Nonlinear Control and Singularities Porquerolles, Var (83), France, - - PDF document

Workshop on

Nonlinear Control and Singularities

Porquerolles, Var (83), France, October 24th − 28th, 2010

nlc@lsis.org

Supported by

Abstracts of the expository talks

Andrey Agrachev On the continuity of the optimal cost We give sharp sufficient conditions for local openness of the endpoint maps for control-affine systems in the Lp-topology. Such an openness implies continuity of the optimal costs for natural integral functionals and the ”weak KAM theorem” for the corresponding Hamiltonians. Thomas Chambrion Approximate controllability of the bilinear Schr¨

• dinger equation

In this talk, we investigate the approximate controllability of the infinite dimensional bilinear Schr¨

• dinger equation describing the evolution of a quantum system. A new sufficient condition

for approximate controllability of the wave functions is presented. The proof relies on geometric control theory techniques applied to finite dimensional Lie groups. The result extends to sufficient conditions for partial output tracking of density matrices. Gr´ egoire Charlot Local study of 2-d almost-Riemannian structures I will present two articles, done in collaboration with B. Bonnard, U. Boscain, G. Janin and

• R. Ghezzi, concerning the local study of tangency points in 2-d almost-Riemannian geometry. In

particular, they concern the construction of a normal form at these points where the singularity of the distribution is related to the Martnet case in 3-d sub-Riemannian geometry. Nataliya Cherbakova Optimal control of a dissipative 2-level quantum system I will present our common work with B. Bonnard, D. Sugny and O. Cots on the optimal control

• f the 2 levels dissipative quantum control system whose dynamics is governed by Kossakowski-

Lindblad equation. I will briefly recall the main features of minimal-time problem, and mainly focus

• n our recent results concerning the energy-minimizing case. The problem possess an interesting

dynamical structure, in particular for certain values of physical parameters it is Liouville integrable. Moreover, in contrast with time-minimal case, the energy-minimizing extremals admit an explicit representation in terms of Jacobi elliptic functions. This fact allowed us to find a complete classifi- cation of the extremal solutions in the integrable case, and in the same time to refine the numerical algorithms used to compute optimal solutions the non-integrable case by mean of continuations methods. Francesca Chittaro Quantum control via adiabatic theory In this talk, we expose a new method for controlling a quantum dynamical system driven by a Hamiltonian depending on two controls. If the Hamiltonian has a discrete spectrum that presents conical intersections between the eigenvalues, we can take advantage of the adiabatic theory to induce transfers of population between the energy levels. This strategy permits to approximately control the occupation probability. Alexey Davydov Generic profit singularities of cyclic processes A cyclic process is modeled by a smooth control system on the circle with positive admissible velocities only and a control parameter belonging to a smooth closed manifold or a disjoint union

• f ones with at least two different points.

An admissible motion is defined as an absolutely continuous map x from a time interval to the circle such that at each moment of its differentiability the velocity ˙ x belongs to the convex hull of the admissible velocities of the system. A cycle with a period T > 0 is defined as a periodic admissible motion x, x(t + T) ≡ x(t). In the applications there is usually a continuous profit density f, and 1

the motion along the cycle collects the respective profit. That leads to the famous optimization problem: how to select a cycle providing the maximum of time averaged profit: 1 T

T

f (x(t))dt → max . This problem was touched by various approaches. V. I. Arnold proposed the one based on the singularity theory achievements. He demonstrated that in a typical case the motion along an

• ptimal cycle uses the maximum and minimum velocities when the profit density is less or greater

than a certain constant, respectively [1], [2], [3] and analyzed some profit singularities. We not

• nly have completed the classification in this case but also have proved analogous results in the

presence of a discount rate of the profit. The work was completed by partial financial support of RFBR grants 06-01-00661-a, 10-01- 91004-ASF-a and NSh-8462.2010.1 References [1] V.I. Arnol’d, Averaged optimization and phase transition in control dynamical systems, Funct.

• Anal. and Appl. 36 (2002), 1-11.

[2] A.A Davydov, Generic singularities in Arnold’s model of cyclic processes, Proc. Steklov Inst.

• Math. 250 (2005), 70-84.

[3] A.A Davydov, H. Mena Matos, Generic phase transition and profit singularities in Arnold’s model, Sbornik: Mathematics 198:1, 17-37, 2007. Remco Duits Left Invariant Evolutions on Lie Groups and their applications to image analysis The case H(3) : Phase covariant evolutions on Gabor transforms The case SE(2): Crossing preserving diffusion via adaptive left-invariant diffusions on invertible

• rientation scores

The case SE(3): Left-Invariant diffusions on diffusion weighted MRI-images Revaz Gamkrelidze Invariant form of the Maximum Principle Extremals in classical calculus of variations are always obtained as solutions of second order dif- ferential equations, with left-hand sides (the Euler-Lagrange derivative) having well known tensorial properties. On the contrary, the extremals of the maximum principle are obtained as solutions of a system

• f equations composed of a Hamiltonian system of differential equations with a parameter and a

finite equation – the “maximum condition”, which dynamically eliminates the control parameter from the joint system in the process of motion of the phase point along the trajectories of the Hamiltonian system, providing us with the extremals of the optimal problem. The Hamiltonian vector field with a parameter defined be the Hamiltonian system is a basic ingredient of the maximum principle. It was discovered by Pontryagin before the maximum principle was formulated, I call it the Pontryagin derivative and denote PX, where X is the controlled vector field describing the optimal problem. I shall give an invariant diff-geometric definition of the vector field PX and derive its basic properties, thus obtaining an invariant formulation of the maximum principle. Fr´ ed´ eric Jean Optimal control models of the goal-oriented human locomotion In recent papers it has been suggested that human locomotion may be modeled as an inverse

• ptimal control problem. In this paradigm, the trajectories are assumed to be solutions of an optimal

control problem that has to be determined. We discuss the modeling of both the dynamical system and the cost to be minimized, and we analyze the corresponding optimal synthesis. The main 2

results describe the asymptotic behavior of the optimal trajectories as the target point goes to infinity. Camille Laurent Local controllability of 1D linear and nonlinear Schr¨

• edinger (joint work with

Karine Beauchard) In this talk, we will discuss about the exact local controllability of the Schrdinger equation on an interval. Thanks to a “regularizing effect”, we are able to prove the bilinear controllability with a proof much simpler as the previous one of K. Beauchard. This proof is enough robust to be extended to the nonlinear Sch¨

• dinger and wave equation.

Paolo Mason About Polynomial Instability of Linear Switched Systems In this talk we consider the particular case of linear switched systems with zero Lyapunov

• exponent. It is well known that such systems can be stable as well as unstable, even if they are

never asymptotically stable nor they admits a trajectory blowing up exponentially fast. We will discuss a necessary condition for such systems to be polynomially unstable. Yuri Sachkov Symmetries and Maxwell strata in the plate-ball problem The problem on rolling of a sphere on a plane without slipping or twisting is considered. One should roll the sphere from one contact configuration to another so that the length of the curve traced by the contact point in the plane was the shortest possible. Extremal trajectories in this problem were studied by A.M. Arthur, G.R. Walsh, and V. Jurdjevic. We construct discrete and continuous symmetries of the problem, and study fixed points of their action in preimage and image of the exponential mapping. On the basis of this analysis we

• btain necessary optimality conditions — upper bounds on cut time along extremal trajectories.

Moreover, global structure of the exponential mapping is studied.

Abstracts of the short talks

Andrei Ardentov Extremal trajectories and Maxwell points in sub-Riemannian problem on the Engel group The left-invariant sub-Riemannian problem on the Engel group is considered. This problem is important as a nilpotent approximation of nonholonomic systems in four-dimensional space with two-dimensional control, for instance of the system describing motion of mobile robot with a

• trailer. For the left-invariant sub-Riemannian problem, Pontryagin’s maximum principle was applied.

Abnormal extremals were found. Adjoint subsystem of the normal Hamiltonian system was reduced to equation of pendulum. Parameterization of normal extremals by Jacobian functions was obtained. Discrete symmetries of the exponential mapping were described. The corresponding Maxwell points were computed. Davide Barilari On the Hausdorff volume in sub-Riemannian geometry For a regular sub-Riemannian manifold we study the Radon-Nikodym derivative of the spherical Hausdorff measure with respect to a smooth volume. We prove that this is the volume of the unit ball in the nilpotent approximation and it is always a continuous function. We then prove that up to dimension 4 it is smooth, while starting from dimension 5, in contact and quasi-contact cases, it is C4 but in general not C5. These results answer to a question addressed by Montgomery about the relation between two intrinsic volumes that can be defined in a sub-Riemannian manifold, namely the Popp and the 3

Hausdorff volume. If the nilpotent approximation depends on the point (that may happen starting from dimension 5), then they are not proportional, in general. Alexei Mashtakov Asymptotics of exponential mapping and limit behavior of Maxwell points in the plate-ball problem The problem on rolling of a sphere on a plane without slipping or twisting is considered. First,

• ptimal control problem is studied. One should roll the sphere from one contact configuration to

another so that the length of the curve traced by the contact point in the plane was the shortest

• possible. Asymptotics of extremal trajectories and behavior of Maxwell time for rolling of sphere

along sinusoids of small amplitude are studied; for such trajectories, bounds of cut time are obtained. Moreover, we address the motion planning problem for the rolling sphere, and consider algorithm for reconfiguration based on nilpotent approximation. It achieves reconfiguration using optimal trajectories of canonical nilpotent five-dimensional sub-Riemannian problem. Tatiana Shutkina Existence and uniqueness of optimal cyclic process with discount 4