Chairmen of the Organizing Committee are Ugo Boscain (Université de Bourgogne and SISSA Trieste) and Ludovic Faubourg (Université de Bourgogne).

Organizers

• Ugo Boscain, S.I.S.S.A, Trieste, Italy and University of Bourgogne, Dijon, France (boscain @sissa.it).
• Ludovic Faubourg, University of Bourgogne, Dijon, France (Ludovic.Faubourg @u-bourgogne.fr.

Conference Topics

The main scientific topics of the Conference are:
• Control Theory,
• Stabilization,
• Some topics about (control of) Quantum Systems.

 

Scientific Program

The Scientific Programme consists of 30-40 minutes invited lectures:
• 02/10 (pm) 5 lectures,
• 03/10 8 lectures,
• 04/10 (am) 5 lectures.

 

Preliminary list of Speakers:

Riccardo Adami, Ugo Boscain, Francesca Ceragioli, Thomas Chambrion, Gregoire Charlot, Ludovic Faubourg, Thierry Floquet, Manuel Guerra, Urszula Ledzewicz, Frederic Mazenc, Sebastien Jacquet, Frederic Jean, Christophe Prieur, Ludovic Rifford, Eugenio Rocha, Heinz Schattler, Mario Sigalotti, Delfim Torres, Emmanuel Trelat.

Titles and Abstracts (for speaker only)

Titles and (short) abstracts should be sent by e-mail no later than July 20, to boscain@sissa.it or to Ludovic.Faubourg@u-bourgogne.fr. Please, write 'Abstract' in position 'Subject'.

Travel and Lodge Information

Hotels reservation will be made by the organizing committee. Please send as soon as possible your arrival and departure datas. For Hotel prices (for nonspeakers only), contact the organizing commitee.
 

Registration Fee

The participation at the conference is free.
 

Proceedings

Proceedings will be pubblished as an internal report of the Laboratoire Analyse Appliquée-Optimisation.
 

Important Dead Lines

• 20 July 2002 (for speakers only): abstract submission.
• 10 September 2002 (for nonspeakers only): registration.

Please contact boscain@sissa.it or Ludovic.Faubourg@u-bourgogne.fr.

Titles and Abstracts:

• Riccardo Adami, Quantum models with point interactions.

First introduced in the 30's by Fermi, during the last decades point interactions turned out to provide a simple tool in order to construct phenomelogical models and to approach sophisticated mathematical problems. In this talk, we will introduce this kind of models and show some applications in various branches of quantum mechanics.
 
• Claudio Altafini, Controllability of finite level quantum systems via structure theory of semisimple compact Lie algebras.

The controllability of the unitary propagator of a finite level quantum system is studied by analyzing the structure theory of its Lie algebra. For the semisimple compact case, the testing of global controllability is the simplest of all the noncommutative Lie algebras. In fact, compactness implies that the accessibility property collapses into (global) controllability, while semisimplicity implies that almost all pairs of vector fields span the corresponding Lie algebra. The first property means that purely algebraic tools, like the Lie algebra rank condition normally used in control theory, provide necessary and sufficient conditions for controllability, while the second property affirms that controllability is generically verified even with a single control field. Genericity is interpreted in terms of regularity of the roots of the Lie algebra and another property, regularity along the control vector field, immediately follows. The main scope of this work is to provide alternative controllability tests to the exhaustive computation of commutators that the Lie algebraic rank condition requires and the main tool we use, together with the regularity of the roots, is the connectivity of the graph of the control vector field. Both properties were "classically" used to analyze controllability of vector fields on semisimple Lie algebras (especially noncompact ones). The conditions we obtain, based only on the a priori knowledge of the two vector fields, are only sufficient but they allow us to avoid any on-line computation of Lie brackets. From the generic case, physically representing a quantum system with all different transition values between its (nondegenerate) energy levels, these tools carry on to the singular case, where some of these levels might be equispaced.
 
• Ugo Boscain, Thomas Chambrion, On the $K+P$ Problem for a Three-level Quantum System

We apply techniques of subriemannian geometry on Lie groups to laser-induced population transfer in a three-level quantum system. The aim is to induce transitions by two laser pulses, of arbitrary shape and frequency, minimizing the pulse energy. We prove that the Hamiltonian system given by the Pontryagin Maximum Principle is completely integrable, since this problem can be stated as a ``$\k\oplus\p$ problem'' on a simple Lie group. Optimal trajectories and controls are exhausted. The main result is that optimal controls correspond to lasers that are ``in resonance''.
 
• Francesca Ceragioli, Nonsmooth Optimal Regulation And Discontinuous Stabilization.

We study the relationship between an optimal regulation problem on the infinite horizon and the stabilization problem for systems affine in the control. This relationship is very well understood in the case of the quadratic regulator for linear systems. The generalization to nonlinear systems has been usually considered in the case the value function is smooth. Here we relax this regularity assumption and study to what extent and in which sense solvability of the optimal regulation problem still implies stabilizability.
 
• Gregoire Charlot, Conjugate Locus in the Quasi-Contact Sub-Riemannian Case.

We consider the distributions D in R^(2n+2) of codimension 1 such that if a 1-form w has kernel D then w\wedge(dw)^2n is not 0. It is said to be "quasi-contact". We suppose given a metric g on D. First we construct a normal form of this structure, equivalent to the one built in the contact case by A. Agrachev and J.-P. Gauthier. Second, using this normal form, we study the first conjugate locus of the exponential mapping in the 4-dimensional case. We give the complete list of its generic singularities.
 
• Ludovic Faubourg, Optimal control with state constraints and the space shuttle re-entry problem.

We present a first step in the classification of local optimal syntheses in dimension 2 or 3 for time minimal problem with state constraints under generic assumptions. We use both necessary conditions of the minimum principle and direct time evaluation on normal forms to obtain the small time minimal syntheses. This work is motivated by the optimal control of the atmospheric arc for the re-entry of the space shuttle, where the trajectory has to minimize the total amount of thermal flux and must satisfy state constraints on the thermal flux and the normal acceleration.
 
• Thierry Floquet, Higher order sliding mode stabilization.

The purpose of this work is to give first some generalities about Higher order sliding modes. These generalize the basic sliding mode idea. They are characterized by a discontinuous control acting on the higher order time derivatives of the sliding variable (which is an auxiliary output variable to be kept identically null) instead of influencing its first time derivative, as it happens in standard sliding modes. Preserving the main advantages of the original approach with respect to robustness and easiness of implementation, they remove the chattering effect and guarantee higher accuracy. The problem of higher order sliding modes stabilization for non linear systems, both from a control and an observation point of view, will be more particularly treated. Some illustrative examples (electro-mechanical systems, rigid body, ...) will be given.
 
• Manuel Guerra, Linear-Quadratic Problems with Time-Variant Order of Singularity.

In the autonomous case, any linear-quadratic optimal control problem that is bounded from below can be solved by extending it into a space of generalized controls that contains distributions of order up to the "order of singularity" of the problem. One important obstacle found in the attempt to generalize this approach to the nonautonomous case is the fact that, in problems of this type, the order of singularity may change along the time interval. We discuss some of the phenomena that may occur at the points of the time interval where the order of singularity changes and their bearing on the structure of the spaces of generalized optimal controls that can be constructed for such problems.
 
• Sebastien Jacquet, Abnormals and cut locus.

We consider an analytic sub-riemannian structure on an open subset of $\R^n$. We call $d$ the sub-Riemannian distance defined by this structure and for $x_0$ fixed, $f= x \mapsto d(x_0 , x)$. The smoothness of $f$ at a point is related with the nature of the minimizers which reach that point. In particular the set of points where $f$ is not $C^1$ is the union of the cut locus and set of points reached by abnormal minimizers. In this talk we will focus on the existence of the cut locus in a neighborhood of a strictly abnormal minimizer.
 
• Frédéric Jean, Measures of Transverse Paths in Sub-Riemannian Geometry

We define a class of lengths of paths in a sub-Riemannian manifold. It includes the length of horizontal paths but it also measures the length of transverse paths. It is obtained by integrating an infinitesimal measure which generalizes the norm on the tangent space. This requires the definition and the study of the metric tangent space (in Gromov sense). As an example we compute those measures in the case of contact sub-Riemannian manifolds.
 
• Urszula Ledzewicz, Sufficient conditions for optimality of controls in biomedical systems.

We consider a general class of optimal control problems of Bolza type which arise in mathematical models of biomedical systems describing treatment protocols for diseases like cancer or HIV over a fixed therapy interval. The controls are the dosages of the drugs which are being administered and the objective can either be of $L_1$ (linear) or of $L_2$-type (quadratic) in the control. Depending on the type chosen various candidates for optimal controls arise like singular or bang-bang controls in the $L_1$-case or continuous controls which ``switch" between control values on the boundary and the interior of the control set in the $L_1$-case or continuous controls which ``switch" between control values on the boundary and the interior of the control set in the $L_2$-case. In this talk we present necessary and sufficient conditions for optimality which are based on high-order conditions for optimality and the method of characteristics. For a class of models for cancer chemotherapy the optimality of singular controls is eliminated and bang-bang and continuous controls are analyzed further.
 
• Frederic Mazenc, Global Asymptotic Output Feedback Stabilization of Feedforward Systems.

The problem of the global asymptotic stabilization by dynamic output feedback of systems belonging to a particular family of feedforward systems is solved. The equations considered depend nonlinearly on the unmeasured part of the state and are nonminimum phase. A family of feedbacks containing elements arbitrarily small in norm is proposed. The result can be applied repeatedly. It is illustrated by the cart-pendulum system with the angle and the position as output.
 
• Christophe Prieur, Stabilization of a tank containing a fluid modeled by the shallow water equations.

We consider a tank containing a fluid. The tank is subjected to a one-dimensional horizontal move and the motion of the fluid is described by the shallow water equations which are hyperbolic partial differential equations. By means of a Lyapunov approach, we deduce control laws to stabilize the fluid's state and the tank's trajectory. Although global asymptotic stability is yet to be proved, we numerically simulate the system and observe the stabilization for different control situations.
 
• Ludovic Rifford, Stabilization of Driftless Homogeneous Controllable Systems in Dimension 3.

Our purpose is to study the stabilization problem for systems of the form \dot x=\sum_{i=1}^{m} u_i f_i(x), where the vector fields f_1,...,f_m are smooth on the Euclidean space of dimension 3, homogeneous along a certain dilatation and where the rank at the origin of the Lie algebra generated by the f_i's is 3: Rank(Lie{f_1,\cdots,f_m}) = 3. Using specific results for the stabilization problem in the plane ( from [3]), we prove that there exists a closed homogeneous repulsive set (in the sense which was defined in [2]) and a stabilizing feedback which is smooth outside this set. Finally, we exploit the main theorem of [1] to derive the construction of a nice time-varying periodic stabilizing control law.
[1] P. Morin, J-B. Pomet, C. Samson. Design of homogeneous time-varying stabilizing control laws for driftless controllable systems via oscillatory approximation of Lie brackets in closed loop. SIAM J. Control Optim., 38(1):22--49, 1999.
[2] L.Rifford. Semiconcave control-Lyapunov functions and stabilizing feedbacks. Submitted to SIAM J. Control Optim.
[3] L.Rifford. The stabilization problem in the plane and related results. Submitted to Discrete and Continuous Dynamical Systems.
 
• Eugénio Rocha, Coordinates of first kind and periodic time-variant feedback stabilization.

We present explicit formulas for the formal Chen-Fliess logarithm. This logarithm seen as coordinates of first kind provides an alternative to Sussmann's exponential product expansion of the Chen-Fliess series. An application is made to the design of a time-variant local asymptotically stabilizer for two bodies rolling (without either slipping or twisting) one over another. We also present conditions for a feedback stabilizer for the truncated logarithm of the nilpotent approximation system to be a stabilizer for the original system.
 
• Heinz Schaettler, On optimal control problems with state constraints arising in the design of bipolar transistors.

The active area in bipolar transistors is called base region and the time needed by the electric charges to cross the base region, the base transit time, is arguably one of the most important parameters related to the speed of the transistors. The base transit time can significantly be improved by inducing an electric field through a distribution of dopants in the base region. Under commonly made simplifying assumptions (like low-level injunction) minimizing the base transit time becomes a finite-dimensional optimal control problem with the doping profile as control. However, physical constraints in terms of state-space constraints need to be taken into account. In the electronics literature optimal solutions have only been given numerically. In fact, there are several incorrect published results which try to give an optimal solution analytically, but are actually slower than numerically found solutions. Based on the Maximum Principle with state-space constraints we derive an explicit analytic solution which is slightly better than numerically found solutions and prove its optimality using synthesis type arguments.
 
• Mario Sigalotti, On the local structure of control functions corresponding to time-optimal trajectories in R^3.

We analyse the structure of a control function u(t) corresponding to a time-optimal trajectory for the system q'=f(q)+ug(q) in a three-dimensional manifold nearby a point where the Lie brackets configuration has a singularity of codimension less or equal than one. The control turns out to be the concatenation of some bang and some singular arc. Studying the index of the second variation of the switching times, the number of such arcs is bounded by four.
 
• Delfim Torres, Carath\'{e}odory-Equivalence, Noether Theorems, and Tonelli Full-Regularity in the Calculus of Variations and Optimal Control

We address, in a unified way, the following questions related to the properties of Pontryagin extremals for optimal control problems with unrestricted controls: i) How the transformations, which define the equivalence of two problems, transform the extremals? ii) How to obtain quantities which are conserved along any extremal? iii) How to assure that the set of extremals include the minimizers predicted by the existence theory? These questions are related to: i) the Carath\'{e}odory method which establishes a correspondence between the minimizing curves of equivalent problems; ii) the interplay between the concept of invariance and the theory of optimality conditions in optimal control, which are the concern of the theorems of Noether; iii) regularity conditions for the minimizers and the work pioneered by Tonelli.
 
• Emmanuel Trelat, Asymptotics of accessibility sets near a singular trajectory and consequences.

We describe precisely, under generic conditions, the contact of the accessibility set at time $T$ with an abnormal direction, first for a single-input affine control system with constraint on the control, and then as an application for a sub-Riemannian system of rank 2. As a consequence we obtain in sub-Riemannian geometry a splitting-up of the sphere near an abnormal minimizer $\gamma$ into two sectors, bordered by the first Pontryagin's cone along $\gamma$, called the $L^\infty$-sector and the $L^2$-sector. Moreover we find again necessary and sufficient conditions of optimality of an abnormal trajectory for such systems, for any optimization problem.