a new efficient method for cases of the singular integral equation of the first kind various cases of cauchy type singular integral equation of the first kind occur rather frequently in mathematical physics and possess very unusual properties these equations are usually difficult to solve analytically and it is required to obtain approximate solutions this paper investigates the numerical solution of various cases of cauchy type singular integral equations using reproducing kernel hilbert space rkhs method the solution u x is represented in the form of a series in the reproducing kernel space afterwards the n term approximate solution u n x is obtained and it is proved to converge to the exact solution u x the major advantage of the method is that it can produce good globally smooth approximate solutions moreover in this paper an efficient error estimation of the rkhs method is introduced finally numerical experiments show that our reproducing kernel method is efficient

the approximate solution of a class of fredholm integral equations with a weakly singular kernel a method for finding the numerical solution of a weakly singular fredholm integral equation of the second kind is presented the taylor series is used to remove singularity and legendre polynomials are used as a basis furthermore the legendre function of the second kind is used to remove singularity in the cauchy type integral equation the integrals that appear in this method are computed in terms of gamma and beta functions and some of these integrals are computed in the cauchy principal value sense without using numerical quadratures four examples are given to show the accuracy of the method

an approximate numerical method for solving cauchy singular integral equations composed of multiple implicit parameter functions with unknown integral limits in contact mechanics abstract cauchy singular integral equations are widely used in physics and mathematics especially in solid contact mechanics the solution of cauchy singular integral equations composed of explicit functions has already been achieved in existing research however when dealing with the contact problem between two solid bodies with irregular surfaces described by implicit parametric functions difficulties arise when trying to solve the cauchy singular integral because it is composed of multiple implicit parameter functions moreover the integral limits are unknown and constrained by physical characteristics to solve this kind of problem an approximate calculation method with high accuracy will be provided in this paper specifically based on the quadrature method and taking the constraint function of the boundary as the convergence criterion both the integral limits satisfying the physical characteristics condition and the solution of the cauchy singular integral equations composed of multiple parameter functions can be derived by an iterative method finally five different examples are calculated using the new method and the absolute errors between the approximate values provided by the new method and the true values are analysed

groups jacobi functions and rigged hilbert spaces this paper is a contribution to the study of the relations between special functions lie algebras and rigged hilbert spaces the discrete indices and continuous variables of special functions are in correspondence with the representations of their algebra of symmetry which induce discrete and continuous bases coexisting on a rigged hilbert space supporting the representation meaningful operators are shown to be continuous on the spaces of test vectors and the dual here the chosen special functions called algebraic jacobi functions are related to the jacobi polynomials and the lie algebra is su 2 2 these functions with m and q fixed also exhibit a su 1 1 symmetry different discrete and continuous bases are introduced an extension in the spirit of the associated legendre polynomials and the spherical harmonics is presented introducing the jacobi harmonics that are a generalization of the spherical harmonics to the three dimensional hypersphere s3

brief paper feedback and invariance under uncertainty via set iterates we examine discrete time control systems under non parametric disturbances sets which a given control feedback makes invariant under the disturbance are analyzed via lifting the feedback operation to the space of sets properties of being an attractor of the disturbed dynamics and being a minimal invariant set are derived from the corresponding notions of the set dynamics yielding in turn useful characterizations and error estimates for numerical algorithms which detect the minimal invariant sets concrete numerics for some examples of practical feedback rules are offered

a universal construction of artstein s theorem on nonlinear stabilization abstract this note presents an explicit proof of the theorem due to artstein which states that the existence of a smooth control lyapunov function implies smooth stabilizability moreover the result is extended to the real analytic and rational cases as well the proof uses a universal formula given by an algebraic function of lie derivatives this formula originates in the solution of a simple riccati equation

an efficient method for approximate solution of a singular integral equation with cauchy kernel abstract in this paper the least squares method and the reproducing kernel space are employed to obtain approximate solution of a general form of singular integral equations with cauchy kernel this approach is based on the construction of the subspace which is dense in the reproducing kernel space the numerical examples illustrate the high accuracy of the method

an efficient lyapunov function for discrete t s models observer design this paper deals with the design of a new observer synthesis for discrete takagi sugeno t s fuzzy models it is well established that quadratic synthesis for discrete t s models and or linear parameter varying systems can be outperformed easily via nonquadratic syntheses several lyapunov functions can be used nevertheless this paper shows that with a small change in the initial lyapunov function a better in the sense of solutions to the linear matrix inequality constraints problem lyapunov function can be reached this one can introduce very important improvements

cauchy type singular integral equation with constant coefficients on the real line abstract in this paper exact solutions of the characteristic and dominant equation with cauchy kernel on the real line are presented next trigonometric polynomials are used to derive approximate solutions of these equations moreover estimations of errors of the approximated solutions are presented and proved

on the numerical treatment of the singular integral equation of the second kind here a numerical treatment for solving the integral equation of the second kind with cauchy kernel is presented the singular term has been removed and the solution in the legendre polynomial form has been used to obtain a system of linear algebraic equation this system is solved numerically

a membership function dependent approach for stability analysis and controller synthesis of takagi sugeno models this paper presents a new approach for stability analysis and controller design of takagi sugeno ts models the analysis considers information derived from existing or induced order relations among the membership functions partitioning of the state space and the use of piecewise lyapunov functions pwlf arise naturally as a consequence of induced order relations conditions under the novel approach can be expressed as linear matrix inequalities lmis facilitating the inclusion of performance design examples are provided to show the advantages over the classical quadratic approach

solution of second kind integral equation with cauchy type kernel using daubechies scale function a method based on the daubechies scale function has been devised to solve numerically a class of second kind integral equations with a cauchy type kernel the convergence associated to the method has been established the method is illustrated with integral equations in this class whose exact solutions are known the numerical results obtained by the present method are seen to differ negligibly from the exact results

asymptotically necessary and sufficient conditions for stability and performance in fuzzy control applications of polya s theorem stability and performance requirements in fuzzy control of takagi sugeno systems are usually stated as fuzzy summations i e sums of terms related to lyapunov functions which are weighted by membership functions or products of them this paper presents an application to fuzzy control of polya s theorems on positive forms on the standard simplex the result provides a set of progressively less conservative sufficient conditions for proving positivity of fuzzy summations such conditions are less and less conservative as a complexity parameter n increases particular cases of such conditions are those in c h fang y s liu s w kau l hong c h lee a new lmi based approach to relaxed quadratic stabilization of t s fuzzy control systems ieee trans fuzzy systems 14 2006 286 397 x liu q zhang new approaches to h controller designs based on fuzzy observers for t s fuzzy systems via lmi automatica 39 9 2003 1571 1582 with n 2 and 3 respectively the proposed conditions are asymptotically exact i e necessary and sufficient when n tends to infinity or equivalently when a tolerance parameter tends to zero

a new efficient method with error analysis for solving the second kind fredholm integral equation with cauchy kernel the main objective of this paper is to give an efficient numerical method for the solution of the second kind fredholm integral equation with cauchy type kernel although numerical treatment of singular integral equations sies has been considered by many researchers and many numerical methods have been proposed for the solution of this equation strong singularity of the cauchy singular integral operator still remains as a challenge to numerical methods most of the previous methods rely on specific quadrature rule or suitable base functions for capturing the singularity here we focus on operator transformation and graded mesh in addition we study error analysis in this regard efficiency of reproducing kernel hilbert space rkhs method using smooth transformation on the graded mesh is improved and compared with some other numerical methods

application of jacobi polynomials to approximate solution of a singular integral equation with cauchy kernel abstract in this paper exact solution of the characteristic equation with cauchy kernel on the interval 1 1 is presented next jacobi polynomials are used to derive approximate solutions of the characteristic and general cauchy type singular integral equations moreover estimations of errors of the approximated solutions are presented and proved

the explicit linear quadratic regulator for constrained systems we present a technique to compute the explicit state feedback solution to both the finite and infinite horizon linear quadratic optimal control problem subject to state and input constraints we show that this closed form solution is piecewise linear and continuous as a practical consequence of the result constrained linear quadratic regulation becomes attractive also for systems with high sampling rates as on line quadratic programming solvers are no more required for the implementation

asymptotically exact stabilisation for constrained discrete takagisugeno systems via set invariance given a takagisugeno ts system this paper proposes a novel methodology to obtain the state feedback controller guaranteeing asymptotically as a polya related complexity parameter grows the largest membership shape independent possible domain of attraction with contraction rate performance based on polyhedral contractive sets from constrained linear systems literature the resulting controller is valid for any realisation of the memberships as usual in most ts literature for a finite complexity parameter an inner estimate of such largest set is obtained the frontier of such approximation can be understood as the level set of a polyhedral control lyapunov function convergence of a proposed iterative algorithm is asymptotically necessary and sufficient for ts system stabilisation for a high enough value of the complexity parameter any conceivable shape independent lyapunov controller design procedure will yield a proven domain of attraction smaller or equal to the algorithm s output

average case optimality for linear problems abstract we introduce an average case model and define general notions of optimal algorithm and optimal information we prove that the same algorithm and information are optimal in the worst and average cases and that adaptive information is not more powerful than non adaptive information

numerical treatment of solving singular integral equations by using sinc approximations abstract in this paper we propose numerical treatment for singular integral equations the methods are developed by means of the sinc approximation with smoothing transformations such approximation is an effective technique against the singularities of the equations and achieves exponential convergence therefore the methods improve conventional results where only polynomial convergence have been reported the resulting algebraic system is solved by least squares approximation and leap frog algorithm estimation of errors of the approximate solution is presented some experimental tests are presented to show the efficient of the proposed methods

lmi based relaxed nonquadratic stabilization conditions for nonlinear systems in the takagi sugeno s form this paper presents the stabilization analysis for a class of nonlinear systems that are represented by a takagi and sugeno ts discrete fuzzy model takagi and sugeno ieee trans systems man cybern 15 1 1985 116 the main result given here concerns their stabilization using new control laws and new nonquadratic lyapunov functions new relaxed conditions and linear matrix inequality based design are proposed that allow outperforming previous results found in the literature two examples are also provided to demonstrate the efficiency of the approaches