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

generalized eigenvalue problems for p q laplacian with indefinite weight abstract this paper presents existence and non existence results on a positive solution for quasilinear elliptic equations of the form r u r u m r x u r 2 u in with 1 r r and 0 under dirichlet boundary condition where is a bounded domain in r n and m r is a weight function in l admitting sign change we show that existence and non existence of a positive solution depend only on the relation between and the first eigenvalue of r laplacian with weight function m r whence it is independent of the operator r and the parameter 0

the beginning of the fucik spectrum for the p laplacian has a nontrivial solution u here 1 p 2p u div u p 2 u u max u 0 u uu in 1 1 denoting by 1 2 the first two eigenvalues of 2p on w 1 p 0 0 cf the end of this introduction 7p clearly contains in particular 1 1 2 2 and the two lines 1 r and r 1 this generalized notion of spectrum was introduced for p 2 i e in the linear case in the 1970s by fuc ik 21 and dancer 15 in connection article id jdeq 1999 3645 available online at http www idealibrary com on

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

classifying c algebras via ordered mod p k theory we introduce an order structure on k 0 9 k0 z p this group may also be thought of as ko 7z z p we exhibit new examples of real rank zero c algebras that are inductive limits of finite dimensional and dimension drop algebras have the same ordered graded k theory with order unit and yet are not isomorphic in fact they are not even stably shape equivalent the order structure on k0 z 3 z p naturally distinguishes these algebras the same invariant is used to give an isomorphism theorem for such realrank zero inductive limits as a corollary we obtain an isomorphism theorem for all real rank zero approximately homogeneous c algebras that arise from systems of bounded dimension growth and torsion free k0 group at the 1980 kingston conference effros posed the problem of finding suitable invariants for use in studying c algebras that are limits of sequences of homogeneous c algebras these are now called almost homogeneous ah c algebras the classification of ah algebras is a rapidly developing field and we will not attempt to summarize all this activity instead we will focus on the growth of the invariants used specifically we consider an ah algebra a that is the direct limit o f a system of the form

unitary orbits in 1 dimensional nccw complexes abstract let a be a 1 dimensional nccw complex in this paper it is shown that the usual distance d u defined on the approximate unitary equivalence classes unitary orbits of self adjoints in a is equal to a distance d w defined on morphisms from the cuntz semigroup of c 0 0 1 to cu a in addition another distance d p on approximately unitary equivalence classes related to spectral information of positive elements in the classes is proved to be equal to d u and d w on simple unital inductive limits of 1 dimensional nccw complexes

c algebras generated by stable relations abstract presentations of certain c algebras such as c0 0 1 mn by generators and relations are proven to be stable by the weakest definition of stability a set p1 pn of star polynomials is stable if operators a1 an which almost satisfy these relations meaning pi a1 an can be perturbed within c a1 an to satisfy these relations exactly homotopy and perturbation results for these c algebras are corollaries in particular we prove some new results about the dimension drop c algebras a n c 0 1 m n 0 1 c i we show that an is semiprojective answering a question of effros and kaminker also we show that whenever an b is a homomorphism approximately contained in a sub c algebra b0 b there is a nearby homomorphism of an into b0 this answers a question raised by elliott we prove technical results regarding mn c0 0 1 which should be useful elsewhere first we show this is projective meaning a star homomorphism mn c0 0 1 b j always lifts to a star homomorphism mn c0 0 1 b secondly we show that for a finitely generated m n a m n c 0 0 1 c 0 0 1 a if the amalgamation is done properly

existence of positive solutions for nonlinear elliptic equations with convection terms abstract we prove the existence of positive solutions for the equation i 1 n x i a i x u u f x u u in under the dirichlet boundary condition where the essential point is the dependence of the terms of the elliptic equation on the solution u and its gradient u we develop an approach based on approximate solutions and on a new strong maximum principle

global stabilization of discrete time homogeneous systems abstract the feedback stabilization of continuous time r homogeneous systems has been studied by several authors see for instance dayawansa and martin proceedings of the 28th c d c tampa florida december 1989 hermes j differential equations 92 1991 76 89 hermes systems control lett 24 1995 7 11 kawski proceedings of the 27th c d c texas 1988 texas kawski control theory and advanced technology vol 6 mita press tokyo 1996 pp 497 516 sepulchre and aeyels math control signal systems 9 1996 34 58 in kawski analysis of controlled dynamical systems birkhauser basel 1991 kawski proceedings of ifac nolcos 1995 kawski gave an intrinsic concept of homogeneity with respect to general smooth dilations in this note we define the homogeneity of transition maps w r t continuous dilations and we show how one can construct a global stabilizing feedback control for discrete time homogeneous systems which are locally asymptotically stabilizable

quasilinear elliptic equations on noncompact riemannian manifolds abstract the existence of solutions to a class of quasilinear elliptic problems on noncompact riemannian manifolds with finite volume is investigated boundary value problems with homogeneous neumann conditions in possibly irregular euclidean domains are included as a special instance a nontrivial solution is shown to exist under an unconventional growth condition on the right hand side which depends on the geometry of the underlying manifold the identification of the critical growth is a crucial step in our analysis and entails the use of the isocapacitary function of the manifold a condition involving its isoperimetric function is also provided

an explicit formula of bounded feedback stabilizers for feedforward systems abstract this work constitutes continuation of a recent result of the same authors which provides an explicit formula for bounded feedback stabilizers for a wide class of triangular nonlinear systems in the present paper the previous result is extended for nonlinear systems having feedforward structure and whose linearization at the equilibrium is in general uncontrollable

inductive limits of interval algebras the tracial state space introduction this is the first in a series of papers where we investigate the structure of the c algebras obtained as inductive limits of interval algebras c 0 1 0m the purpose here is to show that these c algebras comprise a large stock of simple c algebras and that some of them allow endomorphisms with chaotic actions on the tracial state space to be more specific we show that to any dense subgroup g of q containing z and any metrizable choquet simplex s there is a sequence

a classification of simple limits of splitting interval algebras abstract let a be a unital simple limit of finite direct sums of sub homogeneous interval algebras of a certain type cf definition 1 1 it is proved that a can be classified by the scaled ordered group k 0 a the simplex t a and the canonical pairing between them it is also shown that k 0 a might fail to have the riesz decomposition property

finitely generated nilpotent group c algebras have finite nuclear dimension we show that group c algebras of finitely generated nilpotent groups have finite nuclear dimension it then follows from a string of deep results that the c algebra a generated by an irreducible representation of such a group has decomposition rank at most 3 if in addition a satisfies the universal coefficient theorem another string of deep results shows it is classifiable by its ordered ktheory and is approximately subhomogeneous we observe that all c algebras generated by faithful irreducible representations of finitely generated torsion free nilpotent groups satisfy the universal coefficient theorem

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

nuclear dimension and mathcal z stability simple separable unital monotracial and nuclear mathrm c algebras are shown to have finite nuclear dimension whenever they absorb the jiang su algebra mathcal z tensorially this completes the proof of the toms winter conjecture in the unique trace case

sign changing and constant sign solutions for p laplacian problems with jumping nonlinearities abstract by variational methods we prove the existence of a sign changing solution for the p laplacian equation under dirichlet boundary condition with jumping nonlinearity having relation to the fucik spectrum of p laplacian we also provide the multiple existence results for the p laplacian problems

a green s formula for quasilinear elliptic operators abstract we prove a theorem of traces which allows us to define the boundary condition of the neumann problem associated to a quasilinear elliptic operator in divergence form later we prove a green s formula that justifies the variational formulation of the problem

on a simple unital projectionless c algebra we construct a unital separable c algebra z as an analog of the hyperfinite type ii 1 factor besides being nuclear simple projectionless and infinite dimensional z has a unique tracial state and is kk equivalent to inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i the algebra of complex numbers it is shown that unital endomorphisms on z are approximately inner and that z is isomorphic to the infinite tensor product of its replicas it is also shown that a z a for certain interesting classes of unital simple nuclear c algebras a of real rank zero

nuclear dimension mathcal z stability and algebraic simplicity for stably projectionless c algebras the main result here is that a simple separable c algebra is mathcal z stable where mathcal z denotes the jiang su algebra if i it has finite nuclear dimension or ii it is approximately subhomogeneous with slow dimension growth this generalizes the main results of toms invent math 183 2 225 244 2011 and winter invent math 187 2 259 342 2012 to the nonunital setting as a consequence finite nuclear dimension implies mathcal z stability even in the case of a separable c algebra with finitely many ideals algebraic simplicity is established as a fruitful weakening of being simple and unital and the proof of the main result makes heavy use of this concept

homogeneous feedback controls for homogeneous systems let x x j 2n aj x xj be a real analytic n dimensional vector field with x 0 0 which is homogeneous of degree m 0 with respect to a dilation erx er1x1 ernxn and y xj we show that control system x x x uy admits a continuous asymptotically stabilizing feedback control u x then a one dimensional dynamic extension of the system admits a homogeneous asymptotically stabilizing dynamic feedback control u x such that the vector field w x 1 m u x 1 j 2aj x xj describing the extended controlled system is homogeneous of degree m with respect to the extended dilation er x e er1x1 ernxn

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

mathcal c _ 0 x algebras stability and strongly self absorbing mathcal c algebras we study permanence properties of the classes of stable and so called mathcal d stable mathcal c algebras respectively more precisely we show that a mathcal c 0 x algebra a is stable if all its fibres are provided that the underlying compact metrizable space x has finite covering dimension or that the cuntz semigroup of a is almost unperforated a condition which is automatically satisfied for mathcal c algebras absorbing the jiang su algebra mathcal z tensorially furthermore we prove that if mathcal d is a k 1 injective strongly self absorbing mathcal c algebra then a absorbs mathcal d tensorially if and only if all its fibres do again provided that x is finite dimensional this latter statement generalizes results of blanchard and kirchberg we also show that the condition on the dimension of x cannot be dropped along the way we obtain a useful characterization of when a mathcal c algebra with weakly unperforated cuntz semigroup is stable which allows us to show that stability passes to extensions of mathcal z absorbing mathcal c algebras

receding horizon control and discontinuous state feedback stabilization this paper addresses three aspects of receding horizon control in discrete time 1 feedback stabilization of general nonlinear systems with receding horizon control 2 the generation of stabili

the range of a class of classifiable separable simple amenable c algebras we study the range of a classifiable class a of unital separable simple amenable c algebras which satisfy the universal coefficient theorem the class a contains all unital simple ah algebras we show that all unital simple inductive limits of dimension drop circle c algebras are also in the class this unifies some of the previous known classification results for unital simple amenable c algebras we also show that there are many other c algebras in the class we prove that for any partially ordered simple weakly unperforated rationally riesz group g0 with order unit u any countable abelian group g1 any metrizable choquet simplex s and any surjective affine continuous map r s su g0 where su g0 is the state space of g0 which preserves extremal points there exists one and only one up to isomorphism unital separable simple amenable c algebra a in the classifiable class a such that k0 a k0 a 1a k1 a t a a g0 g0 u g1 s r

multiple existence results of solutions for the neumann problems via super and sub solutions by variational methods we provide existence results of multiple solutions for quasilinear elliptic equations under the neumann boundary condition our main result shows the existence of two constant sign solutions and a sign changing solution in the case where we do not impose the subcritical growth condition to the nonlinear term not including derivatives of the unknown function the studied equations contain the p laplacian problems as a special case moreover we give a result concerning a local minimizer in c1 versus w1 p auxiliary results of independent interest are also obtained a density property for the space w1 p a strong maximum principle of zhang s type and a moser s iteration scheme depending on a parameter

the nuclear dimension of c algebras abstract we introduce the nuclear dimension of a c algebra this is a noncommutative version of topological covering dimension based on a modification of the earlier concept of decomposition rank our notion behaves well with respect to inductive limits tensor products hereditary subalgebras hence ideals quotients and even extensions it can be computed for many examples in particular it is finite for all uct kirchberg algebras in fact all classes of nuclear c algebras which have so far been successfully classified consist of examples with finite nuclear dimension and it turns out that finite nuclear dimension implies many properties relevant for the classification program surprisingly the concept is also linked to coarse geometry since for a discrete metric space of bounded geometry the nuclear dimension of the associated uniform roe algebra is dominated by the asymptotic dimension of the underlying space

the generator problem for z stable c algebras the generator problem was posed by kadison in 1967 and it remains open until today we provide a solution for the class of c algebras absorbing the jiang su algebra z tensorially more precisely we show that every unital separable z stable c algebra a is singly generated which means that there exists an element x a that is not contained in any proper sub c algebra of a to give applications of our result we observe that z can be embedded into the reduced group c algebra of a discrete group that contains a non cyclic free subgroup it follows that certain tensor products with reduced group c algebras are singly generated in particular c r f c r f is singly generated

regularity properties in the classification program for separable amenable c algebras we report on recent progress in the program to classify separable amenable c algebras our emphasis is on the newly apparent role of regularity properties such as finite decomposition rank strict comparison of positive elements and z stability and on the importance of the cuntz semigroup we include a brief history of the program s successes since 1989 a more detailed look at the villadsen type algebras which have so dramatically changed the landscape and a collection of announcements on the structure and properties of the cuntz semigroup

the jiang su algebra revisited we give a number of new characterizations of the jiang su algebra z both intrinsic and extrinsic in terms of c algebraic dynamical topological and k theoretic conditions along the way we study divisibility properties of c algebras we give a precise characterization of those unital c algebras of stable rank one that admit a unital embedding of the dimensiondrop c algebra zn n 1 and we prove a cancellation theorem for the cuntz semigroup of c algebras of stable rank one

a strong maximum principle for differential equations with nonstandard p x growth conditions abstract in this paper with some special technics we give a strong maximum principle for the equations with nonstandard p x growth conditions div x u u d x f x u 0 in where x s d x f x u satisfy some conditions

existence results for nonlinear elliptic equations with leray lions operator and dependence on the gradient the existence of positive solutions for nonlinear elliptic problems under dirichlet boundary condition is studied as well as the compactness and directness of the solution set the main novelties consist in the presence of a leray lions operator in the differential part and in the dependence of the reaction term on the gradient of the solution our approach relies on the method of sub supersolution nonlinear regularity theory and strong maximum principle

on resonant neumann problems we consider semilinear neumann problems at resonance and prove existence and multiplicity theorems the existence theorems allow resonance with respect to any eigenvalue of the negative neumann laplacian the multiplicity theorems concern problems resonant at 0 the principal eigenvalue or at the first nonzero eigenvalue our approach uses tools from critical point theory and from morse theory

asymptotic stabilization of planar systems abstract sufficient conditions for the existence of a c1 asymptotically stabilizing feedback control for planar affine systems are given the methods are constructive and yield homogeneous feedback controls for homogeneous systems which allow the results to be extended to more general systems

explicit formulas of feedback stabilizers for a class of triangular systems with uncontrollable linearization for a class of nonlinear systems having triangular structure we provide explicit formulas for feedback controllers that exhibit global stabilization the global part of our analysis is mainly based on a version of sontag s input to state stability property and constitutes a generalization of previous works of the second author

compact group actions on uhf algebras obtained by folding the interval we show that for any integer p greater than one there exists a c algebra which is not af such that mp a mp where mp mp is the uhf algebra of type p using this construction of uhf algebras as inductive limits of subhomogeneous algebras on an interval embedded in each other via a folding operation on the interval we exhibit actions of compact groups on uhf algebras whose fixed point algebras are not af in particular for any compact group g e there exists an action of g on mp whose fixed point algebra is not af

inverse systems of groupoids with applications to groupoid c algebras abstract we define what it means for a proper continuous morphism between groupoids to be haar system preserving and show that such a morphism induces via pullback a morphism between the corresponding convolution algebras we proceed to provide a plethora of examples of haar system preserving morphisms and discuss connections to noncommutative cw complexes and interval algebras we prove that an inverse system of groupoids with haar system preserving bonding maps has a limit and that we get a corresponding direct system of groupoid c algebras an explicit construction of an inverse system of groupoids is used to approximate a compact groupoid g by second countable groupoids if g is equipped with a haar system and 2 cocycle then so are the approximation groupoids and the maps in the inverse system are haar system preserving as an application of this construction we show how to easily extend the maximal equivalence theorem of jean renault to compact groupoids

a complete invariant foradalgebras with real rank zero and bounded torsion ink1 abstract for each integerp 1 we consider an algebraic invariant forc algebras the invariant consists ofk0 k1 thek0 group with z pcoefficients the order structures these groups possess and the natural maps between the three groups we prove that this invariant is complete for the class ofadalgebras of real rank zero ifpannihilates every torsion element ofk1 theadalgebras arec algebras which are inductive limits of finite direct sums ofc algebras of continuous matrix valued functions over the circle or over the interval where in the interval case we require that the value over the endpoints is a scalar multiple of the unit matrix examples show that the condition on the torsion ink1is necessary when only one modp k0 group is considered applying a theorem of d d rlat the result also applies to a subclass of theahalgebras

two global stabilizability results for homogeneous systems from a local stabilizability assumption in this technical note two problems have been investigated the first one concerns the global asymptotic feedback stabilization of homogeneous systems for these systems it is shown that local asymptotic stabilization implies the global dynamic feedback stabilization one the second problem consists to show that under a weak assumption on the observability if a state affine system is locally asymptotically stabilizable then it becomes globally asymptotically stabilizable by using a discontinuous output feedback

an infinite family of non isomorphic c algebras with identical k theory we exhibit a countably infinite family of simple separable nuclear and mutually non isomorphic c algebras which agree on k theory and traces the algebras do not absorb the jiang su algebra z tensorially answering a question of n c phillips they are also pairwise shape and morita equivalent confirming a conjecture from our earlier work the distinguishing invariant is the radius of comparison a non stable invariant of the cuntz semigroup

exponential stabilization of driftless nonlinear control systems using homogeneous feedback this paper focuses on the problem of exponential stabilization of controllable driftless systems using time varying homogeneous feedback the analysis is performed with respect to a homogeneous norm in a nonstandard dilation that is compatible with the algebraic structure of the control lie algebra it can be shown that any continuous time varying controller that achieves exponential stability relative to the euclidean norm is necessarily non lipschitz despite these restrictions we provide a set of constructive sufficient conditions for extending smooth asymptotic stabilizers to homogeneous exponential stabilizers the modified feedbacks are everywhere continuous smooth away from the origin and can be extended to a large class of systems with torque inputs the feedback laws are applied to an experimental mobile robot and show significant improvement in convergence rate over smooth stabilizers

receding horizon control of nonlinear systems without differentiability of the optimal value function abstract in this paper we show that receding horizon control when applied to a class of nonlinear systems has stabilizing properties our assumptions are necessarily fairly strong because they guarantee implicity that a smooth stabilizing feedback can be constructed this is an extension of our previous results which require the finite horizon value function to be continuously differentiable and therefore applies to a more restricted class of system continuous differentiability of the value function is replaced by local lipschitz continuity dini derivatives and a generalization of the invariance principle are employed to establish stability