0000000000450461
AUTHOR
Jérémy Rouot
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…
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.
Optimization of chemical batch reactors using temperature control
International audience
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.
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…
Sub-Riemannian Geometry and swimming at low Reynolds number
International audience
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.
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.
Averaging for minimum time control problems and applications
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.
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.
Optimal control theory, sub-Riemannian geometry and swimming of copepod
International audience
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…
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.
Averaging techniques in the time minimal orbital transfer using low propulsion
International audience
Optimal Control Theory and the Efficiency of the Swimming Mechanism of the Copepod Zooplankton
International audience
Optimal Control Techniques for Sampled-Data Control Systems with Medical Applications
International audience