This paper studies a hybrid discrete-time signed gradient-consensus method with heavy-ball momentum, paired communication delays, and fixed logarithmic quantization. The method preserves the aggregate supply-demand equality at every iteration. For a static connected undirected graph and bounded delays, a Lyapunov-Krasovskii analysis establishes practical stability by isolating the discontinuous quantization error as an additive disturbance and closing a computable invariant-sublevel argument. The resulting ergodic residual bound is quadratic in the quantization resolution, and exact convergence to the smooth penalized equality-constrained optimizer is recovered in the unquantized case. For delay-free unquantized quadratic costs, an exact modal analysis identifies the heavy-ball stability region and optimal parameters, with the predicted rate validated on a fixed sparse network. A reproducible delayed experiment reports all constants and a certificate whose guaranteed steady-state level lies below the reported initialization. A separate mixed generator-battery experiment exercises the signed update directly. Finite smooth penalties approximate, but do not generally enforce, the local box constraints exactly.
This paper proposes a distributed resource allocation algorithm in which each agent communicates only the sign of its local cost gradient, a single bit per directed link per iteration. The algorithm is based on Laplacian-gradient dynamics and preserves the equality (demand) constraint at every iterate, guaranteeing anytime feasibility irrespective of termination time. First, we provide a complete Lyapunov-based convergence analysis for the fixed-step-size regime: the iterates are shown to enter, in at most O(1/ε^2) iterations, a neighborhood of the optimizer whose radius scales as O(τ) in the decision variable and O(τ^2) in the cost residual, where τ is the step-size. Crucially, the step-size bound is a time-invariant network constant (depending only on the graph Laplacian spectrum and the cost curvature bounds), so no adaptive tuning is required. Second, we characterize the invariant set under B-connected, weight-balanced digraphs, strictly weaker than the all-time connectivity and doubly-stochastic weights assumed in most prior work, and show that any odd, sign-preserving nonlinearity yields the same invariant-set and descent structure, with the single-bit protocol being the extreme communication-minimizing instance of a broader nonlinear class. Third, we address local box constraints via smooth penalty relaxation and quantify explicitly how the penalty weight interacts with the step-size bound and the optimality gap. Simulations on economic dispatch (1000 generators) and CPU scheduling (500 servers) demonstrate that the single-bit protocol achieves 5-60 times reduction in total communication cost relative to multi-bit baselines, at the price of a tunable and explicitly bounded optimality gap. A diminishing step-size schedule is shown empirically to eliminate this gap; its formal convergence analysis is identified as an open problem.
This paper establishes an asymptotic ergodic classification for mass-conserving biological switching diffusions subjected to multiplicative common noise and Markovian regime-switching. The inclusion of multiplicative common noise yields a positive predictable quadratic variation density on the spatial intersection manifold whenever the biological mass is active, precluding the application of classical pathwise spatial comparison theorems. To resolve this measure-theoretic bottleneck, we impose constant stoichiometry and Beddington–DeAngelis spatial interference to guarantee the uniform fractional integrability and derive algebraic bounds for the continuous spatial variance terms. The global dynamics are classified into a trichotomy governed by the algebraic sign of the principal continuous-time hybrid Lyapunov exponent, λ. For λ<0, we bypass intersection bounds using a continuous-time logarithmic discrepancy tracking sequence and continuous local martingale limit dichotomies to establish global exponential extinction. At the critical threshold λ=0, we prove biological extinction in expectation by applying the multidimensional generator to the boundary Poisson resolvent. Because the abiotic stochastic flow is pathwise affine, it preserves spatial concavity, yielding a non-positive spatial operator difference. For λ>0, we construct a separable Meyn–Tweedie Foster–Lyapunov mapping to secure positive Harris recurrence and exponential convergence in total variation distance. The theoretical results are illustrated numerically.
Bidirectional associative memories (BAM) have found widespread applications in autoassociative and heteroassociative learning. Although BAM neural networks with various classes of delays have been extensively investigated, studies concerning generalized piecewise constant delay (DEGPCD) remain relatively limited. This paper focuses on the global exponential stability and periodicity of the BAM neural network model with deviation arguments, specifically the effects of generalized piecewise constant delay. We employ an approach based on the construction of an equivalent integral equation to establish the existence and global exponential stability of the unique ω-periodic solution of the BAM neural network model with the DEGPCD system. The linearization method, Banach’s fixed point theorem, a DEGPCD integral inequality of Gronwall type, and inequality techniques are used to establish a new sufficient condition for the existence and global exponential stability of the unique ω-periodic solution. Our research indicates that the generalized piecewise constant delay has a significant effect on the global exponential stability of the BAM neural network model with the DEGPCD system. Two illustrative examples with simulations are presented to validate the results.
This paper is concerned with the asymptotic stabilization of switched systems over a digital communication network, where state sampling and finite-level quantization are required due to limited bandwidth resources. The main challenge of this study lies in how to coordinately handle quantizer saturation, quantization error, sampling error, and network-induced asynchronous switching during the control strategy design. To address the above problem, a novel event-triggered mechanism (ETM), constructed with a zoom-function-related threshold, is first introduced and proven to exclude Zeno behavior. Along with the ETM, a new hybrid system is proposed to generate the zoom function of dynamic quantizer. Compared with the existing results, our ETM no longer need to artificially set an upper bound of triggering intervals; the hybrid update of zoom function can prevent quantizer saturation at triggering instants and regulate the sampling frequency and convergence rate. Furthermore, accounting for the asynchronous switching, a quantization-dependent switching law is established to guarantee the convergence of zoom function and thus the asymptotic stability of closed-loop switched system. Simulation results demonstrate the effectiveness of the proposed method.
