Stability of Bimodal Planar Switched Linear Systems with Both Stable and Unstable Subsystems

The paper derives case-by-case dwell–flee relations for planar two-mode switched linear systems by working in Jordan bases, introducing scaled transition matrices, and proving Schur stability of grouped flow terms; it yields explicit thresholds τ1,2(F), τ2,1(F) and shows stability for S[τ(F),F] and asymptotic stability for S′[τ(F),F] where τ(F)=min{τ1,2,τ2,1} (e.g., see the problem setup and main reduction via (4)–(9) and Remark 1.2, and the summary statement) . The candidate solution uses a different, conservative but valid path: mode-dependent Euclidean norms tied to Jordan forms and a per-cycle product bound V_{k+1} ≤ K χ2(F) χ1(τ) V_k with K a 2-norm condition factor, solving K χ2(F) χ1(τ) ≤ 1 (Lambert W appears only for the defective stable mode). This recovers linear-in-F thresholds (and W-based ones when defective) ensuring stability for S[τ(F),F] and asymptotic stability for S′. It is less sharp (does not exploit the paper’s scaled transition optimization and detailed Schur function bounds) and predicts τ1,2=τ2,1 by symmetry of its coarse bound, whereas the paper often obtains distinct τ1,2 and τ2,1 (e.g., 5.1/5.2 and remarks) . Net: both arguments are logically sound; the paper’s results are tighter, while the model’s thresholds are conservative but correct.