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.     

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.     

Local Fields of Extremals for Optimal Control Problems with State Constraints of Relative Degree 1

A local embedding of a boundary arc into a field of extremals is constructed for single-input optimal control problems with state space constraints given by control-invariant manifolds of relative degree 1. The strong local optimality of the reference trajectory is proved. 2000 Mathematics Subject Classification. 49N35, 49K15, 49L20. Research was partially supported by the National Science Foundation under grants Nos. DMS-0305965 and DMS-0405848.     

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.     

The Step-2 Nilpotent (n, n(n + 1)/2) Sub-Riemannian Geometry

We study the sub-Riemannian structure determined by a left-invariant distribution of rank n on a step-2 simply-connected nilpotent Lie group G of dimension n(n + 1)/2. We describe a transitive group action that leaves invariant the sub-Riemannian structure. By using geometric optimal control theory techniques, we derive necessary conditions for the length-minimality of the sub-Riemannian geodesics. We perform an integration process for the associated Hamiltonian system that yields explicit expressions for the extremal curves and the corresponding sub-Riemannian geodesics, the obtained formulas allow the complete parametrization of the exponential mapping in terms of algebraic invariants of the problem. 2000 Mathematics Subject Classification. 49K30,49Q99,93B29.     

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.     

Conic Distributions and Accessible Sets

Motivated by nonlinear control theory, we introduce the notion of conic distributions on a smooth manifold. We study topological and smoothness aspects of the set of accessible points associated with a conic distribution. We introduce the notion of abnormal paths and study their relation to boundary points of the accessible set. In particular, we provide sufficient conditions for the accessible set to be a maximal integral of the smallest integrable vector distribution containing the conic distribution. Under rather strong conditions, we are able to prove that the accessible set has the structure of a ‘manifold with corners’.      

Hypergeometric Functions and Parabolic Partial Differential Equations

We investigate the summability of formal power series solutions of parabolic linear partial differential equations with exponential coefficients.2000 Mathematics Subject Classification. 35C10, 35C20.This research was supported by a Marie Curie Fellowship of the European Community program Improving Human Research Potential under contract number HPMF-CT-2002-01818.     

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).     

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.     

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.     

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.      

Practical Stability of Nonlinear Time-Varying Cascade Systems

In this paper, we investigate the practical stability problem of nonlinear time-varying cascade systems. We give some sufficient conditions that guarantee practical global uniform asymptotic stability and practical global uniform exponential stability of such dynamical systems.      

Dynamic Inversion for Nonaffine-in-Control Systems via Time-Scale Separation. Part I

This paper presents a new method for approximate dynamic inversion of nonaffine-in-control systems via time-scale separation. The control signal is sought as a solution of the “fast” dynamics and is shown to asymptotically stabilize the original nonaffine system. Sufficient conditions are formulated, which satisfy the assumptions of the Tikhonov theorem in the theory of singular perturbations. Several examples illustrate the theoretical results.      

Universal Covering of Driftless Control Systems

Let M be a finite-dimensional manifold and Σ be a driftless control system on M of full rank. We prove that for a given initial state x ∈ M, the covering space Γ(Σ, x) for a monotonic homotopy of trajectories of Σ which is recently constructed in [1] coincides with the simply connected universal covering manifold of M and that the terminal projection ε _x : Γ(Σ, x) → M given by ε _x ([α]) = α(1) is a covering mapping. Research supported by CNPq, Brasilian Council for Scientific and Technological Development, Grant no. 200809/2005-9.     

Stabilization of Generic Trees of Strings

We study the energy decay of a tree-shaped network of vibrating elastic strings when the pointwise feedback acts in the root of the tree. We show that the strings are not exponentially stable in the energy space. Moreover, we give explicit polynomial decay estimates valid for regular initial data.2000 Mathematics Subject Classification. 35B37, 93B07, 93D15.     

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.     

Global Stabilization for Systems Evolving on Manifolds

We show that any globally asymptotically controllable system on any smooth manifold can be globally stabilized by a state feedback. Since we allow discontinuous feedbacks, we interpret the solutions of our systems in the “sample and hold” sense introduced by Clarke, Ledyaev, Sontag, and Subbotin (CLSS). We generalize their theorem which is the special case of our result for systems on Euclidean space. We apply our result to the input-to-state stabilization of systems on manifolds with respect to actuator errors, under small observation noise. 2000 Mathematics Subject Classification. 93B05.     

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).