Spectral Constraints in Adaptive Control of Graphon-induced SPDEs: Stability and Regularity Preservation

This paper investigates stability and regularity properties of stochastic partial differential equations (SPDEs) arising from graphon-induced interaction structures under adaptive control mechanisms. The dynamics are modeled by linear SPDEs driven by multiplicative noise, with interaction operators defined through a graphon limit framework. Motivated by applications in epidemic management, opinion dynamics, power-grid stabilization, and neuronal regulation, we address the problem of designing control laws that respect the intrinsic spectral geometry of the underlying network rather than treating it as an afterthought. By exploiting spectral properties of graphon operators, we derive sufficient conditions ensuring mean-square stability and preservation of spatial regularity in controlled systems. The proposed spectral constraints provide a tractable criterion linking network structure, noise intensity, and adaptive control gains. Our analysis employs semigroup techniques and stochastic energy estimates, accommodating non-symmetric graphon structures through pseudospectral analysis. We demonstrate that spectrally admissible controls preserve Sobolev regularity and prevent crossing of critical thresholds that separate function-valued from distribution-valued solutions. The framework is compared with traditional Linear Quadratic Regulator (LQR) and robust control methods, showing bounded conservativeness while guaranteeing regularity preservation. Numerical illustrations on block, power-law, and stochastic block model graphons validate the theoretical results. An application to stochastic epidemic control demonstrates practical implementation with online spectral estimation. These findings are further supported by large-scale simulations on networks with more than one thousand nodes, which confirm that the proposed spectral-constraint methodology scales favorably with system size while keeping regularity violations at zero or near-zero levels. The work establishes fundamental principles for responsible control of complex network systems, ensuring control objectives are pursued in harmony with intrinsic network geometry. Overall, the results offer both rigorous theoretical guarantees and practical, verifiable algorithms for practitioners who must design controllers for large stochastic systems evolving on networked and graph-limit structures.