Diagonal Stability of Two-Block Systems with Reciprocal Single-Channel Coupling
- 1Department of Mathematics, College of Science, Qassim University, P.O. Box 6644, Buraydah 51452, Saudi Arabia
Abstract
This paper considers coupled linear subsystems x˙=−Ax−By and y˙=−Cx−Dy, whose local blocks A and D may have different dimensions. For reciprocal single-channel coupling B=βeiejT and C=γejeiT, let δi(A) and δj(D) denote the directional diagonal-Lyapunov margins of the respective blocks: each is the supremum of a normalized coordinatewise Lyapunov reserve over admissible positive diagonal Lyapunov matrices. I prove that the full matrix is diagonally stable if and only if both local blocks are diagonally stable and βγ<δi(A)δj(D). The strict proof does not presume attainment of the local suprema. I also give a general Schur-complement test, constructive relative diagonal-Lyapunov scalings, and a dimension-free criterion for arbitrary one-way coupling. Closed planar formulas recover the classical acyclic threshold, while reproducible examples with cyclic internal graphs illustrate the broader theorem and show that Hurwitz stability need not imply diagonal stability.
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BibTeX
@article{algefary2026diagonal,
title = {{Diagonal Stability of Two-Block Systems with Reciprocal Single-Channel Coupling}},
author = {Ali Algefary},
journal = {Mathematics},
year = {2026},
month = nov,
doi = {10.3390/math14214023},
url = {https://www.mdpi.com/2227-7390/14/21/4023},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Diagonal Stability of Two-Block Systems with Reciprocal Single-Channel Coupling
AU - Ali Algefary
PY - 2026
DA - 2026/11/03
JO - Mathematics
DO - 10.3390/math14214023
UR - https://www.mdpi.com/2227-7390/14/21/4023
AB - This paper considers coupled linear subsystems x˙=−Ax−By and y˙=−Cx−Dy, whose local blocks A and D may have different dimensions. For reciprocal single-channel coupling B=βeiejT and C=γejeiT, let δi(A) and δj(D) denote the directional diagonal-Lyapunov margins of the respective blocks: each is the supremum of a normalized coordinatewise Lyapunov reserve over admissible positive diagonal Lyapunov matrices. I prove that the full matrix is diagonally stable if and only if both local blocks are diagonally stable and βγ<δi(A)δj(D). The strict proof does not presume attainment of the local suprema. I also give a general Schur-complement test, constructive relative diagonal-Lyapunov scalings, and a dimension-free criterion for arbitrary one-way coupling. Closed planar formulas recover the classical acyclic threshold, while reproducible examples with cyclic internal graphs illustrate the broader theorem and show that Hurwitz stability need not imply diagonal stability.
SN - 2227-7390
ER -