control lyapunov razumikhin functions and robust stabilization of time delay systems motivated by control lyapunov functions and razumikhin theorems on stability of time delay systems we introduce the concept of control lyapunov razumikhin functions clrf the main reason for considering clrfs is construction of robust stabilizing control laws for time delay systems most existing universal formulas that apply to clfs are not applicable to clrfs it turns out that the domination redesign control law applies achieving global practical stability and under an additional assumption global asymptotic stability this additional assumption is satisfied in the practically important case when the quadratic part of a clrf is itself a clrf for the jacobian linearization of the system the clrf based domination redesign possesses robustness to input unmodeled dynamics including an infinite gain margin while in general construction of clrfs is an open problem we show that for several classes of time delay systems a clrf can be constructed in a systematic way

direct adaptive nn control of a class of nonlinear systems in this paper direct adaptive neural network nn control is presented for a class of affine nonlinear systems in the strict feedback form with unknown nonlinearities by utilizing a special property of the affine term the developed scheme avoids the controller singularity problem completely all the signals in the closed loop are guaranteed to be semiglobally uniformly ultimately bounded and the output of the system is proven to converge to a small neighborhood of the desired trajectory the control performance of the closed loop system is guaranteed by suitably choosing the design parameters simulation results are presented to show the effectiveness of the approach

design of fuzzy control systems with guaranteed stability abstract this paper addresses the analysis and design of fuzzy control systems the fuzzy systems are represented by a family of local state space models with aggregation the controller is designed by considering each local state feedback controller and a compensating controller the compensating controller is based on the well known variable structure control theory it is shown that this controller guarantees global stability of the closed loop fuzzy system the issues of a state observer for the fuzzy system is also addressed a numerical example and an example of nonlinear system control are given for demonstration of the performance of the method

adaptive fuzzy output feedback control for output constrained nonlinear systems in the presence of input saturation abstract in this paper the problem of adaptive fuzzy output feedback control is investigated for a class of output constrained uncertain nonlinear systems with input saturation and unmeasured states to address output constraint and input constraint a barrier lyapunov function and an auxiliary design system are employed respectively fuzzy logic systems are utilized to approximate the unknown nonlinear functions and a fuzzy state observer is designed to estimate the unmeasured states by utilizing the fuzzy state observer and combining the adaptive backstepping technique with adaptive fuzzy control design an observer based adaptive fuzzy output feedback control approach is developed it is proved that the proposed control approach can guarantee that all the signals in the closed loop system are bounded and the input and output constraints are circumvented simultaneously a numerical example is provided to illustrate the effectiveness of the proposed approach

robust h sub spl infin and mixed h sub 2 h sub spl infin filters for equalization designs of nonlinear communication systems fuzzy interpolation approach generally it is difficult to design equalizers for signal reconstruction of nonlinear communication channels with uncertain noises in this paper we propose the h sub spl infin and mixed h sub 2 h sub spl infin filters for equalization detection of nonlinear channels using fuzzy interpolation and linear matrix inequality lmi techniques first the signal transmission system is described as a state space model and the input signal is embedded in the state vector such that the signal reconstruction estimation design becomes a nonlinear state estimation problem then the takagi sugeno fuzzy linear model is applied to interpolate the nonlinear channel at different operation points through membership functions since the statistics of noises are unknown the fuzzy h sub spl infin equalizer is proposed to treat the state estimation problem from the worst case robust point of view when the statistics of noises are uncertain but with some nominal or average information available the mixed h sub 2 h sub spl infin equalizer is employed to take the advantage of both h sub 2 optimal performance with nominal statistics of noises and the h sub spl infin robustness performance against the statistical uncertainty of noises using the lmi approach the fuzzy h sub 2 h sub spl infin equalizer detector design problem is characterized as an eigenvalue problem evp the evp can be solved efficiently with convex optimization techniques

the classification of complete stable area stationary surfaces in the heisenberg group h1 abstract we prove that any c 2 complete orientable connected stable area stationary surface in the sub riemannian heisenberg group h 1 is either a euclidean plane or congruent to the hyperbolic paraboloid t x y

h sub spl infin nonlinear filtering of discrete time processes this correspondence investigates the problem of h sub spl infin estimation of a discrete time nonlinear process an estimator which may be nonlinear is introduced so that an h sub spl infin norm like of what we call a generalized estimation error is guaranteed to be bounded by a prescribed level conditions for the existence of such an estimator and formulae for its derivation are obtained utilizing a discrete time analog of the hamilton jacobi inequality an approximate filter based on linearization is developed this filter relates to the extended kalman filter in the same way that the linear h sub spl infin filter relates to the kalman filter

two step stability analysis for general polynomial fuzzy model based control systems this paper investigates the stability of a polynomial fuzzy model based pfmb control system formed by a nonlinear plant represented by a polynomial fuzzy model and a polynomial fuzzy controller connected in a closed loop three cases of polynomial fuzzy controllers are proposed for the control process with the consideration of a matched mismatched number of rules and or premise membership functions which demonstrate different levels of controller complexity design flexibility and stability analysis results a general polynomial lyapunov function candidate is proposed to investigate the system stability unlike the published work there is no constraint on the polynomial lyapunov function candidate which is independent of the form of the polynomial fuzzy model thus it can be applied to a wider class of pfmb control systems and potentially produces more relaxed stability analysis results two step stability conditions in terms of sum of squares sos are obtained to numerically find a feasible solution to facilitate the stability analysis and relax the stability analysis result the boundary information of membership functions is taken into account in the stability analysis and incorporated into the sos based stability conditions simulation examples are given to illustrate the effectiveness of the proposed approach

global adaptive neural control for strict feedback time delay systems with predefined output accuracy this paper focuses on the stabilization problem for a class of nonlinear strict feedback systems with disturbances and time delay by constructing some nonnegative functionals a multiswitching based adaptive backstepping neural state feedback controller is designed compared with all the existing control methodologies for the uncertain time delay systems the outstanding merits of the proposed control scheme are presented as follows first the controller guarantees that all the signals in the closed loop system remain globally bounded meanwhile the system output achieves an accuracy that is given a priori according to the practical requirements second in contrast to the classical adaptive backstepping neural control schemes we analyze the convergence of the tracking error by using barbalat s lemma instead of lyapunov stability theory third the main technical novelty is to construct three new nth order continuously differentiable functions which are used to design the actual controller the virtual control variables and the adaptive laws two simulation examples are provided to illustrate the effectiveness and advantage of the proposed control scheme

robust backstepping control of a class of nonlinear systems using fuzzy logic abstract fuzzy logic fl controllers for the robust back stepping control of a class of nonlinear systems in both continuous and discrete time are presented control action is employed to achieve tracking performance for unknown nonlinear system novel parameter tuning algorithms are obtained that are similar to modification in the case of continuous time adaptive control uniform ultimate boundedness of the tracking error and the fuzzy parameter estimates are presented without using the persistency of excitation pe condition initialization of the fuzzy parameters is straightforward simulation results justify the theoretical conclusions

h spl infin deconvolution filtering prediction and smoothing a krein space polynomial approach this correspondence is concerned with the infinite horizon h spl infin deconvolution for discrete time invariant systems described by polynomial equations a simple frequency domain approach in krein space is developed for the deconvolution problem with the help of an arma innovation model solutions to the problems of deconvolution filtering prediction and fixed lag smoothing are obtained the estimators are calculated by performing one j spectral factorization

a method of moments approach for the efficient and accurate modeling of moderately thick cylindrical wire antennas this paper introduces a moment method formulation which is capable of accurately modeling moderately thick cylindrical wire antennas new algorithms are presented for the efficient computation of the cylindrical wire kernel and related impedance matrix integrals these algorithms make use of exact series representations as well as efficient numerical procedures and lead to a significant reduction in overall computation time for thicker wires another major advantage of this moment method technique is that it is no longer restricted by the segment length to radius ratio limitations inherent in past formulations thereby making it possible to achieve solution convergence for a much wider class of wire antenna structures several examples illustrating the superior convergence properties of this new moment method formulation are presented and discussed

on the oscillations appearing in numerical solutions of solvable and nonsolvable integral equations for thin wire antennas differences between certain solvable and nonsolvable ill posed integral equations with the same nonsingular kernel are discussed the main results come from constructing a solvable equation in the context of straight thin wire antennas the kernel of this equation is the usual approximate also called reduced kernel while its exact solution is the familiar sinusoidal current numerical solutions to this solvable equation are compared to corresponding numerical solutions of the usual halle n and pocklington equations with the approximate kernel it is known from previous publications that these last two equations are nonsolvable and that their numerical solutions present severe oscillations when the number of basis functions is sufficiently large it is found that the difficulties encountered in the former solvable equation are much less important compared to those of the nonsolvable ones the same conclusion is brought out from other integral equations arising in different contexts thin wire circular loop antenna method of auxiliary sources and straight wire antenna of infinite length we discuss the consistency of our results with picard s theorem the results in this paper supplement previous publications regarding the difficulties of numerically solving thin wire integral equations with the approximate kernel

corrections to sub riemannian geometry an error in the proof of corollary 6 2 of 1 has been pointed out by gerard ben arous the computation of m x in the case o 0 on p 243 is incorrect because m x 0 when 0 0 and jg x 0 for all k there is also a factor of missing in the formula as stated for 0 0 but this is not significant thus when applying the pontryagin theorem theorem 6 1 it is necessary to consider one additional case o 0 and m x9 0 along the curve the conditions of the theorem in this case lead to the equations

robust spl hscr sub spl infin filtering for uncertain discrete time state delayed systems this paper addresses the problem of robust spl hscr sub spl infin filtering for linear discrete time systems subject to parameter uncertainties in the system state space model and with multiple time delays in the state variables the uncertain parameters are supposed to belong to a given convex bounded polyhedral domain a methodology is developed to design a stable linear filter that assures asymptotic stability and a prescribed spl hscr sub spl infin performance for the filtering error irrespective of the uncertainty and the time delays the proposed design is given in terms of linear matrix inequalities which has the advantage in that it can be implemented numerically very efficiently

adaptive neural control for a class of stochastic nonlinear systems by backstepping approach this paper addresses adaptive neural control for a class of stochastic nonlinear systems which are not in strict feedback form based on the structural characteristics of radial basis function rbf neural networks nns a backstepping design approach is extended from stochastic strict feedback systems to a class of more general stochastic nonlinear systems in the control design procedure rbf nns are used to approximate unknown nonlinear functions and the backstepping technique is utilized to construct the desired controller the proposed adaptive neural controller guarantees that all the closed loop signals are bounded and the tracking error converges to a sufficiently small neighborhood of the origin two simulation examples are used to illustrate the effectiveness of the proposed approach

adaptive neural tracking control for stochastic nonlinear strict feedback systems with unknown input saturation in this paper the problem of adaptive neural tracking control is considered for a class of single input single output siso strict feedback stochastic nonlinear systems with input saturation to deal with the non smooth input saturation nonlinearity a smooth nonaffine function of the control input signal is used to approximate the input saturation function classical adaptive technique and backstepping are used for control synthesis based on the mean value theorem a novel adaptive neural control scheme is systematically derived without requiring the prior knowledge of bound of input saturation it is shown that under the action of the proposed adaptive controller all the signals of the closed loop system remain bounded in probability and the tracking error converges to a small neighborhood around the origin in the sense of mean quartic value two simulation examples are provided to demonstrate the effectiveness of the presented results

robust spl hscr sub spl infin filtering for continuous time varying uncertain systems with deterministic input signals many dynamical systems involve not only process and measurement noise signals but also parameter uncertainty and known input signals when spl lscr sub 2 or spl hscr sub spl infin filters that were designed based on a nominal model of the system are applied the presence of parameter uncertainty will not only affect the noise attenuation property of the filter but also introduce a bias proportional to the known input signal and the latter may be very appreciable we introduce a finite horizon robust spl hscr sub spl infin filtering method that provides a guaranteed spl hscr sub spl infin bound for the estimation error in the presence of both parameter uncertainty and a known input signal this method is developed by using a game theoretic approach and the results generalize those obtained for cases without parameter uncertainty or without a known input signal it is also demonstrated via an example that the proposed method provides significantly improved signal estimates

an exact formulation for the vector potential of a cylindrical antenna with uniformly distributed current and arbitrary radius an exact formulation is given for the vector potential and corresponding electric fields associated with a uniform current cylindrical dipole antenna of arbitrary radius the exact expressions converge rapidly in the induction and near field regions of the antenna they are completely general and independent of the usual restrictions involving the wavelength field point distance dipole radius and length the properties of these expressions allow them to be readily used for the efficient as well as accurate computational modeling of cylindrical wire antennas an exact expression for the cylindrical wire kernel which is analytically as well as numerically convenient to work with is presented it is demonstrated that the integral of the kernel can be expressed in terms of a series which involves a generalized exponential integral and higher order associated integrals several methods for evaluating the generalized exponential integral are discussed including some useful asymptotic expansions a numerically stable forward recurrence relation for the higher order associated integrals is also presented

fuzzy h_ infty filter design for a class of nonlinear discrete time systems with multiple time delays this paper studies the fuzzy hinfin filter design problem for signal estimation of nonlinear discrete time systems with multiple time delays and unknown bounded disturbances first the takagi sugeno t s fuzzy model is used to represent the state space model of nonlinear discrete time systems with time delays next we design a stable fuzzy hinfin filter based on the t s fuzzy model which guarantees asymptotic stability and a prescribed hinfin index for the filtering error system irrespective of the time delays and uncertain disturbances a sufficient condition for the existence of such a filter is established by using the linear matrix inequality lmi approach the proposed lmi problem can be efficiently solved with global convergence guarantee using convex optimization techniques such as the interior point algorithm simulation examples are provided to illustrate the design procedure of the present method

adaptive feedback control of nonlinear time delay systems the lasalle razumikhin based approach this note presents a new method to design adaptive feedback controller for nonlinear time delay systems first by using the lasalle razumikhin theorem a sufficient condition is derived that ensures the convergence of a part of the solution with stability for a class of functional differential equations then using this condition the adaptive stabilization problem is solved for nonlinear time delay systems moreover it is shown that for a class of nonlinear time delay systems with triangular structure the adaptive stabilizing controller can be obtained by recursively constructing the lyapunov razumikhin function it will be shown that the provided recursive design approach which is obviously motivated by the typical backstepping method is not a trivial extension of the existing design method

robust estimation of a single complex sinusoid in white noise h sub spl infin filtering approach a novel robust estimator is proposed for extracting a single complex sinusoid and its parameter frequency from measurements corrupted by white noise this estimator is called an h sub spl infin sinusoidal estimator hse which is derived by applying an h sub spl infin filter to a noisy sinusoidal model with the state space representation simulations demonstrate that the hse is more robust to the nature of observation noise v sub k than the kalman sinusoidal estimator kse which is an improved version of the nonlinear filter previously proposed by the author

area stationary surfaces in the heisenberg group h1 we consider area stationary surfaces perhaps with a volume constraint in the heisenberg group h1 endowed with its carnot caratheodory distance by analyzing the first variation of area we characterize c2 area stationary surfaces as those with mean curvature zero or constant if a volume preserving condition is assumed and such that the characteristic curves meet orthogonally the singular curves moreover a minkowski type formula relating the area the mean curvature and the volume is obtained for volume preserving area stationary surfaces enclosing a given region as a consequence of the characterization of area stationary surfaces we refine the bernstein type theorem given in jih hsin cheng jenn fang hwang andrea malchiodi paul yang minimal surfaces in pseudohermitian geometry ann sc norm super pisa cl sci 5 4 1 2005 129 177 mr 2165405 and nicola garofalo scott pauls the bernstein problem in the heisenberg group arxiv math dg 0209065 to describe all c2 entire area stationary graphs over the xy plane in a calibration argument shows that these graphs are globally area minimizing finally by using the description of the singular set in jih hsin cheng jenn fang hwang andrea malchiodi paul yang minimal surfaces in pseudohermitian geometry ann sc norm super pisa cl sci 5 4 1 2005 129 177 mr 2165405 the characterization of area stationary surfaces and the ruling property of constant mean curvature surfaces we prove our main results where we classify area stationary surfaces in with or without a volume constraint and non empty singular set in particular we deduce the following counterpart to alexandrov uniqueness theorem in euclidean space any compact connected c2 surface in area stationary under a volume constraint must be congruent to a rotationally symmetric sphere obtained as the union of all the geodesics of the same curvature joining two points as a consequence we solve the isoperimetric problem in assuming c2 smoothness of the solutions 13 completely reducible hypersurfaces in a pencil pages 672 688 j v pereira s yuzvinsky preview pdf 324 k related articles 14 modules over motivic cohomology pages 689 727 oliver rondigs paul arne ostvaer preview pdf 303 k related articles 15 the fundamental group of period domains over finite fields pages 728 741 sascha orlik preview pdf 154 k related articles articles 1 15

l2 l filter design for discrete time singular markovian jump systems with time varying delays the problem of l 2 l filter design for discrete time singular systems with markovian jump and time varying delays is investigated in this paper by using the delay partitioning technique a delay dependent condition is established to guarantee the filtering error systems to be stochastically admissible and achieve a prescribed l 2 l performance index based on the derived condition the full order and reduced order filters with mode independent characterization are designed in a unified framework the corresponding filter parameters can be obtained by solving a set of linear matrix inequalities the reported results not only depend upon the delay but also depend upon the partitioning which aims at reducing the conservatism two numerical examples are provided to illustrate the effectiveness of the proposed methods

instability of graphical strips and a positive answer to the bernstein problem in the heisenberg group h1 in the first heisenberg group h 1 with its sub riemannian structure generated by the horizontal subbundle we single out a class of c 2 non characteristic entire intrinsic graphs which we call strict graphical strips we prove that such strict graphical strips have vanishing horizontal mean curvature i e they are h minimal and are unstable i e there exist compactly supported deformations for which the second variation of the horizontal perimeter is strictly negative we then show that modulo left translations and rotations about the center of the group every c 2 entire hminimal graph with empty characteristic locus and which is not a vertical plane contains a strict graphical strip combining these results we prove the conjecture that in h 1 the only stable c 2 hminimal entire graphs with empty characteristic locus are the vertical planes

strongly stable surfaces in sub riemannian 3 space forms abstract a surface of constant mean curvature cmc equal to h in a sub riemannian 3 manifold is strongly stable if it minimizes the functional area 2 h volume up to second order in this paper we obtain some criteria ensuring strong stability of surfaces in sasakian 3 manifolds we also produce new examples of c 1 complete cmc surfaces with empty singular set in the sub riemannian 3 space forms by studying those ones containing a vertical line as a consequence we are able to find complete strongly stable non vertical surfaces with empty singular set in the sub riemannian hyperbolic 3 space m 1 in relation to the bernstein problem in m 1 we discover strongly stable c entire minimal graphs in m 1 different from vertical planes these examples are in clear contrast with the situation in the first heisenberg group where complete strongly stable surfaces with empty singular set are vertical planes finally we analyze the strong stability of cmc surfaces of class c 2 and non empty singular set in the sub riemannian 3 space forms when these surfaces have isolated singular points we deduce their strong stability even for variations moving the singular set

robust stability and stabilizing controller design of fuzzy systems with discrete and distributed delays in this paper we consider delay dependent stability conditions of takagi sugeno fuzzy systems with discrete and distributed delays although many kinds of stability conditions for fuzzy systems with discrete delays have already been obtained almost no stability condition for fuzzy systems with distributed delays has appeared in the literature this is also true in case of the robust stability for uncertain fuzzy systems with distributed delays here we employ a generalized lyapunov functional to obtain delay dependent stability conditions of fuzzy systems with discrete and distributed delays we introduce some free weighting matrices to such a lyapunov functional in order to reduce the conservatism in stability conditions these techniques lead to generalized and less conservative stability conditions we also consider the robust stability of fuzzy time delay systems with uncertain parameters applying the same techniques made on the stability conditions we obtain delay dependent sufficient conditions for the robust stability of uncertain fuzzy systems with discrete and distributed delays moreover we consider the state feedback stabilization based on stability and robust stability conditions we obtain conditions for the state feedback controller to stabilize the fuzzy time delay systems finally we give two examples to illustrate our results delay dependent stability conditions obtained here are shown to guarantee a wide stability region

area stationary surfaces inside the sub riemannian three sphere we consider the sub riemannian metric gh on mathbb s 3 given by the restriction of the riemannian metric of curvature 1 to the plane distribution orthogonal to the hopf vector field we compute the geodesics associated to the carnot caratheodory distance and we show that depending on their curvature they are closed or dense subsets of a clifford torus we study area stationary surfaces with or without a volume constraint in mathbb s 3 g h by following the ideas and techniques by ritore and rosales area stationary surfaces in the heisenberg group mathbb h 1 arxiv math dg 0512547 we introduce a variational notion of mean curvature characterize stationary surfaces and prove classification results for complete volume preserving area stationary surfaces with non empty singular set we also use the behaviour of the carnot caratheodory geodesics and the ruling property of constant mean curvature surfaces to show that the only c2 compact connected embedded surfaces in mathbb s 3 g h with empty singular set and constant mean curvature h such that h sqrt 1 h 2 is an irrational number are clifford tori finally we describe which are the complete rotationally invariant surfaces with constant mean curvature in mathbb s 3 g h

on the application of numerical methods to hallen s equation the so called hallen integral equation for the current on a finite linear antenna center driven by a delta function generator takes two forms depending on the choice of kernel the two kernels are usually referred to as the exact and the approximate or reduced kernel with the approximate kernel the integral equation has no solution nevertheless the same numerical method is often applied to both forms of the integral equation in this paper the behavior of the numerical solutions thus obtained is investigated and the similarities and differences between the two numerical solutions are discussed the numerical method is galerkin s method with pulse functions we first apply this method to the two corresponding forms of the integral equation for the current on a linear antenna of infinite length in this case the method yields an infinite toeplitz system of algebraic equations in which the width of the pulse basis functions enters as a parameter the infinite system is solved exactly for nonzero pulse width the exact solution is then developed asymptotically for the case where the pulse width is small when the asymptotic expressions for the case of the infinite antenna are used as a guide for the behavior of the solutions of the finite antenna the latter problem is greatly facilitated for the approximate kernel the main results of this paper carry over to a certain numerical method applied to the corresponding equation of the pocklington type

single parameter learning based fuzzy fault tolerant output feedback dynamic surface control of constrained input nonlinear systems abstract this paper addresses the problem of adaptive fuzzy fault tolerant dynamic surface control for a class of constrained input nonlinear systems to resolve this problem the design of an observer based single parameter learning spl control method using output feedback is proposed the takagi sugeno t s fuzzy system is used to identify and approximate online the uncertain nonlinear dynamics requiring no knowledge the barriers that restrict the applications of the traditional backstepping and approximation based approach including the explosion of complexity and the dimension curse problems are circumvented via dynamic surface control and spl techniques the merit of the proposed method lies in that only one parameter in the entire control scheme requires online adjustment regardless of the number of parameters in the t s fuzzy system that characterizes the fuzzy rules the calculation burden in this sense is reduced to the extent of the minimum value the truncated adaptation method is used to avoid the chattering and instability caused by constrained input it is shown with rigorous proof using the lyapunov and invariant set theorems that all the closed loop signals are guaranteed semi globally uniformly ultimately bounded the output tracking error is adjustable by means of design parameters in an explicit form and can be adjusted to an arbitrarily small value around zero by appropriately chosen control parameters even under faulty and constrained actuators simulation and comparative results are provided to demonstrate the effectiveness of the proposed control approach

decentralized adaptive output feedback control for a class of nonlinear large scale systems with unknown time varying delayed interactions in this paper the authors investigate a decentralized adaptive output feedback controller design for large scale nonlinear systems with input saturations and time delayed interconnections unmatched in control inputs the interaction terms with unknown time varying delays are bounded by unknown nonlinear bounding functions including all states of subsystems this point is a main contribution of this paper compared with previous output feedback control approaches which assume that the time delayed bounding functions only depend on measurable output variables the bounding functions are compensated by using appropriate lyapunov krasovskii functionals and the function approximation technique based on neural networks the observer dynamic surface design technique is employed to design the proposed memoryless local controller for each subsystem in addition we prove that all signals in the closed loop system are semiglobally uniformly bounded and control errors converge to an adjustable neighborhood of the origin finally an example is provided to illustrate the effectiveness of the proposed control system

direct adaptive fuzzy control for nonlinear systems with time varying delays this paper focuses on the problem of direct adaptive fuzzy control for nonlinear strict feedback systems with time varying delays based on the razumikhin function approach a novel adaptive fuzzy controller is designed the proposed controller guarantees that the system output converges to a small neighborhood of the reference signal and all the signals in the closed loop system remain bounded different from the existing adaptive fuzzy control methodology the fuzzy logic systems are used to model the desired but unknown control signals rather than the unknown nonlinear functions in the systems as a result the proposed adaptive controller has a simpler form and requires fewer adaptation parameters

adaptive fuzzy control for a class of uncertain nonaffine nonlinear systems an adaptive fuzzy control approach is proposed for a class of multiple input multiple output mimo nonlinear systems with completely unknown nonaffine functions the mimo systems are composed of n subsystems and each of subsystems is in the nested lower triangular form it is difficult and complicated to control this class of systems due to the existence of unknown nonaffine functions and the couplings among the nested subsystems this difficulty is overcome by introducing some special type lyapunov functions and taking advantage of the mean value theorem the backstepping design method and the approximation property of the fuzzy systems the proposed control approach can guarantee that all the signals in the closed loop system are bounded a simulation experiment is utilized to verify the feasibility of the proposed approach

robust h control for uncertain fuzzy systems with distributed delays via output feedback controllers this paper investigates the problem of robust h output feedback control for a class of uncertain fuzzy systems with distributed delays the system is described by a state space takagi sugeno fuzzy model with distributed delays and norm bounded parameter uncertainties the aim is to design a fuzzy dynamic output feedback controller which ensures the robust asymptotic stability of the closed loop system and guarantees an h norm bound constraint on disturbance attenuation for all admissible uncertainties in terms of linear matrix inequalities a sufficient condition for the solvability of this problem is presented finally simulation examples are provided to illustrate the effectiveness of the proposed design approach

stable adaptive fuzzy control of nonlinear systems a direct adaptive fuzzy controller that does not require an accurate mathematical model of the system under control is capable of incorporating fuzzy if then control rules directly into the controllers and guarantees the global stability of the resulting closed loop system in the sense that all signals involved are uniformly bounded is developed the specific formula for the bounds is provided so that controller designers can determine the bounds based on their requirements the direct adaptive fuzzy controller is used to regulate an unstable system to the origin and to control the duffing chaotic system to track a trajectory the simulation results show that the controller worked without using any fuzzy control rules and that after fuzzy control rules were incorporated the adaptation speed became much faster it is shown explicitly how the supervisory control forces the state to remain within the constraint set and how the adaptive fuzzy controller learns to regain control

adaptive control for nonlinear mimo time delay systems based on fuzzy approximation this paper focuses on the problem of adaptive control for a class of nonlinear multi input and multi output mimo systems with time delays a state feedback adaptive controller is constructed by backstepping technique fuzzy logic systems are used to approximate unknown nonlinear functions in the process of the controller design the main advantages of the proposed approach are twofold 1 the controller design is independent of the knowledge of the basis functions of fuzzy logic systems and 2 the suggested approach requires only m adaptive laws to control a nonlinear time delay system with m inputs simulation results illustrate the effectiveness of the proposed approach

adaptive fuzzy decentralized dynamics surface control for nonlinear large scale systems based on high gain observer this paper discusses the adaptive fuzzy decentralized output feedback control problem for a class of nonlinear large scale systems the systems under study have unknown nonlinearities mismatched interconnections in control inputs and unmeasurable state variables therefore fuzzy logic systems are employed to approximate the unknown nonlinear functions in the control design and an adaptive high gain observer is established to estimate the unmeasured states based on the backstepping design technique and the dynamic surface control dsc approach a novel adaptive fuzzy decentralized output feedback control scheme is developed it is proved that the proposed control approach can ensure that all the signals of the closed loop system are semi globally uniformly ultimately bounded suub and the observer error and tracking error can all converge to a small neighborhood of the origin the effectiveness of the proposed approach is illustrated by simulation examples with comparisons with several existing methods

new results on delay dependent robust stability criteria of uncertain fuzzy systems with state and input delays this paper focuses on the delay dependent stability analysis and stabilization for uncertain t s fuzzy control systems with state and input delays the key features of the approach include the introduction of uncorrelated augmented matrix items into the lyapunov functional and the use of a tighter bounding technology in fact these techniques lead to more general and less conservative stability conditions that guarantee a wide stability region our delay dependent stability conditions thus obtained are given in terms of linear matrix inequalities lmis finally some examples are included to show that the proposed criteria improve the existing results significantly

adaptive fuzzy output feedback control of mimo nonlinear systems with unknown dead zone inputs this paper is concerned with the problem of adaptive fuzzy tracking control for a class of multi input and multi output mimo strict feedback nonlinear systems with both unknown nonsymmetric dead zone inputs and immeasurable states in this research fuzzy logic systems are utilized to evaluate the unknown nonlinear functions and a fuzzy adaptive state observer is established to estimate the unmeasured states based on the information of the bounds of the dead zone slopes as well as treating the time varying inputs coefficients as a system uncertainty a new adaptive fuzzy output feedback control approach is developed via the backstepping recursive design technique it is shown that the proposed control approach can assure that all the signals of the resulting closed loop system are semiglobally uniformly ultimately bounded it is also shown that the observer and tracking errors converge to a small neighborhood of the origin by selecting appropriate design parameters simulation examples are also provided to illustrate the effectiveness of the proposed approach

fuzzy adaptive robust backstepping stabilization for siso nonlinear systems with unknown virtual control direction in this paper a direct fuzzy adaptive robust control approach is proposed for a class of siso nonlinear systems with completely unknown virtual control directions unknown nonlinearities unmodeled dynamics and dynamic disturbances in the backstepping recursive design fuzzy logic systems are employed to approximate the combined nonlinear uncertainties a dynamic signal and nussbaum gain technique are introduced into the control scheme to dominate the dynamic uncertainties and solve the unknown signs of virtual control directions respectively it is proved that the proposed robust fuzzy adaptive scheme can guarantee the all signals in the closed loop system are semi globally uniformly ultimately bounded the effectiveness of the proposed approach is illustrated via three examples

adaptive fuzzy output feedback tracking backstepping control of strict feedback nonlinear systems with unknown dead zones in this paper an adaptive fuzzy backstepping control approach is considered for a class of nonlinear strict feedback systems with unknown functions unknown dead zones and immeasurable states fuzzy logic systems are utilized to approximate the unknown nonlinear functions and a fuzzy filters state observer is designed to estimate the immeasurable states by using the adaptive backstepping recursive design technique and constructing the dead zone inverse a new adaptive fuzzy backstepping output feedback control approach is developed it is mathematically proved that all the signals of the resulting closed loop adaptive control system are semiglobally uniformly ultimately bounded and the tracking error converges to a small neighborhood of the origin by appropriate choice of design parameters the proposed approach cannot only solve the problem of the dead zones but also cancel the restrictive assumption in the previous literature that the states are all available for measurement two simulation examples are provided to show the effectiveness of the proposed approach

solution of hallen s integral equations by means of nystrom method to solve hallen s integral equations accurately nystrom method is applied in this paper to obtain an exact and finite solution both the exact kernel and a finite gap width must be taken into consideration in the case of a gap feed a modification function that converts the approximate kernel to the exact kernel is derived as the modification function is singular a one point correction method is proposed as a singularity subtraction method using a step function source we can use various quadrature rules to solve the problem the results show small difference between them the validity of this theory is confirmed by comparing the measured results and the calculated results the effect of the feed system is shown to be strong

h control of switched delayed systems with average dwell time this paper considers the problems of stability analysis and h controller design of time delay switched systems with average dwell time in order to obtain less conservative results than what is seen in the literature a tighter bound for the state delay term is estimated based on the scaled small gain theorem and the model transformation method an improved exponential stability criterion for time delay switched systems with average dwell time is formulated in the form of convex matrix inequalities the aim of the proposed approach is to reduce the minimal average dwell time of the systems which is made possible by a new lyapunov krasovskii functional combined with the scaled small gain theorem it is shown that this approach is able to tolerate a smaller dwell time or a larger admissible delay bound for the given conditions than most of the approaches seen in the literature moreover the exponential h controller can be constructed by solving a set of conditions which is developed on the basis of th

state observer based adaptive fuzzy output feedback control for a class of uncertain nonlinear systems this paper aims to develop state observer based adaptive fuzzy control techniques for controlling a class of uncertain nonlinear systems with bounded external disturbances an adaptive fuzzy observer is proposed to estimate the system state variables it is shown that the observation errors obtained from the observer are uniformly ultimately bounded applying the estimated system state for design of an output feedback controller the uniformly ultimate boundedness of the tracking errors for the resulting closed loop system can be guaranteed a typical robot arm system is employed in our simulation studies and the results demonstrate the usefulness and effectiveness of the proposed techniques for controlling nonlinear systems with bounded external disturbances

robust decentralized direct adaptive output feedback fuzzy control for a class of large sale nonaffine nonlinear systems in the previous work of huang et al a decentralized direct adaptive fuzzy h tracking controller of large scale nonaffine nonlinear systems is obtained predicated upon the assumption that the mismatching error dynamics stay squared integrable in this note we focus in the absence of the conservative assumption upon developing a robust decentralized direct adaptive output feedback fuzzy controller by combination of a state observer a fuzzy inference system and robust control technique the previous controller design is modified and no a priori knowledge of bounds on lumped uncertainties is required all the signals of the closed loop large scale system are proved to be uniformly ultimately bounded the effectiveness of the developed scheme is demonstrated through the simulation results of interconnected inverted pendulums

adaptive neural control for a class of nonlinearly parametric time delay systems in this paper an adaptive neural controller for a class of time delay nonlinear systems with unknown nonlinearities is proposed based on a wavelet neural network wnn online approximation model a state feedback adaptive controller is obtained by constructing a novel integral type lyapunov krasovskii functional which also efficiently overcomes the controller singularity problem it is shown that the proposed method guarantees the semiglobal boundedness of all signals in the adaptive closed loop systems an example is provided to illustrate the application of the approach

fuzzy basis functions comparisons with other basis functions fuzzy basis functions fbf s which have the capability of combining both numerical data and linguistic information are compared with other basis functions because a fbf network is different from other networks in that it is the only one that can combine numerical and linguistic information comparisons are made when only numerical data is available in particular a fbf network is compared with a radial basis function rbf network from the viewpoint of function approximation their architectural interrelationships are discussed additionally a rbf network which is implemented using a regularization technique is compared with a fbf network from the viewpoint of overcoming ill posed problems a fbf network is also compared with specht s probabilistic neural network and his general regression neural network grnn from an architectural point of view a fbf network is also compared with a gaussian sum approximation in which gaussian functions play a central role finally we summarize the architectural relationships between all the networks discussed in this paper

adaptive fuzzy output feedback control of uncertain nonlinear systems with unknown backlash like hysteresis in this paper the problem of adaptive fuzzy output feedback control is investigated for a class of uncertain nonlinear systems with unknown backlash like hysteresis and unmeasured states the fuzzy logic systems are used to approximate the nonlinear system functions and a fuzzy state observer is designed to estimate the unmeasured states by utilizing the fuzzy state observer and combining the adaptive backstepping technique with adaptive fuzzy control design an observer based adaptive fuzzy output feedback control approach is developed it is proved that the proposed control approach can guarantee that all the signals in the closed loop system are semi globally uniformly ultimately bounded suub and both observer error and tracking error can converge to a small neighborhood of the origin two simulations are included to illustrate the effectiveness of the proposed approach

fundamentals of thin wire integral equations with the finite gap generator part ii we discuss certain fundamental properties of hallen s and pocklington s equations with the finite gap generator fgg which has long been used as a feed model for cylindrical antennas for the case of the exact kernel we focus on whether important quantities are finite or infinite and compare to corresponding quantities obtained using the delta function generator dfg for the approximate kernel we focus on the unsolvability of the two equations and the relation of unsolvability to the important phenomenon of unphysical oscillations we also extend to the fgg the so called effective current method this is an easy to apply remedy for the oscillatory solutions that allows one to obtain reasonable answers for important quantities such as current distributions and the input impedance it is shown here that these answers are close to those obtained with the exact kernel we show that in a certain sense the method is more advantageous to apply in the present fgg case than in the dfg case our discussions especially in part ii of this paper bring out similarities to 1 the phenomenon of superdirectivity and 2 recently discussed numerical phenomena occurring within the method of auxiliary sources

regularity theory and superalgebraic solvers for wire antenna problems we consider the problem of evaluating the current distribution j z that is induced on a straight wire antenna by a time harmonic incident electromagnetic field the scope of this paper is twofold one of its main contributions is a regularity proof for a straight wire occupying the interval 1 1 in particular for a smooth time harmonic incident field this theorem implies that j z i z sqrt 1 z 2 where i z is an infinitely differentiable function the previous state of the art in this regard placed i in the sobolev space w 1 p p 1 the second focus of this work is on numerics we present three superalgebraically convergent algorithms for the solution of wire problems two based on hallen s integral equation and one based on the pocklington integrodifferential equation both our proof and our algorithms are based on two main elements 1 a new decomposition of the kernel of the form g z f 1 z ln z f 2 z where f 1 z and f 2 z are analytic functions on the real line and 2 removal of the end point square root singularities by means of a coordinate transformation the hallen and pocklington based algorithms we propose converge superalgebraically faster than mathcal o n m and mathcal o m m for any positive integer m where n and m are the numbers of unknowns and the number of integration points required for construction of the discretized operator respectively in previous studies at most the leading order contribution to the logarithmic singular term was extracted from the kernel and treated analytically the higher order singular derivatives were left untreated and the resulting integration methods for the kernel exhibit mathcal o m 3 convergence at best a rather comprehensive set of tests we consider shows that in many cases to achieve a given accuracy the numbers n of unknowns required by our codes are up to a factor of five times smaller than those required by the best solvers previously available the required number m of integration points in turn can be several orders of magnitude smaller than those required in previous methods in particular four digit solutions were found in computational times of the order of four seconds and in most cases of the order of a fraction of a second on a contemporary personal computer much higher accuracies result in very small additional computing times

observer based adaptive neural network control for nonlinear stochastic systems with time delay this paper considers the problem of observer based adaptive neural network nn control for a class of single input single output strict feedback nonlinear stochastic systems with unknown time delays dynamic surface control is used to avoid the so called explosion of complexity in the backstepping design process radial basis function nns are directly utilized to approximate the unknown and desired control input signals instead of the unknown nonlinear functions the proposed adaptive nn output feedback controller can guarantee all the signals in the closed loop system to be mean square semi globally uniformly ultimately bounded simulation results are provided to demonstrate the effectiveness of the proposed methods

robust adaptive critic control of nonlinear systems using fuzzy basis function networks an lmi approach abstract this paper proposes an adaptive critic tracking control design for a class of nonlinear systems using fuzzy basis function networks fbfns the key component of the adaptive critic controller is the fbfn which implements an associative learning network aln to approximate unknown nonlinear system functions and an adaptive critic network acn to generate the internal reinforcement learning signal to tune the aln another important component the reinforcement learning signal generator requires the solution of a linear matrix inequality lmi which should also be satisfied to ensure stability furthermore the robust control technique can easily reject the effects of the approximation errors of the fbfn and external disturbances unlike traditional adaptive critic controllers that learn from trial and error interactions the proposed on line tuning algorithm for aln and acn is derived from lyapunov theory thereby significantly shortening the learning time simulation results of a cart pole system demonstrate the effectiveness of the proposed fbfn based adaptive critic controller

delay dependent lmi conditions for stability and stabilization of t s fuzzy systems with bounded time delay this paper studies the stability and stabilization problems for time delay nonlinear systems through takagi sugeno t s fuzzy model approach the time delays are assumed to be bounded but perhaps non smooth delay dependent conditions are presented in terms of lmis illustrative examples are given to show the effectiveness and merits of the present results

fuzzy basis functions universal approximation and orthogonal least squares learning fuzzy systems are represented as series expansions of fuzzy basis functions which are algebraic superpositions of fuzzy membership functions using the stone weierstrass theorem it is proved that linear combinations of the fuzzy basis functions are capable of uniformly approximating any real continuous function on a compact set to arbitrary accuracy based on the fuzzy basis function representations an orthogonal least squares ols learning algorithm is developed for designing fuzzy systems based on given input output pairs then the ols algorithm is used to select significant fuzzy basis functions which are used to construct the final fuzzy system the fuzzy basis function expansion is used to approximate a controller for the nonlinear ball and beam system and the simulation results show that the control performance is improved by incorporating some common sense fuzzy control rules

reduced order h sub spl infin filtering for stochastic systems this paper deals with the reduced order h sub spl infin filtering problem for stochastic systems necessary and sufficient conditions are obtained for the existence of solutions to the continuous time and discrete time problems in terms of certain linear matrix inequalities lmis and a coupling nonconvex rank constraint condition furthermore when these conditions are feasible an explicit parametrization of all desired reduced order filters corresponding to a feasible solution is given in particular when the reduced order filter is restricted to be a static one then simple conditions expressed by lmis only without any rank constraints are derived and a parametrization of all solutions is also given finally an illustrative example is provided to show the effectiveness of the proposed approach

adaptive h control in finite frequency domain for uncertain linear systems this paper is concerned with the problem of adaptive h controller design in finite frequency domains for uncertain linear systems the uncertainties are assumed to be time invariant unknown but bounded which appear affinely in the matrices of system models an adaptive mechanism is introduced to construct a novel finite frequency h controller with time varying gains by using lyapunov theory and parseval s theorem the controller design conditions are given in terms of a set of linear matrix inequalities lmis it is shown that the proposed finite frequency controller with time varying gains can achieve better h performance than the traditional ones with fixed gains finally a numerical example of the f 18 aircraft model is given to illustrate the presented theoretical results

sub riemannian calculus on hypersurfaces in carnot groups we develop a sub riemannian calculus for hypersurfaces in graded nilpotent lie groups we introduce an appropriate geometric framework such as horizontal levi civita connection second fundamental form and horizontal laplace beltrami operator we analyze the relevant minimal surfaces and prove some basic integration by parts formulas using the latter we establish general first and second variation formulas for the horizontal perimeter in the heisenberg group such formulas play a fundamental role in the sub riemannian bernstein problem

new delay dependent non fragile h observer based control for continuous time delay systems the problem of the delay dependent non fragile h observer based control for a class of continuous time delay systems is investigated in this paper the additive gain variations under consideration are contained in both the controller gain and obhsiunserver gain delay dependent criteria are derived to guarantee the stability of the non fragile h observer based control system using the lyapunov functional approach the controller and observer gains are given from the lmi feasible solutions based on the result of this paper the constraint of matrix equality is not necessary for designing the non fragile h observer based controls computer software matlab can be applied to solve all the proposed results finally a numerical example is illustrated to show the improvement of this paper

prescribed performance adaptive fuzzy output feedback dynamic surface control for nonlinear large scale systems with time delays the adaptive fuzzy decentralized output feedback control method is proposed for a class of nonlinear large scale systems the considered nonlinear systems contain unstructured uncertainties unknown time varying delays and immeasurable states fuzzy logic systems are used to approximate the unstructured uncertainties and a fuzzy state observer is designed to estimate the unmeasured states based on the backstepping recursive design technique and the prescribed performance technique a new adaptive fuzzy output feedback control method is developed in order to overcome the problem of explosion of complexity inherent in the backstepping control design the dynamic surface control dsc technique is introduced into the control scheme based on lyapunov krasovskii functional it is proved that all the signals in the closed loop system are bounded and the tracking errors remain an adjustable neighborhood of the origin with the prescribed performance bounds the simulation examples and comparison with the previous control methods are provided to show the effectiveness of the proposed control approach

sliding mode filter design for nonlinear polynomial systems with unmeasured states this paper addresses the mean square and mean module filtering problems for a nonlinear polynomial stochastic system with gaussian white noises the obtained solutions contain a sliding mode term signum of the innovations process it is shown that the designed sliding mode mean square filter generates the mean square estimate which has the same minimum estimation error variance as the estimate given by the conventional mean square polynomial filter basin et al 2008 8 although the gain matrices of both filters are different the designed sliding mode mean module filter generates the mean module estimate which yields a better value of the mean module criterion in comparison to the conventional mean square polynomial filter the theoretical result is complemented with an illustrative example verifying performance of the designed filters it is demonstrated that the estimates produced by the designed sliding mode mean square filter and the conventional mean square polynomial filter yield the same estimation error variance and there is an advantage in favor of the designed sliding mode mean module filter

numerical solution of the helmholtz equation in 2d and 3d using a high order nystrom discretization we show how to solve time harmonic scattering problems by means of a high order nystrom discretization of the boundary integral equations of wave scattering in 2d and 3d the novel aspect of our new method is its use of local corrections to the discretized kernel in the vicinity of the kernel singularity enhanced by local corrections the new algorithm has the simplicity and speed advantages of the traditional nystrom method but also enjoys the advantages of high order convergence for controlling solution error we explain the practical details of implementing a scattering code based on a high order nystrom discretization and demonstrate by numerical example that a scattering code based on this algorithm can achieve high order convergence to the correct answer we also demonstrate its performance advantages over a high order galerkin code

spl hscr sub spl infin filtering for continuous time linear systems with delay the problem of spl hscr sub spl infin filtering for continuous time linear systems with time delayed measurement is investigated the authors develop a methodology for designing linear filters which ensure a prescribed bound on the spl lscr sub 2 induced gain from the noise signals to the estimation error filtering problems for time varying systems over a finite horizon as well as stationary infinite horizon filtering for time invariant systems are tackled in the finite horizon case our estimation procedure entails an overdesign that stems from the last d seconds of the time interval 0 t where d is the delay length this overdesign becomes smaller as t increases and it vanishes in the infinite horizon case

the exact kernel for cylindrical antenna in the numerical calculation for a cylindrical antenna the thin wire kernel is widely used as a convenient approximation to the exact kernel of the electric field equation however this thin wire approximation is valid only if the radius of wire is much less than its length and the wavelength furthermore arbitrarily using this approximation may not obtain the convergent outcomes the expression of the exact kernel for the cylindrical antenna which would be useful in the numerical work is presented

robust control for a class of uncertain nonlinear systems adaptive fuzzy approach based on backstepping a robust adaptive fuzzy control design approach is developed for a class of multivariable nonlinear systems with modeling uncertainties and external disturbances the controller design for the overall systems has been carried out through a number of simpler controller designs for a series of the relevant auxiliary systems based on the backstepping design technique for each auxiliary system an adaptive fuzzy logic system is introduced to learn the behavior of unknown dynamics and then a robust control algorithm is employed to efficiently compensate the approximation error and the external disturbances as well it is shown that the resulting closed loop systems guarantee a satisfactory transient and asymptotic performance a simulation example is also presented to illustrate the design procedure and controller performance

dissipativity analysis for discrete time delay fuzzy neural networks with markovian jumps this paper is concerned with the dissipativity analysis and design of discrete markovian jumping neural networks with sector bounded nonlinear activation functions and time varying delays represented by takagi sugeno fuzzy model the augmented fuzzy neural networks with markovian jumps are first constructed based on estimator of luenberger observer type then applying piecewise lyapunov krasovskii functional approach and stochastic analysis technique a sufficient condition is provided to guarantee that the augmented fuzzy jump neural networks are stochastically dissipative moreover a less conservative criterion is established to solve the dissipative state estimation problem by using matrix decomposition approach furthermore to reduce the computational complexity of the algorithm a dissipative estimator is designed to ensure stochastic dissipativity of the error fuzzy jump neural networks as a special case we have also considered the mixed h infty and passive analysis of fuzzy jump neural networks all criteria can be formulated in terms of linear matrix inequalities finally two examples are given to show the effectiveness and potential of the new design techniques

adaptive fuzzy tracking control for a class of perturbed strict feedback nonlinear time delay systems this paper is concerned with the problem of adaptive fuzzy output tracking for a class of perturbed strict feedback nonlinear systems with time delays and unknown virtual control coefficients fuzzy logic systems in mamdani type are used to approximate the unknown nonlinear functions then the adaptive fuzzy tracking controller is designed by using the backstepping technique and lyapunov krasovskii functionals the proposed adaptive fuzzy controller guarantees that all the signals in the closed loop system are bounded and the system output eventually converges to a small neighborhood of the desired reference signal an advantage of the proposed control scheme lies in that the number of the online adaptive parameters is not more than the order of the original system finally two examples are used to demonstrate the effectiveness of our results proposed in this paper

an observer based approach to controlling time delay chaotic systems via takagi sugeno fuzzy model this study presents a fuzzy model based control criterion for time delay chaotic systems with uncertainty the takagi sugeno t s fuzzy model is adopted for representing a chaotic system possessing a time delay a unified approach for designing a fuzzy observer which integrates parallel distributed compensation with an adaptive updating method is first discussed an observer based fuzzy control scheme is then developed to deal with stabilization and tracking problems based on lyapunov stability analysis sufficient conditions that ensure robust control performance of time delay chaotic systems are derived these conditions are represented in terms of linear matrix inequalities and adaptive updating laws finally the proposed approach is validated through numerical synchronization stabilization and model tracking control simulations

stability of hypersurfaces of constant mean curvature in riemannian manifolds hypersurfaces m n with constant mean curvature in a riemannian manifold overline m n 1 display many similarities with minimal hypersurfaces of overline m n 1 they are both solutions to the variational problem of minimizing the area function for certain variations in the first case however the admissible variations are only those that leave a certain volume function fixed for precise definitions see sect 2 this isoperimetric character of the variational problem associated to hypersurfaces of constant mean curvature introduces additional complications in the treatment of stability of such hypersurfaces

existence and uniqueness for p area minimizers in the heisenberg group in 3 we studied p mean curvature and the associated p minimal surfaces in the heisenberg group from the viewpoint of pde and differential geometry in this paper we look into the problem through the variational formulation we study a generalized p area and associated p minimizers in general dimensions we prove the existence and investigate the uniqueness of minimizers since this is reduced to solving a degenerate elliptic equation we need to consider the effect of the singular set and this requires a careful study we define the notion of weak solution and prove that in a certain sobolev space a weak solution is a minimizer and vice versa we also give many interesting examples in dimension 2 an intriguing point is that in dimension 2 a c 2 smooth solution from the pde viewpoint may not be a minimizer however this statement is true for higher dimensions due to the relative smallness of the size of the singular set