This research is the first study focused on studying the total least squares (TLS) for time-varying (TV) tensor multilinear equations (TE). Obtained results extend existing methods for solving the TLS problem in both constant tensor settings and constant and TV matrix settings. Two models are suggested to solve simultaneous perturbations in both tensor coefficient and right-hand vector inside TV TEs. The first model establishes a continuous-time neural network designed for solving the TLS of TV tensor equations (termed as NNTVTETLS), while the second (termed as discrete neural network for total-least-squares solution of time-varying tensor equations (DNNTVTETLS)) is developed as its discrete counterpart. Unlike existing gradient-based neural networks and least-squares methods, the proposed models address TLS perturbation in a high-order TV multilinear settings. A rigorous theoretical convergence analysis is provided, establishing local convergence for both continuous and discrete models. An efficient iterative scheme based on the DNNTVTETLS scheme is also developed. Extensive numerical experiments demonstrate the efficacy of both continuous and discrete-time models. Numerical tests under various TV, constant, bounded vanishing, and bounded nonvanishing noise, confirm the rapid convergence, stability, and robustness of the proposed methods. The suggested framework is applied to solve the Bellman equation, which emerges in dynamic programming, economics, reinforcement learning, and optimal decision-making problems.

This paper develops several new dynamical designs, based on the gradient neural network (GNN), from the perspective of control theory to solve the time-varying Sylvester equation (TVSE). We start with an adaptive gradient neural network called AGNN-S. Then, based on Lyapunov theory, we propose three improved models: ACGNN, AIGNN, and ABGNN. Among them, the ABGNN model stands out for its fastest convergence speed and strongest robustness to noise. Theoretical analysis confirms that all proposed models solve the TVSE effectively, with ACGNN, AIGNN, and ABGNN converging faster than AGNN-S. Theoretical analysis shows that using certain nonlinear activation functions can further boost convergence speed. A robustness analysis indicates that the AGNN-S, ACGNN, and ABGNN models maintain stable convergence even in the presence of differentiation or model-implementation errors. Comparisons with the classical zeroing neural network (ZNN) and other six state-of-the-art GNN- and ZNN-type models demonstrate the superior accuracy and efficiency of our approaches, especially the ABGNN model. Numerical experiments notably highlight ABGNN’s advantages over other models under certain parameter setting. Finally, we showcase applications in time-varying quadratic programming and robotic arm trajectory tracking to verify the practical value of the models.

The primary focus of this study is to estimate least squares solutions for time-varying multi-linear systems (TVMLS) that involve time-varying (TV) perturbations on the right-hand side. To address this issue, we introduce two neural network models: the hybrid neural network (HNN) and a nonlinear enhancement of the HNN design that uses a modified weighted sign-bi-power (Mwsbp) activation function, referred to as MwsbpHNN. Theoretical findings demonstrate the convergence of both the HNN and MwsbpHNN dynamic models to the exact solutions of underlying TVMLS. Furthermore, we rigorously prove a fixed-time convergence of MwsbpHNN and provide upper bounds for the convergence time. Computational complexity of our theoretical framework is compared with current state-of-the-art methods for solving TVMLS, highlighting its efficiency. Numerical simulation further confirms the robustness of the proposed HNN and MwsbpHNN models. A model arising from discretized high-order Bellman equation is considered.

Calculation of the time-varying (TV) matrix generalized inverse has grown into an essential tool in many fields, such as computer science, physics, engineering, and mathematics, in order to tackle TV challenges. This work investigates the challenge of finding a TV extension of a subclass of inner inverses on real matrices, known as generalized-outer (G-outer) inverses. More precisely, our goal is to construct TV G-outer inverses (TV-GOIs) by utilizing the zeroing neural network (ZNN) process, which is presently thought to be a state-of-the-art solution to tackling TV matrix challenges. Using known advantages of ZNN dynamic systems, a novel ZNN model, called ZNNGOI, is presented in the literature for the first time in order to compute TV-GOIs. The ZNNGOI performs excellently in performed numerical simulations and an application on addressing localization problems. In terms of solving linear TV matrix equations, its performance is comparable to that of the standard ZNN model for computing the pseudoinverse.

Real-time high-frequency trading poses a significant challenge to the classical portfolio allocation problem, demanding rapid computational efficiency for constructing Markowitz model-based portfolios. Building on the principles of arbitrage pricing theory (APT), this study introduces a dynamic neural network model aimed at minimizing investment risk, optimizing portfolio allocation within predefined constraints, and maximizing returns. First, a convex optimization objective function incorporating risk constraints is formulated based on APT principles. This is followed by the introduction of a novel dynamic neural network model designed to solve the convex optimization problem, accompanied by comprehensive theoretical analysis and rigorous proofs. The study uses two distinct datasets sourced from Yahoo Finance, consisting of 30 selected stocks, covering a span of 250 valid trading days to validate the proposed methodology. The results of 30 different stock market scenario experiments indicate that, when the upper limit for investment risk is set at$3.285 \times 10^{-4}$, the expected maximum investment return exceeds the Dow Jones Industrial Average (DJIA) index by 16.2816%. These empirical findings highlight the viability, stability, and efficacy of the proposed approach and framework, demonstrating its potential applicability for real-time, high-frequency trading scenarios. Furthermore, the outcomes suggest policy implications for risk management and portfolio optimization in dynamic financial environments.

High-frequency trading proposes new challenges to classical portfolio selection problems. Especially, the timely and accurate solution of portfolios is highly demanded in financial market nowadays. This article makes progress along this direction by proposing novel neural networks with softmax equalization to address the problem. To the best of our knowledge, this is the first time that softmax technique is used to deal with equation constraints in portfolio selections. Theoretical analysis shows that the proposed method is globally convergent to the optimum of the optimization formulation of portfolio selection. Experiments based on real stock data verify the effectiveness of the proposed solution. It is worth mentioning that the two proposed models achieve 5.50% and 5.47% less cost, respectively, than the solution obtained by using MATLAB dedicated solvers, which demonstrates the superiority of the proposed strategies.

The Markowitz model, a Nobel Prize winning model for portfolio analysis, paves the theoretical foundation in finance for modern investment. However, it remains a challenging problem in the high frequency trading (HFT) era to find a more time efficient solution for portfolio analysis, especially when considering circumstances with the dynamic fluctuation of stock prices and the desire to pursue contradictory objectives for less risk but more return. In this paper, we establish a recurrent neural network model to address this challenging problem in runtime. Rigorous theoretical analysis on the convergence and the optimality of portfolio optimization are presented. Numerical experiments are conducted based on real data from Dow Jones Industrial Average (DJIA) components and the results reveal that the proposed solution is superior to DJIA index in terms of higher investment returns and lower risks.

Application of generalized inverses in solving the inverse model control (IMC)-oriented minimum-energy perfect control design (PCD) problem for linear time-invariant multi-input/multi-output systems governed by the discrete-time$d$-state-space structure is presented in this article. For this reason, an appropriate class of polynomial generalized inverses is investigated. Moreover, it can be stated that the nonunique right$\sigma$-inverse, based on properly selected so-called degrees of freedom (DOFs), outperforms the well-known unique Moore–Penrose (MP) minimum-norm right$T$-inverse in terms of the energy consumption of perfect control (PC) input signals. However, the analytical confirmation of such an intriguing statement has only been established for the special class of the single-delayed plants with a zero reference value. Moreover, because of the complexity of the IMC, the objects with a time delay$d>1$having a nonzero setpoint have never been analytically explored in regard to the PC energy context until now. Thus, the newly introduced analytical methods defined in this article allow us to designate the proper forms of$\sigma$-inverse-related DOFs that guarantee the minimum-energy PCD for the entire set of LTI multivariable nonsquare systems with the delay$d \geq 1$. Moreover, the new original results, supported by numerical examples, strongly contest the well-established control and systems theory canons related to the optimal minimum-energy-originated peculiarity of the MP pseudoinverse.

The problem of solving algebraic Riccati equations (AREs) and certain linear matrix equations which arise from the ARE frequently occur in applied and pure mathematics, science, and engineering applications. In this article, by considering the nonsymmetric ARE (NARE) as a general form of ARE, the time-varying NARE (TV-NARE) problem is proposed and investigated. As a particular case of TV-NARE, the time-invariant NARE (TI-NARE) problem is investigated too. Then, by employing the zeroing (or Zhang) neural dynamics (ZND) design, a ZND TV-NARE (ZNDTV-NARE) model and a ZND TI-NARE (ZNDTI-NARE) model are proposed and investigated. Also, by combining the ZNDTV-NARE model with the frozen-time Riccati equation (FTRE) approach to optimal control of linear time-varying (LTV) systems based on the state-dependent Riccati equation (SDRE) process, a hybrid ZND FTRE control (HZND-FTREC) model is developed and investigated. The effectiveness of the proposed dynamical systems is proven in ten numerical experiments, three of which include applications to LTV and nonlinear systems.

A correlation between fuzzy logic systems (FLS) and zeroing neural networks (ZNN) design is investigated. It is shown that the gain parameter included in ZNN design can be dynamically adjusted over time by means of an appropriate value derived as the output of a properly defined FLS, which includes appropriately defined membership functions and fuzzy logic rules. Dynamical systems which are applicable to time-varying rank-deficient matrices are proposed. Convergence properties are investigated and illustrative simulation experiments are performed. Presented simulation experiments confirm the superiority of the FLS proposed in this article with respect to previously proposed FLS for dynamic adjustment of gain parameters. Furthermore, the superiority of the FLS-based ZNN model over the corresponding ZNN models based on the classical approach in defining the varying-gain parameter is demonstrated.

The problem of solving linear equations is considered as one of the fundamental problems commonly encountered in science and engineering. In this article, the complex-valued time-varying linear matrix equation (CVTV-LME) problem is investigated. Then, by employing a complex-valued, time-varying QR (CVTVQR) decomposition, the zeroing neural network (ZNN) method, equivalent transformations, Kronecker product, and vectorization techniques, we propose and study a CVTVQR decomposition-based linear matrix equation (CVTVQR-LME) model. In addition to the usage of the QR decomposition, the further advantage of the CVTVQR-LME model is reflected in the fact that it can handle a linear system with square or rectangular coefficient matrix in both the matrix and vector cases. Its efficacy in solving the CVTV-LME problems have been tested in a variety of numerical simulations as well as in two applications, one in robotic motion tracking and the other in angle-of-arrival localization.

The Zhang neural network (ZNN) has recently realized remarkable success in solving time-varying problems. Harmonic noise widely exists in industrial applications and can severely affect the solution computed by ZNN models. This article attempts to solve the aforementioned limitations by providing the first ZNN design with an inherent capability to prohibit harmonic noise. Moreover, it opens new opportunities to shift the research on ZNNs in ideal situations to that with theoretical consideration on nonideal working environments. We establish a modified ZNN design formula in a noisy environment by incorporating the dynamics of harmonic signals. Theoretical analysis shows the convergence of the proposed ZNN design. An application case study for the new ZNN model verifies its effectiveness for time-varying matrix inversion in the presence of harmonic noise. The simulation further substantiates the effectiveness, superiority, and application prospects of the proposed ZNN design.

Two complex Zhang neural network (ZNN) models for computing the Drazin inverse of arbitrary time-varying complex square matrix are presented. The design of these neural networks is based on corresponding matrix-valued error functions arising from the limit representations of the Drazin inverse. Two types of activation functions, appropriate for handling complex matrices, are exploited to develop each of these networks. Theoretical results of convergence analysis are presented to show the desirable properties of the proposed complex-valued ZNN models. Numerical results further demonstrate the effectiveness of the proposed models.

Two linear recurrent neural networks for generating outer inverses with prescribed range and null space are defined. Each of the proposed recurrent neural networks is based on the matrix-valued differential equation, a generalization of dynamic equations proposed earlier for the nonsingular matrix inversion, the Moore-Penrose inversion, as well as the Drazin inversion, under the condition of zero initial state. The application of the first approach is conditioned by the properties of the spectrum of a certain matrix; the second approach eliminates this drawback, though at the cost of increasing the number of matrix operations. The cases corresponding to the most common generalized inverses are defined. The conditions that ensure stability of the proposed neural network are presented. Illustrative examples present the results of numerical simulations.

In this letter, we present the dynamical equation and corresponding artificial recurrent neural network for computing the Drazin inverse for arbitrary square real matrix, without any restriction on its eigenvalues. Conditions that ensure the stability of the defined recurrent neural network as well as its convergence toward the Drazin inverse are considered. Several illustrative examples present the results of computer simulations.

This paper presents a recurrent neural network (RNN) for computing the Drazin inverse of a real matrix in real time. This recurrent neural network (RNN) is composed of n independent parts (subnetworks), where n is the order of the input matrix. These subnetworks can operate concurrently, so parallel and distributed processing can be achieved. In this way, the computational advantages over the existing sequential algorithms can be attained in real-time applications. The RNN defined in this paper is convenient for an implementation in an electronic circuit. The number of neurons in the neural network is the same as the number of elements in the output matrix, which represents the Drazin inverse. The difference between the proposed RNN and the existing ones for the Drazin inverse computation lies in their network architecture and dynamics. The conditions that ensure the stability of the defined RNN as well as its convergence toward the Drazin inverse are considered. In addition, illustrative examples and examples of application to the practical engineering problems are discussed to show the efficacy of the proposed neural network.