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

fast and precise fourier transforms many applications of fast fourier transforms ffts such as computer tomography geophysical signal processing high resolution imaging radars and prediction filters require high precision output an error analysis reveals that the usual method of fixed point computation of ffts of vectors of length 2 sup l leads to an average loss of l 2 bits of precision this phenomenon often referred to as computational noise causes major problems for arithmetic units with limited precision which are often used for real time applications several researchers have noted that calculation of ffts with algebraic integers avoids computational noise entirely we combine a new algorithm for approximating complex numbers by cyclotomic integers with chinese remaindering strategies to give an efficient algorithm to compute b bit precision ffts of length l more precisely we approximate complex numbers by cyclotomic integers in z e 2 spl pi i 2 sup n whose coefficients when expressed as polynomials in e 2 spl pi i 2 sup n are bounded in absolute value by some integer m for fixed n our algorithm runs in time o log m and produces an approximation with worst case error of o 1 m 2 sup n 2 1 we prove that this algorithm has optimal worst case error by proving a corresponding lower bound on the worst case error of any approximation algorithm for this task the main tool for designing the algorithms is the use of the cyclotomic units a subgroup of finite index in the unit group of the cyclotomic field first implementations of our algorithms indicate that they are fast enough to be used for the design of low cost high speed high precision fft chips

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

complex approximations using algebraic integers the problem of approximating complex numbers by elements of z omega the algebraic integers of q omega where omega is a primitive n th root of unity is considered the motivating application is to reduce the dynamic range requirements of residue number system implementations of the discrete fourier transform smallest error tolerances for the case of eighth roots of unity are derived using a geometric argument scale factors involved are reduced from alpha to sqrt alpha for this case with roughly the same percentage errors the case of sixteenth roots of unity gives even better range reductions and is considered only briefly

compensating drift vector fields with gradient vector fields for asymptotic submanifold stabilization we derive sufficient conditions on a drift vector field to let an asymptotically stable invariant submanifold of an input affine system without drift remain asymptotically stable for the system with drift in doing so we use the same feedback laws modulo control gain tuning such that no new feedback laws need to be designed for the system with drift our main assumption is that the vector field of the input affine system without drift assumes the form of a gradient vector field for given feedback laws we show how one can assess the performance of the system with drift only via knowledge about the system without drift finally we find that our results are relevant in synchronization problems and backstepping controllers for mechanical systems

scattering matrices and weyl functions for a scattering system a a 0 consisting of self adjoint extensions a and a 0 of a symmetric operator a with finite deficiency indices the scattering matrix s and a spectral shift function are calculated in terms of the weyl function associated with a boundary triplet for a and a simple proof of the krein birman formula is given the results are applied to singular sturm liouville operators with scalar and matrix potentials to dirac operators and to schrodinger operators with point interactions

lyapunov design of stabilizing controllers for cascaded systems the design of a state feedback law for an affine nonlinear system to render a as small as possible compact neighborhood of the equilibrium of interest globally attractive is discussed following z artstein s theorem 1983 the problem can be solved by designing a so called control lyapunov function for systems which are in a cascade form a lyapunov function meeting artstein s conditions is designed assuming the knowledge of a control law stabilizing the equilibrium of the head nonlinear subsystem in particular for planar systems this gives sufficient and necessary conditions for a compact neighborhood of the equilibrium to be stabilized

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

density and current of a dissipative schrodinger operator we regard a current flow through an open one dimensional quantum system which is determined by a dissipative schrodinger operator the imaginary part of the corresponding form originates from robin boundary conditions with certain complex valued coefficients imposed on schrodinger s equation this dissipative schrodinger operator can be regarded as a pseudo hamiltonian of the corresponding open quantum system the dilation of the dissipative operator provides a self adjoint quasi hamiltonian of the system more precisely the hamiltonian of the minimal closed system which contains the open one is used to define physical quantities such as density and current for the open quantum system the carrier density turns out to be an expression in the generalized eigenstates of the dilation while the current density is related to the characteristic function of the dissipative operator finally a rigorous setup of a dissipative schrodinger poisson system is outlined

an algorithm for complex approximations in z e 2 pi i 8 corresp an algorithm is described that approximates complex numbers by elements of the algebraic integers of z e 2 pi i 8 with integer coordinates of at most a prescribed size the motivating application is to reduce the dynamic range requirements of residue number system implementations of the discrete fourier transform the closest points to zero of z e 2 pi i 8 m gor any integer m are determined a particular sequence of such points forms the basis of the algorithm an example of 8 bit z omega m approximations of the 128th roots of unity is considered the algorithm yields m 186 with scaling m is reduced to 18

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

computing the discrete fourier transform using residue number systems in a ring of algebraic integers a new method is described for computing an n r m 2 upsilon m point complex discrete fourier transform that uses quantization within a dense ring of algebraic integers in conjunction with a residue number system over this ring the algebraic and analytic foundations for the technique are derived and discussed the architecture for a radix r fast fourier transform algorithm using a residue number system over z omega where omega is a primitive r th root of unity is developed and range and error estimates for this algorithm are derived

local asymptotic feedback stabilization to a submanifold topological conditions abstract we consider the problem of local asymptotic feedback stabilization of a control system x f x u defined in r n to a compact connected oriented embedded codimension one submanifold p of r n using a continuous feedback law this generalizes the problem of local asymptotic feedback stabilization to a point which has been previously considered in the control theory literature it is natural to expect the topology of p to play a role in deciding whether or not local asymptotic stabilization to p of the system x f x u is feasible via continuous feedback and our aim in this paper is precisely to outline necessary conditions on the topology of p for stabilization via continuous feedback to be achievable

encoding algorithms for complex approximations in z e sup 2 pi i 8 two algorithms are presented that approximate complex numbers by elements of the algebraic integers of q w where w e sup 2 pi i 8 these algorithms substantially reduce the computational burden range and memory requirements are given as a function of the desired accuracy and expressions are obtained for the number of computations required by each algorithm

asymptotic stabilization of submanifolds embedded in riemannian manifolds we study control problems in which systems whose states spaces are riemannian manifolds must be steered towards embedded submanifolds of their state spaces in a stable fashion i e feedback laws rendering the given submanifold asymptotically stable are sought bringing the closed loop to the form of a gradient system with drift we find that the gradient part can be scaled in a fashion so as to guarantee asymptotic stability of the desired submanifold under this circumstance we show that solutions of the system can be brought to an arbitrarily small neighborhood of the submanifold in arbitrary time thereafter we derive an algebraic condition under which the control vector fields can be brought to the desired form lastly we recast our algebraic condition in terms of vertical and horizontal spaces for the case that the submanifold is an equivalence class

conley index condition for asymptotic stability in this paper we use conley index theory to develop necessary conditions for stability of equilibrium and periodic solutions of nonlinear continuous time systems the conley index is a topological generalization of the morse theory which has been developed to analyze dynamical systems using topological methods in particular the conley index of an invariant set with respect to a dynamical system is defined as the relative homology of an index pair for the invariant set the conley index can then be used to examine the structure of the system invariant set as well as the system dynamics within the invariant set including system stability properties efficient numerical algorithms using homology theory have been developed in the literature to compute the conley index and can be used to deduce the stability properties of nonlinear dynamical systems

the cayley transform applied to non interacting quantum transport abstract we extend the landauer buttiker formalism in order to accommodate both unitary and self adjoint operators which are not bounded from below we also prove that the pure point and singular continuous subspaces of the decoupled hamiltonian do not contribute to the steady current one of the physical applications is a stationary charge current formula for a system with four pseudo relativistic semi infinite leads and with an inner sample which is described by a schrodinger operator defined on a bounded interval with dissipative boundary conditions another application is a current formula for electrons described by a one dimensional dirac operator here the system consists of two semi infinite leads coupled through a point interaction at zero

stabilization of a class of nonlinear systems by a smooth feedback control we consider the stabilizability problem for systems of the form x f x ub in the neighborhood of an equilibrium point of f x first by means of center manifold theory a lower order system is introduced if this system is stabilizable then so is the original system results on the stabilization of the low order system are presented no resort is taken to liapunov theory the relation between stabilizability and controllability is discussed

topological obstructions to submanifold stabilization we consider the problem of local asymptotic feedback stabilization via a continuously differentiable feedback law of a control system x f x u defined in euclidean space rn with f being continuously differentiable to a compact connected oriented m dimensional submanifold m of rn with codimension strictly larger than one we obtain necessary conditions on the topology of m for such a stabilizing feedback law to exist this extends the work done in where only the codimension one case was treated we also briefly discuss the case where the control is only assumed continuous