Polyhedral-type reachable set estimation with generalized initial conditions: From positive systems to general linear delay systems

Hyung Tae Choi; Jung Hoon Kim; Hyung Tae Choi; Jung Hoon Kim

doi:10.3934/math.20261332

AIMS Mathematics

2026, Volume 11, Issue 10: 34141-34171. doi: 10.3934/math.20261332

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Polyhedral-type reachable set estimation with generalized initial conditions: From positive systems to general linear delay systems

Hyung Tae Choi ¹,
Jung Hoon Kim ^{2
,
,}

1.
School of Electrical and Electronics Engineering, Chung-Ang University, Seoul 06974, Republic of Korea
2.
Department of Electrical Engineering, Pohang University of Science and Technology (POSTECH), Pohang 37673, Republic of Korea

Received: 21 August 2026 Revised: 16 September 2026 Accepted: 22 September 2026 Published: 20 October 2026
MSC : 34H05, 93C23

This work developed a polyhedral set-based approach for bounding reachable sets of linear systems with nonzero polyhedral initial sets. We first considered linear positive systems and established sufficient conditions for constructing polyhedral bounds of reachable sets in both delay-free and time-varying-delay cases. For systems without delays, the conventional polyhedral set-bounding approach was extended to accommodate nontrivial initial sets given in polyhedral form. For systems with time-varying delays, the conventional auxiliary functional used for reachable set bounding was generalized to account for the effect of time-varying delays. The proposed framework was then extended to general linear systems that are not necessarily positive. In particular, for the case with delays, the auxiliary functional established for positive systems was further generalized by employing absolute values of the state. The resulting sufficient conditions provided polyhedral reachable set bounds for both delay-free and time-varying-delay systems, and all the proposed bounding sets were shown to be computed via linear programming. Several numerical examples are provided to illustrate the effects of nonzero initial sets and time-varying delay parameters on the resulting bounds, and to demonstrate the applicability of the proposed approach to general linear systems.
- positive systems,
- time delay,
- reachable set,
- auxiliary functional,
- polyhedral set
Citation: Hyung Tae Choi, Jung Hoon Kim. Polyhedral-type reachable set estimation with generalized initial conditions: From positive systems to general linear delay systems[J]. AIMS Mathematics, 2026, 11(10): 34141-34171. doi: 10.3934/math.20261332

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Abstract

This work developed a polyhedral set-based approach for bounding reachable sets of linear systems with nonzero polyhedral initial sets. We first considered linear positive systems and established sufficient conditions for constructing polyhedral bounds of reachable sets in both delay-free and time-varying-delay cases. For systems without delays, the conventional polyhedral set-bounding approach was extended to accommodate nontrivial initial sets given in polyhedral form. For systems with time-varying delays, the conventional auxiliary functional used for reachable set bounding was generalized to account for the effect of time-varying delays. The proposed framework was then extended to general linear systems that are not necessarily positive. In particular, for the case with delays, the auxiliary functional established for positive systems was further generalized by employing absolute values of the state. The resulting sufficient conditions provided polyhedral reachable set bounds for both delay-free and time-varying-delay systems, and all the proposed bounding sets were shown to be computed via linear programming. Several numerical examples are provided to illustrate the effects of nonzero initial sets and time-varying delay parameters on the resulting bounds, and to demonstrate the applicability of the proposed approach to general linear systems.

References

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