Stability-Certified ResNN-Based Consensus Control for Multi-Agent Systems via Certificate-by-Construction Training

This paper develops a stability-certified distributed residual neural network (ResNN) controller for leaderless consensus of continuous-time first-order multi-agent systems over undirected connected graphs. Unlikeconventional linear protocols and post-training verification methods, the proposed framework combines nonlinear transient shaping with certificate feasibility throughout learning. Each agent uses relative state differences with neighboring agents and applies a shared ResNN policy composed of a linear diffusive backbone and a bias-free neural residual branch. A computable residual-gain bound yields a sufficient condition for global exponential consensus, with the certified rate determined by the backbone–residual gain margin and the algebraic connectivity of the graph. Spectral-norm projection and gain reparameterization preserve this condition after every training update. A complementary forward-Euler certificate provides an explicit step-size condition for exponentially convergent digital implementation. The controller is trained through differentiable closed-loop simulation using an objective that balances disagreement and control effort. Ournumerical results verify the certified decay bound, demonstrate competitive transient trade-offs against three baselines, and evaluate direct deployment across different graph topologies and network sizes.