Machine learning for securing Cyber–Physical Systems under cyber attacks: A survey

Positivity and Stability of Fractional-Order Coupled Neural Network with Time-Varying Delays

“…For example, in [7], the global stability of complex valued FOCNNs with nodes in unequal dimensions and time delays was analyzed using comparison theory; in [8], the stability analysis of FOCNNs with time delays was investigated; in addition, the stability of two three-dimensional FOCNNs with different ring structures and time delays were analyzed; in [9], the authors studied the quantized output feedback synchronization of FOCNNs with output coupling; in [10], the authors considered the finite-time synchronization of FOCNNs with time-varying delays; the results were proved to be applicable to the FOCNNs without time delays and integer-order neural networks; in [11], the authors innovatively introduced the concept of finite-time passivity for FOCNNs with multiple state coupling or multiple derivative coupling; in [12], based on the existing passivity definition, the authors proposed the concepts of finite-time input strict passivity, finite-time output strict passivity, and finitetime strict passivity for FOCNNs; in addition, novel delay-dependent and order-dependent sufficient conditions ensuring the passivity performances were obtained for FOCNNs. More interesting results can be found in [13][14][15][16][17][18][19].…”

“…Lemma 5. Suppose that β(t; ψ 1 ) and β(t; ψ 2 ) are the trajectories of system (17) with the initial condition ψ 1 and ψ 2 . Then, ψ 1 (ι)…”

“…(Sufficiency) Since β α, the linear system ( 17) and ( 18) must be the positive system. Suppose that β 1 (t; ψ) and β 2 (t; ψ) are the trajectories of system (17) and (18) with the initial condition ψ. Since (I N ⊗ A) + c(L ⊗ G) + (I N ⊗ B)L f is a Hurwitz matrix, based on Lemma 2, it can be concluded that there exists a column vector λ 0, satisfying ((I N ⊗ A) + c(L ⊗ G) + (I N ⊗ B)L f )λ ≺ 0.…”