Bernard Bonnard

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

The objective of this article is to complete preliminary results from the work of Bonnard and Sugny (2009) and Sugny et al. (2007), 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 3D-Hamiltonian depending upon three parameters. We combine geometric techniques with numerical simulations to deduce the optimal solutions.

Optimal Control with State Constraints and the Space Shuttle Re-entry Problem

In this article, we initialize the analysis under generic assumptions of the small time optimal synthesis for single input systems with state constraints. We use geometric methods to evaluate the small time reachable set and necessary optimality conditions. Our work is motivated by the 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. A multiple shooting techniqueis finally applied to the computation of the optimal longitudinal arc.

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.

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.

Geometric Optimal Control of the Contrast Imaging Problem in Nuclear Magnetic Resonance

The objective of this article is to introduce the tools to analyze the contrast imaging problem in Nuclear Magnetic Resonance. Optimal trajectories can be selected among extremal solutions of the Pontryagin Maximum Principle applied to this Mayer type optimal problem. Such trajectories are associated to the question of extremizing the transfer time. Hence the optimal problem is reduced to the analysis of the Hamiltonian dynamics related to singular extremals and their optimality status. This is illustrated by using the examples of cerebrospinal fluid/water and grey/white matter of cerebrum.

Quadratic control systems

We outline a geometric theory for a class of homogeneous polynomial control systems called quadratic systems. We describe an algorithm to compute a minimal realization and study the feedback classification problem. Feedback invariants are related to the singularities of the input-output mapping and canonical forms are exhibited.

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.

GEOMETRIC NUMERICAL METHODS AND RESULTS IN THE CONTRAST IMAGING PROBLEM IN NUCLEAR MAGNETIC RESONANCE

The purpose of this paper is to present numerical methods and results about the contrast imaging problem in nuclear magnetic resonance which corresponds to a Mayer problem in optimal control. The candidates as minimizers are selected among a set of extremals, solutions of a Hamiltonian system given by the Pontryagin Maximum Principle and sufficient second order conditions are described. They form the geometric foundations of the HAMPATH code which combines shooting and continuation methods, see Ref. 9. The main contribution of this paper is to present a numerical analysis of the contrast imaging problem in NMR in the case of deoxygenated/oxygenated blood samples as an application of the aforementioned techniques.

Introduction to nonlinear optimal control

The maximum principle is presented in the weak and general forms. The standard proofs are detailed, and the connection with the shooting method for numerical resolution is made. A brief introduction to the micro-local analysis of extremals is also provided. Regarding second-order conditions, small time-optimality is addressed by means of high order generalized variations. As for local optimality of extremals, the conjugate point theory is introduced both for regular problems and for minimum time singular single input affine control systems. The analysis is applied to the minimum time control of the Kepler equation, and the numerical simulations for the corresponding orbit transfer problems are given. In the case of state constrained optimal control problems, necessary conditions are stated for boundary arcs. The junction and reflection conditions are derived in the Riemannian case.

Singular Trajectories and the Contrast Imaging Problem in Nuclear Magnetic Resonance

In this article, the contrast imaging problem in nuclear magnetic resonance is modeled as a Mayer problem in optimal control whose solutions can be parameterized using Pontryagins maximum principle and analyzed using geometric optimal control. In particular, the optimal problem can be mainly reduced to the analysis of the Hamiltonian dynamics describing the singular trajectories and encoding their optimality status.