Let $A$ be a square matrix with a given structure (e.g. real matrix, sparsity pattern, Toeplitz structure, etc.) and assume that it is unstable, i.e. at least one of its eigenvalues lies in the complex right half-plane. The problem of stabilizing $A$ consists in the computation of a matrix $B$, whose eigenvalues have negative real part and such that the perturbation $\Delta=B-A$ has minimal norm. The structured stabilization further requires that the perturbation preserves the structural pattern of $A$. We solve this non-convex problem by a two-level procedure which involves the computation of the stationary points of a matrix ODE. We exploit the low rank underlying features of the problem by using an adaptive-rank integrator that follows slavishly the rank of the solution. We show the benefits derived from the low rank setting in several numerical examples, which also allow to deal with high dimensional problems.

翻译：设$A$为具有特定结构（如实矩阵、稀疏模式、Toeplitz结构等）的方阵，且假设其不稳定，即至少有一个特征值位于复右半平面。矩阵$A$的镇定问题旨在计算矩阵$B$，使得其特征值均具有负实部，且扰动$\Delta=B-A$的范数最小。结构镇定进一步要求扰动保持$A$的结构模式。我们通过双层过程求解此非凸问题，该过程涉及矩阵ODE驻点的计算。通过采用自适应秩积分器严格跟随解的秩变化，我们充分利用了问题的低秩本质特征。数值算例展示了低秩设置带来的优势，这些算例同样适用于高维问题的处理。