Symmetry-Structured Constrained Allocation via Operator Decomposition in Stochastic Dynamical Systems

We introduce a symmetry-structured allocation framework for constrained distributed control systems subject to stochastic excitation. The central observation is that when the plant admits a discrete reflection symmetry, the control action can be decomposed into invariant (symmetric) and anti-invariant (antisymmetric) subspaces induced by the symmetry operator. We show that performing allocation separately within these orthogonal subspaces yields systematic suppression of residual state excursions under actuator magnitude and rate constraints. Formally, the method constructs parity projectors associated with a reflection operator acting on the state space and induces a corresponding decomposition of the control effectiveness operator. The resulting structured allocation is implemented via damped pseudo-inverse operators under hard actuator constraints. Performance is evaluated under colored stochastic excitation modeled as an AR(1) process, with operational risk quantified by exceedance probabilities and tail distribution functionals of the residual norm. Numerical experiments on a reduced-order aeroelastic model demonstrate consistent reductions in peak excursions, RMS residuals, and high-quantile exceedance probabilities relative to unstructured allocation, including robustness under effectiveness uncertainty. The results indicate that symmetry-induced operator decomposition provides a general structural mechanism for reducing extreme responses in constrained networked dynamical systems admitting discrete symmetry actions.