Toward Geometric Time Minimal Control without Legendre Condition and with Multiple Singular Extremals for Chemical Networks. An Extended Version

This article deals with the problem of maximizing the production of a species for a chemical network by controlling the temperature. Under the socalled mass kinetics assumption the system can be modeled as a single-input control system using the Feinberg-Horn-Jackson graph associated to the reactions network. Thanks to Pontryagin's Maximum Principle, the candidates as minimizers can be found among extremal curves, solutions of a (non smooth) Hamiltonian dynamics and the problem can be stated as a time minimal control problem with a terminal target of codimension one. Using geometric control and singularity theory the time minimal syntheses (closed loop optimal control) can be classified near the terminal manifold under generic conditions. In this article, we focus to the case where the generalized Legendre-Clebsch condition is not satisfied, which paves the road to complicated syntheses with several singular arcs. In particular, it is related to the situation for a weakly reversible network like the McKeithan scheme

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

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

International audienceIn 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 set of medical interventions. Inboth case the optimal control problems can be analyzed using direct and indirect schemesaiming to reach an healthy state. Such methods are tested on toys models in dimension 2and 3 related to the construction of reduced dynamics. Even those simple situations leadto interesting questions of accessibility and integrability issues in relation with the studyof dynamical systems

Optimal control theory and the efficiency of the swimming mechanism of the Copepod Zooplankton

International audienceIn 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 using Stokes theorem and curvature control. Also the concept of graded approximations in SR-geometry is used to evaluate strokes with small amplitudes providing a fixed displacement and minimizing the dissipated energy

The Purcell Three-link swimmer: some geometric and numerical aspects related to periodic optimal controls

International audienceThe maximum principle combined with numerical methods is a powerful tool to compute solutions for optimal control problems. This approach turns out to be extremely useful in applications, including solving problems which require establishing periodic trajectories for Hamiltonian systems, optimizing the production of photobioreactors over a one-day period, finding the best periodic controls for locomotion models (e.g. walking, flying and swimming). In this article we investigate some geometric and numerical aspects related to optimal control problems for the so-called Purcell Three-link swimmer [20], in which the cost to minimize represents the energy consumed by the swimmer. More precisely, employing the maximum principle and shooting methods we derive optimal trajectories and controls, which have particular periodic features. Moreover, invoking a linearization procedure of the control system along a reference extremal, we estimate the conjugate points, which play a crucial role for the second order optimality conditions. We also show how, making use of techniques imported by the sub-Riemannian geometry, the nilpotent approximation of the system provides a model which is integrable, obtaining explicit expressions in terms of elliptic functions. This approximation allows to compute optimal periodic controls for small deformations of the body, allowing the swimmer to move minimizing its energy. Numerical simulations are presented using Hampath and Bocop codes

Lunar perturbation of the metric associated to the averaged orbital transfer

International audienceIn a series of previous article [1,2], 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 a standard perturbation in space mechanics, the lunar attraction. Using Hamiltonian formalism, we describe the effects of the perturbation on the orbital transfers and the deformation of the conjugate and cut loci of the original metric

Direct and indirect methods to optimize the muscular force response to a pulse train of electrical stimulation

Recent force-fatigue mathematical models in biomechanics [7] allow to predict the muscular force response to functional electrical stimulation (FES) and leads to the optimal control problem of maximizing the force. The stimulations are Dirac pulses and the control parameters are the pulses amplitudes and times of application, the number of pulses is physically limited and the model leads to a sampled data control problem. The aim of this article is to present and compare two methods. The first method is a direct optimization scheme where a further refined numerical discretization is applied on the dynamics. The second method is an indirect scheme: first-order Pontryagin type necessary conditions are derived and used to compute the optimal sampling times

Optimal control of slow-fast mechanical systems

National audienceWe consider the minimum time control of dynamical systems with slow and fast state variables. With applications to perturbations of integrable systems in mind, we focus on the case of problems with one or more fast angles, together with a small drift on the slow part modelling a so-called secular evolution of the slow variables. According to Pontrjagin maximum principle, minimizing trajectories are projections on the state space of Hamiltonian curves. In the case of a single fast angle, it turns out that, provided 9 the drift on the slow part of the original system is small enough, time minimizing trajectories can be approximated by geodesics of a suitable metric. 11 As an application to space mechanics, the effect of the J2 term in the Earth potential on the control of a spacecraft is considered. In ongoing work, we 13 also address the more involved question of systems having two fast angles

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 four parameters to compute the optimal synthesis using homotopies on the parameters. It is extended to the 3d–case to provide a geometric frame to direct and indirect numerical schemes