direct approach to the study of soliton perturbations a direct approach in studying the soliton perturbations of nonlinear evolution equations has been developed it is based on the method of the derivative expansions for linearization of the perturbed equations and the separation of variables for solution of the linearized equations if differs substantially from the past direct methods the apparent advantage of this approach is that it relies on no knowledge of the inverse scattering transform besides it is very concise and easy to understand as an example we use it to study the perturbed korteweg de vries equation the results we obtained agree with what other authors have found

an effective matrix geometric mean satisfying the ando li mathias properties we propose a new matrix geometric mean satisfying the ten properties given by ando li and mathias linear alg appl 2004 this mean is the limit of a sequence which converges superlinearly with convergence of order 3 whereas the mean introduced by ando li and mathias is the limit of a sequence having order of convergence 1 this makes this new mean very easily computable we provide a geometric interpretation and a generalization which includes as special cases our mean and the ando li mathias mean

direct perturbation theory for solitons of the derivative nonlinear schrodinger equation and the modified nonlinear schrodinger equation a direct perturbation theory for solitons of the derivative nonlinear schrodinger dnls equation is developed based on a closure of eigenfunctions of the linearized dnls equation around a one soliton solution the slow evolution of soliton parameters and the perturbation induced radiation are obtained under the known simple gaugelike transformation these results are transformed into those for the perturbed modified nonlinear schrodinger mnls equation describing propagation of femtosecond pulses in optical fibers a calculation of the perturbation induced radiation fields for the perturbed dnls and mnls equations is also made our results for the perturbed mnls equation can be reduced perfectly to those for the perturbed nonlinear schrodinger equation in the small nonlinear dispersion limit

mixed mean inequalities for several positive definite matrices on basis of the geometric mean proposed recently by t ando chi kwong li and roy mathias in this paper we present several kinds of mixed means for three or more positive definite matrices and prove some related mixed mean inequalities

new convergence results on functional iteration techniques for the numerical solution of m g 1 type markov chains by performing an accurate analysis of the convergence we give a complete theoretical explanation of the experimental behaviour of functional iteration techniques for the computation of the minimal nonnegative solution g of the matrix equation x sum i 0 infty x ia i arising in the numerical solution of m g 1 type markov chains here the a i s are nonnegative k times k matrices such that the matrix sum i 0 infty a i is column stochastic moreover we introduce a general class of functional iteration methods which includes the standard methods and we give an optimality convergence result in this class

the perturbed korteweg de vries equation considered anew the perturbed korteweg de vries equation is studied in a new way by a green s function formalism without use of inverse scattering methods the green s function is determined by employing the backlund transformation and green s theorem after a thorough analysis of the exact first order solution with regard to secular terms a two time scale expansion leads to the adiabatic approximation and the first order correction in accordance with the results of karpman and maslov contrary to statements in the literature the term tanh2 z in the expression for the modified phase of the perturbed soliton arises as a consequence of the systematically conducted first order perturbation theory

on the existence of a positive definite solution of the matrix equation x atx 1a i the question is raised under which conditions on the real square matrix a the matrix equation x atx 1a i has a real symmetric positive definite solution x both necessary and sufficient solvability conditions on a are derived moreover we give an algorithm to calculate the solution for a number of special cases we also present an analytic solution

on the positive definite solutions of the matrix equations xs atx ta in abstract the two matrix equations x s a t x t a i n and x s a t x t a i n are studied based on the fixed point theory the existence of the symmetric positive definite solutions are proved sensitivity analysis of the maximal solution is presented some elegant estimates of the positive definite solutions are obtained three iterative methods for computing the positive solutions are proposed

efficient computation of the extreme solutions of x a x 1 a q and x a x 1 a q we propose a new quadratically convergent algorithm having a low computational cost per step and good numerical stability properties which allows the simultaneous approximation of the extreme solutions of the matrix equations x a x 1 a q and x a x 1 a q the algorithm is based on the cyclic reduction method

on the exponential metric increasing property a short and simple proof is given for the inequality that shows that positive definite matrices constitute a riemannian manifold of negative curvature the idea of the proof leads to generalisations to non riemannian metrics and to connections with some well known inequalities of mathematical physics

matrix power means and the karcher mean we define a new family of matrix means pt a t 1 1 where and a vary over all positive probability vectors in rn and n tuples of positive definite matrices resp each of these means except t 0 arises as a unique positive definite solution of a non linear matrix equation satisfies all desirable properties of power means of positive real numbers and interpolates between the weighted harmonic and arithmetic means the main result is that the karcher mean coincides with the limit of power means as t 0 this provides not only a sequence of matrix means converging to the karcher mean but also a simple proof of the monotonicity of the karcher mean conjectured by bhatia and holbrook and other new properties which have recently been established by lawson and lim and also bhatia and karandikar using probabilistic methods on the metric structure of positive definite matrices equipped with the trace metric

direct perturbation theory for a nearly integrable nonlinear equation with application to dark soliton solutions since the linearized equation for a nearly integrable nonlinear equation is essentially a linear equation the direct perturbation theory is now developed in the frame of green s function theory of linear differential equations the perturbed nls equation with normal dispersion is treated as an example some obscure points in previous works about this equation are clarified

a remark on the problem of semiglobal nonlinear output regulation this paper shows how the method for semiglobal stabilization by output feedback based on high gain observer and saturation and the method for structurally stable regulation based on system immersion can be effectively coupled in order to solve problems of robust output regulation in the presence of parameter uncertainties ranging over compact sets

differential flatness and absolute equivalence of nonlinear control systems this paper presents a formulation of differential flatness a concept originally introduced by fliess levine martin and rouchon in terms of absolute equivalence between exterior differential systems systems that are differentially flat have several useful properties that can be exploited to generate effective control strategies for nonlinear systems the original definition of flatness was given in the context of differential algebra and required that all mappings be meromorphic functions the formulation of flatness presented here does not require any algebraic structure and allows one to use tools from exterior differential systems to help characterize differentially flat systems in particular it is shown that under regularity assumptions and in the case of single input control systems i e codimension 2 pfaffian systems a system is differentially flat if and only if it is feedback linearizable via static state feedback in higher codimensions our approach does not allow one to prove that feedback linearizability about an equilibrium point and flatness are equivalent one must be careful with the role of time as well as the use of prolongations that may not be realizable as dynamic feedback in a control setting applications of differential flatness to nonlinear control systems and open questions are also discussed

nonlinear inversion based output tracking an inversion procedure is introduced for nonlinear systems which constructs a bounded input trajectory in the preimage of a desired output trajectory in the case of minimum phase systems the trajectory produced agrees with that generated by hirschorn s inverse dynamic system however the preimage trajectory is noncausal rather than unstable in the nonminimum phase case in addition the analysis leads to a simple geometric connection between the unstable manifold of the system zero dynamics and noncausality in the nonminimum phase case with the addition of stabilizing feedback to the preimage trajectory asymptotically exact output tracking is achieved tracking is demonstrated with a numerical example and compared to the well known byrnes isidori regulator rather than solving a partial differential equation to construct a regulator the inverse is calculated using a picard like interaction when preactuation is not possible noncausal inverse trajectories can be truncated resulting in the tracking error transients found in other control schemes

a fixed point mean approximation to the cartan barycenter of positive definite matrices abstract in this paper we define a new weighted inductive geometric mean h n on the riemannian manifold of positive definite matrices of fixed size and present its fixed point mean approximation to the cartan barycenter karcher mean we show that for t 0 1 the weighted geometric mean equation x h n 1 t a 1 a n x where t t 1 t w 1 w n 1 t has a unique positive definite solution that approaches as t 0 to the weighted cartan mean of a 1 a n numerical computations for the stochastic convergence in terms of h n to derive a new contractive barycenter other than the cartan barycenter are presented

on the nonlinear matrix equation x a f x a q solutions and perturbation theory abstract in this paper the nonlinear matrix equation x a f x a q is discussed sufficient conditions for the existence and uniqueness of a positive semidefinite solution are derived also conditions are given under which the solution depends continuously on the matrices a and q

computing the extremal positive definite solutions of a matrix equation an efficient and numerically stable implementation of a known algorithm is suggested for finding the extremal positive definite solutions of the matrix equation x a x 1 a i if such solutions exist the convergence rate is analyzed a new algorithm that avoids matrix inversion is presented numerical examples are given to illustrate the effectiveness of the algorithms

hermitian solutions of the equation x q nx 1n we consider the matrix equation x q nx 1n its hermitian solutions are parametrized in terms of the generalized lagrangian eigenspaces of a certain matrix pencil we show that the equation admits both a largest and a smallest solution the largest solution corresponds to the unique positive definite solution the smallest solution is the unique negative definite solution if and only if n is nonsingular if n is singular no negative definite solution exists an interesting relation between the given equation and a standard algebraic riccati equation of kalman filtering theory is also obtained finally we present an algorithm which converges to the positive definite solution for a wide range of initial conditions

systematic perturbation theory for sine gordon solitons without use of inverse scattering methods the perturbed sine gordon soliton solutions kinks breathers in laboratory coordinates are derived by a green function formalism in a novel way dispensing with any inverse scattering methods besides the backlund transformation a simple approach based on green s theorem is employed for determining the green functions a systematic treatment allows the adiabatic approximation to be derived in a new manner the results are compared to those of the relevant papers in this field and thereby further insight is provided in a simple example a kink scattered by an impurity the calculation of energy radiation is demonstrated

a new neuro fds definition for indirect adaptive control of unknown nonlinear systems using a method of parameter hopping the indirect adaptive regulation of unknown nonlinear dynamical systems is considered in this paper the method is based on a new neuro fuzzy dynamical system neuro fds definition which uses the concept of adaptive fuzzy systems afss operating in conjunction with high order neural network functions fhonnfs since the plant is considered unknown we first propose its approximation by a special form of an fds and then the fuzzy rules are approximated by appropriate honnfs thus the identification scheme leads up to a recurrent high order neural network rhonn which however takes into account the fuzzy output partitions of the initial fds the proposed scheme does not require a priori experts information on the number and type of input variable membership functions making it less vulnerable to initial design assumptions once the system is identified around an operation point it is regulated to zero adaptively weight updating laws for the involved honnfs are provided which guarantee that both the identification error and the system states reach zero exponentially fast while keeping all signals in the closed loop bounded the existence of the control signal is always assured by introducing a novel method of parameter hopping which is incorporated in the weight updating law simulations illustrate the potency of the method and comparisons with conventional approaches on benchmarking systems are given also the applicability of the method is tested on a direct current dc motor system where it is shown that by following the proposed procedure one can obtain asymptotic regulation

iterative solution of two matrix equations we study iterative methods for finding the maximal hermitian positive definite solutions of the matrix equations x a x 1 a q and x a x 1 a q where q is hermitian positive definite general convergence results are given for the basic fixed point iteration for both equations newton s method and inversion free variants of the basic fixed point iteration are discussed in some detail for the first equation numerical results are reported to illustrate the convergence behaviour of various algorithms

flatness and defect of non linear systems introductory theory and examples we introduce flat systems which are equivalent to linear ones via a special type of feedback called endogenous their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of kalman s controllability the distance to flatness is measured by a non negative integer the defect we utilize differential algebra where flatness and defect are best defined without distinguishing between input state output and other variables many realistic classes of examples are flat we treat two popular ones the crane and the car with n trailers the motion planning of which is obtained via elementary properties of plane curves the three non flat examples the simple double and variable length pendulums are borrowed from non linear physics a high frequency control strategy is proposed such that the averaged systems become flat

the power mean and the least squares mean of probability measures on the space of positive definite matrices abstract in this paper we derive properties of the least squares or karcher mean of probability measures on the open cone of positive definite matrices of some fixed dimension endowed with the trace metric that generalize known properties of the weighted least squares mean of finitely many positive definite matrices our approach is based on first defining the t power mean of a probability measure as the unique fixed point of the contractive map x x t z d z with respect to the thompson metric establishing its properties analogous to those of the power mean for a finite number of positive definite matrices and showing the t power means converge to the karcher mean as t 0 we carry out this program first of all for the compactly supported probability measures and show that theory including the monotonicity extends to the general case

on ando li mathias geometric mean equations abstract in this paper we consider a family of nonlinear matrix equations based on the higher order geometric means of positive definite matrices that proposed by ando li mathias we prove that the geometric mean equation x b g a 1 a 2 a m x x x n has a unique positive definite solution depending continuously on the parameters of positive definite ai and positive semidefinite b it is shown that the unique positive definite solutions gn a1 a2 am for b 0 satisfy the minimum properties of geometric means yielding a sequence of higher order geometric means of positive definite matrices

green s function method for perturbed dnls equation with the squared spectrum parameter abstract a general direct perturbation theory for the perturbed dnls equation is developed the completeness relation is given by the green s function method similar to that used for the linear equations the effect of damping on a single soliton is calculated as an example and the result is good with the squared spectrum parameter

perturbation theory for the modified nonlinear schrodinger solitons abstract the perturbation theory based on the riemann hilbert problem is developed for the modified nonlinear schrodinger equation which describes the propagation of femtosecond optical pulses in nonlinear single mode optical fibers a detailed analysis of the adiabatic approximation to perturbation induced evolution of the soliton parameter is given the linear perturbation and the raman gain are considered as examples

positive solutions to x a bx 1 b abstract we study the positive semidefinite solutions to the matrix equation x a bx 1 b under the assumption that a 0 we show that a positive solution exists if and only if a certain block tridiagonal operator is positive in which case the solution is given by the shorted operator or generalized schur complement of that operator

practical output tracking of nonlinear systems with uncontrollable unstable linearization the paper focuses on the problem of having the output of nonlinear systems track a prescribed c sup 1 reference signal it is proved that under mild assumptions global practical output tracking is achievable by smooth state feedback although asymptotic output tracking is usually not possible even locally because the linearized system has uncontrollable modes whose eigenvalues are on the right half plane smooth feedback controllers which solve the problem of global practical output tracking are explicitly constructed based on a modified adding a power integrator approach this feedback design method also leads to solutions to challenging benchmark control problems including practical output tracking of an underactuated unstable two degrees of freedom mechanical system

necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation x a x 1a q abstract we consider the problem of when the matrix equation x a x 1 a q has a positive definite solution here q is positive definite we study both the real and the complex case this equation plays a crucial role in solving a special case of the discrete time riccati equation we present both necessary and sufficient conditions for its solvability this result is obtained by using an analytic factorization approach moreover we present algebraic recursive algorithms to compute the largest and smallest the solution of the equation respectively finally we discuss the number of solutions

on the matrix equation x atx 1a i simple necessary and sufficient conditions for the matrix equation x atx 1a i to have a positive definite solution are given these conditions are used to explain some properties of the equation and to obtain new information finally we prove a conjecture on the extremal solutions

a separation principle for a class of non uco systems this paper introduces a new approach to output feedback stabilization of single input single output systems which unlike other techniques found in the literature does not use quasilinear high gain observers and control input saturation to achieve separation between the state feedback and observer designs rather we show that by using nonlinear high gain observers working in state coordinates together with a dynamic projection algorithm the same kind of separation principle is achieved for a larger class of systems which are not uniformly completely observable by working in state coordinates this approach avoids using knowledge of the inverse of the observability mapping to estimate the state of the plant which is otherwise needed when using high gain observers to estimate the output time derivatives

convex geometric means abstract a class of multivariable weighted geometric means of positive definite matrices admitting jensen type inequalities for geodesically convex functions is considered it is shown that there are infinitely many such geometric means including the weighted inductive bini meini poloni and karcher means and each of these means provides a geometric mean majorization on the space of positive definite matrices some connections between our geometric mean majorizations and classical results of the standard majorization of real numbers are discussed in particular we establish the hardy littlewood polya majorization theorem and also rado s theorem and schur s convexity theorem for the weighted karcher mean

generalized polya szego type inequalities for some non commutative geometric means in this paper we shall show generalized polya szego type inequalities of n positive invertible operators on a hilbert space for any integer n 3 in terms of the following two typical non commutative geometric means that is one is the higher order weighted geometric mean due to lawson lim which is an extension of the ando li mathias geometric mean and the other is the weighted chaotic geometric mean among others the specht ratio plays an important role in our discussion which is the upper bound of a ratio type reverse of the weighted arithmetic geometric mean inequality

output feedback tracking a separation principle approach we study the practical and asymptotic tracking problems for nonlinear systems when only the output of the plant and the reference signal are available for feedback we provide sufficient conditions and a control topology yielding practical tracking in the special case when the reference signal is generated by an exosystem and there exists an internal model satisfying suitable observability properties tracking becomes asymptotic

the inverse mean problem of geometric mean and contraharmonic means abstract in this paper we solve the inverse mean problem of contraharmonic and geometric means of positive definite matrices proposed in w n anderson m e mays t d morley g e trapp the contraharmonic mean of hsd matrices siam j algebra disc meth 8 1987 674 682 a x y 2 x 1 y 1 1 b x y by proving its equivalence to the well known nonlinear matrix equation x t bx 1b where t 1 2 a a a 8 ba 1 b is the unique positive definite solution of x a 2bx 1b the inverse mean problem is solvable if and only if b a