On the One-Step-Bracket-Generating Motion Planning Problem

We consider the general motion planning problem for a sub-Riemannian metric with one-step bracket-generating distribution. Our results generalize earlier results in the corank-one case. Mostly, we completely solve the problem in generic situation for corank smaller or equal to 3. Our results are constructive: we explicitly construct the asymptotically optimal solutions.2000 Mathematics Subject Classification. 53C17, 53C99, 93B29.The second author was partially supported by the Russian Foundation for Basic Research (project No. 0201099) and UR 03099.     

On the Motion Planning Problem, Complexity, Entropy, and Nonholonomic Interpolation

We consider the sub-Riemannian motion planning problem defined by a sub-Riemannian metric (the robot and the cost to minimize) and a non-admissible curve to be ε-approximated in the sub-Riemannian sense by a trajectory of the robot. Several notions characterize the ε-optimality of the approximation: the “metric complexity” MC and the “entropy” E (Kolmogorov-Jean). In this paper, we extend our previous results. 1. For generic one-step bracketgenerating problems, when the corank is at most 3, the entropy is related to the complexity by E = 2πMC. 2. We compute the entropy in the special 2-step bracket-generating case, modelling the car plus a single trailer. The ε-minimizing trajectories (solutions of the “ε-nonholonomic interpolation problem”), in certain normal coordinates, are given by Euler's periodic inflexional elastica. 3. Finally, we show that the formula for entropy which is valid up to corank 3 changes in a wild case of corank 6: it has to be multiplied by a factor which is at most 3/2. 2000 Mathematics Subject Classification. 53C17, 49J15, 34H05. The second author is supported by grants RFBR 050100458 and UR 0401128.     

On the Existence of Codimension-One Nonorientable Expanding Attractors

We prove that a structurally stable diffeomorphism of any closed (2m + 1)-manifold, m ≥ 1, has no codimension one nonorientable expanding attractors. However, given any d ≥ 3, there exists an 9-stable A-diffeomorphism of a closed d-manifold with a codimension one nonorientable expanding attractor. 2000 Mathematics Subject Classification. Primary 37D20; Secondary 37C70, 37C15. This research was partially supported by the Russian Foundation for Basic Research (project Nos. 02-01-00098, 05-01-00501.     

Asymptotics of Solutions of Periodic Problem for Generalized Nonlinear Hamiltonian System Singularly Perturbed by a Matrix

The paper deals with the periodic problem for a generalized nonlinear Hamiltonian system, which appears from the control optimality condition for the periodic problem of the minimization of the functional on trajectories of the equation with the operator A + ɛB standing before the derivative, where the operator A is singular and A + ɛB is invertible for sufficiently small ɛ > 0. Under some conditions, the asymptotic solution of such problem has been constructed in the form of the series with respect to nonnegative integer powers of ɛ. The solvability of the considered problem has also been established for sufficiently small ɛ > 0. This work was supported by the Russian Fundamental Research Foundation (project No. 06–01–00296).     

Conjugate Points in the Euler Elastic Problem

For the classical Euler elastic problem, conjugate points are described. Inflexional elasticas admit the first conjugate point between the first and third inflexion points. All other elasticas do not have conjugate points. As a result, the problem of stability of Euler elasticas is solved. This work was supported by the Russian Foundation for Basic Research, project No. 06-01-00330.     

Invariants of binary differential equations

In this paper, we study binary differential equations a(x, y)dy ² + 2b(x, y) dx dy + c(x, y)dx ² = 0, where a, b, and c are real analytic functions. Following the geometric approach of Bruce and Tari in their work on multiplicity of implicit differential equations, we introduce a definition of the index for this class of equations that coincides with the classical Hopf’s definition for positive binary differential equations. Our results also apply to implicit differential equations F(x, y, p) = 0, where F is an analytic function, p = dy/dx, F _p  = 0, and F _pp  ≠ 0 at the singular point. For these equations, we relate the index of the equation at the singular point with the index of the gradient of F and index of the 1-form ω = dy − pdx defined on the singular surface F = 0. This work was partially supported by Fapesp grant No. 02/09157-5.     

Reachable Sets for the Heisenberg Sub-Lorentzian Structure On ℝ3. An Estimate for the Distance Function

In this paper we compute reachable sets from a point for the Heisenberg sub-Lorentzian metric on ℝ³ and give an estimate (from below) for the distance function. 2000 Mathematics Subject Classification. 53C50.     

Optimal Attitude Control of a Rigid Body Using Geometrically Exact Computations on SO(3)

An efficient and accurate computational approach is proposed for a nonconvex optimal attitude control for a rigid body. The problem is formulated directly as a discrete time optimization problem using a Lie group variational integrator. Discrete time necessary conditions for optimality are derived, and an efficient computational approach is proposed to solve the resulting two-point boundary-value problem. This formulation wherein the optimal control problem is solved based on discretization of the attitude dynamics and derivation of discrete time necessary conditions, rather than development and discretization of continuous time necessary conditions, is shown to have significant advantages. In particular, the use of geometrically exact computations on SO(3) guarantees that this optimal control approach has excellent convergence properties even for highly nonlinear large angle attitude maneuvers. The first and second authors have been partially supported by NSF (project Nos. DMS-0504747 and DMS-0726263). The first and third authors have been partially supported by NSF (project Nos. ECS-0244977 and CMS-0555797.     

Nonhomogeneos Boundary-Value Problem for Semilinear Hyperbolic Equation. Stability

We discuss the solvability of the nonhomogeneous boundary-value problem for the semilinear equation of the vibrating string x _tt (t, y) − Δx(t, y) + f(t, y, x(t, y)) = 0 in a bounded domain and a certain type of superlinear nonlinearity. To this end, we deduce new dual variational method. Next, we discuss the stability of solutions with respect to the boundary control and initial conditions.      

Rigid Carnot Algebras: A Classification

A Carnot algebra is a graded nilpotent Lie algebra L = L₁ ⊕ … ⊕ L_r generated by L₁. The bidimension of the Carnot algebra L is the pair (dim L₁, dim L). A Carnot algebra is said to be rigid if it is isomorphic to any of its small perturbations in the space of Carnot algebras of the prescribed bidimension. In this paper, we give a complete classification of rigid Carnot algebras. In addition to free nilpotent Lie algebras, there are two infinite series and 29 exceptional rigid algebras of 16 exceptional bidimensions. 2000 Mathematics Subject Classification. 58A30,58K50.     

Attraction Region of Planar Linear Systems with One Unstable Pole and Saturated Feedback

The bifurcation of the attraction region for planar systems with one stable and one unstable pole under a saturated linear state feedback is considered. The attraction region can have either an unbounded hyperbolic shape or be bounded by a limit cycle. An analytical condition, under which either of these boundary shapes occurs, is given with a formal proof. This condition is based on the relationship between the stable and unstable manifolds associated with secondary saddle equilibrium points, whose presence is caused by the saturation on the input. 2000 Mathematics Subject Classification. 34C23, 93C10, 34D99.     

Local Regularity of Optimal Trajectories for Control Problems with General Boundary Conditions

Let f and g be two smooth vector fields on a manifold M. Given a submanifold S of M, we study the local structure of time-optimal trajectories for the single-input control-affine system ̇q = f(q) + ug(q) with the initial condition q(0) S. When the codimension s of S in M is small (s 4) and the system has a small codimension singularity at a point q₀ S, we prove that all time-optimal trajectories contained in a sufficiently small neighborhood of q₀ are finite concatenations of bang and singular arcs. The proof is based on an extension of the index theory to the case of general boundary conditions.2000 Mathematics Subject Classification. 49K15, 49K30.     

On the existence of observable linear systems on Lie groups

A vector field on a Lie group is linear if its flow is a oneparameter group of automorphisms. A linear system is obtained by adding left invariant controlled vector fields. The observability of such a system, whenever the output function is a Lie group morphism, was studied by Ayala and Hacibekiroglu. Within this framework, it is shown that no observable systems exist on semisimple groups, and necessary conditions for the existence of such a system on a general Lie group are given. The case where the output morphism is replaced by the projection on a homogeneous space is briefly discussed.      

Oscillation of Linear Ordinary Differential Equations: On a Theorem of A. Grigoriev

We give a simplified proof and an improvement of a recent theorem of A. Grigoriev, placing an upper bound for the number of roots of linear combinations of solutions of systems of linear equations with polynomial or rational coefficients. To Yulij Sergeevich Ilyashenko on his 60th birthday. 2000 Mathematics Subject Classification. 34C08, 34C10, 34M10. Gershon Kekst professorial chair in Mathematics.     

On The Stabilization of Homogeneous Perturbed Systems

In this paper, using some results on manifolds, we establish some conditions for stabilization of single-input homogeneous by dilation systems.      

Generic Singularities of Relay Systems

We discuss a class of nonsmooth vector fields on which are called relay systems. The main result provides a systematic classification of typical singularities which arise in generic one-parameter families of such systems. This is done by means of the notion of the mild equivalence between two relay systems. The theory is used to give a geometric characterization of the phase space in neighborhoods of the singularities, and tools of the singularity theory play a key role.      

Multiple positive solutions of singular dirichlet second-order boundary-value problems with derivative dependence

The existence of multiple positive solutions for the singular Dirichlet boundary-value problem

Nonoscillating Projections for Trajectories of Vector Fields

The properties of oscillation for trajectories of three-dimensional vector fields are “visible” in the sense that they can be detected by means of plane projections. We also recover a three-dimensional version of the classical dichotomy “nonoscillation” versus “spiralling” for plane vector fields.      

Navier-Stokes Equation on the Rectangle: Controllability by Means of Low Mode Forcing

We study controllability issues for the Navier-Stokes equation on a two-dimensional rectangle under so-called Lions boundary conditions. The Navier-Stokes equation is controlled by forcing applied to a small number of harmonic modes. Methods of Geometric/Lie Algebraic Control Theory are used to prove controllability by means of low mode forcing of finite-dimensional Galerkin approximations of this system. Proving the continuity of the “control ↦ solution” mapping in the so-called relaxation metric we use it to prove both solid controllability on the observed component and L ²-approximate controllability of the Navier-Stokes equation (full system) by low mode forcing. 2000 Mathematics Subject Classification. 35Q30, 93C20, 93B05, 93B29. Supported by FCT (Portuguese Foundation for Science and Technology).     

The Poincaré Center Problem

We present two examples of real planar polynomial vector fields with an orbitally linearizable saddle point such that they are neither rationally reversible nor Liouvillian integrable. We show that vector fields from one of these examples form an isolated component of the so-called integrable saddle variety. Next, we discuss the problem of partial duality between real centers and real integrable saddles and the problem of continuous moduli for the center variety. The first author is supported by the Polish MNiSzW Grant No. 1 P03A 015 29, and the second by a MCYT/FEDER grant number MTM2005–06098–C02–01 and by a CIRIT grant number 2005SGR 00550.