Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Zeroing neural dynamics solving time-variant complex conjugate matrix equation $X(τ)F(τ)-A(τ)\overline{X}(τ)=C(τ)$

Jiakuang He, Dongqing Wu

TL;DR

This work tackles time-variant complex conjugate matrix equations (TV-CCME) of the form $X(\tau)F(\tau)-A(\tau)\overline{X}(\tau)=C(\tau)$ by introducing zeroing neural dynamics (ZND) in the complex field. It develops two models, Con-CZND1 and Con-CZND2, and provides convergence proofs leveraging complex-vectorization and a real-field transformation framework, including Lyapunov-based stability arguments. The authors rigorously define a complex Kronecker product, derive error dynamics, and validate the methods through numerical experiments, demonstrating superior performance of the complex-field formulation in several cases. The results indicate that solving TV-CCME directly in the complex domain yields improved accuracy and robustness compared to real-field mappings, advancing the theory and application of ZND for CCME and related antilinear systems.

Abstract

Complex conjugate matrix equations (CCME) are important in computation and antilinear systems. Existing research mainly focuses on the time-invariant version, while studies on the time-variant version and its solution using artificial neural networks are still lacking. This paper introduces zeroing neural dynamics (ZND) to solve the earliest time-variant CCME. Firstly, the vectorization and Kronecker product in the complex field are defined uniformly. Secondly, Con-CZND1 and Con-CZND2 models are proposed, and their convergence and effectiveness are theoretically proved. Thirdly, numerical experiments confirm their effectiveness and highlight their differences. The results show the advantages of ZND in the complex field compared with that in the real field, and further refine the related theory.

Zeroing neural dynamics solving time-variant complex conjugate matrix equation $X(τ)F(τ)-A(τ)\overline{X}(τ)=C(τ)$

TL;DR

This work tackles time-variant complex conjugate matrix equations (TV-CCME) of the form by introducing zeroing neural dynamics (ZND) in the complex field. It develops two models, Con-CZND1 and Con-CZND2, and provides convergence proofs leveraging complex-vectorization and a real-field transformation framework, including Lyapunov-based stability arguments. The authors rigorously define a complex Kronecker product, derive error dynamics, and validate the methods through numerical experiments, demonstrating superior performance of the complex-field formulation in several cases. The results indicate that solving TV-CCME directly in the complex domain yields improved accuracy and robustness compared to real-field mappings, advancing the theory and application of ZND for CCME and related antilinear systems.

Abstract

Complex conjugate matrix equations (CCME) are important in computation and antilinear systems. Existing research mainly focuses on the time-invariant version, while studies on the time-variant version and its solution using artificial neural networks are still lacking. This paper introduces zeroing neural dynamics (ZND) to solve the earliest time-variant CCME. Firstly, the vectorization and Kronecker product in the complex field are defined uniformly. Secondly, Con-CZND1 and Con-CZND2 models are proposed, and their convergence and effectiveness are theoretically proved. Thirdly, numerical experiments confirm their effectiveness and highlight their differences. The results show the advantages of ZND in the complex field compared with that in the real field, and further refine the related theory.
Paper Structure (10 sections, 4 theorems, 39 equations, 13 figures)

This paper contains 10 sections, 4 theorems, 39 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation and Preliminaries
  3. Models, Algorithm and Analyses
  4. Con-CZND1 model
  5. Con-CZND2 model
  6. Numerical Experimentation and Verification
  7. Con-CZND1 model
  8. Con-CZND2 model
  9. Con-CZND1 model vs Con-CZND2 model by comparing error and stability
  10. Conclusion

Key Result

Theorem 1

Where $A(\tau)\in \mathbb{C}^{m\times n}$, $B(\tau)\in \mathbb{C}^{s\times t}$, $X(\tau)\in \mathbb{C}^{n\times s}$, are time-variant matrices, and $\tau \ge 0$ denotes the real-time, the following equation can be obtained:

Figures (13)

  • Figure 1: Solving methods of CCME.
  • Figure 2: Solution $X(\tau)$ computed by Con-CZND1 \ref{['eq.solve.linearerrconcznd1']} model in Example \ref{['eq.example1']}.
  • Figure 3: Solution $X(\tau)$ computed by Con-CZND1 \ref{['eq.solve.linearerrconcznd1']} model in Example \ref{['eq.example2']}.
  • Figure 4: $\left \|X(\tau)-X^*(\tau) \right \|_{\mathrm{F}}$ computed by Con-CZND1 \ref{['eq.solve.linearerrconcznd1']} model in Examples \ref{['eq.example1']} and \ref{['eq.example2']}. \ref{['fig.e1.Con-CZND1.normerror.normal']} Example \ref{['eq.example1']}. \ref{['fig.e2.Con-CZND1.normerror.normal']} Example \ref{['eq.example2']}.
  • Figure 5: Solution $X(\tau)$ computed by Con-CZND2 \ref{['eq.solve.linearerrconcznd2']} model in Example \ref{['eq.example1']}.
  • ...and 8 more figures

Theorems & Definitions (14)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Theorem 3
  • ...and 4 more