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Nonnegativity, Convergence and Bounds of Non-homogeneous Linear Time-Varying Real-Order Systems with Application to Electrical Circuit System

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Abstract

Non-homogeneous real-order systems often arise in the construction of equations arising in many inequality systems. This paper considers non-homogeneous linear systems attached to random initial time and associated with Caputo real-order operators and time-dependent elements in standard and block matrix forms. Is it possible to find the convergence and estimate bounds of such systems? We use the idea of comparison methodology along with establishing the nonnegativity of such systems when the coefficient matrix is Metzler and the non-homogeneous term becomes nonnegative. Using the hypothesis on the existence of a constant Metzler matrix via construction, we prove that non-trivial solutions to such non-homogeneous systems converge to 0 under some reasonable conditions. We use the generalized Laplace transform technique as a tool to establish some reasonable conditions. In short, we prove that, if every root of a characteristic equation associated with such a constant Metzler matrix lies in the open left half complex plane and the non-homogeneous term approaches 0 as time tends to \(\infty \), then the non-trivial solution to the original system must be globally asymptotically convergent to 0. This result is then extended to the non-homogeneous real-order systems associated with coefficient block matrix form. We do not know the bounds and rate decay of the solutions to such systems. As a typical result, we prove that, if there is a constant symmetric Metzler matrix \(\Delta _{M^{+}}\) with maximum eigenvalue \(\lambda _{\textrm{max}}\) negative and the matrix \(\Delta _{{M}^{+}}-\lambda _{\textrm{max}}I\) is negative semi-definite, then there exist different measures for bounds of solutions to such systems. A particular case of the above-mentioned result gives rise to the new concept of \(\gamma \)-Mittag–Leffler asymptotic stability in the absence of a non-homogeneous input term and it thus provides \(\gamma \)-Mittag–Leffler decay rate. We provide an application to an electrical circuit system that illustrates the novel significance of some proposed theoretical results.

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Acknowledgements

Bichitra Kumar Lenka acknowledges Indian Institute of Technology Guwahati for providing Institute Post-Doctoral Fellowship under grant MATH/IPDF/2020-21/BIKL/01. The authors would like to thank the reviewers for their time and insightful comments which have helped us to improve the quality of the manuscript in its current form. The authors are immensely grateful to the Associate Editor and the Editor-in-chief Prof. M.N.S. Swamy for allowing the manuscript revision.

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  1. Swaroop Nandan Bora has equally contributed to this work.

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, Assam, 781039, India

    Bichitra Kumar Lenka & Swaroop Nandan Bora

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  1. Bichitra Kumar Lenka
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Lenka, B.K., Bora, S.N. Nonnegativity, Convergence and Bounds of Non-homogeneous Linear Time-Varying Real-Order Systems with Application to Electrical Circuit System. Circuits Syst Signal Process 42, 5207–5232 (2023). https://doi.org/10.1007/s00034-023-02368-5

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