Sparse Control Synthesis for Uncertain Responsive Loads with Stochastic Stability Guarantees

The paper’s propositions and theorems (Proposition 1, Theorems 1–2) correctly reduce the MSES condition for multiplicative-noise linear systems to convex LMIs via a standard splitting lemma from [23], and they give consistent synthesis/equivalence steps, the K0 construction under full-state, the two-stage design under partial observation, the allowable-uncertainty bounds, and the nominal deterministic certificate (A Q* + Q* A^T ≺ −γ2* I) (see equations (10)–(14), (16)–(21), and (24)–(26) in the paper: ). By contrast, the candidate solution’s key bounding step is incorrect: it asserts the matrix inequality C_i^T B_i^T Q B_i C_i ⪯ (C_i Q C_i^T) B_i B_i^T to upper-bound the diffusion term; this fails in general (e.g., take any x with B_i^T x = 0 but C_i x ≠ 0). The paper correctly avoids this by introducing auxiliary scalars α_i and using the equivalence in [23, Lemma 12] to split AQ + QA^T + Σ σ_i^2 Bi Ci Q C_i^T B_i^T ≺ 0 into AQ + QA^T + Σ α_i Bi B_i^T ≺ 0 and σ_i^2 C_i Q C_i^T < α_i (). The candidate also overstates the decay rate by writing d/dt E[V] ≤ −γ2* E[V]; the LMI yields d/dt E[V] ≤ −γ2* E[‖z‖^2], which implies exponential decay after accounting for λmax(Q*), but not with the unscaled factor γ2*. Despite these proof errors, the candidate’s final claims (K0 construction, allowable σi, and nominal certificate) align with the paper’s results. Therefore: paper correct; model’s proof flawed.