Bilinear optimal control of a second-order-in-time fourth-order dynamical system with advection effects

This paper addresses a bilinear optimal control problem for a second-order-in-time fourth-order dynamical system with advection effects. Although bilinear control problems for parabolic and wave-type equations have been widely studied, the optimal control analysis of second-order-in-time systems involving fourth-order diffusion and drift-type bilinear controls remains less developed. This motivates the present work. The main contribution is the development of a complete optimal control framework for this class of systems. We first prove the well-posedness of the controlled state equation and establish the existence of an optimal control by using the direct method of calculus of variations. Then, we derive the first-order derivative of the reduced cost functional through a perturbation argument and characterize the optimal control by means of the associated adjoint system. We also discuss second-order necessary and sufficient optimality conditions, which provide a refined characterization of local optimality. For the numerical approximation, we construct a finite difference scheme for the state and adjoint equations and combine it with a gradient-based iterative algorithm to approximate the optimal control. The numerical experiments illustrate the decrease of the distributed and terminal tracking errors, as well as the boundedness of the control energy. These results confirm the effectiveness of the proposed approach and show its ability to steer the system toward prescribed targets with moderate control effort.