Optimal control for time fractional fluid–solid interaction model

This manuscript investigates the optimal control of a time-fractional fluid–structure interaction (FSI) model. The fluid dynamics are governed by a time-fractional Stokes system with a Riemann–Liouville derivative of order α∈(0,1). The solid deformation follows a time-fractional linear elasticity equation with a Riemann–Liouville derivative of order β∈(1,2). The two subsystems are interconnected through interface conditions that mandate the continuity of velocity and the equilibrium of stress across the common boundary. A variational formulation of the coupled system is established, and the well-posedness of the solution is affirmed utilizing suitable functional settings via the Lax–Milgram lemma. An optimal control framework is proposed, incorporating a quadratic cost functional, and first-order necessary conditions for optimality are derived through a variational inequality methodology. The adjoint systems, which are defined in a backward temporal context and incorporate Caputo fractional derivatives, are formulated, resulting in a gradient-based characterization of the optimal control strategy. This analysis offers a robust mathematical framework for controlling fractional-order FSI systems, with potential applications in interactions involving viscoelastic fluids and solid structures.