solution bounds of the continuous riccati matrix equation a new approach is proposed for estimating the solution of the continuous algebraic riccati equation care upper and lower solution bounds of the care are presented comparisons show that the present bounds are more general and or tighter than existing results

the discriminance for a special class of circulant matrices and their diagonalization abstract this paper deals with a new type of matrices the structure of this class of matrices is similar to circulant matrix we call it h circulant matrix furthermore the sum the difference the product the inverse and the adjoint matrix of these matrices are also h circulant matrixes we not only give six discriminance for h circulant matrices but also discuss the diagonalization and nonsingularity of these matrices finally author gives a numerical example

new upper solution bounds of the continuous algebraic riccati matrix equation by utilizing some linear algebraic techniques new upper bounds of the solution of the continuous algebraic riccati equation care are derived according to the present bounds iterative procedures are also developed for obtaining more precise estimations comparing with existing results the obtained bounds are less restrictive

bounds in algebraic riccati and lyapunov equations a survey and some new results this paper summarizes and investigates the relations of a number of bounds for the solutions of the algebraic riccati and lyapunov equations that have been reported during the last two decades also presented are bounds for the unified riccati equation using the delta operator and it is shown that some bounds for the continuous and discrete riccati equations can be unified by them

lower summation bounds for the discrete riccati and lyapunov equations lower eigenvalue summation including trace bounds for the solution of the discrete algebraic riccati and lyapunov matrix equations are presented these are tighter than or supplement existing results

homoclinic solutions of an infinite dimensional hamiltonian system we consider the system begin equation left begin array l partial t u delta x u v x u h v t x u v 3mm partial t v delta x v v x v h u t x u v end array qquad text for t x in mathbb r times mathbb r n right end equation which is an unbounded hamiltonian system in l 2 mathbb r n mathbb r 2m we assume that the constant function u 0 v 0 equiv 0 0 in mathbb r 2m is a stationary solution and that h and v are periodic in the t and x variables we present a variational formulation in order to obtain homoclinic solutions z u v satisfying z t x to 0 as t x to infty it is allowed that v changes sign and that delta v has essential spectrum below and above 0 we also treat the case of a bounded domain omega instead of mathbb r n with dirichlet boundary conditions

neural network approach to computing matrix inversion a neural network for matrix inversion is proposed we show both analytically and by simulations that this network is guaranteed to be stable and to provide results arbitrarily close to the accurate inversion of a matrix within an elapsed time of only a few characteristic time constants of the network the parameters such as the interconnection strengths of this network can be obtained from the known matrix without any computations

upper matrix bound of the solution for the discrete riccati equation a new upper matrix bound of the solution for the discrete algebraic matrix riccati equation is developed this matrix bound is then used to derive bounds on the eigenvalues trace and determinant of the same solution it is shown that these eigenvalue bounds are less restrictive than previous results

recurrent neural networks for computing pseudoinverses of rank deficient matrices three recurrent neural networks are presented for computing the pseudoinverses of rank deficient matrices the first recurrent neural network has the dynamical equation similar to the one proposed earlier for matrix inversion and is capable of moore penrose inversion under the condition of zero initial states the second recurrent neural network consists of an array of neurons corresponding to a pseudoinverse matrix with decaying self connections and constant connections in each row or column the third recurrent neural network consists of two layers of neuron arrays corresponding respectively to a pseudoinverse matrix and a lagrangian matrix with constant connections all three recurrent neural networks are also composed of a number of independent subnetworks corresponding to the rows or columns of a pseudoinverse the proposed recurrent neural networks are shown to be capable of computing the pseudoinverses of rank deficient matrices

a recurrent neural network for solving sylvester equation with time varying coefficients presents a recurrent neural network for solving the sylvester equation with time varying coefficient matrices the recurrent neural network with implicit dynamics is deliberately developed in the way that its trajectory is guaranteed to converge exponentially to the time varying solution of a given sylvester equation theoretical results of convergence and sensitivity analysis are presented to show the desirable properties of the recurrent neural network simulation results of time varying matrix inversion and online nonlinear output regulation via pole assignment for the ball and beam system and the inverted pendulum on a cart system are also included to demonstrate the effectiveness and performance of the proposed neural network

a neural network for linear matrix inequality problems gradient type hopfield networks have been widely used in optimization problems solving the paper presents a novel application by developing a matrix oriented gradient approach to solve a class of linear matrix inequalities lmis which are commonly encountered in the robust control system analysis and design the solution process is parallel and distributed in neural computation the proposed networks are proven to be stable in the large representative lmis such as generalized lyapunov matrix inequalities simultaneous lyapunov matrix inequalities and algebraic riccati matrix inequalities are considered several examples are provided to demonstrate the proposed results to verify the proposed control scheme in real time applications a high speed digital signal processor is used to emulate the neural net based control scheme

solving simultaneous linear equations using recurrent neural networks abstract simultaneous linear algebraic equations can be found in many mathematical model formulations in monitoring and control of dynamic systems there is often a need for solving simultaneous linear algebraic equations in real time in this paper recurrent neural networks for solving simultaneous linear algebraic equations are proposed the asymptotic stability of the proposed neural networks and solvability of simultaneous linear equations by using the neural networks are substantiated a circuit schematic for realizing the neural networks is described the results of numerical simulations are discussed via illustrative examples an extension of the recurrent neural networks for solving quadratic programming problems subject to equality constraints is also discussed

matrix bounds of the discrete are solution this paper provides new lower and upper matrix bounds of the solution to the discrete algebraic riccati equation the lower bound always works if the solution exists the upper bounds are presented in terms of the solution of the discrete lyapunov equation and its upper matrix bound the upper bounds are always calculated if the solution of the lyapunov equation exists a numerical example shows that the new bounds are tighter than previous results in many cases

eigenvalue bounds for algebraic riccati and lyapunov equations the majorization result of wimmer 2 relating the eigenvalues of the matrices involved in the lyapunov equation is extended to the algebraic riccati equation this result coupled with certain results on the eigenvalue bounds for sum and product of matrices yields several lower and upper bounds for the eigenvalues of the solution matrix of the algebraic riccati and lyapunov equations

bounds for the solution of the lyapunov matrix equation a unified approach abstract in recent years several bounds have been reported for the solution of the continuous and the discrete lyapunov equations using the unified lyapunov equation we give in this paper bounds for the solution of this equation in the limiting cases the bounds reduce to existing bounds for both the continuous and discrete lyapunov equations

matrix bounds of the solutions of the continuous and discrete riccati equations a unified approach in this paper a new scheme is introduced to measure the matrix bounds of the continuous and discrete riccati equations by estimating upper and lower matrix bounds of the solution of the unified algebraic riccati equation uare the same measurements for the solutions of the continuous and discrete riccati equations respectively can be obtained in limiting cases according to these obtained matrix bounds several eigenvalue bounds are also defined all the proposed results for the uare are new and more general than previous work some obtained results are compared with those of the literature via numerical examples it is shown that in some cases the presented results are tighter than the existing ones

a note on the discrete lyapunov and riccati matrix equations abstract inequalities which are satisfied by the determinants of the positive definite solutions of the discrete algebraic riccati and lyapunov matrix equations are presented the results give lower bounds for the product of the eigenvalues of the matrix solutions also for a discrete lyapunov equation an algorithm is presented to determine under what conditions a positive diagonal solution will exist if all the conditions are satisfied the algorithm also provides such a diagonal solution

a note on eigenvalue bounds in algebraic riccati equation in this note an upper bound on the maximum eigenvalue of the solution matrix k of the algebraic riccati equation is established the approach outlined also results in several lower bounds which are more general than those derived in 1 for some of the largest eigenvalues of k

multiple solutions for a class of nonlinear schrodinger equations abstract in this paper we find new conditions to ensure the existence of infinitely many homoclinic type solutions for the schrodinger equation u v x u g x u for x r n assuming v x and g x u depend periodically on x we deal with the situations where g x u is as u asymptotically linear or superlinear with certain hypothesis different from ones used in previous related study our approach is variational and we use the cerami condition instead of the palais smale one for deformation arguments

three dimensional structured networks for matrix equation solving two three dimensional structured networks are developed for solving linear equations and the lyapunov equation the basic idea of the structured network approaches is to first represent a given equation solving problem by a 3 d structured network so that if the network matches a desired pattern array the weights of the linear neurons give the solution to the problem then train the 3 d structured network to match the desired pattern array using some training algorithms and finally obtain the solution to the specific problem from the converged weights of the network the training algorithms for the two 3 d structured networks are proved to converge exponentially fast to the correct solutions simulations were performed to show the detailed convergence behaviors of the 3 d structured networks

iterative matrix bounds and computational solutions to the discrete algebraic riccati equation bilateral matrix bounds for the solution of the discrete algebraic riccati equation dare are presented they are new or tighter than the existing bound computational algorithms to solve the dare follow

applications of local linking to critical point theory we prove the existence of nontrivial critical points of functionals with a local linking at 0 applications to elliptic boundary value problems and to hamiltonian systems are given c 1995 academic press inc

a new approach for upper bound estimations of the solution of the continuous riccati equation by utilizing some linear algebraic techniques a new approach for measuring upper bounds of the solution of the continuous algebraic riccati equation care is developed two upper matrix bounds are developed comparing with those appeared in the literature the obtained bounds are less restrictive or more concise therefore they can be applied easily to deal with control problems

closed form solutions to sylvester conjugate matrix equations some explicit closed form solutions of homogeneous and nonhomogeneous sylvester conjugate matrix equations are provided in this paper one of the solutions is expressed in terms of controllability matrices and observability matrices the proposed approach does not require all the coefficient matrices to be in any canonical forms and the solutions provide a significant degree of freedom which is represented by an arbitrarily chosen parameter matrix by specifying the solutions of the homogeneous sylvester conjugate equation some new expressions of the solutions of the normal sylvester normal sylvester conjugate and sylvester equations are given this fact reveals that the sylveter conjugate matrix equations are a more general class of some previously investigated matrix equations

new upper solution bounds of the discrete algebraic riccati matrix equation in this note we present upper matrix bounds for the solution of the discrete algebraic riccati equation dare using the matrix bound of theorem 2 2 we then give several eigenvalue upper bounds for the solution of the dare and make comparisons with existing results the advantage of our results over existing upper bounds is that the new upper bounds of theorem 2 2 and corollary 2 1 are always calculated if the stabilizing solution of the dare exists whilst all existing upper matrix bounds might not be calculated because they have been derived under stronger conditions finally we give numerical examples to demonstrate the effectiveness of the derived results

error bound for transfer function order reduction using frequency weighted balanced truncation abstract an error bound for transfer function order reduction is derived when frequency weighted balanced truncation is the order reduction method the bound is valid for both one sided input or output and two sided weighted balancing approximations with stable weights which can otherwise be arbitrary the error bound formula is valid for both discrete time and continuous time problems examples are studied to demonstrate effectiveness of the error bound

improved bounds for the eigenvalues of solutions of lyapunov equations bounds on the solution and on the eigenvalues of the solution of the discrete and of the continuous matrix lyapunov equations are presented these bounds are nontrivial in the case of system matrices having multiple eigenvalues especially in the latter case these bounds are better than bounds found in the literature further these bounds are sharp i e there exist systems such that the lower and upper bounds coincide yielding the solution or its eigenvalues respectively

on bounds of the riccati and lyapunov matrix equations various bounds for the traces of the solutions of the algebraic riccati and lyapunov matrix equations are established in the continuous and discrete domain respectively the presented results give lower bounds for the sum of the eigenvalue of the matrix solutions

a recurrent neural network for real time semidefinite programming the semidefinite programming problem is an important optimization problem that has been extensively investigated a real time solution method for solving such a problem however is still not yet available the paper proposes a recurrent neural network for this purpose first an auxiliary cost function is introduced to minimize the duality gap between the admissible points of the primal problem and the corresponding dual problem then a dynamical system is constructed to drive the duality gap to zero exponentially along any trajectory by modifying the gradient of the auxiliary cost function furthermore a subsystem is developed to circumvent the computation of matrix inverse so that the resulting overall dynamical system can be realized using a recurrent neural network the architecture of the resulting neural network is discussed the operating characteristics and performance of the proposed approach are demonstrated by means of simulation results

periodic solutions for a class of non autonomous hamiltonian systems abstract we consider the existence of nontrivial periodic solutions for a superlinear hamiltonian system h j u a t u h t u 0 u r 2 n t r we prove an abstract result on the existence of a critical point for a real valued functional on a hilbert space via a new deformation theorem different from the works in the literature the new deformation theorem is constructed under the cerami type condition instead of palais smale type condition in addition the main assumption here is weaker than the usual ambrosetti rabinowitz type condition 0 h t u u h t u 2 u r 0 this result extends theorems given by li and willem j math anal appl 189 1995 6 32 and li and szulkin j differential equations 112 1994 226 238

the inverse of circulant matrix abstract a direct method is proposed to get the inverse matrix of circulant matrix that find important application in engineering the elements of the inverse matrix are functions of zero points of the characteristic polynomial g z and g z of circulant matrix four examples to get the inverse matrix are presented in the paper

explicit solution and eigenvalue bounds in the lyapunov matrix equation an explicit solution to the algebraic lyapunov matrix equation is obtained in terms of the controllability matrix of the pair of coefficient matrices this enables us to determine the number of positive eigenvalues of the positive semidefinite solution through the controllability matrix based on this explicit formula upper and lower bounds for each eigenvalue of the solution are derived which always give nontrivial estimates

trace bounds on the solution of the algebraic matrix riccati and lyapunov equation lower and upper bounds on the trace of the positive semidefinite solution of the algebraic matrix riccati and lyapunov equation are derived the upper trace bound obtained in this note in many cases results in a tighter bound as compared to the upper bound for the maximal eigenvalue proposed in 1 and 2

eigenvalue bounds for the discrete lyapunov matrix equation upper and lower bounds for every eigenvalue of the positive definite solution to the discrete lyapunov matrix equation are provided the obtained bounds have an advantage that they can give nontrivial lower bounds even when some existing results yield trivial ones

comments on upper matrix bounds for the discrete algebraic riccati matrix equation with author s reply in the above paper h h choi ieee trans automat contr vol 46 pp 504 8 2001 new methods to obtain upper matrix bounds for a discrete algebraic riccati equation dare have been established but the author assumes that the system matrix a is nonsingular and dare has a positive definite solution in this note some similar results can be obtained when a solution is positive semidefinite and there are no extra assumption that a is nonsingular in reply the original author explains some of the underlying assumptions and comments on the computational burdens in the inequality in the commenters note

upper bounds for the eigenvalues of the solution of the lyapunov matrix equation eigenvalue upper bounds for the solution of the lyapunov matrix equation are presented for single eigenvalues summations of eigenvalues including the trace and products of eigenvalues including the determinant these bounds are compared to those for the trace and the extreme eigenvalues

a brief summary of the bounds on the solution of the algebraic matrix equations in control theory abstract in recent years several bounds have been reported for different measures of the extent or size of the solution of the algebraic matrix equations arising in control theory such as the riccati equation and the lyapunov equation this paper collects the bounds that have been presented up to now and summarizes them in an unified form this will prove particularly convenient for those wishing to get a ready estimate of the solution while solving the equations numerically or to develop theoretical results that rely on these bounds

on the computation of weighted moore penrose inverse using a high order matrix method we propose a matrix method for calculating generalized weighted moore penrose inverse numerically it is proved that the scheme is convergent with high order via theoretical analysis some tests are also presented to show the applicability and the superiority of the new method

two finite time convergent zhang neural network models for time varying complex matrix drazin inverse abstract this paper concerns the computation of the drazin inverse of a complex time varying matrix based on two zhang functions constructed from two limit representations of the drazin inverse we present two complex zhang neural network znn models with the li activation function for computing the drazin inverse of a complex time varying square matrix we prove that our znn models globally converge in finite time in addition upper bounds of the convergence time are derived analytically via the lyapunov theory our simulation results verify the theoretical analysis and demonstrate the superiority of our znn models over the gradient based gnn models

iterative solutions to the kalman yakubovich conjugate matrix equation abstract two operations are introduced for complex matrices in terms of these two operations an infinite series expression is obtained for the unique solution of the kalman yakubovich conjugate matrix equation based on the obtained explicit solution some iterative algorithms are given for solving this class of matrix equations convergence properties of the proposed algorithms are also analyzed by using some properties of the proposed operations for complex matrices

recurrent neural networks for solving linear matrix equations abstract recurrent neural networks for solving linear matrix equations are proposed the proposed recurrent neural networks consist of two bidirectionally connected layers and each layer consists of an array of neurons the proposed recurrent neural networks are shown to be asymptotically stable in the large and capable of computing inverse matrices and solving lyapunov matrix equations the operating characteristics of the proposed recurrent neural networks are demonstrated via several illustrative examples

recurrent neural network for computing the w weighted drazin inverse abstract two gradient based recurrent neural networks rnns for computing the w weighted drazin inverse of a real rectangular matrix are proposed and considered usage of the first rnn is limited by a specific constraint on the spectrum of a certain matrix the second rnn is usable without restrictions the stability of the recurrent neural networks as well as their convergence are considered numerical examples are given to show the efficiency of the proposed neural networks

qualitative analysis and decentralized controller synthesis for a class of large scale systems with symmetrically interconnected subsystems abstract a number of large scale interconnected systems often encountered in practice are composed of subsystems with similar dynamics interconnected in a symmetrical fashion and the synthesis of controllers for such systems must exploit the special structural properties in order to avoid overly conservative designs and to take advantage of the possible beneficial effects of the interconnections an analysis of some important qualitative properties of such symmetrically interconnected systems focussing on the spectrum characterization controllability and observability and the solutions of the algebraic riccati equation and the matrix lyapunov equation is conducted in this paper and procedures for constructing the solutions to the analysis problems at the overall system level from the computationally simple subsystem level solutons are developed a decentralized controller design procedure is presented as an illustration of the utilization of the available structural information in addressing synthesis problems numerical examples are included to demonstrate the superiority of the presented designs over the use of existing approaches which do not take full advantage of the structural knowledge in these large scale systems

upper solution bounds of the continuous and discrete coupled algebraic riccati equations in this paper we propose upper bounds for the sum of the maximal eigenvalues of the solutions of the continuous coupled algebraic riccati equation ccare and the discrete coupled algebraic riccati equation dcare which are then used to infer upper bounds for the maximal eigenvalues of the solutions of each riccati equation by utilizing the upper bounds for the maximal eigenvalues of each equation we then derive upper matrix bounds for the solutions of the ccare and dcare following the development of each bound an iterative algorithm is proposed which can be used to derive tighter upper matrix bounds finally we give numerical examples to demonstrate the effectiveness of the proposed results making comparisons with existing results

critical points theorems concerning strongly indefinite functionals and infinite many periodic solutions for a class of hamiltonian systems based on new deformation theorems concerning strongly indefinite functionals we give some new min max theorems which are useful in looking for critical points of functionals which are strongly indefinite and satisfy cerami condition instead of palais smale condition as one application of abstract results we study existence of multiple periodic solutions for a class of non autonomous first order hamiltonian system hs z jh z t x z t x rx w where d dt w r nn 1 is a bounded domain with smooth boundary w z p q r mxr m r 2 m j 0i i0 and h t x z c 1 rx wxr 2 m r

on the discrete riccati equation several bounds have been reported recently for the trace of the solution to the discrete algebraic matrix riccati equation this note adds an alternative one to them

on some bounds in the algebraic riccati and lyanupov equations some bounds for the trace and determinant of the solution to the algebraic riccati and lyapunov matrix equations are presented these bounds give information on mean size of the solution to the equations in a simple way the obtained determinant bound for the lyapunov equation shows the improvement over the previously reported one

analog approach for the eigen decomposition of positive definite matrices abstract this paper proposes an analog approach for performing the eigen decomposition of positive definite matrices we show analytically and by simulations that the proposed circuit is guaranteed to converge to the desired eigenvectors and eigenvalues of positive definite matrices

positive integer powers of complex skew symmetric circulant matrices abstract in j rimas on computing of arbitrary positive integer powers for one type of odd order symmetric circulant matrices i applied mathematics and computation 165 2005 137 141 j rimas on computing of arbitrary positive integer powers for one type of even order symmetric circulant matrices i applied mathematics and computation 172 2006 86 90 rimas derived a general expression for the entries of the qth power q n of the n n real symmetric circulant matrix circ n 0 1 0 0 0 0 1 for all n n in this paper we present an extension of that interesting work deriving a similar expression for the entries of the positive integer powers of any complex symmetric circulant matrix

a note on the lyapunov matrix equation in 5 bound for the determinant of the solution to the lyapunov matrix equation was reported this note gives an another bound for this value

some results on symmetric circulant matrices and on symmetric centrosymmetric matrices we derive a readily computable sufficient condition for the existence of a nonnegative symmetric circulant matrix having a prescribed spectrum moreover we prove that any set 1 2 n is the spectrum of real symmetric centrosymmetric matrices s1 and s2 such that for s1 an eigenvector corresponding to 1 is the all ones vector and for s2 an eigenvector corresponding to 1 is the vector with components 1 and 1 alternately the proof is constructive then we derive an improved condition on k k 1n such that s1 1e where e is a stochastic symmetric centrosymmetric matrix finally we propose an algorithm to compute the eigenvalues of some real symmetric centrosymmetric matrices all the numerical procedures are based on the use of the fast fourier transform

solutions of the lyapunov matrix equation bx xa c the main purpose of this correspondence is to establish an expression for the solution of the lyapunov matrix equation in terms of the principal indempotents and nilpotents of the coefficient matrices the coefficient matrices are not necessarily of the same size and may have common characteristic roots and have elements belonging to the field of complex numbers

lower bounds on the solution of lyapunov matrix and algebraic riccati equations in this note various lower bounds for all the eigenvalues of the solution matrix k of the lyapunov matrix equation are established a special case of this result is a generalization of that presented in 1 3 where lower bounds for the maximum and minimum eigenvalues of k are given moreover the approach used here enables one to establish various lower bounds for some of the largest eigenvalues of the solution matrix of the algebraic riccati equation

stability analysis of a family of matrices the authors extend the use of the lyapunov equation method to check the stability of a family of matrices the family of interval matrices is defined as the convex hull of a finite number of matrices and is a special case where each element has independent perturbation while the stability of interval matrices cannot be inferred from the stability of its vertex matrices the authors show that the family of matrices will be stable if the symmetric matrices associated with the vertex matrices have matrix measures less than two

the existence uniqueness and the fixed iterative algorithm of the solution for the discrete coupled algebraic riccati equation in this article applying the properties of m matrix and non negative matrix utilising eigenvalue inequalities of matrix s sum and product we firstly develop new upper and lower matrix bounds of the solution for discrete coupled algebraic riccati equation dcare secondly we discuss the solution existence uniqueness condition of the dcare using the developed upper and lower matrix bounds and a fixed point theorem thirdly a new fixed iterative algorithm of the solution for the dcare is shown finally the corresponding numerical examples are given to illustrate the effectiveness of the developed results

eigenvalue decay bounds for solutions of lyapunov equations the symmetric case abstract we present two new bounds for the eigenvalues of the solutions to a class of continuous and discrete time lyapunov equations these bounds hold for lyapunov equations with symmetric coefficient matrices and right hand side matrices of low rank they reflect the fast decay of the nonincreasingly ordered eigenvalues of the solution matrix

bounds for the eigenvalues of the solution of the discrete riccati and lyapunov equations and the continuous lyapunov equation we present some bounds for the eigenvalues and certain sums and products of the eigenvalues of the solution of the discrete riccati and lyapunov matrix equations and the continuous lyapunov matrix equation nearly all of our bounds for the discrete riccati equation are new the bounds for the discrete and continuous lyapunov equations give a completion of some known bounds for the extremal eigenvalues and the determinant and the trace of the solution of the respective equation