A three-operator proximal splitting approach for sparse consensus control
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We address sparse stabilization in multi-agent systems by designing control strategies that promote consensus with minimal intervention. Using a second-order nonlinear framework, we incorporate $\ell_1$-norm penalization into the cost functional to induce sparsity in the controls. The resulting non-smooth optimization problem is tackled using a three-operator splitting (TOS) algorithm based on proximal methods. To mitigate computational complexity while preserving the essential features of the microscopic model, we derive a mean-field approximation of the dynamics. The proposed approach is validated through a combination of particle-based simulations and model predictive control (MPC), applied to the mean-field formulation.