无限维离散时间代数Riccati方程的非负解

NONNEGATIVE SOLUTIONS OF INFINITE DIMENSIONAL DISCRETE TIME ALGEBRAIC RICCATI EQUATIONS

本文研究了无限维离散时间代数Ｒｉｃｃａｔｉ方程（ＤＡＲＥ）的非负自伴解，给出了（ＤＡＲＥ）有非负 自伴解的充要条件．对幂可稳定化的离散时间系统∑ｄ（Ａ，Ｂ，－），若Ａ是可逆的，Ｂ是紧的，给出 了（ＤＡＲＥ）的非负解集的参数化刻画，并以Ａ的有限维的含于反稳定的不可观察子空间中的不变子 空间为参数．该结果把［５］中关于有限维系统∑ｄ（Ａ，Ｂ，－）的结果推广到了一般的系统∑ｄ（Ａ，Ｂ，－） 中．最后，还给出了∑ｄ（Ａ，Ｂ，－）具有非负稳定化解的充要条件．

The non-negative solutions of the infinite dimensional discrete time algebraic Riccati equation (DARE) are studied. Sufficient and Necessary conditions are given for (DARE) to have a non-negative solution. For power stabilizable discrete time system ∑d(A, B, -) with A being invertible, B being compact, the set of all non-negative solutions of (DARE) is parameterized in terms of finite dimensional A-invariant subspaces contained in the anti-stable non-observable subspaces. This result generalizes the main results in [5] on finite dimensional system (DARE) to general case. Finally, sufficient and necessary conditions for (DARE) to have a stabilizable non-negative solution was given.