Distributed Nash Equilibrium Seeking for Constrained Aggregative Games over Jointly Connected and Weight-Balanced Switching Networks

The paper proves exponential convergence of the projected gradient plus PI dynamic average consensus algorithm over jointly connected, weight-balanced switching graphs with exactly the same step-size region and constants k1(δ1), k2(δ1), k3(δ1), and M as in the candidate solution, culminating in δ2 < k1/(k1 k3 + k2^2) and the limit (x*, Pn φ(x*), α P⊥n φ(x*)) (see Theorem 1 and (31)–(32) in the paper ). The algorithm used in the paper explicitly includes the −v compensator in ṡ (their (14b)), matching the candidate’s setup and steady-state target (their (14b)–(14c) with ∑ v_i(0)=0) . Both proofs share the same structure: (i) a key projection inequality yielding k1(δ1) (cf. (33)–(36) in the paper) , (ii) an ISS-type bound for the DAC block with M = 2 p l sqrt(α^2+1) (cf. (37)–(39)) , (iii) an ẋ bound identical to (38) , and (iv) a composite Lyapunov inequality leading to a 2×2 matrix condition with the same determinant test (40)–(41) and the same δ2*(δ1) . The only methodological nuance is that the paper uses a time-varying Lyapunov function H(t) solving a differential Lyapunov equation for the switching DAC subsystem (Lemma 1 and (29)–(30)) , while the model phrases the same effect as a uniform ISS bound; these are equivalent in spirit for establishing the same inequality used in the small-gain step. The paper also explicitly identifies the limiting values of s and v via Proposition 1 and (23)–(25) using the zero-sum initial condition on v, which matches the model’s limit characterization .