Optimal control on a metric graph for a damped linear fractional hyperbolic problem

The optimal control of fractional PDEs has been extensively studied in standard domains, but the existence and uniqueness of optimal controls in metric graphs, particularly for hyperbolic equations, remain less explored. Most studies focus on classical damping (e.g., viscous damping) or integer-order damping in hyperbolic problems, whereas the impact of fractional-order damping on control and optimization in metric graphs has received limited attention. Given the potential applications of these results to real-world problems such as pollution transport in river networks, traffic flow control, and heat propagation in branched structures, this presents a significant and promising research gap. This paper addresses a quadratic control problem involving a damped linear fractional hyperbolic equation subject to Dirichlet and Neumann boundary conditions. The considered fractional derivative is a composition of the right Caputo fractional derivative and the left Riemann–Liouville fractional derivative. We first give some existence and uniqueness results on an open bounded real interval, prove the existence of solutions to a quadratic optimal control problem and provide a characterization via optimality systems. We then investigate the analogous problems for a fractional Damped hyperbolic problem on a metric graph with mixed Dirichlet and Neumann boundary controls. The paper’s motivation likely arises from the desire to advance mathematical theory and control theory, especially in the context of complex systems represented by metric graphs. The potential impact or applications of these results span a wide range of fields, from engineering and network control to medical imaging and environmental science, where understanding and optimizing damped hyperbolic systems are essential.