the convergence of spline collocation for strongly elliptic equations on curves most boundary element methods for two dimensional boundary value problems are based on point collocation on the boundary and the use of splines as trial functions here we present a unified asymptotic error analysis for even as well as for odd degree splines subordinate to uniform or smoothly graded meshes and prove asymptotic convergence of optimal order the equations are collocated at the breakpoints for odd degree and the internodal midpoints for even degree splines the crucial assumption for the generalized boundary integral and integro differential operators is strong ellipticity our analysis is based on simple fourier expansions in particular we extend results by j saranen and w l wendland from constant to variable coefficient equations our results include the first convergence proof of midpoint collocation with piecewise constant functions i e the panel method for solving systems of cauchy singular integral equations

relative equilibria of point vortices on the hyperbolic sphere relative equilibria of point vortices on a hyperbolic sphere with constant negative curvature is formulated and considered known symmetries for the vortex motion on the hyperbolic sphere are expounded by setting up a geometrical formulation of dynamics in particular the configuration of rings of vortices is investigated in detail

on stability of fuzzy systems expressed by fuzzy rules with singleton consequents addresses the issue of stability of a fuzzy system described by fuzzy rules with singleton consequents it first presents two canonical forms of a fuzzy system a parametric expression and a state space expression a fuzzy system with singleton consequents is found to be a piecewise polytopic affine system then the paper gives stability theorems for a discrete time system as well as for a continuous time system it also gives a brief survey on the stability issues of fuzzy control systems

local error estimates for some petrov galerkin methods applied to strongly elliptic equations on curves on etudie les proprietes de convergence locales de quelques methodes de petrov galerkin appliquees aux equations fortement elliptiques sur des courbes regulieres du plan on etablit des estimations d erreurs locales pour une grande echelle de normes de sobolev

thomson s heptagon a case of bifurcation at infinity abstract vortex modeling has a long history descartes 1644 used it as a model for the solar systems j j thomsom 1883 used it as a model for the atom we consider point vortex systems which can be regarded as discrete solutions of the euler equation their dynamics is described by a hamiltonian system of equations we are interested in polygonal configurations and how their stability depends upon various dynamical variables in the plane a polygon with seven vortices has been shown to be a special boundary case polygons with n 7 vortices are linearly and nonlinearly stable while polygons with n 7 vortices are unstable why should n 7 be special celestial mechanics helped us to simplify a problem that has been studied for over a century and to show that the case of thomson s heptagon is actually a case of bifurcation at infinity this becomes particularly clear when considering the corresponding problem of a ring on a sphere with two polar vortices of variable intensities n and s at the north and south pole respectively

point vortices on the hyperbolic plane we investigate the dynamical system of point vortices on the hyperboloid this system has non compact symmetry sl 2 r and a coadjoint equivariant momentum map the relative equilibrium conditions are found and the trajectories of relative equilibria with non zero momentum value are described we also provide the classification of relative equilibria and the stability criteria for a number of cases focusing on 2 and 3 vortices unlike the system on the sphere this system has relative equilibria with non compact momentum isotropy subgroup and these are used to illustrate the different stability types of relative equilibria

local convergence of the boundary element method on polyhedral domains the local behavior of the lowest order boundary element method on quasi uniform meshes for symm s integral equation and the stabilized hyper singular integral equation on polygonal polyhedral lipschitz domains is analyzed we prove local a priori estimates in l 2 for symm s integral equation and in h 1 for the hyper singular equation the local rate of convergence is limited by the local regularity of the sought solution and the sum of the rates given by the global regularity and additional regularity provided by the shift theorem for a dual problem

integrable and chaotic motions of four vortices i the case of identical vortices it is shown that the three vortex problem in two dimensional hydrodynamics is integrable whereas the motion of four identical vortices is not a sequence of canonical transformations is obtained that reduces the n degree of freedom hamiltonian which describes the interaction of n identical vortices to one with n 2 degrees of freedom for n 3 a reduction to a single degree of freedom is obtained and this problem can be solved in terms of elliptic functions for n 4 the reduction procedure leads to an effective hamiltonian with two degrees of freedom of the form found in problems with coupled nonlinear oscillators resonant interaction terms in this hamiltonian suggest non integrable behaviour and this is verified by numerical experiments explicit construction of a solution that corresponds to a heteroclinic orbit in phase space is possible the relevance of the results obtained to fundamental problems in hydrodynamics such as the question of integrability of euler s equation in two dimensions is discussed the paper also contains a general exposition of the hamiltonian and poisson bracket formalism for point vortices

green s function for the laplace beltrami operator on a toroidal surface green s function for the laplacebeltrami operator on the surface of a three dimensional ring torus is constructed an integral ingredient of our approach is the stereographic projection of the toru

the n vortex problem on a rotating sphere i multi frequency configurations the problem of n point vortices moving on a rotating unit sphere is considered through a sequence of linear coordinate transformations which takes into account the orientation of the centre of vorticity vector with respect to the axis of rotation we show how to reduce the problem to that on a non rotating sphere where the centre of vorticity vector is aligned with the z axis as a consequence we prove that integrability on the rotating sphere is the same as on the non rotating sphere namely the three vortex problem on the rotating sphere is integrable for all vortex strengths while the four vortex problem is integrable in the special case where the centre of vorticity is zero rigid multi frequency configurations that retain their shape while rotating about two independent axes with two independent frequencies are obtained and necessary conditions for one and two frequency motions are derived examples including dipoles which exhibit global wobbling and tumbling dynamics rings and platonic solid configurations are shown to undergo either periodic or quasi periodic evolution on the rotating sphere depending on the ratio of the solid body rotational frequency to the rotational frequency associated with the rigid structure

fuzzy adaptive control for a class of nonlinear systems an adaptive tracking control architecture is proposed for a class of continuous time nonlinear dynamic systems for which an explicit linear parameterization of the uncertainty in the dynamic is either unknown or impossible the architecture employs fuzzy systems to approximate nonlinear functions uses fuzzy sliding controller to compensate for the plant uncertainties global asymptotic stability is established in the lyapunov sense with the tracking errors converging to a neighborhood of zero

comments on robust stabilization of a class of uncertain nonlinear systems via fuzzy control quadratic stabilizability h sup spl infin control theory and linear matrix inequalities this paper presents stability analysis for a class of uncertain nonlinear systems and a method for designing robust fuzzy controllers to stabilize the uncertain nonlinear systems first a stability condition for takagi and sugeno s fuzzy model is given in terms of lyapunov stability theory next new stability conditions for a generalized class of uncertain systems are derived from robust control techniques such as quadratic stabilization h sup spl infin control theory and linear matrix inequalities the derived stability conditions are used to analyze the stability of takagi and sugeno s fuzzy control systems with uncertainty which can be regarded as a generalized class of uncertain nonlinear systems the design method employs the so called parallel distributed compensation important issues for the stability analysis and design are remarked finally three design examples of fuzzy controllers for stabilizing nonlinear systems and uncertain nonlinear systems are presented

on the asymptotic convergence of collocation methods we prove quasioptimal and optimal order estimates in various sobolev norms for the approximation of linear strongly elliptic pseudodifferential equations in one independent variable by the method of nodal collocation by odd degree polynomial splines the analysis pertains in particular to many of the boundary element methods used for numerical computation in engineering applications equations to which the analysis is applied include fredholm integral equations of the second kind certain first kind fredholm equations singular integral equations involving cauchy kernels a variety of integro differential equations and two point boundary value problems for ordinary differential equations the error analysis is based on an equivalence which we establish between the collocation methods and certain nonstandard galerkin methods we compare the collocation method with a standard galerkin method using splines of the same degree showing that the galerkin method is quasioptimal in a sobolev space of lower index and furnishes optimal order approximation for a range of sobolev indices containing and extending below that for the collocation method and so the standard galerkin method achieves higher rates of convergence

n body dynamics on closed surfaces the axioms of mechanics a major challenge for our understanding of the mathematical basis of particle dynamics is the formulation of n body and n vortex dynamics on riemann surfaces in this paper we show how the two problems are in fact closely related when considering the role played by the intrinsic geometry of the surface this enables a straightforward deduction of the dynamics of point masses using recently derived results for point vortices on general closed differentiable surfaces m endowed with a metric g we find generally that kepler s laws do not hold what is more even newton s first law the law of inertia fails on closed surfaces with variable curvature e g the ellipsoid

local error estimates for the galerkin method applied to strongly elliptic integral equations on open curves saranen math comp 48 1987 pp 485 502 proved local estimates in sobolev norms for the galerkin method applied to strongly elliptic equations on smooth closed curves in the plane we extend his results to the case of open curves of particular interest are weakly singular and hypersingular integral equations on a slit an interesting outcome from our result is that we can judge the orders of convergence of the local errors in some norms that do not exist in the global sense

robust fuzzy logic control of mechanical systems an approach for the design of fuzzy control laws for tracking control of a large class of mechanical systems is proposed the approach employs the framework of lyapunov s stability theory to formulate a class of control laws that guarantee convergence of the tracking errors to within specification limits in presence of bounded parameter uncertainties and input disturbances the proposed control laws possess a large number of parameters and functional relationships to be chosen by the designer according to a methodology developed in the paper the large number of design degrees of freedom makes the approach suitable for fuzzy logic implementation a number of fuzzy implementations of the proposed control methodology are provided all implementations guarantee tracking error convergence to within prespecified performance limits an extensive simulation study using a model of a two degree of freedom robot manipulator was conducted fuzzy and non fuzzy implementations of the proposed methodology were compared to control laws designed using other design methods simulation study results indicate a superiority of the proposed control methodology compared to other approaches the study also demonstrates better performance of the fuzzy control implementation compared to its non fuzzy counterpart

numerical description of the motion of a point vortex pair on ovaloids this paper presents a numerical study on the dynamics of point vortex pairs moving on convex closed surfaces ovaloids approximated by triangular icosahedral meshes firstly both a discretized conformal mapping between the ovaloid and the unit sphere and a least squares fitting for this approximated conformal map in terms of a spherical harmonic expansion are obtained then an approximation of the conformal spherical metric of the ovaloid is derived by obtaining a spherical harmonic expansion of the corresponding conformal factor and its gradient on the equations of motion for a point vortex pair on with this quasi conformal metric are integrated by using the classical gaussian collocation method for some particular models

stability analysis and design of fuzzy control systems abstract the stability analysis and the design technique of fuzzy control systems using fuzzy block diagrams are discussed first we show the concept of fuzzy blocks and consider the connection problems of fuzzy blocks diagrams we derive some theorems and corollaries with respect to two basic types of connections of fuzzy blocks in order to preserve some properties in a connection of fuzzy blocks continuous piecewise polynomial membership functions are defined secondly a sufficient condition which guarantees the stability of fuzzy systems is obtained in terms of lyapunov s direct method we give an important fact based on this condition thirdly we propose a new design technique of a fuzzy controller the fuzzy block diagrams and the stability analysis are applied to the design problems of a model based fuzzy controller

fuzzy lyapunov based approach to the design of fuzzy controllers in this paper we extend the classical lyapunov synthesis method to the domain of computing with words this new approach is used to design fuzzy controllers assuming minimal knowledge about the plant to be controlled the proposed method enables us to systematically derive the fuzzy rules that constitute the rule base of the controller we demonstrate the approach by designing mamdani type and takagi sugeno kang type fuzzy controllers for two well known plants

on the asymptotic convergence of collocation methods with spline functions of even degree we investigate the collocation of linear one dimensional strongly elliptic integro dif ferential or more generally pseudo differential equations on closed curves by even degree polynomial splines the equations are collocated at the respective midpoints subject to uniform nodal grids of the even degree s splines we prove quasioptimal and optimal order asymptotic error estimates in a scale of sobolev spaces the results apply in particular to boundary element methods used for numerical computations in engineering applications the equations considered include fredholm integral equations of the second and the first kind singular integral equations involving cauchy kernels and integro differential equations having convolutional or constant coefficient principal parts respectively the error analysis is based on an equivalence between the collocation and certain varia tional methods with different degree splines as trial and as test functions we further need to restrict our operators essentially to pseudo differential operators having convolutional prin cipal part this allows an explicit fourier analysis of our operators as well as of the spline spaces in terms of trigonometric polynomials providing babuska s stability condition based on strong ellipticity our asymptotic error estimates extend partly those obtained by d n arnold and w l wendland from the case of odd degree splines to the case of even degree splines 1 introduction in this paper we investigate the asymptotic convergence of the collocation method using even degree polynomial splines applied to strongly elliptic systems of pseudo differential equations on closed curves with convolutional prin cipal part the collocation here employs the gauss points of one point integration as collocation points which corresponds to the usual boundary element collocation in applications 5 12 17 19 26 31 34 this is in contrast to the method investigated by g schmidt in 29 where one collocates at the break points of even order splines the asymptotic convergence properties for the standard galerkin method with splines of arbitrary orders are well known 22 for the collocation method however asymptotic convergence up to now has been shown for strongly elliptic systems only in the case of odd order splines by d n arnold and w l wendland 7

three ways to solve the poisson equation on a sphere with gaussian forcing motivated by the needs of vortex methods we describe three different exact or approximate solutions to the poisson equation on the surface of a sphere when the forcing is a gaussian of the three dimensional distance 2 j exp 2 e 2 1 cos q c g a u s s e more precisely the forcing is a gaussian minus the gauss constraint constant c g a u s s this subtraction is necessary because j is bounded for any type of forcing only if the integral of the forcing over the sphere is zero y kimura h okamoto vortex on a sphere j phys soc jpn 56 1987 4203 4206 d g dritschel contour dynamics surgery on the sphere j comput phys 79 1988 477 483 the legendre polynomial series is simple and yields the exact value of the gauss constraint constant but converges slowly for large e the analytic solution involves nothing more exotic than the exponential integral but all four terms are singular at one or the other pole cancelling in pairs so that j is everywhere nice the method of matched asymptotic expansions yields simpler uniformly valid approximations as series of inverse even powers of e that converge very rapidly for the large values of e e 40 appropriate for geophysical vortex computations the series converges to a nonzero o exp 4 e 2 error everywhere except at the south pole where it diverges linearly with order instead of the usual factorial order

the k operator and the galerkin method for strongly elliptic equations on smooth curves local estimates superconvergence in the l2 norm for the galerkin approximation of the integral equation lu f is studied where i is a strongly elliptic pseudodifferential operator on a smooth closed or open curve let uf be the galerkin approximation to u by using the operator an operator that averages the values of uh we will construct a better approximation than uh itself that better approximation is a legacy of the highest order of convergence in negative norms for symm s equation on a slit the same order of convergence can be recovered if the mesh is suitably graded in this paper we shall study a way of increasing the order of local convergence in the l2 norm for the galerkin approximation to the solution of strongly ellip tic pseudodifferential equations on a smooth closed or open curve in r2 this kind of integral equation is of importance in solving interior or exterior bound ary value problems of potential theory the most common example is symm s first kind integral equation with logarithmic kernel see 6 7 12 other appli cations are hypersingular integral equations and singular integral equations of cauchy type which occur e g in elasticity see 18 for general petrov galerkin methods when smoothest splines are used as trial and test functions local error estimates were proved in 11 for smooth closed curves and in 16 for smooth open curves consider for example symm s equation with piecewise constant functions used as trial and test functions it was proved that the local l2 error converges with order 0 h in the case of smooth closed curves 11 and with order 0 hxi2 in the case of smooth open curves 16 however it is well known that the highest orders of global convergence achieved in negative norms are 0 h3 for the closed smooth case 5 and 0 h for the open smooth case 4 13 the purpose of this article is to construct from the galerkin solution a better approximate solution which inherits the highest possible orders of global convergence to give best local con vergence in the l2 norm e g in the example mentioned above order 0 hy for the closed case and 0 h for the open case can be achieved locally in the l2 norm that better approximation is constructed by averaging the values of the galerkin solution using the k operator

high order fuzzy sliding manifold control the design of high order sliding fuzzy control strategies is addressed and carried out by using rigorous mathematical tools of the singular perturbation theory the resulting fuzzy controller implements an output feedback to solve output tracking problems and has the double benefit to be robust and low order namely the controller order is the plant relative degree simulations are presented to show the effectiveness of the proposed strategy

the method of images and green s function for spherical domains motivated by problems in electrostatics and vortex dynamics we develop two general methods for constructing green s function for simply connected domains on the surface of the unit sphere we prove a riemann mapping theorem showing that such domains can be conformally mapped to the upper hemisphere we then categorize all domains on the sphere for which green s function can be constructed by an extension of the classical method of images we illustrate our methods by several examples such as the upper hemisphere geodesic triangles and latitudinal rectangles we describe the point vortex motion in these domains which is governed by a hamiltonian determined by the dirichlet green s function

theorectical analysis of a fuzzy logic controller with unequally spaced triangular membership functions abstract the mathematical analysis concerning the explicit input output relation of the fuzzy logic controller flc is addressed in this article under the assumption of simple rule mapping and the use of arbitrarily located triangular fuzzy partitions as the membership functions the inferred output of the flc can be decomposed into two terms the global multilevel relay and the local nonlinear compensator the ultimate control behavior of this flc and the equivalence to nonlinear proportional integral pi controller as the number of linguistic terms are approaching to infinity are also investigated the local stability criteria for pi type flcs are derived and the global stability properties for pd type flcs are established this analysis is a generalization to previous related research works where the flc is characterized by equally spaced triangular fuzzy partitions on the universal sets

vortex motion on surfaces with constant curvature vortex motion on two dimensional riemannian surfaces with constant curvature is formulated by way of the stereographic projection the relation and difference between the vortex motion on a sphere s 2 and on a hyperbolic plane h 2 can be clearly analysed the hamiltonian formalism is presented for the motion of point vortices on s 2 and h 2 the set of first integrals for each hamiltonian shows a corresponding algebraic property in terms of the poisson bracket defined respectively for s 2 and h 2 as an example of analytic solutions the motion of a vortex pair dipole is considered it is shown that a dipole draws a geodesic curve as its trajectory on s 2 and h 2

some boundary element methods using dirac s distributions as trial functions in this article a boundary element method using dirac s distributions as trial functions is proposed as test functions we use smoother splines the computational complexity of this method is comparable with the collocation method and it converges with less regularity requirements on the boundary data asymptotic error analysis is derived an example considering solution of the potential problem is discussed in some detail finally a numerical example demonstrates the applicability of the method in practical calculations

on the analysis of fuzzy logic controllers abstract in this paper we address the issue of analyzing the properties of the fuzzy logic controller flc we introduce two basic models of the flc the first representation uses the concept of the quasilinear fuzzy model the flc is approximated by a virtual pid controller consisting of linear controller connected in parallel the second interpretation of the flc is from the point of view of nonlinear control the flc is analyzed as a sliding mode like controller based on these two alternative representations are derived rules for designing and tuning fuzzy logic controllers

the motion of point vortices on closed surfaces we develop a mathematical framework for the dynamics of a set of point vortices on a class of differentiable surfaces conformal to the unit sphere when the sum of the vortex circulations is non ze

adaptive control of robot manipulators using fuzzy logic systems under actuator constraints in this paper a stable fuzzy adaptive controller for trajectory tracking is developed for robot manipulators without velocity measurements taking into account the actuator constraints the controller is based on structural knowledge of the dynamics of the robot and measurements of link positions only the gravity torque including system uncertainty like payload variation etc is estimated by a fuzzy logic system fls the adaptive controller represents an amalgamation of a filtering technique to eliminate velocity measurements and the theory of function approximation using fls to estimate the gravity torque the proposed controller ensures the local asymptotic stability and the convergence of the position error to zero the proposed controller is robust not only to structured uncertainty such as payload parameter variation but also to unstructured one such as disturbances the validity of the control scheme is shown by simulations on a two link robot manipulator

piecewise quadratic stability of fuzzy systems presents an approach to stability analysis of fuzzy systems the analysis is based on lyapunov functions that are continuous and piecewise quadratic the approach exploits the gain scheduling nature of fuzzy systems and results in stability conditions that can be verified via convex optimization over linear matrix inequalities examples demonstrate the many improvements over analysis based on a single quadratic lyapunov function special attention is given to the computational aspects of the approach and several methods to improve the computational efficiency are described

mathematical foundation of adaptive boundary element methods abstract in this paper we review some mathematical aspects of adaptive boundary element methods in brief and give some a posteriori local error estimates for the h version and the p version