Quadratic optimal control and feedback stabilization of bilinear systems

In this work, we investigate the quadratic bilinear optimal control. We first review the case of finite-time interval, and then focus on the case of infinite-time horizon. The main difficulty in solving a quadratic optimal control for bilinear systems is the non-convexity of the cost function, which due to the fact that the dependence of the state with respect to the control is highly nonlinear. Then we provide a class of bilinear systems, including the commutative case, for which the optimal control can be expressed as a time-varying feedback control. We further show that under a controllability inequality, the obtained optimal control guarantees the strong stability of the resulting system. The techniques rely on linear semigroup theory and conditions of optimality. Applications to transport and heat equations are also presented.