Stability of Bimodal Planar Linear Switched Systems

Swapnil Tripathi, Nikita Agarwal

correctmedium confidence

Category: Not specified
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves stability on S_τ and asymptotic stability on S′_τ by (i) introducing real-Jordan reductions, (ii) scaled transition matrices M_{D1,D2}, and (iii) contraction of the 2-step maps for t,s above case-dependent thresholds τ_{1,2}, τ_{2,1}, with equality only at the corner, implying uniform contraction away from it; see (5)–(9) and Remark 1.1 in the paper. The candidate solution uses the same core construction (mode-dependent norms via D_iP_i, the exact cycle maps T_{1→1}, T_{2→2}, continuity, and a uniform margin away from (τ,τ)) to obtain the same stability conclusions. The proofs are the same in substance, though the paper further computes τ’s explicitly by Schur tests in each Jordan-case, which the model does not attempt.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The paper presents a coherent, correct, and practically relevant analysis for bimodal planar switched systems using real Jordan forms and diagonal scalings. The core stability mechanism is standard and well-executed; the novelty lies in systematic case-by-case Schur analyses and improved dwell-time bounds. Minor clarifications would strengthen readability and rigor (explicit existence of optimal scalings; a concise uniform-contraction argument away from the corner).