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Continuous and discrete zeroing neural dynamics handling future unknown-transpose matrix inequality as well as scalar inequality of linear class

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Abstract

In this paper, the future unknown-transpose matrix inequality (FUTMI) as well as scalar inequality is formulated and investigated for the first time. This particular class of inequality may be encountered in scientific researches or engineering fields. In order to solve this intricate and complex problem, the corresponding continuous unknown-transpose matrix inequality (CUTMI) is formulated and discussed. In addition, a method transforming inequality into equality equivalently is introduced. Then, zeroing neural dynamics (ZND) is applied to the CUTMI solving, and a novel ZND model termed continuous-time ZND (CTZND) model is proposed and investigated. Furthermore, by adopting the Euler forward formula to discretize CTZND model, a novel discrete-time ZND (DTZND) model for solving FUTMI is derived and analyzed. Theoretical analysis indicates that the proposed DTZND model is zero-stable, consistent, and convergent. Finally, numerical experiment results further substantiate the good effectiveness and accuracy of the proposed CTZND and DTZND models.

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Acknowledgments

This work is supported by the National Natural Science Foundation of China (with number 61473323) and also by Shenzhen Science and Technology Plan Project (with number JCYJ20170818154936083). Besides, kindly note that both authors of the paper are jointly of the first authorship.

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Authors and Affiliations

  1. School of Data and Computer Science, Sun Yat-sen University, Guangzhou, 510006, China

    Jianrong Chen & Yunong Zhang

  2. Research Institute of Sun Yat-sen University in Shenzhen, Shenzhen, 518057, China

    Jianrong Chen & Yunong Zhang

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  1. Jianrong Chen
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  2. Yunong Zhang
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Correspondence to Yunong Zhang.

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Appendix

Appendix

In the Appendix, the following four results [37, 38] for a linear N-step method are provided.

Result 1: :

A linear N-step method \(\sum _{i = 0}^{N}\alpha _ix_{k+i}=\tau \sum _{i = 0}^{N}\kappa _i\psi _{k+i}\) can be checked for 0-stability by determining the roots of its characteristic polynomial \(P_N(\varsigma )=\sum _{i = 0}^{N}\alpha _i\varsigma ^i\). If all roots denoted by ς of the polynomial PN(ς) satisfy |ς|≤ 1 with |ς| = 1 being simple, then the linear N-step method is 0-stable (i.e., has 0-stability).

Result 2: :

A linear N-step method is said to be consistent (i.e., has consistency) of order p if the truncation error for the exact solution is of order O(τp+ 1) where p > 0.

Result 3: :

A linear N-step method is convergent, i.e., x[t/τ] → x∗(t), for all t ∈ [0, tf], as τ → 0, if and only if the method is 0-stable and consistent. That is, 0-stability plus consistency means convergence, which is also known as Dahlquist equivalence theorem.

Result 4: :

A linear 0-stable consistent method converges with the order of its truncation error.

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Chen, J., Zhang, Y. Continuous and discrete zeroing neural dynamics handling future unknown-transpose matrix inequality as well as scalar inequality of linear class. Numer Algor 83, 529–547 (2020). https://doi.org/10.1007/s11075-019-00692-z

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  • DOI: https://doi.org/10.1007/s11075-019-00692-z

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