Sparse stabilization and optimal control of the Cucker-Smale model
Marco Caponigro Massimo Fornasier Benedetto Piccoli Emmanuel Trélat
This article is mainly based on the work [7], and it is dedicated to the 60th anniversary of B. Bonnard, held in Dijon in June 2012.
    We focus on a controlled Cucker--Smale model in finite dimension. Such dynamics model self-organization and consensus emergence in a group of agents. We explore how it is possible to control this model in order to enforce or facilitate pattern formation or convergence to consensus. In particular, we are interested in designing control strategies that are componentwise sparse in the sense that they require a small amount of external intervention, and also time sparse in the sense that such strategies are not chattering in time. These sparsity features are desirable in view of practical issues.
    We first show how very simple sparse feedback strategies can be designed with the use of a variational principle, in order to steer the system to consensus. These feedbacks are moreover optimal in terms of decay rate of some functional, illustrating the general principle according to which ``sparse is better''. We then combine these results with local controllability properties to get global controllability results. Finally, we explore the sparsity properties of the optimal control minimizing a combination of the distance from consensus and of a norm of the control.
keywords: Cucker--Smale model $l_1$-norm minimization local controllability sparse optimal control. consensus emergence sparse stabilization
Optimal control of a space shuttle, and numerical simulations
Emmanuel Trélat
We study the Earth re-entry problem of 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. The optimal solution is approximated by a concatenation of bang and boundary arcs, and is numerically computed using a multiple-shooting code.
keywords: Control of the atmospheric arc Optimal control with state constraints Multiple-shooting techniques.
J.-B. Caillau Monique Chyba Dominique Sugny Emmanuel Trélat
Life is singular in many ways, and Prof. Bernard Bonnard is aware of this better than anybody else. Throughout his work and career, he has shown us that following a singular path is not only exciting and interesting but most importantly that it is often the most efficient way to accomplish a goal!

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Optimal sampled-data control, and generalizations on time scales
Loïc Bourdin Emmanuel Trélat
In this paper, we derive a version of the Pontryagin maximum principle for general finite-dimensional nonlinear optimal sampled-data control problems. Our framework is actually much more general, and we treat optimal control problems for which the state variable evolves on a given time scale (arbitrary non-empty closed subset of $\mathbb{R}$), and the control variable evolves on a smaller time scale. Sampled-data systems are then a particular case. Our proof is based on the construction of appropriate needle-like variations and on the Ekeland variational principle.
keywords: Optimal control sampled-data Pontryagin maximum principle time scale.
Optimal design of sensors for a damped wave equation
Yannick Privat Emmanuel Trélat
In this paper we model and solve the problem of shaping and placing in an optimal way sensors for a wave equation with constant damping in a bounded open connected subset $\Omega$ of $\mathbb{R}^n$. Sensors are modeled by subdomains of $\Omega$ of a given measure $L|\Omega|$, with $0 < L < 1$. We prove that, if $L$ is close enough to $1$, then the optimal design problem has a unique solution, which is characterized by a finite number of low frequency modes. In particular the maximizing sequence built from spectral approximations is stationary.
keywords: shape optimization Damped wave equation observability.
Jean-Baptiste Caillau Maria do Rosário de Pinho Lars Grüne Emmanuel Trélat Hasnaa Zidani
This special volume gathers a number of new contributions addressing various topics related to the field of optimal control theory and sensitivity analysis. The field has a rich and varied mathematical theory, with a long tradition and a vibrant body of applications. It has attracted a growing interest across the last decades, with the introduction of new ideas and techniques, and thanks to various new applications.

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Controllability of couette flows
Michael Schmidt Emmanuel Trélat
In this article, we investigate the problem of controlling Navier-Stokes equations between two infinite rotating coaxial cylinders. We prove that it is possible to move from a given Couette flow, that is a special stationary solution, to another one, by controlling the rotation velocity of the outer cylinder.
keywords: partial differential equations. Navier-Stokes equations controllability Couette flows
Planar tilting maneuver of a spacecraft: Singular arcs in the minimum time problem and chattering
Jiamin Zhu Emmanuel Trélat Max Cerf
In this paper, we study the minimum time planar tilting maneuver of a spacecraft, from the theoretical as well as from the numerical point of view, with a particular focus on the chattering phenomenon. We prove that there exist optimal chattering arcs when a singular junction occurs. Our study is based on the Pontryagin Maximum Principle and on results by M.I. Zelikin and V.F. Borisov. We give sufficient conditions on the terminal values under which the optimal solutions do not contain any singular arc, and are bang-bang with a finite number of switchings. Moreover, we implement sub-optimal strategies by replacing the chattering control with a fixed number of piecewise constant controls. Numerical simulations illustrate our results.
keywords: singular control Pontryagin Maximum Principle Spacecraft planar tilting maneuver chattering arcs sub-optimal strategy. minimum time control
High order variational integrators in the optimal control of mechanical systems
Cédric M. Campos Sina Ober-Blöbaum Emmanuel Trélat
In recent years, much effort in designing numerical methods for the simulation and optimization of mechanical systems has been put into schemes which are structure preserving. One particular class are variational integrators which are momentum preserving and symplectic. In this article, we develop two high order variational integrators which distinguish themselves in the dimension of the underling space of approximation and we investigate their application to finite-dimensional optimal control problems posed with mechanical systems. The convergence of state and control variables of the approximated problem is shown. Furthermore, by analyzing the adjoint systems of the optimal control problem and its discretized counterpart, we prove that, for these particular integrators, dualization and discretization commute.
keywords: high order Optimal control commutation property. direct methods geometric integration Runge-Kutta mechanical systems variational integrator
Optimal control of infinite-dimensional piecewise deterministic Markov processes and application to the control of neuronal dynamics via Optogenetics
Vincent Renault Michèle Thieullen Emmanuel Trélat

In this paper we define an infinite-dimensional controlled piecewise deterministic Markov process (PDMP) and we study an optimal control problem with finite time horizon and unbounded cost. This process is a coupling between a continuous time Markov Chain and a set of semilinear parabolic partial differential equations, both processes depending on the control. We apply dynamic programming to the embedded Markov decision process to obtain existence of optimal relaxed controls and we give some sufficient conditions ensuring the existence of an optimal ordinary control. This study, which constitutes an extension of controlled PDMPs to infinite dimension, is motivated by the control that provides Optogenetics on neuron models such as the Hodgkin-Huxley model. We define an infinite-dimensional controlled Hodgkin-Huxley model as an infinite-dimensional controlled piecewise deterministic Markov process and apply the previous results to prove the existence of optimal ordinary controls for a tracking problem.

keywords: Piecewise Deterministic Markov Processes optimal control semilinear parabolic equations dynamic programming Markov Decision Processes Optogenetics

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