Submodularity conjecture for noncooperative ACC H₂ performance

Let $S$ be the set of autonomous vehicles in the mixed traffic system, let $J_1(S)=-\lVert \mathbf{G}_1(S)\rVert_2^2$ be the negative squared $\mathcal{H}_2$ norm of the transfer function from disturbances to the weighted spacing- and velocity-error output, and let $\Delta_{J_1}(e\mid S)$ denote the marginal improvement of adding vehicle $e$ to $S$. Submodularity conjecture. The $\mathcal{H}_2$ performance $J_1(S)$ under the noncooperative ACC controller is submodular. The conjecture is based on numerical Monte Carlo experiments in which the tested marginal-improvement sequences were non-increasing; no general analytical proof is provided.