a necessary condition for feedback stabilization abstract we give a new necessary condition for the existence of a static or dynamic continuous asymptotically stabilizing feedback control for the system x dot f x u we apply our condition to give an example of a system in r 3 which is small time locally controllable satisfies brockett s second and third condition and cannot be locally asymptotically stabilized by employing a static or dynamic continuous feedback law

nilpotent and high order approximations of vector field systems this paper is concerned with the study of structurally stable properties associated with systems described by one or several real analytic vector fields via approximating systems which retain inf

representations of control systems in the fliess algebra and in the algebra of nonlinear power moments in this paper we analyze a representation of affine nonlinear systems in the form of series of nonlinear power moments a difference between such a representation and the representation as the fliess series is caused by a difference in the structure of corresponding algebras it leads to a difference between the conditions of realizability in the form of n dimensional affine systems for these two types of series

a general theorem on local controllability we prove a general sufficient condition for local controllability of a nonlinear system at an equilibrium point earlier results of brunovsky hermes jurdjevic crouch and byrnes sussmann and grossmann are shown to be particular cases of this result also a number of new sufficient conditions are obtained all these results follow from one simple general principle namely that local controllability follows whenever brackets with certain symmetries can be neutralized in a suitable way by writing them as linear combinations of brackets of a lower degree both the class of symmetries and the definition of degree can be chosen to suit the problem

fine level set structure of flat isometric immersions a result by pogorelov asserts that c 1 isometric immersions u of a bounded domain s subset mathbb r 2 into mathbb r 3 whose normal takes values in a set of zero area enjoy the following regularity property the gradient f nabla u is developable in the sense that the nondegenerate level sets of f consist of straight line segments intersecting the boundary of s at both endpoints motivated by applications in nonlinear elasticity we study the level set structure of such f when s is an arbitrary bounded lipschitz domain we show that f can be approximated by uniformly bounded maps with a simplified level set structure we also show that the domain s can be decomposed up to a controlled remainder into finitely many subdomains each of which admits a global line of curvature parametrization

singularities of maximal surfaces we show that the singularities of spacelike maximal surfaces in lorentz minkowski 3 space generically consist of cuspidal edges swallowtails and cuspidal cross caps the same result holds for spacelike mean curvature one surfaces in de sitter 3 space to prove these we shall give a simple criterion for a given singular point on a surface to be a cuspidal cross cap

the generic local time optimal stabilizing controls in dimension 3 this paper studies the control system dot x t x x t y x t u t qquad x p 0 0 u t leqq 1 where x and y are mathcal c infty vector fields on a 3 dimensional manifold mathcal m under generic assumptions on x y the structure of the time optimal stabilizing controls is completely determined in a neighborhood of p 0 the proofs rely on a systematic use of a local asymptotic approximation of x and y by means of vector fields which generate a nilpotent lie algebra

fliess series a generalization of the ree s theorem and an algebraic approach to a homogeneous approximation problem in this paper an algebraic technique originated by m fliess is developed and applied to a homogeneous approximation problem for nonlinear affine control systems the central role is played by a generalization of the r ree s theorem on a connection of lie elements and the shuffle product operation in a free associative algebra

brief paper on state representations of time varying nonlinear systems this work considers a nonlinear time varying system described by a state representation with input u and state x a given set of functions v which is not necessarily the original input u of the system is the new input candidate the main result provides necessary and sufficient conditions for the existence of a local classical state space representation with input v these conditions rely on integrability tests that are based on a derived flag as a byproduct one obtains a sufficient condition of differential flatness of nonlinear systems

local realizations of nonlinear causal operators we give necessary and sufficient conditions for a causal operator to have a local realization of class c k k 1 cdots infty omega of the form dot x f x u quad y h x we show that two minimal local realizations of the same response map are locally difleomorphic

on the support of graph lie algebras we deal with the support of graph lie algebras and we characterize the set of independence alphabets for which the noncommutative result can naturally be extended we also provide an algorithm which decides if an independence alphabet fulfills the preceding condition

comments on the stabilization of the angular velocity of a rigid body abstract this paper discusses the stabilizability by smooth feedback of the angular velocity of a rigid body first we show how to asymptotically stabilize the system with a single control then it is shown that a single control aligned with a principal axis cannot asymptotically stabilize the system finally we present some new results on stabilization by means of scalar feedback control

improper affine maps we study a class of improper affine spheres with singularities called improper affine maps new examples of genus 0 and 1 are described and the classification of complete improper affine maps with at most two embedded ends will be given

feedback stabilization of planar nonlinear systems abstract the stabilization of second order control systems by smooth state feedback is addressed a reduction procedure proposed in 8 is applied in order to reduce the problem to the stabilization of a first order control system by a c 2 feedback function generic sufficient conditions are given for the solution of the last problem and by the reduction procedure of the first problem as well counter examples show that the conditions are not necessary and which kind of pathologies may arise

lie brackets and local controllability a sufficient condition for scalar input systems we develop a general formalism based on exponential lie series which can be used to study control variations we apply the formalism to prove a result conjectured by h hermes that gives a sufficient condition for small time local controllability for single input systems

stabilization of nonlinear systems in the plane abstract it is shown that every small time locally controllable system in the plane can locally be asymptotically stabilized by employing locally holder continuous feedback laws as essentially was conjectured by e sontag an explicit algorithm for the construction of such feedback laws is given

asymptotic stabilization of a class of smooth two dimensional systems this paper studies the asymptotic stabilizability of two dimensional control systems the class under consideration includes c infty systems that satisfy a certain genericity assumption and all real analytic systems necessary and sufficient conditions for feedback stabilization using continuous feedback and a sufficient condition for c 1 feedback stabilization are given this latter condition is given in terms of an inequality involving two indices if the direction of the inequality is changed an obstruction to c infty feedback stabilizability is obtained a subclass of polynomial systems is also studied and given complete necessary and sufficient conditions for global asymptotic stabilization using c 1 feedback

approximation of time optimal control problems via nonlinear power moment problems in this paper we continue the construction of the time optimal control problems classification for nonlinear affine analytic systems in a neighborhood of the equilibrium the moment approach is the basis of this investigation all possible asymptotics for solutions of the nonlinear time optimal control problems are described in terms of special right ideals in the algebra of nonlinear power moments we demonstrate the possible asymptotic behavior of the time optimal control for a certain class of essentially nonlinear three dimensional systems

nilpotent approximations of control systems and distributions a constructive method is given to approximate the vector fields in a nonlinear control system which is linear in the controls by a system of similar form and on the same state space with the describing vector fields of the approximating system generating a lie algebra which has a certain relevant sub algebra nilpotent given a k dimensional distribution on r n say locally defined near zero the method leads to an approximating k distribution which has nilpotent basis agrees with the original distribution at zero and each derived distribution of the approximating distribution and original distribution also agree at zero

homogeneous time varying systems robustness analysis the problem of stability robustness with respect to time varying perturbations of a given frequency spectrum is studied applying homogeneity framework the notion of finite time stability over time intervals of finite length i e short finite time stability is introduced and used for that purpose the results are applied to analyze the super twisting algorithm sta behavior under time varying perturbations some simulation examples illustrate these robustness properties

non causal fliess operators and their shuffle algebra fliess operators as a class of non linear operators have been well studied in several respects they have a well developed realization theory and convenient representations in terms of directed infinite products of exponential lie series their interconnection as subsystems has been studied as has their relationship to rational systems they find applications in such diverse areas as discretization methods for controls systems optimal control neural network analysis and the numerical solution of stochastic differential equations one issue concerning fliess operators however that has received little attention is their possible generalization to the non causal case examples of such operators appear implicitly in the literature addressing hilbert adjoints of causal non linear operators and system inversion for the purpose of output tracking but a general systematic treatment of the subject has not appeared in this paper a non causal extension of a fliess operator is developed with the primary focus being on local convergence continuity the associated shuffle algebra and computing adjoint operators

existence of realizations of nonlinear analytic input output maps we are interested in the existence of realizations of nonlinear input output maps when they are defined for positive times and when the set of control values is finite first we give examples that point out the nature of the problems which may appear in the attempt to find global results in realization theory then we state a necessary and sufficient condition for the existence of realizations and we give a sketch of the proof

solvable approximations to control systems this paper is concerned with extending certain results of rothschild and stein acts math 137 1976 pp 247 320 and goodman lecture notes in mathematics 562 springer verlag new york 1976 in which a finite set of vector fields g 1 cdots g m are lifted to vector fields approximated by generators of a free nilpotent lie algebra we wish to add a vector field f to the set g 1 cdots g m and lift these vector fields to ones on a finite dimensional vector space v approximated by generators f g 1 cdots g m of a solvable lie algebra in which ad j f g i generate a nilpotent but not free ideal this procedure is accomplished in the context of a nonlinear control system with outputs in which f vanishes at the initial state and in such a way that the output functions lift to the state space v to define a system whose input output map is the same as the original system the approximating system is obtained from a suitable realization of a truncation of the volterra series

asymptotically stabilizing feedback controls and the nonlinear regulator problem continuous asymptotically stabilizing feedback controls are constructed for two dimensional and certain three dimensional small time locally controllable affine systems the lie products which de

orthogonal projection onto the free lie algebra abstract as an answer to a question of garsia we give a universal formula in a normal form for the orthogonal projection of free associative algebra onto free lie algebra

the shuffle algebra and its derivations abstract the shuffle algebra on a partially commutative alphabet and its derivations are studied the derivation of the shuffle algebra is characterized in terms of admissible mappings the module of all derivations of the shuffle algebra is proved to be isomorphic to the free partially commutative power series lie algebra with the coefficient ring extended the orthogonality between the lie elements and the proper shuffles is established

the hamilton jacobi bellman equation for time optimal control in this paper several assertions concerning viscosity solutions of the hamilton jacobi bellman equation for the optimal control problem of steering a system to zero in minimal time are proved first two rather general uniqueness theorems are established asserting that any positive viscosity solution of the hjb equation must in fact agree with the minimal time function near zero if also a boundary condition introduced by bardi siam j control optim 27 1988 pp 776 785 is satisfied then the agreement is global additionally the holder continuity of any subsolution of the hjb equation is proved in the case where the related dynamics satisfy a hormander type hypothesis this last assertion amounts to a half derivative analogue of a theorem of crandall and lions traps amer math soc 277 1 983 pp 1 42 concerning lipschitz viscosity solutions

homogeneous state feedback stabilization of homogenous systems we show that for any asymptotically controllable homogeneous system in euclidean space not necessarily lipschitz at the origin there exists a homogeneous control lyapunov function and a homogeneous possibly discontinuous state feedback law stabilizing the corresponding sampled closed loop system if the system satisfies the usual local lipschitz condition on the whole space we obtain semiglobal stability of the sampled closed loop system for each sufficiently small fixed sampling rate if the system satisfies a global lipschitz condition we obtain global exponential stability for each sufficiently small fixed sampling rate the control lyapunov function and the feedback are based on the lyapunov exponents of a suitable auxiliary system and admit a numerical approximation

moment approach to nonlinear time optimality the time optimal control problem to a stationary point for an affine analytic nonlinear system is considered as a certain nonlinear power markov moment min problem conditions for a solution to be asymptotically close to a solution of a linear time optimal control problem are obtained in terms of lie brackets of the vector fields appearing in the system

entire solutions of the degenerate monge ampere equation with a finite number of singularities we determine the global behavior of every c2 solution to the two dimensional degenerate monge ampere equation uxxuyy uxy2 0 over the finitely punctured plane with this we classify every solution in the once or twice punctured plane moreover when we have more than two singularities if the solution u is not linear in a half strip we obtain that the singularities are placed at the vertices of a convex polyhedron p and the graph of u is made by pieces of cones outside of p which are suitably glued along the sides of the polyhedron finally if we look for analytic solutions then there is at most one singularity and the graph of u is either a cylinder no singularity or a cone one singularity

series expansions for the evolution of mechanical control systems this paper presents a series expansion that describes the evolution of a mechanical system starting at rest and subject to a time varying external force mechanical systems are presented as second order systems on a configuration manifold via the notion of affine connections the series expansion is derived by exploiting the homogeneity property of mechanical systems and the variations of constant formula a convergence analysis is obtained using some analytic functions and combinatorial analysis results this expansion provides a rigorous means of analyzing locomotion gaits in robotics and lays the foundation for the design of motion control algorithms for a large class of underactuated mechanical systems

brief paper stability analysis of planar systems with nilpotent non zero linear part a simple necessary and sufficient condition for asymptotic stability of the origin is derived for a large class of planar nonlinear systems that cannot be studied by means of their linearization which is nilpotent and not zero the condition is based on the use of darboux polynomials to derive a lyapunov function for the first approximation of the system with respect to a vector function representing a dilation results by andreev 1958 on the qualitative behaviour of equilibrium points and the belitskii normal form are used in the proofs the proposed condition is applied to the asymptotic stabilization of the origin for a class of planar systems whose linearization cannot be stabilized

graded approximations and controllability along a trajectory graded approximations of an affine control system are defined and their properties are investigated in particular it is proved that the local controllability along a reference trajectory of an approximating system implies the local controllability along the corresponding reference trajectory of the original system using graded approximations sufficient conditions of local controllability along a reference trajectory that generalize some known results are given

fliess operators on lp spaces convergence and continuity fliess operators as input output mappings are particularly useful in a number of fundamental problems concerning nonlinear realization theory in the classical analysis of these operators certain growth conditions on the coe3cients in their series representations insure uniform and absolute convergence provided every input is uniformly bounded by some 4xed upperbound in some emerging applications however it is more natural to consider other classes of inputs in this paper lp function spaces are considered in particular it is shown that the classic growth conditions also provide su3cient conditions for convergence and continuity when the admissible inputs are from a ball in lp t0 t 0 t where t is bounded and p 1 in addition stronger global growth conditions are given that apply even for the case where t is unbounded when the coe3cients of a fliess operator have a state space representation it is shown that the state space model will always locally realize the correspondinginput output map on lp t0 t 0 t for su3ciently small t 0 if certain well posedness conditions are satis4ed then the state space model will globally realized the input output mapping for unbounded t when the coe3cients satisfy the global growth condition c 2002 elsevier science b v all rights reserved

control systems which generate decomposable lie algebras let x y be real analytic vector fields on an n manifold m ad x y x y denote the lie product inductively ad k 1 x y x ad k x y and 1 ad v x y v 0 the attainable set at time t denoted download full size image t for the control system 1 x x x u t y x x 0 p u t 1 is contained in the integral manifold through p of the lie algebra l 1 generated by 1 l 1 is said to be decomposable at p if essentially when dim l 1 p n the solution of 1 can be written as exp tx exp f 1 t u v 1 exp f n t u v n p with v 1 v n l 1 this is true for example at any p if l 1 is nilpotent the f i t u are shown to satisfy a system on n of the form 2 x i u t g i t x 1 x i 1 x i 0 0 i 1 n where if l 1 is nilpotent each g i is a polynomial in x 1 x i 1 in this case reachability properties of 2 for controls being countably additive measures impluses reduce to the study of algebraic equations homogeneity allows these to be considered on projective space n giving local results by bezout s theorem similar results hold for controls in the unit ball of krener s results on approximating systems of the form 1 by systems which generate nilpotent lie algebras can then be used to remove the condition that l 1 be nilpotent the method applies to show that if m denotes the set of all products of k elements of 1 with k m and dim span m 1 p dim span m for all odd m then exp tx p int download full size image t for all t 0

global properties of the triangular systems in the singular case we introduce a new class of the triangular multi input and multi output control systems of o d e which are not feedback linearizable and investigate its global behavior the triangular form introduced is a generalization of the classes of triangular systems considered before for our class we solve the problem of global robust controllability combining our main result with that of f h clarke yu s ledyaev e d sontag a i subbotin asymptotic controllability implies feedback stabilization ieee trans automat control 42 1997 1394 1407 we obtain a corollary on the global discontinuous sampled stabilization an example showing that global smooth stabilization can be irrelevant to the singular case is considered to prove our main result we apply a certain back stepping algorithm and combine the technique proposed in v i korobov s s pavlichkov w h schmidt global robust controllability of the triangular integro differential volterra systems j math anal appl 309 2005 743 760 with solving a specific problem of global practical stabilization by means of a discontinuous time varying feedback law