Resilient time-varying formation containment for multi-agent systems with cooperative-adversarial interactions and communication delays

Distributed coordination of multi-agent systems must remain reliable despite communication delays and the coexistence of cooperative and adversarial interactions induced by resource contention or misbehaving nodes. The central difficulty beyond conventional formation containment is that, once cooperative and adversarial couplings coexist on a sampled network with nonuniform delays, the closed-loop comparison dynamics lose row-stochasticity and individual transition steps become expansive, so neither standard delayed-consensus analysis nor a single Lyapunov or spectral certificate applies; the contribution is a transformation and a row-sum contraction argument that recover a closed-form resilience guarantee in this regime. This paper investigates resilient time-varying formation containment for second-order discrete-time multi-agent systems with cooperative-adversarial interactions modeled by signed graphs and subject to nonuniform delays. A distributed protocol combining each follower’s instantaneous self-state with delayed neighbor and leader states is designed; an explicit model transformation recasts the closed-loop dynamics as an augmented nonnegative comparison system whose row sums can exceed one due to competition. Using sub-stochastic and super-stochastic matrix theory, closed-form design conditions and a quantitative input-to-state stability bound are established, relating the steady-state formation error to the corrected leader- and sign-induced residual and exposing the trade-off among controller gains, sampling period, and cooperation-competition balance. Numerical experiments up to 144 agents confirm graceful error degradation with competitive intensity, bounded behavior in the super-stochastic regime, divergence when the competition margin is violated, and a clear advantage over cooperative-only baselines.