Optimal Illumination via Joint Movement and Phase Optimization for Movable Antenna-RIS Configuration

Reconfigurable intelligent surfaces (RIS) enable programmable control of wireless propagation but remain vulnerable to persistent deep fades in static deployments. This paper introduces a Movable Antenna-enhanced RIS (MA-RIS) architecture where antenna elements physically reposition to sample independent spatial channels, enabling mobility-induced diversity. We model antenna motion using a Stochastic Differential Equation (SDE) framework capturing controlled drift and environmental diffusion. It^o calculus-based analysis characterizes steady-state antenna distributions, spatial decorrelation, and outage probability, revealing fundamental trade-offs between control strength and mobility randomness. To maximize long-term SNR while accounting for control overhead, we propose an overhead-aware Two-timescale framework separating slow antenna trajectory control from fast phase adaptation. The stochastic optimal control problem is solved via predictive approximation of the Hamilton-Jacobi-Bellman (HJB) formulation, enabling real-time implementation. Simulations validate theoretical predictions: the Two-timescale strategy achieves up to 36 dB steady-state SNR with remarkable stability, outperforming position-only control by up to 15 dB and uncontrolled baselines by over 30 dB. Despite experiencing a lower SNR than Active RIS, the proposed approach delivers up to 16 times higher energy efficiency (EE) across varying system scales, establishing a new paradigm of mobility-enabled channel adaptation for resilient wireless systems.