Optimal control of the reaction–diffusion process on directed networks

Reaction-diffusion processes organized in networks have attracted much interest in recent years due to their applications across a wide range of disciplines. As one type of most studied solutions of reaction-diffusion systems, patterns broadly exist and are observed from nature to human society. So far, the theory of pattern formation has made significant advances, among which a novel class of instability, presented as wave patterns, has been found in directed networks. Such wave patterns have been proved fruitful but significantly affected by the underlying network topology, and even small topological perturbations can destroy the patterns. Therefore, methods that can eliminate the influence of network topology changes on wave patterns are needed but remain uncharted. Here, we propose an optimal control framework to steer the system generating target wave patterns regardless of the topological disturbances. Taking the Brusselator model, a widely investigated reaction-diffusion model, as an example, numerical experiments demonstrate our framework's effectiveness and robustness. Moreover, our framework is generally applicable, with minor adjustments, to other systems that differential equations can depict.