Invariant Cones and Convex Sets For Bilinear Control Systems and Parabolic Type of Semigroups

This paper studies invariant cones for bilinear control systems and relates them to the controllability of the system. The full picture is provided by the parabolic type of a semigroup.     

Local Controllability for Linear Control Systems on Lie Groups

Abtract  In this work, we study controllability properties of linear control systems on Lie groups. Using Lie theory of semigroups, we obtain local controllability results for this type of systems. In addition, properties for the flow of the drift vector field X and for the reachable set of the system are presented. Finally, an example on the Heisenberg Lie group is considered, and its properties are proved using the theory developed. 2000 Mathematics Subject Classification. Primary 93B05; Secondary 93C25, 93B27. The authors want to thank the Fachbereich Mathematik of the Technische Universit?t Darmstadt for warm hospitality during the production of this work. The first author is pleased to thank Professor Guillermo Ferreyra and the department of mathematics at LSU for their support.     

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.      

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’.      

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.     

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.     

Geometric structure-preserving optimal control of a rigid body

In this paper, we study a discrete variational optimal control problem for a rigid body. The cost to be minimized is the external torque applied to move the rigid body from an initial condition to a pre-specified terminal condition. Instead of discretizing the equations of motion, we use the discrete equations obtained from the discrete Lagrange–d’Alembert principle, a process that better approximates the equations of motion. Within the discrete-time setting, these two approaches are not equivalent in general. The kinematics are discretized using a natural Lie-algebraic formulation that guarantees that the flow remains on the Lie group SO(3) and its algebra . We use the Lagrange method for constrained problems in the calculus of variations to derive the discrete-time necessary conditions. We give a numerical example for a three-dimensional rigid body maneuver.      

On Complex Codimension-One Foliations Transverse Fibrations

We study codimension-one holomorphic foliations which are transverse to fibrations and, therefore, are conjugate to suspensions of groups of M?bius transformations. For the general case, we assume that there exists an invariant transverse measure. The existence of such an invariant transverse measure is proved for the case where there is a leaf with the subexponential growth or the basis has an amenable fundamental group. This result applies, e.g., for the study of amenable Lie group actions on complex projective spaces. 2000 Mathematics Subject Classification. Primary 37F75; Secondary 57R30, 53C12.     

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.     

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.     

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.     

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).     

Regularity for semilinear nonlocal problems with boundary input maps

In this paper, we establish the well-posedness of solutions and regularity properties for the boundary control systems with nonlocal conditions with the aid of an intermediate property and the contraction mapping principle. A numerical example illustrating our equivalence results is given.      

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.      

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.     

The dynamic interpolation problem: On Riemannian manifolds,Lie groups,and symmetric spaces

We consider the dynamic interpolation problem for nonlinear control systems modeled by second-order differential equations whose configuration space is a Riemannian manifoldM. In this problem we are given an ordered set of points inM and would like to generate a trajectory of the system through the application of suitable control functions, so that the resulting trajectory in configuration space interpolates the given set of points. We also impose smoothness constraints on the trajectory and typically ask that the trajectory be also optimal with respect to some physically interesting cost function. Here we are interested in the situation where the trajectory is twice continuously differentiable and the Lagrangian in the optimization problem is given by the norm squared acceleration along the trajectory. The special cases whereM is a connected and compact Lie group or a homogeneous symmetric space are studied in more detail.Work supported in part by NSF grant No. 89-14643 and NATO project CRG 910926.     

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.      

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.     

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