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

Workshop on Nonlinear Control and Singularities Porquerolles, Var (83), France, October 24 th − 28 th , 2010 nlc@lsis.org Supported by

Abstracts of the expository talks On the continuity of the optimal cost Andrey Agrachev We give sharp sufficient conditions for local openness of the endpoint maps for control-affine systems in the L p -topology. Such an openness implies continuity of the optimal costs for natural integral functionals and the ”weak KAM theorem” for the corresponding Hamiltonians. Approximate controllability of the bilinear Schr¨ odinger equation Thomas Chambrion In this talk, we investigate the approximate controllability of the infinite dimensional bilinear Schr¨ odinger 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. Local study of 2-d almost-Riemannian structures Gr´ egoire Charlot 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 of 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 on 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 of 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: � T 1 f ( x ( t )) dt → max . T 0 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 optimal 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 only 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 orientation 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 of 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 P X , where X is the controlled vector field describing the optimal problem. I shall give an invariant diff-geometric definition of the vector field P X 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 optimal 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