Stabilization of a matrix via a low-rank-adaptive ODE

Abstract 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 all negative real part and such that the perturbation $$\Delta =B-A$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mo>=</mml:mo> <mml:mi>B</mml:mi> <mml:mo>-</mml:mo> <mml:mi>A</mml:mi> </mml:mrow> </mml:math> has minimal norm. The structured stabilization further requires that the perturbation preserves the structural pattern of A . This non-convex problem is solved by a two-level procedure which involves the computation of the stationary points of a matrix ODE. It is possible to exploit the underlying low-rank features of the problem by using an adaptive-rank integrator that follows rigidly the rank of the solution. Some benefits derived from the low-rank setting are shown in several numerical examples. These computational advantages also allow to deal with high dimensional problems.