Frequency-Domain Asymptotic Synchronization Criteria for Delayed Fuzzy BAM Neural Networks Under Novel Controllers

This paper studies the asymptotic synchronization problem for drive-response delayed fuzzy bidirectional associative memory (BAM) neural networks via a frequency-domain approach. The network model explicitly retains fuzzy logic operations—fuzzy AND and fuzzy OR—in its connection weights, as originally proposed in the literature. Two linear feedback controllers are designed: Controller I uses only instantaneous error feedback, while Controller II additionally incorporates a delayed error term to compensate for transmission delays. By transforming the closed-loop error dynamics into a Lur’e-type system, two novel frequency-domain synchronization criteria are derived from the multivariable circle criterion and Parseval’s theorem. The proofs are fully self-contained and detailed, using signal energy arguments to rigorously handle truncation and initial conditions without constructing Lyapunov–Krasovskii functionals. The criteria only require evaluating the minimum eigenvalue of a frequency-dependent Hermitian matrix over a bounded interval. Numerical simulations confirm the theoretical results and show that the delayed-feedback controller significantly enlarges the synchronizable region. The potential applicability of the proposed framework to associative memory, secure communication, and cooperative control is also discussed.