Optimal control for infinite-horizon bilinear systems subject to a set of vector-valued functions

Abstract

This paper investigates the optimal control of an abstract bilinear system in a Gelfand triple of Hilbert spaces, where the system dynamics are governed by an unbounded operator and controlled by vector-valued functions. The control process is analyzed over an infinite time horizon. We establish well-posedness of both state and adjoint state equations, and prove the existence of optimal controls. Through analysis of the cost functional's differentiability properties, we derive necessary and sufficient optimality conditions. We provide stability analysis of optimal controls with respect to perturbations in the desired state and regularization parameter, establishing explicit convergence rates. Applications to the Fokker-Planck equation and a fourth-order parabolic equation, along with numerical simulations, serve to illustrate the theoretical results.