发表机构
Missouri University of Science and Technology（密苏里科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过预解族的标量不变量，给出了无需辅助矩阵的Riccati稳定性刻画，该不变量基于协方差平衡，结合严格有界实引理得到仅含Hurwitz条件和单个标量不等式的充要条件。
AI 中文摘要
在2004年的文集《数学系统与控制理论中的未解决问题》中，Erik Verriest提出了刻画Riccati稳定性“不借助额外矩阵”的问题。我们通过预解族的一个标量不变量给出了这样的刻画。该不变量由协方差平衡定义，最多使用n^2+1个频率-方向对。一个有限维分离论证表明，该协方差半径等于预解族的最优公共椭球范数。结合严格有界实引理，这给出了Riccati稳定性的一个充要条件，仅涉及第一个矩阵的Hurwitz性质和单个标量不等式。该不变量对于单个矩阵退化为普通谱半径，介于预解族的逐点谱半径和未缩放小增益水平之间，并且对于自然预解函数空间，与Shalit和Shamovich的算子空间谱半径一致。
英文摘要
In the 2004 collection \emph{Unsolved Problems in Mathematical Systems and Control Theory}, Erik Verriest posed the problem of characterizing Riccati stability ``without invoking additional matrices.'' We give such a characterization through a scalar invariant of the resolvent family. The invariant is defined by covariance balances using at most $n^2+1$ frequency--direction pairs. A finite-dimensional separation argument shows that this covariance radius equals the optimal common ellipsoidal norm of the resolvent family. Combined with the strict bounded real lemma, this yields a necessary and sufficient condition for Riccati stability involving only the Hurwitz property of the first matrix and a single scalar inequality. The invariant reduces to the ordinary spectral radius for one matrix, lies between the pointwise spectral-radius and unscaled small-gain levels of the resolvent family, and coincides with the operator-space spectral radius of Shalit and Shamovich for the natural resolvent function space.
