Uniform Exponential Practical Stability of Impulsive Perturbed Systems

In this paper, we provide some sufficient conditions for the uniform exponential stability of perturbed impulsive systems by using the Lyapunov second method. Practical stability is also investigated for a class of perturbed impulsive systems.      

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.     

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.     

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.     

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.     

Generic Nonoccurrence of the Lavrentiev Phenomenon for a Class of Optimal Control Problems

In this paper, we study nonoccurrence of the Lavrentiev phenomenon for a large class of nonconvex optimal control problems. We establish that for most problems (in the sense of Baire category) the infimum over the full admissible class of trajectory-control pairs is equal to the infimum over a subclass of trajectory-control pairs with bounded controls.      

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.     

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.     

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.     

Regularity of an Euler-Bernoulli Equation with Neumann Control and Collocated Observation

This paper studies the regularity of an Euler-Bernoulli plate equation on a bounded domain of ℝⁿ, n ≥ 2, with partial Neumann control and collocated observation. It is shown that the system is not only well posed in the sense of D. Salamon but also regular in the sense of G. Weiss. It is also shown that the corresponding feedthrough operator is zero. 2000 Mathematics Subject Classification. 93C20, 34G10, 93D25, 35L20. This work was supported by the National Natural Science Foundation of China and the National Research Foundation.     

Haar Wavelet-Based Approach for Optimal Control of Second-Order Linear Systems in Time Domain

In this paper, a Haar wavelet-based method for optimal control of the second-order linear systems with respect to a quadratic cost function for any length of time is proposed. A Haar wavelet integral operational matrix and properties of the Kronecker product are used in finding the approximate solutions of optimal trajectories and optimal control by solving only two algebraic equations instead of solving the Riccati differential equation. Numerical results of a typical example are presented to illustrate the advantage of the approach.2000 Mathematics Subject Classification. 53A04, 30C15.This research was partially supported under Research Grant 8101004-1-1 provided by university of Tehran and partially by the German Academic Exchange Service (DAAD).     

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.     

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.     

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.     

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).     

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

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).     

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.     

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.