Distributed Optimal Control under Stochastic Sleep Scheduling via Trajectory-Based Gradient Estimation

This letter studies distributed optimal control for interconnected systems under synchronized stochastic sleep scheduling. A dual-mode controller is considered. During wake intervals, cooperative state feedback is applied using neighboring states. During sleep intervals, each subsystem uses its local state and buffered past input for local compensation. The resulting closed-loop system is modeled as a mode-dependent stochastic system. The corresponding optimal controller design problem is formulated as a finite-horizon expected quadratic optimization problem. To solve this problem, a trajectory-based gradient method is developed. A sample-path cost is introduced first. Then adjoint variables are used to derive local gradient formulas for the cooperative gain and the sleep-mode compensation gains. These gradients are averaged over sampled trajectories and used to update the admissible controller parameters. We establish an expected stationarity-neighborhood bound for the iterates generated by the proposed trajectory-based gradient algorithm. We also derive a sufficient condition for the mean-square stability of the synthesized closed-loop system. Numerical results on a multi-zone building thermal benchmark verify the sufficient mean-square stability condition for all tested sleep ratios.