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.

中文翻译：

---

双线性系统的二次最优控制和反馈镇定

在这项工作中，我们研究了二次双线性最优控制。我们首先回顾有限时间间隔的情况，然后关注无限时间范围的情况。求解双线性系统的二次最优控制的主要困难是成本函数的非凸性，这是由于状态对控制的依赖性是高度非线性的。然后我们提供一类双线性系统，包括交换情况，其最优控制可以表示为时变反馈控制。我们进一步表明，在可控性不等式下，获得的最优控制保证了所得系统的强稳定性。这些技术依赖于线性半群理论和最优条件。还介绍了在传输和热方程中的应用。