抽象微分代数方程的线性二次最优控制
Linear-quadratic optimal control for abstract differential-algebraic equations
AI总结：
该研究将基于Popov算子的经典线性二次最优控制方法推广到希尔伯特空间的抽象线性微分代数方程，推导了最优控制的反馈形式与有界Riccati算子描述，并刻画了求解无限时域问题所需的指数稳定性。
摘要：
本文将基于Popov算子（波波夫算子）的经典线性二次（LQ）最优控制方法，推广到希尔伯特空间中的抽象线性微分代数方程（ADAEs）。为保证解的存在性，我们假设所研究的微分代数方程在伪预解式意义下为一阶指数。由此可得到一个退化半群，用于定义本系统的Popov算子。研究表明，在Popov算子满足合适的强制性假设下，最优代价可由一个有界Riccati算子（黎卡提算子）描述，且最优控制输入为反馈形式。此外，我们还刻画了ADAEs的指数稳定性，这是求解无限时域LQ问题的必要条件。
英文摘要：
In this paper, we extend a classical approach to linear quadratic (LQ) optimal control via Popov operators to abstract linear differential-algebraic equations (ADAEs) in Hilbert spaces. To ensure existence of solutions, we assume that the underlying differential-algebraic equation has index one in the pseudo-resolvent sense. This leads to the existence of a degenerate semigroup that can be used to define a Popov operator for our system. It is shown that under a suitable coercivity assumption for the Popov operator the optimal costs can be described by a bounded Riccati operator and that the optimal control input is of feedback form. Furthermore, we characterize exponential stability of ADAEs which is required to solve the infinite horizon LQ problem.
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