Bilinear optimal control problem with minimum energy for a fourth-order reaction diffusion system

This work was devoted to a minimum-energy optimal control problem governed by a fourth-order reaction–diffusion equation in which the control acts through a drift term. Such bilinear control problems arise naturally in transport phenomena and higher-order diffusion models, but their analysis is challenging because of the nonlinear coupling between the state and the control, as well as the difficulty of characterizing controls that are truly optimal with respect to energy. To overcome this difficulty, we formulated a family of penalized optimal control problems and characterized the minimum energy control as the limit of their solutions. The analysis was based on the associated optimality system, involving the state and adjoint equations, together with tools from convex optimization. Under appropriate assumptions, we established the existence and uniqueness of the optimal control, which provides a well-posed framework for the problem under consideration. For the numerical approximation, the resulting optimality system was solved by means of a conjugate gradient algorithm that requires, at each iteration, only the solution of the state equation and the adjoint equation. Numerical simulations showed a clear decrease in the tracking error and indicated that the computed state approaches the desired configuration with a small relative error after convergence. These results confirmed the stability, accuracy, and efficiency of the proposed approach for the minimum-energy control of fourth-order advection–reaction–diffusion systems.