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Reachability analysis of linear systems

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Abstract

In this paper, we propose a decision procedure of reachability for a linear system \(\xi '=A\xi +u\), where the matrix \(A's\) eigenvalues can be arbitrary algebraic number and the input u is a vector of trigonometric-exponential polynomials. If the initial set contains only one point, the reachability problem under consideration is reduced to the decidability of the sign of trigonometric-exponential polynomial and then achieved by being reduced to verification of a series of univariate polynomial inequalities through Taylor expansions of the related exponential functions and trigonometric functions. If the initial set is open semi-algebraic, we will propose a decision procedure based on OpenCAD and an algorithm of real root isolation derived from the sign-deciding procedure for the trigonometric-exponential polynomials. The experimental results indicate the efficiency of our approach. Under the assumption of Schanuel’s Conjecture, the above procedures are complete for bounded time except for several cases.

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Authors and Affiliations

  1. Sichuan Vocational College of Finance and Economics, Chengdu, 610101, Sichuan, China

    Shiping Chen

  2. Chengdu Institute of Computer Application, Chinese Academy of Sciences, Chengdu, 610213, Sichuan, China

    Xinyu Ge

  3. University of Chinese Academy of Sciences, Beijing, 100049, China

    Xinyu Ge

Authors
  1. Shiping Chen
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  2. Xinyu Ge
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CS wrote the main manuscript text and GX presented the examples. All authors reviewed the manuscript.

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Correspondence to Shiping Chen.

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Chen, S., Ge, X. Reachability analysis of linear systems. Acta Informatica (2024). https://doi.org/10.1007/s00236-024-00458-8

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