Controlling the dynamics of large-scale networks is essential for a macroscopic reduction of overall consumption and losses in the context of energy supply, finance, logistics, and mobility. We investigate the optimal control of semilinear dynamical systems on asymptotically infinite networks, using the notion of graphons. Graphons represent a limit object of a converging graph sequence and serve as a generalization of graphs. This notion enables the systematic analysis of converging graph sequences as the number of vertices tends to infinity. Based on the theory of graphons and optimal control, we derive novel convergence results for the states, adjoints, and controls for dynamical systems on converging graph sequences. In other words, 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 dense but finite graph, which is sampled from the limit object of the converging graph sequence. Numerical experiments support our theoretical results and verify our derived convergence rates.
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