necessary and sufficient small gain conditions for integral input to state stable systems a lyapunov perspective this paper is concerned with conditions for the stability of interconnected nonlinear systems consisting of integral input to state stable iiss systems with external inputs the treatment of iiss and input to state stable iss systems is unified both necessary conditions and sufficient conditions are investigated using a lyapunov formulation in the presence of model uncertainty this paper proves that for the stability of the interconnected system at least one subsystem is necessarily iss which is a stronger stability property in the set of iiss the necessity of a small gain type property is also demonstrated this paper proposes a common form of smooth lyapunov functions which can establish the iiss and the iss of the interconnection comprising iiss and iss subsystems whenever the small gain type condition is satisfied the result covers situations more general than the earlier study and removes technical conditions assumed in the previous literature global asymptotic stability is discussed as a special case

a small gain theorem for switched interconnected nonlinear systems and its applications a small gain theorem for switched interconnected nonlinear systems is set up which provides a tool for analyzing the behavior of switched systems also a method exploiting an average dwell time and the small gain technique is proposed based on the small gain theorem this technical note investigates the problem of global stabilization of switched nonlinear systems in strict feedback form whose all zero dynamics are input to state stable iss or only some are iss and others are not a sufficient condition for the global stabilization problem under a class of switching signals with average dwell time is derived by constructing state feedback controllers for subsystems the result is also applied to a class of switched nonlinear network control systems

normal scalar curvature conjecture and its applications abstract in this paper we proved the normal scalar curvature conjecture and the bottcher wenzel conjecture we developed a new bochner formula and it becomes useful with the first conjecture we proved using the results we established some new pinching theorems for minimal submanifolds in spheres

the second pinching theorem for hypersurfaces with constant mean curvature in a sphere we generalize the second pinching theorem for minimal hypersurfaces in a sphere due to peng terng wei xu zhang and ding xin to the case of hypersurfaces with small constant mean curvature let m n be a compact hypersurface with constant mean curvature h in s n 1 denote by s the squared norm of the second fundamental form of m we prove that there exist two positive constants gamma n and delta n depending only on n such that if h le gamma n and beta n h le s le beta n h delta n then s equiv beta n h and m is one of the following cases i s k big sqrt frac k n big times s n k big sqrt frac n k n big 1 le k le n 1 ii s 1 big frac 1 sqrt 1 mu 2 big times s n 1 big frac mu sqrt 1 mu 2 big here beta n h n frac n 3 2 n 1 h 2 frac n n 2 2 n 1 sqrt n 2h 4 4 n 1 h 2 and mu frac n h sqrt n 2h 2 4 n 1 2

robust stability and stabilization of linear switched systems with dwell time sufficient conditions are given for the stability of linear switched systems with dwell time and with polytopic type parameter uncertainty a lyapunov function in quadratic form which is non increasing at the switching instants is assigned to each subsystem during the dwell time this function varies piecewise linearly in time after switching occurs it becomes time invariant afterwards this function leads to asymptotic stability conditions for the nominal set of subsystems that can be readily extended to the case where these subsystems suffer from polytopic type parameter uncertainties the method proposed is then applied to stabilization via state feedback both for the nominal and the uncertain cases

adaptive fuzzy tracking control of switched uncertain nonlinear systems with unstable subsystems this paper is concerned with the problem of adaptive fuzzy tracking control for a class of switched uncertain nonlinear systems where the solvability of the fuzzy tracking control problem for subsystems is unnecessary fuzzy logic systems are utilized to approximate the unknown nonlinear functions and a new adaptive fuzzy tracking control technique for the problem under study is set up based on the technique above we design a switching law and construct adaptive fuzzy controllers of subsystems explicitly by a recursive design algorithm to guarantee that all the signals in the resulting closed loop system remain bounded and the system output of the closed loop system converges to a small neighborhood of the reference signal multiple lyapunov functions are exploited to get out of the conservativeness caused by adoption of a common lyapunov function for all subsystems which is commonly required when applying the backstepping like recursive design scheme a mass spring damper system is provided to demonstrate the effectiveness of the proposed design method

on the convergence of linear switched systems this paper investigates sufficient conditions for the convergence to zero of the trajectories of linear switched systems we provide a collection of results that use weak dwell time dwell time strong dwell time permanent and persistent activation hypothesis the obtained results are shown to be tight by counterexample finally we apply our result to the three cell converter

a small gain condition for interconnections of iss systems with mixed iss characterizations we consider interconnected nonlinear systems with external inputs where each of the subsystems is assumed to be input to state stable iss sufficient conditions of small gain type are provided guaranteeing that the interconnection is iss with respect to the external input to this end we extend recently obtained small gain theorems to a more general type of interconnections the small gain theorem provided here is applicable to situations where the iss conditions are formulated differently for each subsystem and are either given in the maximization or the summation sense furthermore it is shown that the conditions are compatible in the sense that it is always possible to transform sum formulations to maximum formulations without destroying a given small gain condition an example shows the advantages of our results in comparison with the known ones

global output feedback stabilization for a class of switched uncertain nonlinear systems this paper is mainly concerned with the problem of global output feedback stabilization for a class of switched nonlinear systems with uncertain control coefficients first of all it is shown that the unknown control coefficients are lumped together via a common coordinate transformation thus the original switched nonlinear system is transformed into a new switched system for which control design is feasible second by constructing a common lyapunov function relying on the designed state observer the common output feedback controller independent of switching signals is designed based on the backstepping method so as to guarantee the global asymptotic stability of the resulting closed loop switched system under arbitrary switchings two examples are included for verifying the effectiveness of the method proposed

weyl titchmarsh functions of vector valued sturm liouville operators on the unit interval the matrix valued weyl titchmarsh functions m of vector valued sturm liouville operators on the unit interval with the dirichlet boundary conditions are considered the collection of the eigenvalues i e poles of m and the residues of m is called the spectral data of the operator the complete characterization of spectral data or equivalently n n weyl titchmarsh functions corresponding to n n self adjoint square integrable matrix valued potentials is given if all n eigenvalues of the averaged potential are distinct

non fragile reduced order dynamic output feedback h control for switched systems with average dwell time switching abstractthis paper is concerned with the problem of non fragile reduced order dynamic output feedback h control for both continuous and discrete time switched systems with average dwell time switching first lyapunov conditions for the analysis of asymptotic stability and weighted h performance are presented then sufficient conditions for the solvability of the problem of controller design are developed in terms of linear matrix inequalities the design methods are suitable for general switched systems without the need to impose extra constraints on the system matrices finally numerical examples are presented to illustrate the effectiveness of the proposed methods

switched observer based adaptive neural control of mimo switched nonlinear systems with unknown control gains in this paper the problem of adaptive neural output feedback control is addressed for a class of multi input multioutput mimo switched uncertain nonlinear systems with unknown control gains neural networks nns are used to approximate unknown nonlinear functions in order to avoid the conservativeness caused by adoption of a common observer for all subsystems an mimo nn switched observer is designed to estimate unmeasurable states a new switched observer based adaptive neural control technique for the problem studied is then provided by exploiting the classical average dwell time adt method and the backstepping method and the nussbaum gain technique it effectively handles the obstacle about the coexistence of multiple nussbaum type function terms and improves the classical adt method since the exponential decline property of lyapunov functions for individual subsystems is no longer satisfied it is shown that the technique proposed is able to guarantee semiglobal uniformly ultimately boundedness of all the signals in the closed loop system under a class of switching signals with adt and the tracking errors converge to a small neighborhood of the origin the effectiveness of the approach proposed is illustrated by its application to a two inverted pendulum system

a new characterization of the clifford torus via scalar curvature pinching abstract let m n be a compact hypersurface with constant mean curvature h in s n 1 denote by s the squared norm of the second fundamental form of m we prove that there exists an explicit positive constant n depending only on n such that if h n and n h s n h n 23 then s n h and m is one of the following cases i s k k n s n k n k n 1 k n 1 ii s 1 1 1 2 s n 1 1 2 here n h n n 3 2 n 1 h 2 n n 2 2 n 1 n 2 h 4 4 n 1 h 2 and n h n 2 h 2 4 n 1 2 for n 6 12 24 we give examples to show that the assumption h n cannot be removed

brief paper input to state stability of switched systems and switching adaptive control in this paper we prove that a switched nonlinear system has several useful input to state stable iss type properties under average dwell time switching signals if each constituent dynamical system is iss this extends available results for switched linear systems we apply our result to stabilization of uncertain nonlinear systems via switching supervisory control and show that the plant states can be kept bounded in the presence of bounded disturbances when the candidate controllers provide iss properties with respect to the estimation errors detailed illustrative examples are included

an integral type multiple lyapunov functions approach for switched nonlinear systems an integral type multiple lyapunov functions mlfs approach for switched nonlinear systems is set up for the first time which gives a more general condition for analyzing the behavior of switched nonlinear systems since the branicky s nonincreasing condition is no longer assumed and the generalized mlfs condition is a special case of the condition provided meanwhile based on the integral type mlfs approach global stabilization for a class of switched nonlinear systems in p normal form is achieved by constructing state feedback controllers of subsystems and a proper switching law where the solvability of the problem under study for individual subsystems is not assumed two examples are also provided to demonstrate the effectiveness of the proposed design method

weyl titchmarsh m function asymptotics local uniqueness results trace formulas and borg type theorems for dirac operators we explicitly determine the high energy asymptotics for weyl titchmarsh matrices associated with general dirac type operators on half lines and on r we also prove new local uniqueness results for dirac type operators in terms of exponentially small differences of weyl titchmarsh matrices as concrete applications of the asymptotic high energy expansion we derive a trace formula for dirac operators and use it to prove a borg type theorem

input output to state stability and state norm estimators for switched nonlinear systems in this paper the concepts of input output to state stability ioss and state norm estimators are considered for switched nonlinear systems under average dwell time switching signals we show that when the average dwell time is large enough a switched system is ioss if all of its constituent subsystems are ioss moreover under the same conditions a non switched state norm estimator exists for the switched system furthermore if some of the constituent subsystems are not ioss we show that still ioss can be established for the switched system if the activation time of the non ioss subsystems is not too big again under the same conditions a state norm estimator exists for the switched system however in this case the state norm estimator is a switched system itself consisting of two subsystems we show that this state norm estimator can be constructed such that its switching times are independent of the switching times of the switched system it is designed for

dissipativity theory for switched systems a framework of dissipativity theory for switched systems using multiple storage functions and multiple supply rates is set up each subsystem of a switched system is associated with a storage function to describe the energy stored in the subsystem and is associated with a supply rate that represents energy coming from outside the subsystem when the subsystem is active the exchange of energy between the active subsystem and an inactive subsystem is characterized by cross supply rates stability is reached when all supply rates can be made negative as long as the total exchanged energy between the active subsystem and any inactive subsystems is finite in some sense two special forms of dissipativity passivity and l 2 gain are addressed for both cases asymptotic stability is guaranteed under certain negative output feedback plus asymptotic zero state detectability switched passivity conditions and switched l 2 gain inequalities are respectively derived which are generalizations of classical ones feedback invariance of passivity and a small gain theorem are also given

on chern s conjecture for minimal hypersurfaces and rigidity of self shrinkers using the generalized yau parameter method and the sylvester theory we verify that if m is a compact minimal hypersurface in sn 1 whose squared length of the second fundamental form satisfies 0 a 2 n n22 then a 2 n and m is a clifford torus moreover we prove that if m is a complete self shrinker with polynomial volume growth in rn 1 and if the squared length of the second fundamental form of m satisfies 0 a 2 1 121 then a 2 1 and m is a round sphere or a cylinder our results improve the rigidity theorems due to ding and xin 21 22

a simplified small gain theorem for time varying nonlinear systems the small gain theorem as established in two previous papers has been given in terms of a quite general interconnection of two nonlinear subsystems this note gives a simplified version and a more straightforward proof for the small gain theorem for a simpler interconnection of two nonlinear subsystems by using a different characterization of the input to state stable iss concept the resulting small gain condition features the familiar contraction mapping inequality for time invariant systems moreover we have also incorporated some type of model uncertainties into the system description thus making this special version directly applicable to the global robust stabilization and output regulation problems

embedded minimal tori in s3 and the lawson conjecture we show that any embedded minimal torus in s 3 is congruent to the clifford torus this answers a question posed by h b lawson jr in 1970

the first eigenvalue of the laplacian of an isoparametric minimal hypersurface in a unit sphere soit m n 1 une variete de riemann connexe compacte a n 1 dimensions sans bord et son laplacien agissant sur des fonctions lisses sur m a un spectre discret 0 0 m 1 m 2 m k m x soit f une immersion minimale de m n 1 dans la sphere unite a n dimensions s n 1 de l espace euclidien on etudie la premiere valeur propre du laplacien d une hypersurface isoparametrique minimale fermee dans s n 1

state dependent scaling problems and stability of interconnected iiss and iss systems this paper addresses the problem of establishing stability of nonlinear interconnected systems this paper introduces a mathematical formulation of the state dependent scaling problems whose solutions directly provide lyapunov functions proving stability properties of interconnected dissipative systems in a unified manner stability criteria are interpreted as sufficient conditions for the existence of solutions to the state dependent scaling problems computing solutions to the problems is straightforward for systems covered by classical stability criteria it however could be too difficult for systems with strong nonlinearity the main purpose of this paper is to demonstrate the effectiveness beyond formal applicability by focusing on interconnected integral input to state stable iiss systems and input to state stable iss systems this paper derives small gain type theorems for interconnected systems involving iiss systems from the state dependent scaling formulation this paper provides solutions and lyapunov functions explicitly the new framework seamlessly generalizes the iss small gain theorem and classical stability criteria such as the lp small gain theorem the passivity theorems the circle and popov criteria state dependence of the scaling is crucial for effective treatment of essential nonlinearities while constants are sufficient for classical nonlinearities

embedded weingarten tori in s3 abstract in this paper we show that an embedded weingarten surface in s 3 of genus 1 must be rotationally symmetric provided that certain structure conditions are satisfied the argument involves an adaptation of our proof of lawson s conjecture for embedded minimal tori

technical communique stabilizing controllers design for switched nonlinear systems in strict feedback form this paper considers the stabilization problem for a class of switched nonlinear systems under arbitrary switching based on the backstepping method and the control lyapunov function approach it is shown that under a simultaneous domination assumption a switched nonlinear system in strict feedback form can be globally uniformly asymptotically stabilized by a continuous state feedback controller a universal formula for constructing stabilizing feedback laws is presented one example is included for verifying the obtained results

h_ infty control of switched nonlinear systems in p normal form using multiple lyapunov functions the problem of h control of switched nonlinear systems in p normal form is investigated in this technical note where the solvability of the h control problem for individual subsystems is unnecessary using the generalized multiple lyapunov functions method and the adding a power integrator technique we design a switching law and construct continuous state feedback controllers of subsystems explicitly by a recursive design algorithm to produce global asymptotical stability and a prescribed h performance level multiple lyapunov functions are exploited to reduce the conservativeness caused by adoption of a common lyapunov function for all subsystems which is usually required when applying the backstepping like recursive design scheme an example is provided to demonstrate the effectiveness of the proposed design method

multiple lyapunov functions based small gain theorems for switched interconnected nonlinear systems multiple lyapunov functions mlfs based small gain theorems are presented for switched interconnected nonlinear systems with unstable subsystems which extend the small gain technique from its original non switched nonlinear version to a switched nonlinear version each low dimensional subsystem does not necessarily have the input to state stability iss property in the whole state space and it only has individual iss property in some subregions of the state space the novelty of this paper is that integral type mlfs and small gain techniques are utilized to establish some mlfs based small gain theorems for switched interconnected nonlinear systems which derive various stability results under some novel switching laws designed and construct integral type mlfs the small gain theorems proposed cover several recent results as special cases which also permit removal of a common restriction in which all low dimensional subsystems in switched interconnected systems are iss or only some are iss and others are not finally two illustrative examples are presented to demonstrate the effectiveness of the results provided

closed curve solutions and limit cycles in a class of second order switched nonlinear systems this paper studies the problem of finding the initial states for which the solution of a class of switched systems consisting of unstable second order nonlinear subsystems is convergent a method is described and applied to establish the regions in the plane where it is possible to define a switching law such that the solution of a class of switched nonlinear systems converges to the origin we prove that under certain conditions these regions are delimited by closed curve solutions of the switched system furthermore a sufficient condition for the closed curve solution to be a limit cycle is presented finally a numerical example is included in order to illustrate the results

on chernʼs problem for rigidity of minimal hypersurfaces in the spheres abstract for a compact minimal hypersurface m in s n 1 with the squared length of the second fundamental form s we confirm that there exists a positive constant n depending only on n such that if n s n n then s n i e m is a clifford minimal hypersurface in particular when n 6 the pinching constant n n 23

embedded constant mean curvature tori in the three sphere we prove that any constant mean curvature embedded torus in the three dimensional sphere is axially symmetric and use this to give a complete classification of such surfaces for any given value of the mean curvature

inverse spectral problems for dirac operators on a finite interval abstract we consider the direct and inverse spectral problems for dirac operators that are generated by the differential expressions t q 1 i i 0 0 i d d x 0 q q 0 and some separated boundary conditions here q is an r r matrix valued function with entries belonging to l 2 0 1 c and i is the identity r r matrix we give a complete description of the spectral data eigenvalues and suitably introduced norming matrices for the operators under consideration and suggest an algorithm of reconstructing the potential q from the corresponding spectral data

a nonlinear small gain theorem for the analysis of control systems with saturation a nonlinear small gain theorem is presented that provides a formalism for analyzing the behavior of certain control systems that contain or utilize saturation the theorem is used to show that an iterative procedure can be derived for controlling systems in a general nonlinear feedforward form this result in turn is applied to the control of 1 linear systems stable and unstable with inputs subject to magnitude and rate saturation and time delays 2 the cascade of globally asymptotically stable nonlinear systems with certain linear systems those that are stabilizable right invertible and such that all of their invariant zeros have nonpositive real part 3 the inverted pendulum on a cart and 4 the planar vertical takeoff and landing aircraft

robust adaptive fuzzy control of a class of uncertain switched nonlinear systems with mismatched uncertainties this paper studies the adaptive fuzzy control problem for a class of uncertain switched nonlinear systems with mismatched uncerta inties and external disturbances a new switched adaptive control technique for the problem studied is set up such that the considered switched systems can be stabilized under a class of switching signals characterized by the generalized average dwell time condition first the adaptive state feedback controller is designed based on tuning the estimations of the bounds on switching parameters instead of estimating the switching parameters directly next by using a switched observer to reconstruct the states of the system an output feedback controller is also presented in the design procedure fuzzy logic systems are used to approximate the unknown nonlinear functions to overcome the undesirable chattering phenomenon both control laws are developed using the continuous control strategy it is shown that both controllers can guarantee that all the signals in the resulting closed loop system are bounded finally two simulation examples are presented to demonstrate the effectiveness of the proposed control results

on an inverse scattering problem for a class of dirac operators with spectral parameter in the boundary condition abstract in this work we consider the inverse scattering problem for a class of one dimensional dirac operators on the semi infinite interval with the boundary condition depending polynomially on a spectral parameter the scattering data of the given problem is defined and its properties are examined the main equation is derived its solvability is proved and it is shown that the potential is uniquely recovered in terms of the scattering data a generalization of the marchenko method is given for a class of dirac operator

geometrical quasi dissipativity and boundedness property for switched discrete time nonlinear systems this paper studies geometrical quasi dissipativity and boundedness properties for switched discrete time nonlinear systems first a new concept of geometrical quasi dissipativity for a switched di

a generalization of the small gain theorem for nonlinear feedback systems abstract a generalization is given of the usual small gain theorem which allows greater flexibility in use of gain bounds for stability analysis of nonlinear feedback systems

exponential stabilization of singularly perturbed switched systems subject to actuator saturation this paper considers the problem of exponential stabilization for a class of singularly perturbed switched systems subject to actuator saturation a time dependent switching law is designed to obtain the exponential stabilization conditions of the switched systems with unstabilizable subsystems based on the multiple lyapunov functions method depending on singular perturbation parameters a domain of attraction is constructed furthermore in order to maximize an estimate of the domain of attraction convex optimization problems are formulated finally a numerical example is presented in order to illustrate the effectiveness of the established results

a polynomial approach for stability analysis of switched systems a polynomial approach to deal with the stability analysis of switched non linear systems under arbitrary switching using dissipation inequalities is presented it is shown that a representation of the original switched problem into a continuous polynomial system allows us to use dissipation inequalities for the stability analysis of polynomial systems with this method and from a theoretical point of view we provide an alternative way to search for a common lyapunov function for switched non linear systems

stability and stabilizability of switched linear systems a survey of recent results during the past several years there have been increasing research activities in the field of stability analysis and switching stabilization for switched systems this paper aims to briefly survey recent results in this field first the stability analysis for switched systems is reviewed we focus on the stability analysis for switched linear systems under arbitrary switching and we highlight necessary and sufficient conditions for asymptotic stability after a brief review of the stability analysis under restricted switching and the multiple lyapunov function theory the switching stabilization problem is studied and a variety of switching stabilization methods found in the literature are outlined then the switching stabilizability problem is investigated that is under what condition it is possible to stabilize a switched system by properly designing switching control laws note that the switching stabilizability problem has been one of the most elusive problems in the switched systems literature a necessary and sufficient condition for asymptotic stabilizability of switched linear systems is described here

switched adaptive control of switched nonlinearly parameterized systems with unstable subsystems this paper investigates the problem of adaptive stabilization for a class of switched nonlinearly parameterized systems where the solvability of the adaptive stabilization problem for subsystems is unnecessary a new switched adaptive control technique for the problem studied is set up by exploiting the generalized multiple lyapunov functions method and the parameter separation technique different update laws are designed to reduce the conservativeness caused by adoption of a common update law for all subsystems also the proposed technique permits removal of a common restriction in which the parameterization in the switched systems is restricted to a linear parameterization as an application of the proposed design technique stabilization for a class of non triangular switched systems with nonlinear parameterization is achieved by design of adaptive controllers since no subsystem of such a switched system is assumed to be adaptively stabilizable the problem under study cannot be handled by the existing methods a two inverted pendulums as a practical example is also provided to demonstrate the effectiveness of the proposed design method

decentralized adaptive fuzzy output feedback control of switched large scale nonlinear systems in this paper a decentralized adaptive fuzzy output feedback control problem is discussed for a class of switched large scale nonlinear systems without the measurements of the system states fuzzy logic systems are used to approximate the unknown nonlinear functions and a switched observer is designed to avoid the conservativeness caused by adoption of a common observer for all subsystems a common coordinate transformation of all subsystems is constructed to get rid of individual coordinate transformations for subsystems that are required when applying the backstepping recursive design scheme meanwhile since the exponential decline property of lyapunov functions for individual subsystems is no longer satisfied in this paper the classical average dwell time method cannot be directly applied based on this a switched observer based decentralized adaptive fuzzy output feedback control technique is set up by exploiting the average dwell time method and backstepping it is proved that all the signals of the resulting closed loop system are semiglobally uniformly ultimately bounded under a class of switching signals with average dwell time while the tracking errors converge to a small neighborhood of the origin the effectiveness of the proposed control scheme is illustrated by its applications to a two inverted pendulum system and a quadruple tank process system

hypersurfaces with constant scalar curvature in space forms let rn l c be an n 1 dimensional riemannian manifold with constant sectional curvature c we also call it space form when c 1 r l c s l i e n 1 dimensional unit sphere space when c o rn t c e n l i e n 1 dimensional euclidean space let m be an n dimensional compact hypersurface in rn l c and el e a local orthonormal frame field on m col con its dual coframe field then the second fundamental form of m is

a lyapunov based small gain theorem for interconnected switched systems abstract stability of an interconnected system consisting of two switched systems is investigated in the scenario where in both switched systems there may exist some subsystems that are not input to state stable non iss we show that providing the switching signals neither switch too frequently nor activate non iss subsystems for too long a small gain theorem can be used to conclude global asymptotic stability gas of the interconnected system for each switched system with the constraints on the switching signal being modeled by an auxiliary timer a correspondent hybrid system is defined to enable the construction of a hybrid iss lyapunov function apart from justifying the iss property of their corresponding switched systems these hybrid iss lyapunov functions are then combined to establish a lyapunov type small gain condition which guarantees that the interconnected system is globally asymptotically stable