sub-Riemannian Problems Research Articles

Left-invariant optimal control problems on Lie groups: classification and problems integrable by elementary functions

AbstractLeft-invariant optimal control problems on Lie groups are an important class of problems with a large symmetry group. They are theoretically interesting because they can often be investigated in full and general laws can be studied by using these model problems. In particular, problems on nilpotent Lie groups provide a fundamental nilpotent approximation to general problems. Also, left-invariant problems often arise in applications such as classical and quantum mechanics, geometry, robotics, visual perception models, and image processing.The aim of this paper is to present a survey of the main concepts, methods, and results pertaining to left-invariant optimal control problems on Lie groups that can be integrated by elementary functions. The focus is on describing extremal trajectories and their optimality, the cut time and cut locus, and optimal synthesis. Questions concerning the classification of left-invariant sub-Riemannian problems on Lie groups of dimension three and four are also addressed.Bibliography: 91 titles.

Time-optimal control of quantum lambda systems in the KP configuration

We consider the problem of time-optimal control for a three-level quantum system where one level is coupled by the control field to the lowest two, which are not coupled to each other. A bound is assumed on the norm of the control at every time. Such a problem belongs to the class of KP sub-Riemannian problems for which we can perform a symmetry reduction and reduce to a Riemannian problem on the associated quotient space. We prove several properties of such a quotient space in our case, including the fact that it is an example of an almost-Kähler manifold that is not Kähler. We provide the explicit form of the optimal controls for any unitary transformation on the lowest two levels and discuss the geometric and practical implications of this result.

Sub-Riemannian Geodesics in $SU(n)/S(U(n-1) \times U(1))$ and Optimal Control of Three Level Quantum Systems

We study the time optimal control problem for the evolution operator of an $n$ -level quantum system. For the considered models, the control couples all the energy levels to a given one and is assumed to be bounded in Euclidean norm. The resulting problem is a sub-Riemannian $K\hbox{--}P$ problem, (as introduced in articles by U. Boscain and by V. Jurdjevic), whose underlying symmetric space is $SU(n)/S(U(n-1) \times U(1))$ . Following a method introduced by F. Albertini and D. D'Alessandro, we consider the action of $S(U(n-1) \times U(1))$ on $SU(n)$ as a conjugation $X \rightarrow KXK^{-1}$ . This allows us to do a symmetry reduction and consider the problem on a quotient space. We give an explicit description of such a quotient space which has the structure of a stratified space. We prove several properties of sub-Riemannian problems with the given structure. We derive the explicit optimal control for the case of three level quantum systems where the desired operation is on the lowest two energy levels ( $\Lambda$ -systems). We reduce the latter problem to an integer quadratic optimization problem with linear constraints, which we solve completely for a specific set of final data.

Liouville integrability of sub-Riemannian problems on Carnot groups of step 4 or greater

One of the main approaches to investigating sub-Riemannian problems is Mitchell's theorem on nilpotent approximation, which reduces the analysis of a neighbourhood of a regular point to the analysis of the left-invariant sub-Riemannian problem on the corresponding Carnot group. Usually, the in-depth investigation of sub-Riemannian shortest paths is based on integrating the Hamiltonian system of Pontryagin's maximum principle explicitly. We give new formulae for sub-Riemannian geodesics on a Carnot group with growth vector and prove that left-invariant sub-Riemannian problems on free Carnot groups of step 4 or greater are Liouville nonintegrable. Bibliography: 30 titles.

Liouville nonintegrability of sub-Riemannian problems on free Carnot groups of step 4

One of the main approaches to the study of the Carnot–Caratheodory metrics is the Mitchell–Gromov nilpotent approximation theorem, which reduces the consideration of a neighborhood of a regular point to the study of the left-invariant sub-Riemannian problem on the corresponding Carnot group. A detailed analysis of sub-Riemannian extremals is usually based on the explicit integration of the Hamiltonian system of Pontryagin’s maximum principle. In this paper, the Liouville nonintegrability of this system for left-invariant sub-Riemannian problems on free Carnot groups of step 4 and higher is proved.

Superintegrability of left-invariant sub-Riemannian structures on unimodular three-dimensional Lie groups

We consider left-invariant sub-Riemannian problems on three-dimensional unimodular Lie groups. We show that the Hamiltonian system of the Pontryagin maximum principle for such problems is Liouville integrable and even superintegrable (i.e., has four independent integrals, three of which are in involution).

Cut time in sub-riemannian problem on engel group

The left-invariant sub-Riemannian problem on the Engel group is considered. The problem gives the nilpotent approximation to generic rank two sub-Riemannian problems on four-dimensional manifolds. The global optimality of extremal trajectories is studied via geometric control theory. The global diffeomorphic structure of the exponential mapping is described. As a consequence, the cut time is proved to be equal to the first Maxwell time corresponding to discrete symmetries of the exponential mapping.

Solutions sous-analytiques globales de certaines équations d'Hamilton–Jacobi

In the 1980 Crandall and Lions introduced the concept of viscosity solution in order to get existence and/or unicity results for Hamilton–Jacobi equations. In this Note we focus on the Dirichlet problem for Hamilton–Jacobi equations stemming from calculus of variations, and assert that if the data are analytic then the viscosity solution is moreover subanalytic. We extend this result to generalized eikonal equations, stemming from sub-Riemannian geometry problems. To cite this article: E. Trélat, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

Elastic Curves as Solutions of Riemannian and Sub-Riemannian Control Problems

We consider the nonlinear dynamic interpolation problem on Riemannian manifolds and, in particular, on connected and compact Lie groups. Basically we force the dynamic variables of a control system to pass through specific points in the configuration space, while minimizing a certain energy function, by a suitable choice of the controls. The energy function we consider depends on the velocity and acceleration along trajectories. The solution curves can be seen as generalizations of the classical splines in tension for the Euclidean space. The relations with sub-Riemannian optimal control problems are explained.