When Mean Age Is Not Enough: Distribution-Aware Scheduling for Networked LQR Control

We study finite-horizon linear-quadratic regulation of a scalar linear system with intermittent state feedback under an average communication-rate constraint. In this setting, the scheduling policy and controller are generally coupled through the dual effect: transmission decisions shape future estimation errors, while control actions influence the information available for scheduling. Existing treatments often recover tractability by restricting attention to symmetric scheduling policies, but the optimality of this restriction has remained unclear. We show that, for i.i.d. zero-mean disturbances, symmetric policies are optimal. Consequently, the communication-constrained LQR problem admits a separation structure. The optimal controller is a linear feedback law independent of the scheduling policy, while the optimal scheduler is obtained from a dynamic program. We further show that the optimal scheduling rule is a symmetric threshold policy in the accumulated disturbance since the most recent update.

Safety-critical IoT systems such as industrial closed-loop control, V2X coordination, and remote teleoperation require every sensor's peak Age of Information (peak AoI, also abbreviated PAoI) to stay below a hard per-slot deadline, not merely an average bound. Existing approaches meet this requirement only under restrictive assumptions: stochastic channels for Whittle-index AoI, simulator rollouts for deep reinforcement learning, or sublinear cumulative violation for long-term constrained online convex optimization. Under adversarial coefficients, OCO-PAoI-Hard guarantees zero per-slot violation of the modeled AoI state under one-step viability and O(sqrt(T)) regret against any static safe comparator; packet-level safety requires stronger service assumptions. Our key observation is that the fractional peak-AoI deadline collapses exactly to an affine half-space constraint on the resource-allocation vector, turning hard real-time scheduling into time-varying constrained online convex optimization over a polyhedral safe set. A strictly causal proposal-shield-update loop enforces feasibility through one Euclidean projection per slot, the gradient step preserves no-regret behavior, and the classical virtual queue is reduced to an a-posteriori certificate. We establish closed-form static and dynamic regret bounds, a matching Omega(sqrt(T)) minimax lower bound, a margin-safe variant against execution noise, and a deadline-induced competitive ratio. On a four-sensor adversarial fluid-model trap channel, OCO-PAoI-Hard attains zero modeled-state deadline violations across all ten seeds, while four representative baselines miss between 1.65 percent and 64.0 percent of slots, and the empirical normalized regret stays below the theoretical envelope across two orders of magnitude in T.

Modern network policy control maps intent to sequential placement-control decisions. Bellman-style policy optimization primarily asks which action to optimize, while constraints are commonly handled through penalty, barrier, or Lagrangian mechanisms. We observe that before a value function can certify the best deployment, intermediate signals may already identify many candidates that should be excluded from further optimization. This motivates a complementary direction: \emph{Learning Not to Optimize}. Before a value function is accurate enough to select the best placement-control decision, intermediate signals may already show that candidates are equivalent under state--intent relabeling (quotienting), lead to a uniformly worse future state (dominance), or violate executable network laws (residual screening). \LNOQRD{} uses these computed or learned signals as a shadow process to reshape the domain on which primal policy optimization is performed, thereby reducing the action space. We prove lossless quotienting and dominance under explicit equivariance and monotonicity conditions, bound frontier size and ranking cost, and quantify losses from approximate certificates and primal estimates. Experiments show that \LNOQRD{} reduces small-instance candidates by $75.9\%$ while retaining $90.8\%$ near-oracle coverage and, on large instances, achieves the highest utility and intent satisfaction, the lowest hard-law violation and post-generation latency, and a $73.0\%$ average reduction among candidate-based baselines.