Construction Of Invariant Sets Research Articles

Identification and stabilization of faulty Boolean control networks

Fault-tolerant control is a fundamental branch in the modern control theory, and has wide applications such as aerospace, automotive technology and nuclear engineering. Particularly, the study of faulty Boolean control networks (BCNs) is meaningful to the disease treatment. This paper focuses on both stuck-at fault and bridging fault in BCNs, and investigates the identification and stabilization of BCNs subject to these two faults. The basic mathematical tool is semi-tensor product (STP) of matrices, which is used to determine the algebraic formulation of faulty BCNs. Through the construction of invariant sets corresponding to the faulty nodes, the relations between these two faults and state transition matrices are presented, which is helpful to identify the faulty nodes. In addition, the robust stabilization of BCNs subject to these two faults is discussed and several new criteria are derived. Finally, the obtained results are applied to analyze the stabilization of oxidative stress response pathways.

Necessary and Sufficient Razumikhin-Type Conditions for Stability of Delay Difference Equations

This technical note considers stability analysis of time-delay systems described by delay difference equations (DDEs). All existing analysis methods for DDEs that rely on the Razumikhin approach provide sufficient, but not necessary conditions for asymptotic stability. Nevertheless, Lyapunov-Razumikhin functions are of interest because they induce invariant sets in the underlying state space of the dynamics. Therefore, we propose a relaxation of the Razumikhin conditions and prove that the relaxed conditions are necessary and sufficient for asymptotic stability of DDEs. For linear DDEs, it is shown that the developed conditions can be verified by solving a linear matrix inequality. Moreover, it is indicated that the proposed relaxation of Lyapunov-Razumikhin functions has an important implication for the construction of invariant sets for linear DDEs.

Construction of Invariant Sets for Anosov Diffeomorphisms

Construction of Invariant Sets for Anosov Diffeomorphisms