Xiao
Meng
Hong Kong Baptist University
Well-posedness and optimal control of a PDE-ODE spatial-network model on metric graphs and sub-domains
Abstract.
Mathematical modeling of dynamics on metric graphs arises in various contexts, from epidemic dynamics to chemical transport in fractured media. Diffusion processes and interactions within complex topological structures pose significant analytical challenges. In practical scenarios, effective intervention strategies are also critical. In this work, we analyze a coupled PDE-ODE system defined on a hybrid structure. The model is formulated as a nonlinear system with junction conditions that capture diffusion in subdomains, along edges, and at vertices. The well-posedness of the system, including the existence, uniqueness, and regularity of solutions, is established via Galerkin approximations and energy estimates. Building on these coupled dynamics, we extend our approach to optimal control. In this framework, a quadratic cost functional penalizes deviations of the state variables from desired targets while accounting for the energetic cost of control actions. Our analysis demonstrates that the state-to-control mapping is Fréchet differentiable, and we derive the corresponding linearized system and adjoint system, along with the first-order optimality conditions.