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Based on practical applications of the complex-valued Sylvester equation and the effectiveness of zeroing neural network (ZNN) in solving time-varying problems, two discrete nonlinear and noise-tolerant ZNN (DNN-TZNN) models are proposed to solve the time-varying augmented complex Sylvester (TACS) equation by using the Adams-Bashforth formula to discretize the continuous nonlinear and noise-tolerant ZNN (CNN-TZNN) model. The presented DNN-TZNN models are divided into two types: DNN-TZNK and DNN-TZNU models, according to whether the derivative information is known or not in the CNN-TZNN design formula. Compared with the continuous ZNN methods, the proposed DNN-TZNN models are more innovative, and compared with other discrete ZNN methods, the convergence speed of the DNN-TZNN models is much faster and more accurate. Through the theoretical analysis, the superior convergence and strong robustness of the DNN-TZNN models are guaranteed. At last, the simulative experiments not only verify the correctness of the theoretical analysis but also demonstrate the availability of the DNN-TZNN models in solving the TACS problem.

In this article, two Adams-Bashforth-type integration-enhanced discrete-time zeroing neural dynamic (ADTIZD) models are proposed to solve the time-varying complex Sylvester equation (TVCSE) problem in the first time. In ADTIZD models, Adams-Bashforth discrete formulas as novel discrete formulas are used, giving our ADTIZD models higher accuracy [truncation error being O(τ⁵)] but less time and space complexity than the ordinary multi-instant models. Enhanced by the integration part, the ADTIZD models can resist large additive noises, where even constant noises cannot decrease their precision. All convergence and robustness performance conclusions about our ADTIZD models are supported by rigorous theoretical proofs and numerical experiments. More comparisons between ADTIZD models and other discrete-time zeroing neural network models are shown in these experiments too. The efficacy of ADTIZD models is finally been validated in the simulation of adopting them in controlling a robotic manipulator.

Exponential time integrators are well-established discretization methods for time semilinear systems of ordinary differential equations. These methods use (Formula presented.) functions, which are matrix functions related to the exponential. This work introduces an algorithm to speed up the computation of the (Formula presented.) function action over vectors for two-dimensional (2D) matrices expressed as a Kronecker sum. For that, we present an auxiliary exponential-related matrix function that we express using Kronecker products of one-dimensional matrices. We exploit state-of-the-art implementations of (Formula presented.) functions to compute this auxiliary function's action and then recover the original (Formula presented.) action by solving a Sylvester equation system. Our approach allows us to save memory and solve exponential integrators of 2D+time problems in a fraction of the time traditional methods need. We analyze the method's performance considering different linear operators and with the nonlinear 2D+time Allen–Cahn equation.

Sylvester equation is often applied to various fields, such as mathematics and control systems due to its importance. Zeroing neural network (ZNN), as a systematic design method for time-variant problems, has been proved to be effective on solving Sylvester equation in the ideal conditions. In this paper, in order to realize the predefined-time convergence of the ZNN model and modify its robustness, two new noise-tolerant ZNNs (NNTZNNs) are established by devising two novelly constructed nonlinear activation functions (AFs) to find the accurate solution of the time-variant Sylvester equation in the presence of various noises. Unlike the original ZNN models activated by known AFs, the proposed two NNTZNN models are activated by two novel AFs, therefore, possessing the excellent predefined-time convergence and strong robustness even in the presence of various noises. Besides, the detailed theoretical analyses of the predefined-time convergence and robustness ability for the NNTZNN models are given by considering different kinds of noises. Simulation comparative results further verify the excellent performance of the proposed NNTZNN models, when applied to online solution of the time-variant Sylvester equation.

This paper presents the numerical solution of the Black–Scholes partial differential equation (PDE) for the evaluation of European call and put options. The proposed method is based on the finite difference and Legendre wavelets aproximation scheme. We derive a matrix structure for the Legendre wavelets integral operator which has been widely used so far. Moreover, in order to use the payoff function, another operational matrix is derived. By the proposed combined method, the solving Black–Scholes PDE problem reduces to those of solving a Sylvester equation. The proposed algorithms show that in compared to literature methods, the proposed method is easy to be implemented and have high execution speed. Furthermore, we prove that the obtained Sylvester equation has a unique solution. In addition, the effect of the finite difference space step size to the computational accuracy is studied. For having suitable solution, the numerical solutions show that there is no need to select very small step size. Also only a small number of basis functions in the Legendre wavelets series is needed. The numerical results demonstrate efficiency and capability of the proposed method.

Model order reduction (MOR) has emerged as a de-facto solution for simulation, control, and optimization of large-scale and complex dynamical systems from the past decades. This paper demonstrates two comprehensive MOR approaches to overcome the overall computational cost required to simulate the nonlinear, viscous Burger’s equation. The former scheme is the approximate proper orthogonal decomposition (APOD), which involves capturing the approximate snapshot ensemble of the states by performing multiple linearizations along the state-trajectory. The latter technique is based on the notion of nonlinear moment matching (NLMM), in which the reduced projection basis is generated by numerically solving the simplified, nonlinear, Sylvester equation. Both the reduction schemes are tested on the high-fidelity finite difference (FD) formulation of 1D and the 2D Burgers’ equation. The results show substantial computational savings in extracting the orthonormal projection basis required to generate the reduced manifold. Furthermore, a comparative study of both methods is presented with proper orthogonal decomposition (POD) for various test signals.

This paper proposes the modified generalization of the HSS (MGHSS) to solve a large and sparse continuous Sylvester equation, improving the efficiency and robustness. The analysis shows that the MGHSS converges to the unique solution of AX + XB = C unconditionally. We also propose an inexact variant of the MGHSS (IMGHSS) and prove its convergence under certain conditions. Numerical experiments verify the efficiency of the proposed methods.

In this paper, the Incomplete Global GMERR algorithm and the Global GMERR algorithm are used to solve the Sylvester equation. The numerical experiment is given to compare the CPU run time and the number of iterations of the two methods.

Based on the second order differential equation, this paper investigates finite-time stability, finite-time convergence criterions and estimates of convergence time. The main contributions of this paper lie in the fact that two new finite-time convergence criterions are proposed through the property of the second order differential equation and their upper bound of the convergence time is derived. In addition, our finite-time stability criterions are used to a recurrent neural network for solving time-varying Sylvester equation and Pseudo-Inverse of a Matrix. At last, a numerical example and a Pseudo-Inverse of a Matrix demonstrate the effectiveness of our method.

The time-varying complex-valued Sylvester equation (TVCVSE) often appears in many fields such as control and communication engineering. Classical recurrent neural network (RNN) models (e.g., gradient neural network (GNN) and zeroing neural network (ZNN)) are often used to solve such problems. This paper proposes an adaptive coefficient and non-convex projection zeroing neural network (ACNPZNN) model for solving TVCVSE. To enhance its adaptability as residual error decreasing as time, an adaptive coefficient is designed based on residual error. Meanwhile, this paper breaks the convex constraint by constructing two complex-valued non-convex projection activation functions from two different aspects. Moreover, the global convergence of the proposed model is proved, the anti-noise performance of the ACNPZNN model under different noises is theoretically analyzed. Finally, simulation experiments are provided to compare the convergence performance of different models, which simultaneously verifies the effectiveness and superiority of the proposed model.

In this paper, we study several constrainted least squares solutions of quaternion Sylvester matrix equation. We first propose a real vector representation of quaternion matrix and study its properties. By using this real vector representation, semi-tensor product of matrices, swap matrix and Moore-Penrose inverse, we derive compatible conditions and the expressions of several constrainted least squares solutions of quaternion Sylvester equation.

There are two equivalent methods for dealing with the nonlinearity of complex-valued problems. The first method is to handle the real part and imaginary part of complex inputs, and the second method is to handle the modulus of complex inputs. Based on these two methods, this paper explores a superior nonlinear activation function and proposes two improved finite-time zeroing neural network (IFTZNN) models for time-varying complex Sylvester equation solving. Regarding the existing neural model activated by the sign-bi-power (SBP) activation function, the convergence upper bounds of the IFTZNN models are much smaller, and thus we can estimate their convergence time more accurately. Furthermore, the detailed theoretical analysis of the IFTZNN models is provided to show their effectiveness. Comparative simulation results also verify the advantages of our proposed IFTZNN models for complex Sylvester equation solving.

A novel finite-time convergent zeroing neural network (ZNN) based on varying gain parameter for solving time-varying (TV) problems is presented. The model is based on the improvement and generalization of the finite-time ZNN (FTZNN) dynamics by means of the varying-parameter ZNN (VPZNN) dynamics, and termed as VPFTZNN. More precisely, the proposed model replaces fixed and large values of the scaling parameter by an appropriate time-dependent gain parameter, which leads to a faster and bounded convergence of the error function in comparison to previous ZNN methods. The convergence properties of the proposed VPFTZNN dynamical evolution in its generic form is verified. Particularly, VPFTZNN for solving linear matrix equations and for computing generalized inverses are investigated theoretically and numerically. Moreover, the proposed design is applicable in solving the TV matrix inversion problem, solving the Lyapunov and Sylvester equation as well as in approximating the matrix square root. Theoretical analysis as well as simulation results validate the effectiveness of the introduced dynamical evolution. The main advantages of the proposed VPFTZNN dynamics are their generality and faster finite-time convergence with respect to FTZNN models.

In this study, the authors design distributed algorithms for solving the Sylvester equation A X + X B = C in the sense of least squares over a multi-agent network. In the problem setup, every agent in the interconnected system only has local information of some columns or rows of data matrices A, B and C, and exchanges information among neighbour agents. They propose algorithms with mainly focusing on a specific partition case, whose designs can be easily generalised to other partitions. Three distributed continuous-time algorithms aim at two cases for seeking a least-squares/regularisation solution from the viewpoint of optimisation. Due to the equivalence between an equilibrium point of each system under discussion and an optimal solution to the corresponding optimisation problem, the authors make use of semi-stability theory and methods in convex optimisation to prove convergence theorems of proposed algorithms that arrive at a least-squares/regularisation solution.

A generalized Sylvester equation is a special formulation containing the Sylvester equation, the Lyapunov equation and the Stein equation, which is often encountered in various fields. However, the time-variant generalized Sylvester equation (TVGSE) is rarely investigated in the existing literature. In this article, we propose a noise-suppressing recurrent neural network (NSRNN) model activated by saturation-allowed functions to solve the TVGSE. For comparison, the existing zeroing neural network (ZNN) models and some improved ZNN models are introduced. Additionally, theoretical analysis on the convergence and robustness of the NSRNN model is given. Furthermore, computer simulations on illustrative examples and applications to robots and acoustic source localization are carried out. Validation results synthesized by the NSRNN model and other ZNN models are provided to illustrate the ability in solving the TVGSE and dealing with noises of the NSRNN model, and the inaction of other ZNN models to noises.

In this article, a neural dynamics model is constructed and investigated for solving time-dependent Sylvester equation with matrix inversion involved in the solving process. Besides, to eliminate the matrix inversion in the model, the quasi-Newton Broyden–Fletcher–Goldfarb–Shanno method is leveraged to construct a new model. Moreover, the global convergence performance and the effectiveness of the two discrete computational models are testified by providing theoretical analyses and numerical experiments with comparisons to the existing solutions, respectively. Two applications to robotics and the multiple-input multiple-output system are given to elucidate the feasibility of the proposed models for solving time-dependent Sylvester equation.

For obtaining better convergence performance when zeroing neural network (ZNN) is applied to solve time-varying Sylvester equation, three different types of adaptive design coefficients for the sign-bi-power nonlinear activation function are developed and investigated in this paper. Based on these adaptive coefficients, three new adaptive ZNN models are proposed to improve the convergence performance of the standard ZNN model. For better analysis, the effect of two parts constituting the sign-bi-power activation function on the convergence of the standard ZNN model is first discussed. Then, detailed theoretical derivations and proofs are provided to verify excellent performance of the proposed adaptive ZNN models. Finally, illustrative comparison experiments are presented to show the enhanced finite-time convergence performance of the proposed adaptive ZNN models for solving time-varying Sylvester equation.

The ★ -congruence Sylvester equation is the matrix equation A X + X ⋆ B = C , where A ∈ F m × n , B ∈ F n × m and C ∈ F m × m are given, whereas X ∈ F n × m is to be determined. Here, F = R or C , ...

In scientific and engineering fields, the solutions to many problems can be transformed into finding the solutions to Sylvester equation, for which various computational methods (e.g., recurrent neural network, RNN) have been presented and investigated. RNN models are frequently used to solve computational problems due to the prevalent exploitation of the gradient-based RNN. However, the overlong convergent time and the too large residual error restrict the widespread applications of the RNN model in solving time-varying problems. Further, a special type of RNN named zeroing neural network (ZNN) is able to solve the time-varying Sylvester equation, which breaks the limitations mentioned above, but fails to handle complex time-varying problems owing to the sharp increment of the calculated amount in matrix inversion involved. To remedy the limitation, a modified gradient-based RNN (MGRNN) model is proposed to generate more accurate computational solutions with less convergent time and adaptive coefficients for solving the time-varying Sylvester equation, which replaces the matrix inversion problem with the matrix transposition problem. Besides, theoretical analyses and mathematical verifications are presented to validate the efficiency and superiority of the proposed MGRNN model compared with the traditional gradient-based recurrent neural network (GRNN) and ZNN models. Furthermore, simulation experiments are conducted to substantiate the properties of the newly proposed MGRNN model for solving the time-varying Sylvester equation.

In this paper, a parametric approach to design a Luenberger functional observer for linear time-varying (LTV) systems with time-delay is investigated. Based on the solution to generalized Sylvester equation (GSE), the complete general parametric expressions for the functional observer gain matrices are established with the time-varying coefficient matrices, the time-varying closed-loop system and a group of arbitrary parameters. With the parametric method, the observation error system can be transformed into a linear system with the expected eigenstructure. Finally, a numerical simulation is provided to illustrate the effectiveness of the parametric approach.

In this paper we apply the results obtained in [3] to establish some outcomes of the study of the behaviour of a class of linear operators, which include the Sylvester ones, acting on slice semi-regular functions. We first present a detailed study of the kernel of the linear operator ℒ f,g (when not trivial), showing that it has dimension 2 if exactly one between f and g is a zero divisor, and it has dimension 3 if both f and g are zero divisors. Afterwards, we deepen the analysis of the behaviour of the -product, giving a complete classification of the cases when the functions fv , gv and fv gv are linearly dependent and obtaining, as a by-product, a necessary and sufficient condition on the functions f and g in order their *-product is slice-preserving. At last, we give an Embry-type result which classifies the functions f and g such that for any function h commuting with f + g and f * g, we have that h commutes with f and g, too.

Projection methods based on Krylov subspaces are an effective tool for solving low-rank Sylvester matrix equations in the large scale cases. The initial problem is projected onto a sequence of nested subspaces. At each step of the process, a reduced Sylvester equation, whose coefficient matrices are restrictions of the original coefficient matrices to the projection subspaces, are solved by a direct method. In this paper, after recalling some important properties about the extended block Arnoldi process, we propose an alternative approach that enables us to determine approximate solutions by solving projected and reduced Sylvester equations whose coefficient matrices are restrictions of the inverse of the original matrices. Some numerical experiments are given in order to illustrate the performance of our approach.

In this work, we present a new cross-gramian. The proposed gramian is a truncated gramian which supports bilinear systems. It is proven that the truncated cross-gramian can be obtained by solving ordinary Sylvester equation. It is computationally less expensive to compute the truncated cross-gramian compared to the standard cross-gramians which are solutions to the generalized Sylvester equations. A projection based model reduction method is proposed based upon the truncated crossgramian. The proposed method is among the first cross-gramian based methods - if not the only one- which is developed for bilinear systems. This model reduction is computationally more efficient than model reduction techniques which are derived from standard gramains or the truncated controllability and observability gramians.

In this study, based on the double-step scale splitting (DSS) iteration method for solving complex Sylvester matrix equation, we propose two corresponding lopsided DSS iteration methods. These new methods, LDSS1 and LDSS2, are proved to be convergent under some suitable conditions. Besides, we try to minimize the spectral radii of the iteration matrices. We compare the new methods to the original methods in terms of the spectral radii of the iteration matrices. In the experiment results, we found that LDSS1 and LDSS2 methods are superior in iteration steps and CPU time when Sylvester equation satisfies some certain conditions.

In this paper, we propose a space-time least-squares isogeometric method to solve parabolic evolution problems, well suited for high-degree smooth splines in the space-time domain. We focus on the linear solver and its computational efficiency: thanks to the proposed formulation and to the tensor-product construction of space-time splines, we can design a preconditioner whose application requires the solution of a Sylvester-like equation, which is performed efficiently by the fast diagonalization method. The preconditioner is robust w.r.t. spline degree and mesh size. The computational time required for its application, for a serial execution, is almost proportional to the number of degrees-of-freedom and independent of the polynomial degree. The proposed approach is also well-suited for parallelization., 29 pages, 8 figures

Discrete versions for nonlinear Schrodinger equation and modified nonlinear Schrodinger equation are presented, using the method of generalized Cauchy matrix approach based on a Sylvester equation. By solving the determining equation set, some exact solutions including soliton solutions, Jordan block solutions and mixed solutions for the resulting discrete equations are generated. The associated semi-discrete equations as well as continuous equations are also discussed, applying appropriate continuum limits.

In this study, based on the MHSS (Modified Hermitian and skew-Hermitian splitting) method, we will present a generalized MHSS approach for solving large sparse Sylvester equation with non-Hermitian and complex symmetric positive definite/semi-definite matrices. The new method (GMHSS) is a four-parameter iteration procedure where the iterative sequence is unconditionally convergent to the unique solution of the Sylvester equation. Then to improve the GMHSS method, the inexact version of the GMHSS iterative method (IGMHSS) will be described and will be analyzed. Also, by using a new idea, we try to minimize the upper bound of the spectral radius of iteration matrix. Two test problems are given to illustrate the efficiency of the new approach.

The objective of this paper is to solve the time-fractional Schrodinger and coupled Schrodinger differential equations (TFSE) with appropriate initial conditions by using the Haar wavelet approximation. For the most part, this endeavor is made to enlarge the pertinence of the Haar wavelet method to solve a coupled system of time-fractional partial differential equations. As a general rule, piecewise constant approximation of a function at different resolutions is presentational characteristic of Haar wavelet method through which it converts the differential equation into the Sylvester equation that can be further simplified easily. Study of the TFSE is theoretical and experimental research and it also helps in the development of automation science, physics, and engineering as well. Illustratively, several test problems are discussed to draw an effective conclusion, supported by the graphical and tabulated results of included examples, to reveal the proficiency and adaptability of the method.

Complex-valued time-varying Sylvester equation (CVTVSE) has been successfully applied into mathematics and control domain. However, the computation load of solving CVTVSE will rise significantly with the increase of sampling rate, and it is a challenging job to tackle the CVTVSE online. In this paper, a new sign-multi-power activation function is designed. Based on this new activation function, an improved complex-valued Zhang neural network (ICZNN) model for tackling the CVTVSE is established. Furthermore, the strict proof for the maximum time of global convergence of the ICZNN is given in detail. A total of two numerical experiments are employed to verify the performance of the proposed ICZNN model, and the results show that, as compared with the previous Zhang neural network (ZNN) models with different nonlinear activation functions, this ICZNN model with the sign-multi-power activation function has a faster convergence speed to tackle the CVTVSE.

We propose a deterministic identification method that involves a priori information characterized as moments of a transfer function. The moments are expressed in terms of the solution to a Sylvester matrix equation . The Sylvester equation is incorporated with a conventional subspace identification method, and a problem for moment-constrained subspace identification is formulated. Since the identification problem is in a class of nonlinear optimization problems, it cannot be efficiently solved in numerical computation. Application of a change-of-variable technique reduces the problem to least squares optimization, and the solution provides a state-space model that involves the prespecified moments. Finally, the effectiveness of the proposed method is illustrated in numerical simulations.

In this paper, to accelerate the convergence speed of Zhang neural network (ZNN), two finite-time recurrent neural networks (FTRNNs) are presented via devising two novel design formulas. For verifying the advantages of the proposed FTRNN models, a solution application to time-varying Sylvester equation (TVSE) is given. Compared with the conventional ZNN model, the presented new FTRNN models in this paper are theoretically proved to have better convergence performance, and they are more effective for online solving TVSE within finite time. At last, the superiority and effectiveness of the new FTRNN models for solving TVSE are verified by numerical simulations.

Some new Frobenius norm bounds of the unique solution to certain structured Sylvester equation are derived. Based on the derived norm upper bounds, new multiplicative perturbation bounds are provided both for subunitary polar factors and positive semi-definite polar factors. Some previous results are then improved., Comment: 22 pages. Accepted for publication in Applied Mathematics and Computation

The article considers an approach for partial eigenvalue assignment in singular second order systems via proportional controller plus derived control plus output feedback controller. It is shown that the problem is closely related to a so-called second-order Sylvester matrix equation. This study presents an approach to partial the eigenstructure assignment for the descriptor system where an algorithm is presented for calculated the output feedback matrix by Sylvester equation. The paper is proposed two complete parametric methods for an approach to partial eigenstructure assignment problems. Both methods are respectively given simples complete parametric expressions for feedback gains and the closed-loop eigenvector matrices. The first one mainly depends on a series of singular value decompositions. The second one utilizes methodology for the system's factorization and enables the closed-loop eigenvalues to be set undetermined and search via specific optimization procedures. The theorems are presented using the Sylvester equations. Two algorithms are implemented using the Sylvester equation, and examples are presented with their conclusions.

We derive an iterative procedure for solving a generalized Sylvester matrix equation$AXB+CXD = E$AXB+CXD=E, where$A,B,C,D,E$A,B,C,D,Eare conforming rectangular matrices. Our algorithm is based on gradients and hierarchical identification principle. We convert the matrix iteration process to a first-order linear difference vector equation with matrix coefficient. The Banach contraction principle reveals that the sequence of approximated solutions converges to the exact solution for any initial matrix if and only if the convergence factor belongs to an open interval. The contraction principle also gives the convergence rate and the error analysis, governed by the spectral radius of the associated iteration matrix. We obtain the fastest convergence factor so that the spectral radius of the iteration matrix is minimized. In particular, we obtain iterative algorithms for the matrix equation$AXB=C$AXB=C, the Sylvester equation, and the Kalman–Yakubovich equation. We give numerical experiments of the proposed algorithm to illustrate its applicability, effectiveness, and efficiency.

The Sylvester equation and its particular case, the Lyapunov equation, are widely used in image processing, control theory, stability analysis, signal processing, model reduction, and many more. We present basis-free solution to the Sylvester equation in Clifford (geometric) algebra of arbitrary dimension. The basis-free solutions involve only the operations of Clifford (geometric) product, summation, and the operations of conjugation. To obtain the results, we use the concepts of characteristic polynomial, determinant, adjugate, and inverse in Clifford algebras. For the first time, we give alternative formulas for the basis-free solution to the Sylvester equation in the case $n=4$, the proofs for the case $n=5$ and the case of arbitrary dimension $n$. The results can be used in symbolic computation., Comment: 19 pages

We discuss the use of a matrix-oriented approach for numerically solving the dense matrix equation AX + XAT + M1XN1 + … + MℓXNℓ = F, with ℓ ≥ 1, and Mi, Ni, i = 1, …, ℓ of low rank. The approach relies on the Sherman–Morrison–Woodbury formula formally defined in the vectorized form of the problem, but applied in the matrix setting. This allows one to solve medium size dense problems with computational costs and memory requirements dramatically lower than with a Kronecker formulation. Application problems leading to medium size equations of this form are illustrated and the performance of the matrix-oriented method is reported. The application of the procedure as the core step in the solution of the large-scale problem is also shown. In addition, a new explicit method for linear tensor equations is proposed, that uses the discussed matrix equation procedure as a key building block.

This paper is concerned with the modification of a generalization of the Hermitian and skew-Hermitian splitting iteration method for solving large sparse continuous Sylvester equations. The analysis shows that the MGHSS iteration method converges unconditionally to the unique solution of AX+XB=C. An inexact variant of the GHSS iteration method (IMGHSS) has been presented and the analysis of its convergence property in detail has been discussed. Numerical examples are reported to confirm the efficiency of the proposed methods.

Owing to the parallelism feature and convenient hardware implementation of recurrent neural network (RNN), many RNN models have been proposed to solve linear and nonlinear algebra problems. There into, Zhang neural network (ZNN) and gradient neural network (GNN) have received considerable attention and been exploited to investigate various dynamic models for solving various issues. However, the ZNN and GNN are depicted in quite rigorous formulations, which may limit the development of methodology of dynamic model design. This paper considers developing a new RNN design method named simplified Zhang neural network (SZNN) with more general formulation and versatile instances for statics. The comparisons between SZNN and two conventional methods are analyzed through theoretical results and simulation experiments of linear equation. Furthermore, to show the effectiveness and versatility of SZNN, Sylvester equation and matrix inversion are solved by the proposed SZNN models, and the simulation results substantiate the excellent convergence of the SZNN models.

We consider the problem of stabilizing PDE-ODE cascade systems in which the input is applied to the PDE system whose output drives the ODE system. We also consider the dual problem of constructing an observer for ODE-PDE cascade systems in which the output of the ODE system drives the PDE system, whose output is measured. The PDE in these problems is stable and the ODE is unstable. While the ODE system models the plant in both the problems, the PDE system models the actuator in the stabilization problem and the sensor in the dual problem. In the literature, these problems have been solved for specific PDE models using the backstepping approach. In contrast, in the present work we consider these problems in an abstract framework by letting the PDE system be any regular linear system. Using a state transformation obtained by solving a Sylvester equation with unbounded operators, we first diagonalize the state operator corresponding to the cascade systems. We then solve the stabilization problem and the dual estimation problem, provided they are solvable, by solving certain finite-dimensional counterparts. We also derive necessary and sufficient conditions for verifying the solvability of these problems. We show that the controller which solves the stabilization problem is robust to certain unbounded perturbations. We illustrate our theory by designing a stabilizing controller for a PDE-ODE cascade in which the PDE is a 1D diffusion equation and an observer for a ODE-PDE cascade in which the PDE is a 1D wave equation., 26 pages, 3 figures, in review

A family of varying-parameter finite-time zeroing neural networks (VPFTZNN) is introduced for solving the time-varying Sylvester equation (TVSE). The convergence speed of the proposed VPFTZNN family is analysed and compared with the traditional zeroing neural network (ZNN) and the finite-time zeroing neural network (FTZNN). The behaviour of the proposed neural networks under various activation functions is proved theoretically and verified experimentally. In addition, the stability and noise resistance of the proposed VPFTZNN family are discussed. Further, the proposed VPFTZNN models are applied in the computation of current flows in an electrical network.

Recently, Xiao et al. presented a new noise-tolerant zeroing neural network (NNTZNN) model that can achieve fixed-time convergence even in the presence of some noise to solve the time-varying Sylvester equation . In this brief paper, we extend it to more general case through introducing a power parameter k while retaining its fixed-time convergence and noise-tolerant performance. The new proposed model is here named as the accelerating noise-tolerant zeroing neural network (ANTZNN) model since its fixed convergence time can be shorter than that of the NNTZNN model in certain model parameter ranges. These parameter ranges and the convergence of the ANTZNN model are theoretically analyzed in detail. Numerical experiments are performed to confirm the theoretical results, including the numerical comparisons with the known NNTZNN model under different parameter settings. Furthermore, the ANTZNN model is also applied to the control of robot manipulator , thus showing the applicability of the proposed ANTZNN model.

In this paper, we consider the robust partial quadratic eigenvalue assignment problem for the damped vibroacoustic system by active feedback control. Based on the special structures of the mass and stiffness matrices of the damped vibroacoustic system, we provide a new orthogonality relation and give solution to partial quadratic eigenvalue assignment problem which only needs a few unwanted eigenvalues of the open-loop pencil and associated eigenvectors. With the QR decomposition of the control matrix, we propose a new method to characterize the solution to the quadratic eigenvalue assignment problem, which does not need to solve the Sylvester equation. Finally, we propose two different gradient-based methods for the robust and minimum norm partial quadratic eigenvalue assignment problems, in which the closed-loop eigenvalue sensitivity and the feedback norms can be minimized simultaneously. Numerical experiments demonstrate the effectiveness of our methods.

The Sylvester equation plays a fundamental role in the control system, e.g, which can be applied to inverse-kinematic motion control problem in robot manipulators through a certain conversion. Considering the incompatibility of most existing Sylvester equation solving schemes on noise and the inevitability of noise in real life, by defining a new matrix-valued error function, an integration-enhanced noise-resistant recurrent neural network (IENRRNN) is generalize to the time-varying Sylvester equation solving in this paper. The convergence of the IENRRNN model under both the model implementation error and differential error of coefficient matrices are investigated. What’s more, from both convergence speed and convergence quality, effects of three activation functions on the computational errors achieved by the IENRRNN are evaluated. The influences of different design parameters on them are also discussed. Finally, with MATLAB, the effectiveness of the IENRRNN model for online solving the considered equation and its superiority compared to the traditional zeroing neural network (ZNN) model are demonstrated by simulative results.

In this paper, a structure-improved first-order observer is proposed to more accurately estimate the state and its derivative in the second-order system. Firstly, under the time-invariant change of coordinates, the novel structure of an improved first-order observer is given to deal with the problem where the derivative of the position estimation is not equal to the estimation of state vector for all time, and further, the sufficient conditions for the existence of such an observer is obtained as a type of the full-actuated second-order Sylvester equations, directly related to the second-order system coefficient matrices. Secondly, on the basis of the proposed existence conditions, the parametric method of the improved first-order observer is presented by adopting the solutions of the Sylvester equation. Finally, the correctness of the proposed method is illustrated through numerical simulation about a simple flexible-joint mechanism.

In this paper we solve the Sylvester equation $$AX-XB=C$$ , where A and B are closed densely defined self-adjoint operators, and C is a linear operator. We obtain sufficient conditions for the existence of infinitely many solutions and we manage to classify them. These results generalize the previously known results regarding singular Sylvester equations. Finally, we illustrate our results on an operator equation that appears in quantum mechanics, called the position-momentum operator equation.

This work is about a balanced truncation type order reduction method which is developed for stable and unstable large-scale continuous-time systems. In this method, a quantitative measure criterion for choosing the dominant eigenvalues helps in determining the steady-state and transient information of the dynamical system. These dominant eigenvalues are used to form a new substructure matrix that retains the dominant modes (or may desirable mode) of the original system. Retaining the dominant eigenvalues in the reduced mode assures stability and results in greater accuracy as the retained eigenvalues provides a physical link to the real system. In the quest to preserve the dominant eigenvalue of the real system, the proposed technique uses Sylvester equation for system transformation. Having obtained transformed model, the reduced model has been achieved by truncating the non-dominant eigenvalues using the singular perturbation approximation method. The efficiency and accuracy of the proposed method has been demonstrated by the benchmark test systems which were from the state-of-the-art models.

Density matrix perturbation theory (DMPT) is known as a promising alternative to the Rayleigh-Schr\"odinger perturbation theory, in which the sum-over-state (SOS) is replaced by algorithms with perturbed density matrices as the input variables. In this article, we formulate and discuss three types of DMPT, with two of them based only on density matrices: the approach of Kussmann and Ochsenfeld [J. Chem. Phys.127, 054103 (2007)] is reformulated via the Sylvester equation, and the recursive DMPT of A.M.N. Niklasson and M. Challacombe [Phys. Rev. Lett. 92, 193001 (2004)] is extended to the hole-particle canonical purification (HPCP) from [J. Chem. Phys. 144, 091102 (2016)]. Comparison of the computational performances shows that the aformentioned methods outperform the standard SOS. The HPCP-DMPT demonstrates stable convergence profiles but at a higher computational cost when compared to the original recursive polynomial method, Comment: accepted to J. Chem. Phys