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.     

Constructive Stabilization of Quadratic-Input Nonlinear Systems with Bounded Controls

In this paper, the stabilization of quadratic-input nonlinear systems with bounded controls is considered. According to the type of quadratic-input forms, two cases, namely, positive definite and positive semi-definite, are considered. For the case of positive definiteness, a universal formula for bounded stabilizers is given via a known Lyapunov control function. For the case of positive semidefiniteness, a constructive parametrization of bounded stabilizers is proposed under the assumption that there exists a known Lyapunov control function with respect to a smaller control set than the admissible control set. This work is supported partly by NNSF of China under Grants 60221301 and 60334040, partly by Sida-VR Swedish Research Links Grant 348-2002-6936.     

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.      

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.     

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.     

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.      

Controllability for Semilinear Retarded Control Systems in Hilbert Spaces

This paper deals with well-posedness and L ²-regularity properties for a class of semilinear retarded functional differential equations. A relation between the reachable set of a semilinear system and that of the corresponding linear system is proved. We also show that the Lipschitz continuity and the uniform boundedness of the nonlinear term can be considerably weakened. Finally, a simple example is given, to which our main result can be applied. This work was supported by the Brain Korea 21 Project in 2006.     

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.      

Robust Gradient-Like Property and Controller Design for Uncertain Pendulum-Like Systems

In this paper the gradient-like behavior for the pendulum-like systems with additive, multiplicative, and H _∞ uncertainties is analyzed. Then some conditions ensuring existence of controllers for uncertain pendulum-like systems are given. The corresponding controller design problem is transformed into H _∞ control problem. Two examples are given to illustrate the effectiveness of the method. 2000 Mathematics Subject Classification. 93C10. This work was partially supported by National Science Foundation of China under Grant 60334030, 10472001, 10272001, 60204007.     

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.     

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).     

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).     

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.      

A New Stabilizing Control Law with Respect to a Control Lyapunov Function and Construction of Control Lyapunov Function for Particular Nonaffine Nonlinear Systems

In this paper, a new stabilizing control law with respect to a control Lyapunov function (CLF) is presented. This control law is similar to the pointwise min-norm control law. This control law is designed to maximize the angle between the gradient of the control Lyapunov function and the time derivative of the state vector at the state trajectory, which is defined in what follows as the “pointwise maximum angle control law.” A comparison with the pointwise minnorm control law is provided. A criterion of the stability performance of control laws that are designed with respect to a CLF is presented. Also, by proposing the concept of the “eigen-angle” for real square nonsingular matrices, the stabilization of some nonaffine nonlinear systems, and the construction of a CLF for such systems are reduced to the construction of CLFs for affine nonlinear (linear) systems. Finally, simulation results are provided to show the effectiveness of the proposed methodologies.     

Stabilization of Linear Uncertain Delay Systems with Antisymmetric Stepwise Configurations

This paper investigates the problem of designing a linear memoryless state feedback control to stabilize a class of linear uncertain systems with state delays. Each uncertain parameter and each delay under consideration might vary with time in an arbitrarily large range. In such a situation, the locations of uncertain elements in the system matrices play an important role. Wei introduced the concept of antisymmetric stepwise configuration (ASC) and proved that it is a necessary and sufficient condition for linear uncertain systems to be quadratically stabilizable using linear state feedback control to have this configuration. However, his method is inapplicable to systems that contain delays in the state variables. On the other hand, Amemiya developed conditions for the stabilization of linear uncertain systems with state delays using linear memoryless state feedback control. This paper presents development of the conditions of this problem that have been obtained to date. Fundamentally, it is proved that having an ASC is also a sufficient condition for the stabilization of linear uncertain delay systems. For systems satisfying the stabilizability conditions, a simple control design procedure is also provided and illustrated by an example.      

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.     

On Topological Equivalence of Linear Lows with Applications to Bilinear Control Systems

This paper classifies continuous linear flows using concepts and techniques from topological dynamics. Specifically, the concepts of equivalence and conjugacy are adapted to flows on vector bundles, and the Lyapunov decomposition is characterized using the induced flows on the Grassmann and the flag bundles. These results are then applied to bilinear control systems, for which their behavior in , on the projective space , and on the Grassmannians is characterized. This research was partially supported by Proyecto FONDECYT No. 1060981 and Proyecto FONDECYT de Incentivo a la Cooperación Internacional No. 7020439.     

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.     

Adaptive Dynamic Inversion for Nonaffine-in-Control Uncertain Systems via Time-Scale Separation. Part II

This paper presents an adaptive dynamic inversion method for uncertain nonaffine-in-control dynamical systems. Online approximation of the adaptive dynamic inversion controller is performed using time-scale separation. The resulting control signal is sought as a solution of a “fast” dynamical equation, which inverts a series-parallel model, whose state is shown to track the state of the original nonaffine-in-control system. The design methodology is constructive, and numerically verifiable sufficient conditions on the system parameters are given. A simulation example, motivated by aerospace applications, illustrates the theoretical results.      

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).