affine and bilinear systems on lie groups abstract in this paper we study affine and bilinear systems on lie groups we show that there is an intrinsic connection between the solutions of both systems such relation allows us to obtain some preliminary controllability results of affine systems on compact and solvable lie groups we also show that the controllability property of bilinear systems is very restricted and may only be achieved if the state space g is an euclidean space

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

on the positive output controllability of linear time invariant systems this paper considers the output controllability of autonomous linear control systems that are subject to non negative input constraints based on the evaluation of the geometric properties of the system necessary and sufficient conditions are proposed for the positive output controllability of continuous linear time invariant systems to aid in the practical evaluation of positive output controllability additional sufficient conditions are derived for which efficient numerical techniques exist these conditions are evaluated over a set of numerical examples which support the theoretical results

controllability of linear systems on lie groups with finite semisimple center this paper studies controllability for a given linear system sigma on a connected lie group g by taking into consideration the eigenvalues of an associated derivation mathcal d if we assume that the lie group g has finite center and for some tau 0 the identity element of g is an interior point of its reachable set at time tau then the system is controllable if mathcal d has only eigenvalues with zero real part

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

controllability properties of affine systems this paper deals with transitivity controllability of affine families of vector fields on a finite dimensional vector space v in particular we focus on affine families whose corresponding families of linear fields are transitive on v 0 and which in addition have no fixed points in v we show that such families are necessarily transitive on v and we also show that they remain transitive under small perturbations in general however affine families need not remain transitive under small perturbations for example small affine perturbations of transitive linear systems are not necessarily transitive since any affine system mathcal f naturally defines a system mathcal f r of right invariant vector fields on the semi direct product of v with gl v we also investigate transitivity properties mathcal f r our result is that if mathcal f is an affine family satisfying the preceding conditions then mathcal f r generates the full lie algebra on the semi direct product

asymptotically stabilizing feedback controls abstract the problem considered is the existence and construction of an asymptotically stabilizing feedback control for a rest solution of an n dimensional single input affine control system the method is to construct high order homogeneous approximations of the vector fields defining the control system which then describe an approximating control system such that an asymptotically stabilizing control of the latter system is a local asymptotically stabilizing control of the original system unlike linear approximations there are many possible nonlinear homogeneous approximations these are constructed from lie algebra filtrations obtained by assigning various weights to the defining vector fields one seeks a weighting hence an approximating system which satisfies the various known necessary conditions for the existence of an asymptotically stabilizing feedback control and for which such a control can be constructed as the solution to a homogeneous hamilton jacobi bellman equation associated with an optimization regulator problem

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

approximation properties for group algebras and group von neumann algebras let g be a locally compact group let c g resp vn g be the c algebra resp the von neumann algebra associated with the left regular representation i of g let a g be the fourier algebra of g and let moa g be the set of completely bounded multipliers of a g with the completely bounded norm moa g is a dual space and we say that g has the approximation property ap if there is a net ua of functions in a g with compact support such that ua a 1 in the associated weak topology in particular g has the ap if g is weakly amenable a g has an approximate identity that is bounded in the completely bounded norm for a discrete group f we show that f has the ap x c f has the slice map property for subspaces of any c algebra x vn jt has the slice map property for a weakly closed subspaces of any von neumann algebra property se the semidirect product of weakly amenable groups need not be weakly amenable we show that the larger class of groups with the ap is stable with respect to semidirect products and more generally this class is stable with respect to group extensions we also obtain some results concerning crossed products for example we show that the crossed product m a g of a von neumann algebra m with property s by a group g with the ap also has property s

further results on the bellman equation for optimal control problems with exit times and nonnegative lagrangians abstract in a series of papers we proved theorems characterizing the value function in exit time optimal control as the unique viscosity solution of the corresponding bellman equation that satisfies appropriate side conditions the results applied to problems which satisfy a positivity condition on the integral of the lagrangian this positive integral condition assigned a positive cost for remaining outside the target on any interval of positive length in this note we prove a new theorem which characterizes the exit time value function as the unique bounded from below viscosity solution of the bellman equation that vanishes on the target the theorem applies to problems satisfying an asymptotic condition on the trajectories including cases where the positive integral condition is not satisfied our results are based on an extended version of barb lat s lemma we apply the theorem to variants of the fuller problem and other examples where the lagrangian is degenerate

normal local controllability of order one this paper gives a sufficient condition of normal local controllability at a point x0 for a family of analytical vector fields the families considered are of the following where is a bounded subset of r the condition is of the first order that is it depends only on the values of the fields and their first derivatives at x0 it is proved that this condition is the most general condition of this type in fact if this condition is not satisfied there are families of vector fields which at the first order coincide with the given family and which are not normally locally controllable at x0

state steering by learning for a class of nonlinear control systems abstract the problem of steering the state of nonlinearly perturbed linear systems by learning is investigated a family of algorithms which compute the steering control by means of successive trials on the real plant is presented convergence in the face of a class of nonlinear plant perturbations is proven state feedback linearizable systems are shown to be addressable by the presented algorithms two examples illustrate the applicability of algorithms

configuration controllability of mechanical systems underactuated by one control we investigate local configuration controllability for mechanical control systems within the affine connection formalism we rely on previous results on controllability and series expansions for the evolution of mechanical systems starting from rest extending the work by lewis for the single input case we are able to characterize local configuration controllability for systems with n degrees of freedom and n 1 input forces

controllability of linear systems on generalized heisenberg groups this paper is devoted to the study of global controllability of linear systems on higher dimensional heisenberg groups hn necessary conditions and sufficient ones are provided they are compared to the known necessary and sufficient conditions on h1 stated in dath and jouan 2016 controllability of linear systems on low dimensional nilpotent and solvable lie groups journal of dynamical and control systems 22 2 207 225 the notion of decoupled systems is introduced in order to get examples and counter examples and more precise controllability criteria are stated for them

high order variations for families of vector fields sufficient conditions involving lie brackets of arbitrarily high order are obtained for local controllability of families of vector fields after providing a general framework for the generation of high order control variations we propose a specific method for generating such variations the theory is applied to a number of nontrivial examples

tensor products of operator spaces abstract in this paper we lay the foundations for a systematic study of tensor products of subspaces of c algebras to accomplish this various notions of duality are introduced and employed elementary proofs of the complete injectivity of the haagerup norm and of the extension theorem for completely bounded maps are given pisier s gamma norms are examined and found to be special cases of the haagerup norm we identify the greatest operator space cross norm and show that the spatial tensor norm is the least operator space cross norm in an appropriate sense indeed most of the elementary theory of banach space tensor norms generalizes to the category of operator spaces

controllability of linear systems on solvable lie groups linear systems on lie groups are a natural generalization of linear systems on euclidian spaces for such systems this paper studies controllability by taking into consideration the eigenvalues of an associated derivation mathcal d when the state space is a solvable connected lie group controllability of the system is guaranteed if the reachable set of the neutral element is open and the derivation mathcal d has only pure imaginary eigenvalues for bounded systems on nilpotent lie groups such conditions are also necessary

remarks on complemented subspaces of von neumann algebras in this note we include two remarks about bounded not necessarily contractive linear projections on a von neumann algebra we show that if m is a von neumann subalgebra of b h which is complemented in b h and isomorphic to m m then m is injective or equivalently m is contractively complemented we do not know how to get rid of the second assumption on m in the second part we show that any complemented reflexive subspace of a c algebra is necessarily linearly isomorphic to a hilbert space

morse inequalities for r constructible sheaves this note aims at generalizing of classical morse inequalities for betti numbers of compact manifolds cf lo 1 2 in this paper we work with r constructible sheaves instead and encounter the tight relation between morse theory and microlocal analysis of sheaves see witten 111 hellter and sjiistrand s 61 and henniart 7 for another approach to morse inequalities via microlocal analysis and also goresky and macpherson 3 4 who introduced the stratified morse theory this paper may be considered as a variation on kashiwara s index theorem 8 and in fact our proof is a slight modification of his

controllability is harder to decide than accessibility this article compares the difficulties of deciding controllability and accessibility these are standard properties of control systems but complete algebraic characterizations of controllability have proved elusive the article shows in particular that for subsystems of bilinear systems accessibility can be decided in polynomial time but controllability is np hard

multipliers of the fourier algebras of some simple lie groups and their discrete subgroups for any amenable locally compact group g the space of multipliers ma g of the fourier algebra a g coincides with the space b g of functions on g that are linear combinations of continuous positive definite functions we prove that ma g b g 0 for many non amenable connected groups more specifically we prove that moa g b g 0 for the classical complex lie groups and the general lorentz groups soo n 1 n 2 moa g is a certain subspace of ma g which we call the space of completely bounded multipliers of a g unlike ma g the space moa g has nice stability properties with respect to direct products of groups it is known that the fourier algebra of the free group on n generators n 2 2 admits an unbounded approximate unit pn which is bounded in the multiplier norm we extend this result to any closed subgroup of the general lorentz group soo n 1 moreover we show that for these groups pn can be chosen to be bounded with respect to the moa g norm by a duality argument we obtain that the reduced c algebra of every discrete subgroup of soo n 1 has the completely bounded approximation property in particular this property holds for c f2 the reduced c algebra of the free group on two generators we also prove

geometric conditions for local controllability abstract small time local controllability of a c1 affine control system with a generic drift vector field and n 1 linearly independent input vector fields on an n dimensional connected c1 manifold m is studied showing that a necessary condition namely the drift vector field belonging to the convex hull of the control vector fields at the given point is also sufficient on an open dense subset h1 of the set of points where it is verified local controllability at the points in the interior of h 1 is also considered when the input vector fields generate an involutive distribution and a criterion based on the type of contract of h with the hyperplane spanned by the input vector fields is proved an example shows a situation where the sufficient condition for small time local controllability of h sussmann siam j control option 25 1987 158 194 cannot be applied but in which the method now presented distinguishes the possible cases

the slice map problem and approximation properties we show that a weakly closed subspace j of the bounded operators bh on an infinite dimensional hilbert space h has property s the fubini product lt l for all weakly closed subspaces t b h if and only if j satisfies a certain approximation property which we call the weak approximation property the weak approximation property is implied by but does not imply the weak completely bounded approximation property moreover if j has the weak approximation property then the predual j has the approximation property for banach spaces we also prove analagous results for property s and variations of property s for c algebras as an application of our characterization of subspaces with property s we show that the reflexive algebra tensor product formula alg l1 alg l2 alg l1 l2 is not always valid in fact we show that for each of the types ii1 ii and iii 0 1 there is a separably acting factor m alg l1 of that type and a reflexive algebra a alg l2 such that alg l1 alg l2 is strictly contained in alg l1 l2

proximal analysis and the minimal time function under general hypotheses on the target set s and the dynamics of the system we show that the minimal time function ts cdot is a proximal solution to the hamilton jacobi equation uniqueness results are obtained with two different kinds of boundary conditions a new propagation result is proven and as an application we give necessary and sufficient conditions for ts cdot to be lipschitz continuous near s a petrov type modulus condition is also shown to be sufficient for continuity of ts cdot near s

local controllability for families of diffeomorphisms abstract we study groups and semigroups which are generated by analytic families of diffeomorphisms the central notion is that of local controllability of a family of diffeomorphisms at a given point of the state manifold which generalizes the familiar notion of local controllability of control systems with continuous as well as discrete time lie theory methods are used we systematically exploit the so called fast switching variations and properties of the jet spaces of curves on the state manifold

control and stabilization of nonholonomic dynamic systems a class of inherently nonlinear control problems has been identified the nonlinear features arising directly from physical assumptions about constraints on the motion of a mechanical system models are presented for mechanical systems with nonholonomic constraints represented both by differential algebraic equations and by reduced state equations control issues for this class of systems are studied and a number of fundamental results are derived although a single equilibrium solution cannot be asymptotically stabilized using continuous state feedback a general procedure for constructing a piecewise analytic state feedback which achieves the desired result is suggested

motion planning for control affine systems satisfying low order controllability conditions abstractthis paper is devoted to the motion planning problem for control affine systems by using trigonometric polynomials as control functions the class of systems under consideration satisfies the controllability rank condition with the lie brackets up to the second order the approach proposed here allows to reduce a point to point control problem to solving a system of algebraic equations the local solvability of that system is proved and formulas for the parameters of control functions are presented our local and global control design schemes are illustrated by several examples

a new theoretical approach to absorbing layers we consider the construction of absorbing boundary layers using asymptotic expansions the small parameter being the width of the layer this allows us to define the order of the layer we compute layers of various orders and analyze their stability properties finally we perform some numerical tests to evaluate the practical utility of our layers

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

low order controllability and kinematic reductions for affine connection control systems controllability and kinematic modeling notions are investigated for a class of mechanical control systems first low order controllability results are given for the class of mechanical control systems second a precise connection is made between those mechanical systems which are dynamic i e have forces as inputs and those which are kinematic i e have velocities as inputs interestingly and surprisingly these two subjects are characterized and linked by a certain intrinsic vector valued quadratic form that can be associated to an affine connection control system

completely bounded multipliers of the fourier algebra of a simple lie group of real rank one the fourier algebra a g of a locally compact group g is the space of matrix coefficients of the regular representation and is the predual of the yon neumann algebra vn g generated by the regular representation of g on l 2 g a multiplier m of a g is a bounded operator on a g given by pointwise multiplication by a function on g also denoted m we say m is a completely bounded multiplier ofa g if the transposed operator on vn g is completely bounded definition below it may be possible to find a net ofa g functions m i ie i say such that mi tends to

ein operatorwertiger hahn banach satz abstract we generalize arveson s extension theorem for completely positive mappings 1 to a hahn banach principle for matricial sublinear functionals with values in an injective c algebra or an ideal in b h we characterize injective w algebras by a matricial order condition we illustrate the matricial hahn banach principle by three applications 1 let a b b be unital c algebras b a subalgebra of a and b b injective if a b is a completely bounded self adjoint b bihomomorphism then it can be expressed as the difference of two completely positive b bihomomorphism 2 let m be a w algebra containing 1 h on a hilbert space h if m is finite and hyperfinite there exists an invariant expectation mapping p of b h onto m p is an extension of the center trace 3 combes 7 proved that a lower semicontinuous scalar weight on a c algebra is the upper envelope of bounded positive functionals we generalize this result to unbounded completely positive mappings with values in an injective w algebra

sheaf quantization of hamiltonian isotopies and applications to nondisplaceability problems let i be an open interval m be a real manifold t m its cotangent bundle and phi phi t t in i a homogeneous hamiltonian isotopy of t m defined outside the zero section let lambda be the conic lagrangian submanifold associated with phi lambda is a subset of t m x t m x t i we prove the existence and unicity of a sheaf k on mxmxi whose microsupport is contained in the union of lambda and the zero section and whose restriction to t 0 is the constant sheaf on the diagonal of mxm we give applications of this result to problems of non displaceability in contact and symplectic topology in particular we prove that some strong morse inequalities are stable by hamiltonian isotopies and we also give results of non displaceability for positive isotopies in the contact setting in this new version we suppress one hypothesis in the main theorem and we extend the result of non displaceability for positive isotopies

minimum time with bounded energy minimum energy with bounded time necessary and sufficient conditions for the regularity of the minimum time function and minimum energy function for a control system with controls in l p 0 infty bbb r m and p ge 1 are given in terms of topological properties of the reachable sets in particular standard local controllability assumptions are sufficient to yield the continuity of both value functions for linear systems and p ge1 and the holder continuity of the minimum time function for nonlinear systems and p 1

a polynomial approach to discrete time non linear system controllability this paper uses a polynomial approach to present a necessary and sufficient condition for local controllability of single input single output siso discrete time non linear systems the condition is presented in terms of common factors of a non commutative polynomial expression this extends the result on controllability of linear systems and the recent result on controllability of continuous time non linear systems to expose controllability of discrete time non linear systems in the input output framework the controllability condition yields a finite step computing procedure for examining controllability of discrete time input output non linear systems

the complexity of deciding controllability abstract it is shown that the problem of deciding whether a control system is small time locally controllable or not is np hard this is an extension of a recent paper by sontag to single input systems and to bounded controls

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

local equilibrium controllability of multibody systems controlled via shape change we study local equilibrium controllability of shape controlled multibody systems the multibody systems are defined on a trivial principal fiber bundle by a lagrangian that characterizes the base body motion and shape dynamics a potential dependent on an advected parameter e g uniform gravitational potential is considered this potential breaks base body symmetries but a symmetry subgroup is assumed to exist symmetric product formulas are derived and important properties are obtained for symmetric products of horizontal shape control vector fields and a potential vector field that is dependent on an advected parameter based on these properties sufficient conditions for local equilibrium controllability and local fiber equilibrium controllability are developed these results are applied to two classes of shape controlled multibody systems in a uniform gravitational field multibody attitude systems and neutrally buoyant multibody underwater vehicles

controllability distributions and systems approximations a geometric approach given a nonlinear system we determine a relation at an equilibrium between controllability distributions defined for a nonlinear system and a taylor series approximation of it the value of such a relation is appreciated if we recall that the solvability conditions as well as the solutions to some control synthesis problems can be stated in terms of geometric concepts like controlled invariant controllability distributions the relation between these distributions at the equilibrium will help us to decide when the solvability conditions of this kind of problems are equivalent for the nonlinear system and its approximation