Perturbation Analysis for Finite-Time Stability and Stabilization of Probabilistic Boolean Networks

Area of Science:

• Control Theory
• Computational Biology
• Network Science

Background:

• Probabilistic Boolean networks (PBNs) are widely used to model complex biological systems.
• Understanding the stability and control of PBNs under perturbations is critical for reliable system analysis.
• Existing stability concepts for PBNs require refinement to address finite-time dynamics and perturbations.

Purpose of the Study:

• To analyze the impact of function perturbations on the finite-time stability of PBNs.
• To differentiate and establish criteria for finite-time stability with probability one (FTSPO) and asymptotic stability with probability one (ASPO).
• To develop robust state-feedback stabilization strategies for perturbed probabilistic Boolean control networks (PBCNs).

Main Methods:

• Decomposition of PBN stability into FTSPO and ASPO concepts.
• Introduction of a new criterion for verifying ASPO in PBNs.
• Construction of a parameterized set to characterize stability conditions under function perturbation.
• Application of stability analysis to design robust controllers for PBCNs.

Main Results:

• PBNs maintain FTSPO if the perturbation avoids a specific parameterized set.
• PBNs achieve ASPO if the perturbation falls within the parameterized set.
• A robust state-feedback stabilization method is proposed for PBCNs with function perturbations.

Conclusions:

• Function perturbations have a clear impact on PBN stability, dictating transitions between FTSPO and ASPO.
• The proposed criteria and methods enable precise analysis and control design for perturbed PBNs.
• The findings are validated through applications to biological networks like the WNT5A signaling pathway and the E. coli lac operon.