation

A normal form of the Riemannian metric arising when averaging the coplanar controlled Kepler equation is given. This metric is parameterized by two scalar invariants which encode its main properties. The restriction of the metric to $\SS^2$ is shown to be conformal to the flat metric on an oblate ellipsoid of revolution, and the associated conjugate locus is observed to be a deformation of the standard astroid. Though not complete because of a singularity in the space of ellipses, the metric has convexity properties that are expressed in terms of the aforementioned invariants, and related to surjectivity of the exponential mapping. Optimality properties of geodesics of the averaged controll…

Generic properties of singular trajectories

Abstract Let M be a σ-compact C∞ manifold of dimension d ≥ 3. Consider on M a single-input control system : x (t) = F 0 (x(t)) + u(t) F 1 (x(t)) , where F0, F1 are C∞ vector fields on M and the set of admissible controls U is the set of bounded measurable mappings u : [0Tu]↦ R , Tu > 0. A singular trajectory is an output corresponding to a control such that the differential of the input-output mapping is not of maximal rank. In this article we show that for an open dense subset of the set of pairs of vector fields (F0, F1), endowed with the C∞-Whitney topology, all the singular trajectories are with minimal order and the corank of the singularity is one.

Weak Maximum Principle and Application to Swimming at Low Reynolds Number

We refer to [9, 42, 46] for more details about the general concepts and notations introduced in this section.

Geometric optimal control of elliptic Keplerian orbits

This article deals with the transfer of a satellite between Keplerian orbits. We study the controllability properties of the system and make a preliminary analysis of the time optimal control using the maximum principle. Second order sufficient conditions are also given. Finally, the time optimal trajectory to transfer the system from an initial low orbit with large eccentricity to a terminal geostationary orbit is obtained numerically.

TIME-MINIMAL CONTROL OF DISSIPATIVE TWO-LEVEL QUANTUM SYSTEMS: THE INTEGRABLE CASE

The objective of this article is to apply recent developments in geometric optimal control to analyze the time minimum control problem of dissipative two-level quantum systems whose dynamics is governed by the Lindblad equation. We focus our analysis on the case where the extremal Hamiltonian is integrable.

Optimization of chemical batch reactors using temperature control

International audience

Two Applications of Geometric Optimal Control to the Dynamics of Spin Particles

The purpose of this article is to present the application of methods from geometric optimal control to two problems in the dynamics of spin particles. First, we consider the saturation problem for a single spin system and second, the control of a linear chain of spin particles with Ising couplings. For both problems the minimizers are parameterized using Pontryagin Maximum Principle and the optimal solution is found by a careful analysis of the corresponding equations.

One-parameter family of Clairaut-Liouville metrics

Riemannian metrics with singularities are considered on the $2$-sphere of revolution. The analysis of such singularities is motivated by examples stemming from mechanics and related to projections of higher dimensional (regular) sub-Riemannian distributions. An unfolding of the metrics in the form of an homotopy from the canonical metric on $\SS^2$ is defined which allows to analyze the singular case as a limit of standard Riemannian ones. A bifurcation of the conjugate locus for points on the singularity is finally exhibited.

Conjugate times for smooth singular trajectories and bang-bang extremals

Abstract In this paper we discuss the problem of estimating conjugate times along smooth singular or bang-bang extremals. For smooth extremals conjugate times can be defined in the generic case by using the intrinsic second order derivative or the exponential mapping. An algorithm is given which was implemented in the SR-case to compute the caustic [1] or in recent applied problems [5],[9]. We investigate briefly the problem of using this algorithm in the bang-bang case by smoothing the corners of extremals

Geometric optimal control and two-level dissipative quantum systems

International audience; The objective of this article is to present techniques of geometric time-optimal control developed to analyze the control of two-level dissipative quantum systems. Combined with numerical simulations they allow to compute the time-minimal control using a shooting method. The robustness with respect to initial conditions and dissipative parameters is also analyzed using a continuation method.

Classification générique de synthèses temps minimales avec cible de codimension un et applications

In this article we consider the problem of constructing the optimal closed loop control in the time minimal control problem, with terminal constraints belonging to a manifold of codimension one, for systems of the form v = X + uY, v ϵ R2, R3, |u| ≤ 1 under generic assumptions. The analysis is localized near the terminal manifold and is motivated by the problem of controlling a class of chemical systems.

Second order optimality conditions in the smooth case and applications in optimal control

International audience; The aim of this article is to present algorithms to compute the first conjugate time along a smooth extremal curve, where the trajectory ceases to be optimal. It is based on recent theoretical developments of geometric optimal control, and the article contains a review of second order optimality conditions. The computations are related to a test of positivity of the intrinsic second order derivative or a test of singularity of the extremal flow. We derive an algorithm called COTCOT (Conditions of Order Two and COnjugate times), available on the web, and apply it to the minimal time problem of orbit transfer, and to the attitude control problem of a rigid spacecraft. …

Optimal Control Theory and the Swimming Mechanism of the Copepod Zooplankton

In this article, the model of swimming at low Reynolds number introduced by D. Takagi (2015) to analyze the displacement of an abundant variety of zooplankton is used as a testbed to analyze the motion of symmetric microswimmers in the framework of optimal control theory assuming that the motion occurs minimizing the energy dissipated by the fluid drag forces in relation with the concept of efficiency of a stroke. The maximum principle is used to compute periodic controls candidates as minimizing controls and is a decisive tool combined with appropriate numerical simulations using indirect optimal control schemes to determine the most efficient stroke compared with standard computations usi…

Maximum Principle and Application to Nuclear Magnetic Resonance and Magnetic Resonance Imaging

In this section we state the Pontryagin maximum principle and we outline the proof. We adopt the presentation from Lee and Markus [64] where the result is presented into two theorems.

Lunar perturbation of the metric associated to the averaged orbital transfer

International audience; In a series of previous article we introduced a Riemannian metric associated to the energy minimizing orbital transfer with low propulsion. The aim of this article is to study the deformation of this metric due to the perturbation caused by the lunar attraction. Using Hamiltonian formalism, we describe the effects of the perturbations on the orbital transfers and the deformation of the conjugate and cut loci of the original metric.

Nuclear magnetic resonance: The contrast imaging problem

Starting as a tool for characterization of organic molecules, the use of NMR has spread to areas as diverse as pharmacology, medical diagnostics (medical resonance imaging) and structural biology. Recent advancements on the study of spin dynamics strongly suggest the efficiency of geometric control theory to analyze the optimal synthesis. This paper focuses on a new approach to the contrast imaging problem using tools from geometric optimal control. It concerns the study of an uncoupled two-spin system and the problem is to bring one spin to the origin of the Bloch ball while maximizing the modulus of the magnetization vector of the second spin. It can be stated as a Mayer-type optimal prob…

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

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…

Geometric Optimal Control of the Generalized Lotka-Volterra Model of the Intestinal Microbiome

We introduce the theoretical framework from geometric optimal control for a control system modeled by the Generalized Lotka-Volterra (GLV) equation, motivated by restoring the gut microbiota infected by Clostridium difficile combining antibiotic treatment and fecal injection. We consider both permanent control and sampled-data control related to the medical protocols.

Geometric and numerical techniques in optimal control of two and three- body problems

International audience

Energy minimization of single input orbit transfer by averaging and continuation

AbstractThis article deals with the transfer between Keplerian coplanar orbits using low propulsion. We focus on the energy minimization problem and compute the averaged system, proving integrability and relating the corresponding trajectories to a three-dimensional Riemannian problem that is analyzed in details. The geodesics provide approximations of the extremals of the energy minimization problem and can be used in order to evaluate the optimal trajectories of the time optimal and the minimization of the consumption problems with continuation methods. In particular, minimizing trajectories for transfer towards the geostationary orbit can be approximated in suitable coordinates by straig…

Time minimization versus energy minimization in the one-input controlled Kepler problem with weak propulsion

International audience

Classification of local optimal syntheses for time minimal control problems with state constraints

This paper describes the analysis under generic assumptions of the small \textit{time minimal syntheses} for single input affine control systems in dimension $3$, submitted to \textit{state constraints}. We use geometric methods to evaluate \textit{the small time reachable set} and necessary optimality conditions. Our work is motivated by the \textit{optimal control of the atmospheric arc for the re-entry of a space shuttle}, where the vehicle is subject to constraints on the thermal flux and on the normal acceleration.

Optimality results in orbit transfer

Abstract The objective of this Note is to present optimality results in orbital transfer. Averaging of the energy minimization problem is considered, and properties of the associated Riemannian metric are discussed. To cite this article: B. Bonnard, J.-B. Caillau, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

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

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 particle…

Riemannian metric of the averaged controlled Kepler equation

International audience

Sub-Riemannian Geometry and swimming at low Reynolds number

International audience

Time-Minimal Control of Dissipative Two-Level Quantum Systems: The Generic Case

International audience; The objective of this article is to complete preliminary results from [5], [17] concerning the time-minimal control of dissipative two-level quantum systems whose dynamics is governed by the Lindblad equation. The extremal system is described by a 3-D-Hamiltonian depending upon three parameters. We combine geometric techniques with numerical simulations to deduce the optimal solutions.

Numerical Approach to the Optimal Control and Efficiency of the Copepod Swimmer

International audience

Optimal control of an ensemble of Bloch equations with applications in MRI

International audience; The optimal control of an ensemble of Bloch equations describing the evolution of an ensemble of spins is the mathematical model used in Nuclear Resonance Imaging and the associated costs lead to consider Mayer optimal control problems. The Maximum Principle allows to parameterize the optimal control and the dynamics is analyzed in the framework of geometric optimal control. This lead to numerical implementations or suboptimal controls using averaging principle.

Two applications of geometric optimal control to the dynamics of spin particle

To appear in a volume of "Math and Industry", Springer-Verlag; The purpose of this article is to present the application of methods from geometric optimal control to two problems in the dynamics of spin particles. First, we consider the saturation problem for a single spin system and second, the control of a linear chain of spin particles with Ising couplings. For both problems the minimizers are parameterized using Pontryagin Maximum Principle and the optimal solution is found by a careful analysis of the corresponding equations.

Optimal control of the atmospheric arc of a space shuttle and numerical simulations with multiple-shooting method

This article, continuation of previous works, presents the applications of geometric optimal control theory to the analysis of the Earth re-entry problem for a space shuttle where the control is the angle of bank, the cost is the total amount of thermal flux, and the system is subject to state constraints on the thermal flux, the normal acceleration and the dynamic pressure. Our analysis is based on the evaluation of the reachable set using the maximum principle and direct computations with the boundary conditions according to the CNES research project\footnote{The project is partially supported by the Centre National d'Etude Spatiales.}. The optimal solution is approximated by a concatenat…

Historical Part—Calculus of Variations

The calculus of variations is an old mathematical discipline and historically finds its origins in the introduction of the brachistochrone problem at the end of the 17th century by Johann Bernoulli to challenge his contemporaries to solve it. Here, we briefly introduce the reader to the main results.

Méthodes géométriques et analytiques pour étudier l'application exponentielle, la sphère et le front d'onde en géométrie sous-riemannienne dans le cas Martinet

Consider a sub-riemannian geometry (U,D,g) where U is a neighborhood of 0 in R 3 , D is a Martinet type distribution identified to ker ω , ω being the 1-form: , q=(x,y,z) and g is a metric on D which can be taken in the normal form : , a=1+yF(q) , c=1+G(q) , . In a previous article we analyze the flat case : a=c=1 ; we describe the conjugate and cut loci , the sphere and the wave front . The objectif of this article is to provide a geometric and computational framework to analyze the general case. This frame is obtained by analysing three one parameter deformations of the flat case which clarify the role of the three parameters in the gradated normal form of order 0 where: , . More generall…

Sub-Riemannian geometry: one-parameter deformation of the Martinet flat case

Riemannian metric of the averaged energy minimization problem in orbital transfer with low thrust

Abstract This article deals with the optimal transfer of a satellite between Keplerian orbits using low propulsion and is based on preliminary results of Epenoy et al. (1997) where the optimal trajectories of the energy minimization problem are approximated using averaging techniques. The averaged Hamiltonian system is explicitly computed. It is related to a Riemannian problem whose distance is an approximation of the value function. The extremal curves are analyzed, proving that the system remains integrable in the coplanar case. It is also checked that the metric associated with coplanar transfers towards a circular orbit is flat. Smoothness of small Riemannian spheres ensures global opti…

Comparison of Numerical Methods in the Contrast Imaging Problem in NMR

International audience; In this article, the contrast imaging problem in nuclear magnetic resonance is modeled as a Mayer problem in optimal control. A first synthesis of locally optimal solutions is given in the single-input case using geometric methods based on Pontryagin's maximum principle. We then compare these results using direct methods and a moment-based approach, and make a first step towards global optimality. Finally, some preliminary results are given in the bi-input case.

Feedback equivalence and the contrast problem in nuclear magnetic resonance imaging

International audience; The theoretical analysis of the contrast problem in NMR imaging is mainly reduced, thanks to the Maximum Principle, to the analysis of the so-called singular trajectories of the control system modeling the problem: a coupling of two Bloch equations representing the evolution of the magnetization vector of each spin particle. They are solutions of a constrained Hamiltonian equation. In this article we describe feedback invariants related to the singular flow to distinguish the different cases occurring in physical experiments.

Une approche géométrique du contrôle optimal de l'arc atmosphérique de la navette spatiale

L'objectif de ce travail est de faire quelques remarques géométriques et des calculs préliminaires pour construire l'arc atmosphérique optimal d'une navette spatiale (problème de rentrée sur Terre ou programme d'exploration de Mars). Le système décrivant les trajectoires est de dimension 6, le contrôle est l'angle de gîte cinématique et le coût est l'intégrale du flux thermique. Par ailleurs il y a des contraintes sur l'état (flux thermique, accélération normale et pression dynamique). Notre étude est essentiellement géométrique et fondée sur une évaluation de l'ensemble des états accessibles en tenant compte des contraintes sur l'état. On esquisse une analyse des extrémales du Principe du …

Cotcot: short reference manual

Technical report; This reference introduces the Matlab package COTCOT designed to compute extremals in the case of smooth Hamiltonian systems, and to obtain the associated conjugate points with respect to the index performance of the underlying optimal control problem.

Averaging and optimal control of elliptic Keplerian orbits with low propulsion

This article deals with the optimal transfer of a satellite between Keplerian orbits using low propulsion. It is based on preliminary results of Geffroy [Generalisation des techniques de moyennation en controle optimal, application aux problemes de rendez-vous orbitaux a poussee faible, Ph.D. Thesis, Institut National Polytechnique de Toulouse, France, Octobre 1997] where the optimal trajectories are approximated using averaging techniques. The objective is to introduce the appropriate geometric framework and to complete the analysis of the averaged optimal trajectories for energy minimization, showing in particular the connection with Riemannian problems having integrable geodesics.

Conjugate and cut loci of a two-sphere of revolution with application to optimal control

Abstract The objective of this article is to present a sharp result to determine when the cut locus for a class of metrics on a two-sphere of revolution is reduced to a single branch. This work is motivated by optimal control problems in space and quantum dynamics and gives global optimal results in orbital transfer and for Lindblad equations in quantum control.

Geometric optimal control of spin systems

International audience

Feedback Classification and Optimal Control with Applications to the Controlled Lotka-Volterra Model

Let M be a σ-compact C^∞ manifold of dimension n ≥ 2 and consider a single-input control system: ẋ(t) = X (x(t)) + u(t) Y (x(t)), where X , Y are C^∞ vector fields on M. We prove that there exist an open set of pairs (X , Y ) for the C^∞ –Whitney topology such that they admit singular abnormal rays so that the spectrum of the projective singular Hamiltonian dynamics is feedback invariant. It is applied to controlled Lotka–Volterra dynamics where such rays are related to shifted equilibria of the free dynamics.

The energy minimization problem for two-level dissipative quantum systems

In this article, we study the energy minimization problem of dissipative two-level quantum systems whose dynamics is governed by the Kossakowski–Lindblad equations. In the first part, we classify the extremal curve solutions of the Pontryagin maximum principle. The optimality properties are analyzed using the concept of conjugate points and the Hamilton–Jacobi–Bellman equation. This analysis completed by numerical simulations based on adapted algorithms allows a computation of the optimal control law whose robustness with respect to the initial conditions and dissipative parameters is also detailed. In the final section, an application in nuclear magnetic resonance is presented.

Optimal Control of the Lotka-Volterra Equations with Applications

In this article, the Lotka-Volterra model is analyzed to reduce the infection of a complex microbiote. The problem is set as an optimal control problem, where controls are associated to antibiotic or probiotic agents, or transplantations and bactericides. Candidates as minimizers are selected using the Maximum Principle and the closed loop optimal solution is discussed. In particular a 2d-model is constructed with 4 parameters to compute the optimal synthesis using homotopies on the parameters.

The transcendence required for computing the sphere and wave front in the Martinet sub-Riemannian geometry

Consider a \it{sub-Riemannian geometry} $(U,D,g)$ where $U$ is a neighborhood of $O$ in $\mathbb{R}^3$, $D$ is a \it{Martinet type distribution} identified to $Ker \,\omega$, $\omega =dz-\f{y^2}{2}dx$, $q=(x,y,z)$ and $g$ is a \it{metric on $D$} which can be taken in the normal form : \mbox{$a(q)dx^2+c(q)dy^2$}, \mbox{$a=1+yF(q)$}, \mbox{$c=1+G(q)$}, \mbox{$G_{|x=y=0}=0$}. In a previous article we analyzed the \it{flat case} : \mbox{$a=c=1$} ; we showed that the set of geodesics is integrable using \it{elliptic integrals} of the \it{first and second kind} ; moreover we described the sphere and the wave front near the abnormal direction using the \it{\mbox{exp-log} category}. The objective o…

Stratification du secteur anormal dans la sphère de Martinet de petit rayon

L’objectif de cet article est de fournir le cadre geometrique pour faire une analyse de la singularite de l’application exponentielle le long d’une direction anormale en geometrie sous-Riemannienne. Il utilise les calculs de [9], [12], et conduit dans le cas Martinet a une stratification de la singularite en secteurs Lagrangiens.

Optimal control theory, sub-Riemannian geometry and swimming of copepod

International audience

Determinantal sets, singularities and application to optimal control in medical imagery

International audience; Control theory has recently been involved in the field of nuclear magnetic resonance imagery. The goal is to control the magnetic field optimally in order to improve the contrast between two biological matters on the pictures. Geometric optimal control leads us here to analyze mero-morphic vector fields depending upon physical parameters , and having their singularities defined by a deter-minantal variety. The involved matrix has polynomial entries with respect to both the state variables and the parameters. Taking into account the physical constraints of the problem, one needs to classify, with respect to the parameters, the number of real singularities lying in som…

Optimal Control of the Controlled Lotka-Volterra Equations with Applications - The Permanent Case

In this article motivated by the control of complex microbiota in view to reduce the infection by a pathogenic agent, we introduce the theoretical frame from optimal control to analyze the problem. Two complementary approaches can be applied in the analysis: one is the so-called permanent case, where no digital constraints are concerning the control (taken as a measurable mapping) versus the sampled-data control case taking into account the logistic constraints, e.g. frequency of the medical interventions. The model is the n-dimensional Lotka-Volterra equation controlled using either probiotics or antibiotic agents or transplantation and bactericides. In the permanent case the Maximum princ…

Time Versus Energy in the Averaged Optimal Coplanar Kepler Transfer towards Circular Orbits

International audience; The aim of this note is to compare the averaged optimal coplanar transfer towards circular orbits when the costs are the transfer time transfer and the energy consumption. While the energy case leads to analyze a 2D Riemannian metric using the standard tools of Riemannian geometry (curvature computations, geodesic convexity), the time minimal case is associated to a Finsler metric which is not smooth. Nevertheless a qualitative analysis of the geodesic flow is given in this article to describe the optimal transfers. In particular we prove geodesic convexity of the elliptic domain.

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

International audience

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

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.

Optimal control of two-level dissipative quantum systems

International audience

Averaging techniques in the time minimal orbital transfer using low propulsion

International audience

Remarks on quadratic Hamiltonians in spaceflight mechanics

A particular family of Hamiltonian functions is considered. Such functions are quadratic in the moment variables and arise in spaceflight mechanics when the averaged system of energy minimizing trajectories of the Kepler equation is computed. An important issue of perturbation theory and averaging is to provide integrable approximations of nonlinear systems. It turns out that such integrability properties hold here.

Optimal Control Theory and the Efficiency of the Swimming Mechanism of the Copepod Zooplankton

International audience

Geometric and numerical techniques to compute conjugate and cut loci on Riemannian surfaces

International audience; We combine geometric and numerical techniques - the Hampath code - to compute conjugate and cut loci on Riemannian surfaces using three test bed examples: ellipsoids of revolution, general ellipsoids, and metrics with singularities on S2 associated to spin dynamics.

Optimal Control Techniques for Sampled-Data Control Systems with Medical Applications

International audience

Note on singular Clairaut-Liouville metrics

Computations on Clairaut-Liouville metrics on S^2 with a finite order singularity.

Computation of conjugate times in smooth optimal control: the COTCOT algorithm

Conjugate point type second order optimality conditions for extremals associated to smooth Hamiltonians are evaluated by means of a new algorithm. Two kinds of standard control problems fit in this setting: the so-called regular ones, and the minimum time singular single-input affine systems. Conjugate point theory is recalled in these two cases, and two applications are presented: the minimum time control of the Kepler and Euler equations.