We investigate optimal control of semilinear dynamical systems on asymptotically infinite networks, using the notion of graphons. A graphon represents a limit object of a converging graph sequence and serves as a generalization of a finite graph, enabling a systematic analysis of large-scale networks. We analyze semilinear graphon dynamical systems and establish state convergence along converging graph sequences by employing semigroup theory. Moreover, we show uniform convergence of the cost function and prove convergence of the corresponding control. In particular, we can approximate the optimal system behavior on the infinite-dimensional limit object up to arbitrary precision by solving an optimal control problem subject to dynamics on a finite graph. Numerical experiments support our theoretical results and verify our derived convergence properties.
Citation
M. T. Köhler, A. Makarow and C. Kirches, "Optimal Control for Graphon Dynamical Systems," in IEEE Control Systems Letters, doi: 10.1109/LCSYS.2026.3699407.