Consensus of Double Integrator Multiagent Systems under Nonuniform Sampling and Changing Topology

The paper proves consensus under nonuniform sampling and switching balanced graphs via a uniform contraction test on per-mode maps S(h,λ) after a similarity transform, then specializes to double integrators with an explicit T and inequalities on k1,k2 that hold for all h∈(0, h̄) and λ∈[λ2,λN]. The candidate solution reproduces the same framework: the stacked discrete map, balanced-graph diagonalization, the same T, the same transformed 2×2 block, and the same uniform-contraction inequalities—arrived at with an equivalent norm-bound argument—plus the same feasibility condition. Concretely, the paper’s Theorem 3–4 contraction criterion σ̄(T^{-1}S(h,λ)T)<1 (eqs. (19)–(21)) matches the model’s Part I, and for double integrators the T in (22), the 2×2 Ŝ with entries s11,s12,s21,s22 and the bound (23) lead to the final constraints (24)–(26) and feasibility (27), which coincide with the model’s (I1)–(I3) and its existence test. Minor differences are purely notational or in the chosen singular-value bounds; the logical structure and resulting conditions are the same. See the paper’s derivation of the discretized subsystem (17), the contraction criterion (19)–(21), and the double-integrator design with T (22), inequalities (24)–(26) and existence condition (27) for direct correspondence.