伴随矩阵Lyapunov问题的对偶重构
A Duality Reformulation of the Companion-Matrix Lyapunov Problem
- Università di Padova（帕多瓦大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过伴随性对偶重构，证明了实Hurwitz伴随矩阵的Lyapunov方程解同时满足正半定性与逐项非负性，解决了相关猜想，并建立了与耗散系统及能量解释的联系。
AI中文摘要：
我们研究了连续时间Lyapunov方程解的正半定性与逐项非负性之间的关系。对于实Hurwitz矩阵$A$，我们证明：对于每个对称逐项非负的$Q$，方程$AP+PA^\ op=-Q$的解$P$是正半定的，当且仅当对于每个正半定的$R$，方程$A^\ op X+XA=-R$的解$X$是逐项非负的。这一等价性源于两个解算子的伴随性，并推广到所有实无混合矩阵。随后，我们证明这两个性质对所有实Hurwitz伴随矩阵均成立，从而解决了先前在实谱附加假设下建立的猜想。证明结合了可控性Gramian归一化与实增广矩阵伴随多项式系数的成对正性定理。后者通过二元多项式的系数符号性质、在两个开右半平面乘积上不消失的辅助行列式以及秩一扰动论证获得。协方差与能量解释将这些结果与耗散实现、阻尼二阶系统以及输入Gramian之间的比较联系起来。
英文摘要：
We study the relation between positive semidefiniteness and entrywise nonnegativity for solutions of continuous-time Lyapunov equations. For a real Hurwitz matrix $A$, we show that the solution of $AP+PA^\top=-Q$ is positive semidefinite for every symmetric entrywise nonnegative $Q$ if and only if the solution of $A^\top X+XA=-R$ is entrywise nonnegative for every positive semidefinite $R$. This equivalence follows from the adjointness of the two solution operators and extends to all real unmixed matrices. We then prove that both properties hold for every real Hurwitz companion matrix, settling a conjecture previously established under the additional assumption of a real spectrum. The proof combines a controllability-Gramian normalization with a pairwise positivity theorem for the coefficients of the adjugate polynomial of a real accretive matrix. The latter is obtained from a coefficient-sign property of bivariate polynomials, an auxiliary determinant that does not vanish on the product of two open right half-planes, and a rank-one perturbation argument. Covariance and energy interpretations connect these results with dissipative realizations, damped second-order systems, and comparisons between input Gramians.
