Hilbert空间中具无界控制的无限时区线性二次最优控制问题

An Infinite Horizon Linear Quadratic Problem with Unbounded Controls in Hilbert Space

本文研究了Hilbert空间中一类由解析半群支配的具无界控制的无限时区线性二次最优控制问题,其中指标中的控制项加权算子要求强制而状态项加权算子可允许为不定号.在指数能稳条件下,证明了任意的最优控制及其最优轨线必定连续,建立了正实引理作为此问题唯一可解的充要条件,并用代数Riccati方程的解给出了最优控制的闭环综合。

An infinite horizon linear quadratic optimal control problem for analytic semigroup with unbounded controls in Hilbert space is considered. The state weight operator is allowed to be indfinite while the control weight operator is coercive. Under the exponential stabilization condition, it is proved that any optimal control and its optimal trajectory are continuous. The positive real lemma as a necessary and sufficient condition for the unique solvability of this problem is established. The closed-loop synthesis of optimal control is given via the solution to the algebraic Riccati equation.