unknown time varying input delay compensation for uncertain nonlinear systems abstract a tracking controller is developed for a class of uncertain nonlinear systems subject to unknown time varying input delay and additive disturbances a novel filtered error signal is designed using the past states in a finite integral over a constant estimated delay interval the maximum tolerable error between unknown time varying delay and a constant estimate of the delay is determined to establish uniformly ultimately bounded convergence of the tracking error to the origin the controller development is based on an approach which uses lyapunov krasovskii functionals to analyze the effects of unknown sufficiently slowly time varying input delays a stability analysis is provided to prove ultimate boundedness of the tracking error signals numerical simulation results illustrate the performance of the developed robust controller

a system of matrix equations and a linear matrix equation over arbitrary regular rings with identity abstract in this paper we consider the classical system of matrix equations a 1 xb 1 c 1 a 2 xb 2 c 2 over r an arbitrary regular ring with identity necessary and sufficient conditions for the existence and the expression of the general solution to the system are derived as an application the linear matrix equation axb cyd e over r is considered

the drazin inverse as a gradient the coefficients in the expansion of adj i a are expressed as gradients and some new representations are given for the drazin inverse of a matrix over an arbitrary field these results are then combined to express the drazin inverse as a gradient of a function of the entries of the matrix

the generalized inverse of a sum with radical element applications abstract an expression for the generalized inverse of a sum with radical element in a ring with unity is generalized to the case of a sum in an additive category c of a morphism with a reflexive von neumann regular inverse 1 2 and a morphism of c which is such that 1x 1 2 is invertible also if c is an additive category with an involution a morphism with moore penrose inverse and such that 1 x 1 x 1 x 1 x 1 1 x 1 x 1 and 1 y 1 y 1 y 1 1 y 1 y 1 are invertible then 1 y 1 x 1 1 x has a moore penrose inverse given by 1 x 1 1 x 1 y 1 relations with results of wynn roth and nashed are discussed

the inverse along a lower triangular matrix abstract in this paper we investigate the recently defined notion of inverse along an element in the context of matrices over a ring precisely we study the inverse of a matrix along a lower triangular matrix under some conditions

the perturbation theory for the drazin inverse and its applications abstract let a and e be n n matrices and b a e denote the drazin inverse of a by ad we give an upper bound for the relative error b d a d a d under certain circumstances an error bound for the solution for the singular equations ax b b r a is also considered

wave equation on spherically symmetric lorentzian metrics wave equation on a general spherically symmetric spacetime metric is constructed noether symmetries of the equation in terms of explicit functions of and are derived subject to certain differential constraints by restricting the metric to flat friedman case the noether symmetries of the wave equation are presented invertible transformations are constructed from a specific subalgebra of these noether symmetries to convert the wave equation with variable coefficients to the one with constant coefficients

generalized inverses of a sum with a radical element abstract given a von neumann regular element a and an element j in the jacobson radical j r of a ring r with unity a necessary and sufficient condition is given for a j to be von neumann regular moreover the moore penrose invertibility and group invertibility of the element a j are considered an application is given for matrices over semiprimary sbi rings

drazin invertibility for matrices over an arbitrary ring abstract characterizations are given for existence of the drazin inverse of a matrix over an arbitrary ring moreover the drazin inverse of a product paq for which there exist a p and q such that p pa a aqq can be characterized and computed this generalizes recent results obtained for the group inverse of such products the results also apply to morphisms in additive categories as an application we characterize drazin invertibility of companion matrices over general rings

the 2 2 0 drazin inverse problem abstract we consider the additive drazin problem and we study the existence of the drazin inverse of a two by two matrix with zero 2 2 entry

symmetry classifications and reductions of some classes of 2 1 nonlinear heat equation abstract the 2 1 nonlinear heat equation u t f u u x x u y y 0 is considered a symmetry classification of the equation using lie group method is presented and reduction to the first or second order ordinary differential equations is provided

symmetry reductions and exact solutions of a class of nonlinear heat equations abstract classical and nonclassical symmetries of the nonlinear heat equation ut uxx f u are considered the method of differential grobner bases is used both to find the conditions on f u under which symmetries other than the trivial spatial and temporal translational symmetries exist and to solve the determining equations for the infinitesimals a catalogue of symmetry reductions is given including some new reductions for the linear heat equation and a catalogue of exact solutions of the nonlinear heat equation for cubic f u in terms of the roots of f u 0

some additive results on drazin inverses in this paper some additive results on drazin inverse of a sum of drazin invertible elements are derived some converse results are also presented

a novel error compensation control for a class of high order nonlinear systems with input delay a novel tracking error compensation based adaptive neural control scheme is proposed for a class of high order nonlinear systems with completely unknown nonlinearities and input delay in the tracking errors of existing papers there exist the following difficulties first output curve always lags behind the desired trajectory second some big peak errors cause a decrease in tracking precision and third a big initial value of the modified tracking error can make the closed loop system unstable to tackle them three corresponding error compensation terms are constructed including a prediction and compensation term an auxiliary signal produced by the constructed auxiliary system and a damping term however inequality amplification caused by high order will weaken the effectiveness of the proposed error compensation scheme and the control precision will decrease under an assumption that the lower bounds of the unknown control coefficients should be exactly known to overcome aforementioned difficulties in the derivation of the first virtual control law the radial basis function neural network is used to approximate a hybrid term online constructed by unknown nonlinearities a lumped control coefficient achieved by state transformation and the dynamic of the proposed error compensation terms and desired signal meanwhile input delay is coped with a robust compensation signal constructed based on a finite integral of the past control values finally it is proven that all the closed loop signals are semiglobally uniformly ultimately bounded simulation results demonstrate the effectiveness of the proposed method

the moore penrose inverse of von neumann regular matrices over a ring abstract necessary and sufficient conditions are given in order that a von neumann regular matrix a over an arbitrary ring and also the matrices arising in certain factorizations of a be moore penrose invertible

saturated control of an uncertain nonlinear system with input delay this paper examines saturated control of a general class of uncertain nonlinear systems with time delayed actuation and additive bounded disturbances the bound on the control is known a priori and can be adjusted by changing the feedback gains a lyapunov based stability analysis utilizing lyapunov krasovskii lk functionals is provided to prove uniformly ultimately bounded tracking despite uncertainties in the dynamics a numerical example is presented to demonstrate the performance of the controller

continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems we consider a dynamic system containing uncertain elements only the set of possible values of these uncertainties is known based on this information a class of state feedback controls is proposed in order to guarantee uniform ultimate boundedness of every system response within an arbitrarily small neighborhood of the zero state these feedback controls are continuous functions of the state

adaptive posicast controller for time delay systems with relative degree n 2 in this paper we present an adaptive posicast controller that deals with parametric uncertainties in linear systems with delays it is assumed that the plant has no right half plane zeros and the delay is known the adaptive controller is based on the smith predictor and finite spectrum assignment with time varying parameters adjusted online a novel lyapunov krasovskii functional is used to show semi global stability of the closed loop error equations the controller is applied to engine fuel to air ratio control the implementation results show that the adaptive posicast controller significantly improves the closed loop performance when compared to the case with the existing baseline controller

predictor based control for an uncertain euler lagrange system with input delay controlling a nonlinear system with actuator delay is a challenging problem because of the need to develop some form of prediction of the nonlinear dynamics developing a predictor based controller for an uncertain system is especially challenging in this paper tracking controllers are developed for an euler lagrange system with time delayed actuation parametric uncertainty and additive bounded disturbances the developed controllers represent the first input delayed controllers developed for uncertain nonlinear systems that use a predictor to compensate for the delay the results are obtained through the development of a novel predictor like method to address the time delay in the control input lyapunov krasovskii functionals are used within a lyapunov based stability analysis to prove semi globally uniformly ultimately bounded tracking experimental results illustrate the performance and robustness of the developed control methods

new additive results for the g drazin inverse the paper studies additive properties of the generalized drazin inverse g drazin inverse in a banach algebra and finds an explicit expression for the g drazin inverse of the sum a b in terms of a and b and their g drazin inverses under fairly mild conditions on a and b

representations for the drazin inverses of the sum of two matrices and some block matrices abstract in this paper we give some formulas of the drazin inverses of the sum of two matrices under the conditions p 2 q 0 q 2 p 0 and p 3 q 0 qpq 0 qp 2 q 0 respectively and we also give some representations for the drazin inverse of block matrix a b c d a and d are square under some conditions

some additive results on drazin inverse some additive perturbation results for drazin inverses are given in particular a formula is given for the drazin inverse of a sum of two matrices when one of the products of these matrices vanishes some special applications of this are also considered

backstepping boundary control for first order hyperbolic pdes and application to systems with actuator and sensor delays abstract we consider a problem of boundary feedback stabilization of first order hyperbolic partial differential equations pdes these equations serve as a model for physical phenomena such as traffic flows chemical reactors and heat exchangers we design controllers using a backstepping method which has been recently developed for parabolic pdes with the integral transformation and boundary feedback the unstable pde is converted into a delay line system which converges to zero in finite time we then apply this procedure to finite dimensional systems with actuator and sensor delays to recover a well known infinite dimensional controller analog of the smith predictor for unstable plants we also show that the proposed method can be used for the boundary control of a korteweg de vries like third order pde the designs are illustrated with simulations

a symmetry classification for a class of 2 1 nonlinear wave equation abstract in this paper a symmetry classification of a 2 1 nonlinear wave equation u t t f u u x x u y y 0 where f u is a smooth function on u using lie group method is given the basic infinitesimal method for calculating symmetry groups is presented and used to determine the general symmetry group of this 2 1 nonlinear wave equation

diagonalizing triangular matrices via orthogonal pierce decompositions a class of sufficient conditions is given to ensure that the sum a b in a ring r is equivalent to a sum x y which is an orthogonal pierce decomposition this is then used to show that a lower triangular matrix with a regular diagonal is equivalent to its diagonal iff the matrix admits a lower triangular von neumann inverse

on generalized inverses and green s relations we study generalized inverses on semigroups by means of green s relations we first define the notion of inverse along an element and study its properties then we show that the classical generalized inverses group inverse drazin inverse and moore penrose inverse belong to this class

the inverse symmetry problem for a 2d generalized second order evolutionary equation abstract the paper is devoted to the lie symmetries of 2 d nonlinear dynamical systems described by second order partial differential equations by imposing the invariance condition of the equations under the action of the lie symmetry operator we obtained a determining system which could be solved in two directions i to find the symmetries of a concrete equation ii to obtain all the compatible equations with an imposed form of symmetry algebra this paper will pay attention to the indirect problem ii and will use the algorithm for determining the most general 2 d equations which belong following their symmetries to the same class with two interesting and important physical models the nonlinear heat equation and the transfer equation with power law nonlinearities

determinantal representations of the drazin inverse over the quaternion skew field with applications to some matrix equations abstract within the framework of the theory of the column and row determinants we obtain determinantal representations of the drazin inverse both for hermitian and arbitrary matrices over the quaternion skew field using the obtained determinantal representations of the drazin inverse we get explicit representation formulas analogs of cramer s rule for the drazin inverse solutions of a quaternion matrix equation axb d and consequently ax d and xb d in two cases if a b are hermitian or arbitrary

the representation of the drazin inverse of anti triangular operator matrices based on resolvent expansions abstract this paper deals with the anti triangular operator matrix m a b c 0 with a 2 a and ca b 0 using the resolvent expansion technique we obtain the explicit representation of the drazin inverse of m in terms of its entries and the drazin inverses of the entries and their compositions the result extends the main results in bu et al 2011 2 and castro gonzalez and dopazo 2005 6 and the results on group inverses in bu et al 2008 3 and liu and yang 2012 14 as an application a new additive result is given of the drazin inverse for two matrices p q c n n with p 2 p and pq 2 0

global asymptotic stabilization of feedforward systems with delay in the input the problem of globally uniformly asymptotically and locally exponentially stabilizing a family of nonlinear feedforward systems when there is a delay in the input is solved no limitation on the size of the delay is imposed explicit expressions of bounded control laws are determined

separation of variables of a generalized porous medium equation with nonlinear source this paper considers a general form of the porous medium equation with nonlinear source term ut d u u n x f u n 1 the functional separation of variables of this equation is studied by using the generalized conditional symmetry approach we obtain a complete list of canonical forms for such equations which admit the functional separable solutions as a consequence some exact solutions to the resulting equations are constructed and their behavior are also investigated 2002 elsevier science usa all rights reserved

global asymptotic stabilization for chains of integrators with a delay in the input the problem of the global uniform asymptotic stabilization by bounded feedback of a chain of integrators with a delay in the input is solved no limitation on the size of the delay is imposed to validate the approach a third order example is presented

a note on the formulas for the drazin inverse of the sum of two matrices abstract in this paper we give the explicit formulas for the drazin inverse of p q under some conditions furthermore we give the representations for the drazin inverse of some block matrices a b c d a and d are square under some conditions

linear systems with delayed controls a reduction linear systems with delayed control action are transformed into systems without delays under an absolute continuity condition the new system is an ordinary differential control equation in the general case the new system is a measure differential control system it is shown how the controllability stabilization and various optimization problems can be analyzed via the reduced systems

generalized inverses of a sum of morphisms abstract let c be an additive category suppose that x x is a morphism of c with group inverse and x x is a morphism of c such that 1 x is invertible let e 1 x 1 x 1 x 1 x where 1 x 1 1 x 1 then f e has a group inverse if and only if 1 x 1 x are invertible similar result for mp inverse is also given which completed huylebrouck s result

on the perturbation of the group inverse and oblique projection for given matrices a and e which are n matrices and matrices and e we discuss upper bounds for b bb b a a and bb aa aa also a lower bound for b is derived

the drazin inverse in multibody system dynamics the numerical analysis of multibody system dynamics is based on the equations of motion as differential algebraic systems a thorough analysis of the linearized equations and their solution theory leads to an equivalent system of ordinary differential equations which gives deeper insight into the derivation of integration schemes and into the stabilization approaches the main tool is the drazin inverse a generalized matrix inverse which preserves the eigenvalues the results are illustrated by a realistic truck model finally the approach is extended to the nonlinear index 2 formulation

about the von neumann regularity of triangular block matrices abstract necessary and sufficient conditions are given for the von neumann regularity of triangular block matrices with von neumann regular diagonal blocks over arbitrary rings this leads to the characterization of the von neumann regularity of a class of triangular toeplitz matrices over arbitrary rings some special results and a new algorithm are derived for triangular toeplitz matrices over commutative rings finally the drazin invertibility of some companion matrices over arbitrary rings is considered as an application

additive perturbation results for the drazin inverse abstract in this paper some new additive results for the drazin inverse are presented we give a formula for the drazin inverse of a sum of two matrices under conditions on the matrices less restrictive than those imposed in the corresponding theorem given by hartwig et al linear algebra appl 322 2001 207 217 we consider some aplications of our results to the perturbation of the drazin inverse and analyze a number of special cases

on compensating long actuator delays in nonlinear control we are interested in finite escape open loop unstable plants that are globally stabilizable in the absence of actuator delay but require controller redesign in the presence of delay the simplest such plant is z t z t 2 u t d where d is actuator delay of arbitrary length for this system we present a control law that compensates the delay and achieves feedback linearization of the entire ode delay infinite dimensional cascade however even though exponential stability is achieved the result is not global because the plant can have a finite escape with an initial condition z 0 ges 1 d before the feedback control reaches it at t d we prove a stability result whose region of attraction is essentially z 0 1 d and for which we derive an asymptotic stability bound in terms of the system norm z t 2 intt t d u thetas 2dthetas

on the drazin inverses involving power commutativity abstract we explore the drazin inverses of bounded linear operators with power commutativity p q q m p in a hilbert space conditions on drazin invertibility are formulated and shown to depend on spectral properties of the operators involved moreover we prove that p q is drazin invertible if p and q are dual power commutative p q q m p and q p p n q and show that the explicit representations of the drazin inverse p q d depend on the positive integers m n 2

representations for the drazin inverses of 2 2 block matrices abstract let m denote a 2 2 block complex matrix a b c d where a and d are square matrices not necessarily with the same orders in this paper explicit representations for the drazin inverse of m are presented under the condition that bd i c 0 for i 0 1 n 1 where n is the order of d

connections between razumikhin type theorems and the iss nonlinear small gain theorem a razumikhin type theorem that guarantees input to state stability for functional differential equations with disturbances is established using the nonlinear small gain theorem the result is used to show that input to state stabilizability for nonlinear finite dimensional control systems is robust in an appropriate sense to small time delays at the input also relaxed razumikhin type conditions guaranteeing global asymptotic stability for differential difference equations are given

explicit formulas for determinantal representations of the drazin inverse solutions of some matrix and differential matrix equations the drazin inverse solutions of the matrix equations ax b xa b and axb d are considered in this paper we use both the determinantal representations of the drazin inverse obtained earlier by the author and in the paper we get analogs of the cramer rule for the drazin inverse solutions of these matrix equations and using their for determinantal representations of solutions of some differential matrix equations x ax b and x xa b where the matrix a is singular